Technical University of Cluj-Napoca [632194]

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Technical University of Cluj-Napoca [632194]

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Technical University of Cluj-Napoca
North University Center of Baia Mare
Faculty of Sciences
Ph.D. Thesis
FIXED POINT THEOREMS FOR LOCAL ALMOST
CONTRACTIONS WITH APPLICATIONS
Scientiﬁc Advisor: Ph.D. Student: [anonimizat]. Univ. Dr. Vasile Berinde Monika Zakany
Baia Mare
2020

Contents
Chapter 1. INTRODUCTION………………………………………….. 1
1. Theoretical/conceptual framework: contractions, Picard operators and
almost contractions………………………………………………. 1
2. Almost contractions………………………………………………. 2
3. Considerations about local contractions …………………………….. 5
4. Motivation………………………………………………………. 7
5. Structure of the Thesis, Main Results ………………………………. 9
6. Acknowledgments………………………………………………… 10
Chapter 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN
UNIFORM SPACES………………………………………… 12
1. Preliminaries…………………………………………………….. 12
2. Theoretical background …………………………………………… 17
2.1. Stability and data dependence of the ﬁxed points for ALC-s………… 17
2.2. Continuity of almost local contractions…………………………… 19
3. New classes of almost local contractions…………………………….. 22
3.1. Almost local ϕ-contractions…………………………………….. 34
3.2. (B)-almost local contractions …………………………………… 42
3.3. Approximate ﬁxed points………………………………………. 44
3.4. Almost local contractions in b-pseudometric spaces………………… 57
3.5.ϕ-contractions in b-pseudometric spaces………………………….. 59
3.6. Fixed points of almost local contractions in b-pseudometric spaces ….. 66
Chapter 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS………… 80
1. Multivalued self almost local contractions …………………………… 80
2. Non-self multivalued almost local contractions……………………….. 92
Chapter4. NON-SELFSINGLEVALUEDALMOSTLOCALCONTRACTIONS101
1. Introduction …………………………………………………….. 101
2. Complex valued b-metric spaces and rational local contractions ……….. 109
Chapter 5. Application of almost local contractions in dynamic programming…121
1. Introduction …………………………………………………….. 121
2. Examples……………………………………………………….. 126
3. Main results …………………………………………………….. 127
1

2 CONTENTS
Bibliography ………………………………………………………….. 134

CHAPTER 1
INTRODUCTION
1. Theoretical/conceptual framework: contractions, Picard operators and
almost contractions
The Banach’s Contraction Principle represents probably the most important tool
in nonlinear analysis. In a complete metric space it was ﬁrst established by Stefan
Banach (1922). In the following, we present the form of that theorem given by Berinde
in [11]:
Theorem 1.1.1.(see[11]) Let (X,d)be a complete metric space and T:X→X
be a map satisfying
(1.1.1) d(Tx,Ty )≤a·d(x,y),∀x,y∈X,
where 0≤a<1is constant. Then:
(i) T has a unique ﬁxed point pinX;
(ii) The Picard iteration {xn}∞
n=0deﬁned by
(1.1.2) xn+1=Txn, n = 0,1,2···
converges to p, for anyx0∈X;
(iii) The following a priori and a posteriori error estimates hold:
(1.1.3) d(xn,p)≤an
1−ad(x0,x1),n= 0,1,2…
(1.1.4) d(xn,p)≤a
1−ad(xn−1,xn),n= 1,2…
(iv) The rate of convergence of Picard iteration is given by
(1.1.5) d(xn,p)≤a·d(xn−1,p),n= 1,2…
A map satisfying (i) and (ii) is said to be a Picard operator , see [86], [88],[89].
A mapping satisfying (1.1.1) is usually called a strict contraction or an a-contraction.
Hence, in essence, Theorem 1.1.1 shows that any contraction is a Picard operator.
A mapping satisfying condition (1.1.1) is always continuous; this fact lead researchers
to look up for discontinuous classes of such kind of mappings for which conclusions of
Theorem 1.1.1 still hold.
In 1968, R. Kannan [ 58] found a positive answer to this problem by proving a ﬁxed
1

2 1. INTRODUCTION
point theorem that extends Theorem 1.1.1 to mappings that need not to be continuous,
by replacing condition (1.1.1) by the following one: there exists 0≤b<1
2such that
(1.1.6) d(Tx,Ty )≤b[d(x,Tx ) +d(y,Ty )],∀x,y∈X.
2. Almost contractions
Definition 1.2.1.([58]) Let (X,d)be a complete metric space. A mapping T:
X→Xsatisfying the condition: there exists 0≤b <1/2such that condition (1.1.6)
holds, is called Kannan mapping.
Definition 1.2.2.([36]) Let (X,d)be a complete metric space. Any mapping
T:X→Xsatisfying the contractive condition: there exists 0≤c<1
2such that
d(Tx,Ty )≤c·[d(x,Ty ) +d(y,Tx )],∀x,y∈X,
is called Chatterjea contraction.
One of the most general contractive conditions related to Banach contractions has
been introduced by Ćirić (see [ 39]):
Definition 1.2.3.([39]) Let (X,d)be a complete metric space. The mapping
T:X→Xsatisfying the contractive condition:
there exists 0≤h<1such that
(1.2.1)d(Tx,Ty )≤h·max{d(x,y),d(x,Tx ),d(y,Ty ),d(x,Ty ),d(y,Tx )},∀x,y∈X
is called quasi-contraction.
Definition 1.2.4.(see[15]) Let (X,d)be a metric space. A mapping T:X→X
is called almost contraction or (δ,L)-contraction if there exist a constant δ∈(0,1)and
someL≥0such that
(1.2.2) d(Tx,Ty )≤δ·d(x,y) +L·d(y,Tx ),∀x,y∈X.
Remark 1.2.5.The term of almost contraction is the same as weak contraction,
and it was ﬁrst introduced by V. Berinde in [18].
Remark 1.2.6.Any strict contraction satisﬁes (1.2.2), withδ=aandL= 0,
therefore it is an almost contraction with a unique ﬁxed point.
Other examples of almost contractions are given in [ 11], [12], [18] and [10]. There
are various other examples of contractive conditions which implies the almost contrac-
tiveness condition (1.2.2), see for example Taskovic [ 100], Rus [87].
In the following, we present an existence result (Theorem 1.2.7), as well as an
existence and uniqueness result (Theorem 1.2.8) for almost contractions, as they were
established in [ 11], see also [ 18] and [10]. Their main merit is that they extend

2. ALMOST CONTRACTIONS 3
Banach’s Contraction Principle [ 6], Kannan [ 58], Chatterjea [ 36] and Zamﬁrescu’s
ﬁxed point theorems [ 106]. These results provide a method for approximating the
ﬁxed point, for which both a priori and a posteriori error estimates are available.
Theorem 1.2.7.([11]) Let (X,d)be a complete metric space and T:X→X
almost contraction. Then
(1)Fix(T) ={x∈X:Tx=x}/negationslash=φ;
(2) For any x0∈X, the Picard iteration {xn}∞
n=0, given byxn+1=Txn, converges
to somex∗∈Fix(T);
(3) The following estimates
(1.2.3) d(xn,x∗)≤δn
1−δd(x0,x1), n = 0,1,2…
(1.2.4) d(xn,x∗)≤δ
1−δd(xn−1,xn), n = 1,2…
hold, where δis the constant appearing in (1.2.2).
Theorem 1.2.8.([11]) Let (X,d)be a complete metric space and T:X→Xbe
an almost contraction for which there exist θ∈(0,1)and someL1≥0such that
(1.2.5) d(Tx,Ty )≤θ·d(x,y) +L1·d(x,Tx ),∀x,y∈X.
Then
(1)Thas a unique ﬁxed point, i.e. Fix(T) ={x∗};
(2) For any x0∈X, the Picard iteration {xn}∞
n=0converges to x∗;
(3) The a priori and a posteriori error estimates
d(xn,x∗)≤δn
1−δd(x0,x1), n = 0,1,2…
d(xn,x∗)≤δ
1−δd(xn−1,xn), n = 1,2…
hold.
(4) The rate of convergence of the Picard iteration is given by
(1.2.6) d(xn,x∗)≤θ·d(xn−1,x∗), n = 1,2…
The study of almost contractions emerged as an area of intense research activity,
thanks to the further development of mappings satisfying the condition (1.2.2).
Remark 1.2.9.(1) The error estimates in Theorems 1.1.1, 1.2.7 and 1.2.8 can be
merged in a single estimate, see BRRRRRRR;
(2) A large number of examples of almost contractions were given by V. Berinde in
[11]and[18]. For example, it was proved that:
– any Zamﬁrescu mapping from Theorem Z in [106]with constants α∈[0,1),k∈
[0,1
2),c∈[0,1
2)is an almost contraction (see [72]) withδ= max/braceleftBig
α,k
1−k,c
1−c/bracerightBig
and

4 1. INTRODUCTION
L= max/braceleftBig
2k
1−k,2c
1−c/bracerightBig
;
– any quasi-contraction with 0<h<1
2is an almost contraction;
– any Kannan mapping (see [58]) is an almost contraction with
δ=b
1−b,L=2b
1−b,0<b<1
2.
(3) The ﬁxed point x∗of an almost contraction attained by the Picard iteration depends
on the initial guess x0.
Definition 1.2.10 .(see[87],[89]) A mapT:X→Xis called weakly Picard
operator if the sequence {Tnx0}∞
n=0converges for all initial values x0∈Xand the
limits are ﬁxed of T.
Therefore, the class of almost contractions provides a large class of weakly Picard
operators. Condition (1.2.2) is related to another important concept in ﬁxed point
theory, that is, the so called Banach orbital condition
d(Tx,T2x)≤a·d(x,Tx ),∀x∈X, 0≤a<1,
studied by various authors in the context of ﬁxed point theorems, see for example Hicks
and Rhoades [ 51], Ivanov [ 54], Rus [86] and Taskovic [ 100].
Example 1.2.11 .[18]LetT: [0,1]→[0,1]be the identity map: Tx=x, for all
x∈[0,1]. ThenTis an almost contraction with θ∈(0,1)arbitrary and L≥1−θ.
The set of ﬁxed points of Trepresents the interval [0,1], therefore Fix(T) = [0,1],
which means that Thas an inﬁnite set of ﬁxed points.
The next theorem shows that an almost contraction is continuous at any ﬁxed point
of it, according to [ 9].
Theorem 1.2.12 .([9]) Let (X,d)be a complete metric space and T:X→Xbe
an almost contraction. Then Tis continuous at p, for anyp∈Fix(T).
From the various generalizations and extensions of almost contractions we mention
the following one, due to Suzuki [ 99]
Definition 1.2.13 .(see[99]) LetTbe a mapping on a metric space (X,d). Then
Tis called a generalized Berinde mapping if there exist a constant r∈[0,1)and a
functionb:X→[0,∞)such that
(1.2.7) d(Tx,Ty )≤r·d(x,y) +b(y)·d(y,Tx ),∀x,y∈X.
Definition 1.2.14 .([39]) Let (X,d)be a metric space. Any mapping T:X→X
is called Ćirić-Reich-Rus contraction if it satisﬁes the condition:
(1.2.8) d(Tx,Ty )≤α·d(x,y) +β·[d(x,Tx ) +d(y,Ty )],∀x,y∈X,

3. CONSIDERATIONS ABOUT LOCAL CONTRACTIONS 5
whereα,β∈R+andα+ 2β <1.
3. Considerations about local contractions
TheconceptoflocalcontractionwasﬁrstintroducedbyIncón-ZapateroandRodrigues-
Palmero (2003, 2007) [ 83], in order to study discounted stochastic dynamic program-
ming models with unbounded returns in the setting of metric spaces of functions with
a metric constructed from a countable family of seminorms. That idea was introduced
by Martins da Rocha and Filipe Vailakis in [ 64] (2010). They studied the existence
and uniqueness of ﬁxed points for the local contractions in the setting of a semimetric
space.
We begin with the deﬁnition of a semimetric space.
Definition 1.3.1.(see[107],[108]) LetXbe a nonempty set. The function
d:X×X→R+is said to be a semimetric on Xif the following hold:
(1)d(x,y) = 0if and only if x=y;
(2)d(x,y) =d(y,x),∀x,y∈X.
Note that the triangle inequality is not necessarily satisﬁed in this case.
Definition 1.3.2.(see[64]) LetFbe a set and letD= (dj)j∈Jbe a family of
semimetrics deﬁned on F. We letτbe the weak topology on Fdeﬁned by the family
D. Denote by Ja family of indices (which frequently can be considered as a subset of
N) and letr:J→J.
IfAis a nonempty subset of F, then for each hinA, we denote
dj(h,A)≡inf{dj(h,g) :g∈A},j∈J
the semidistance between the point h∈Fand the set A. An operator T:A→Ais
called a local contraction with respect to ( D,r) if, for every j∈J, there exists βj∈[0,1)
such that
dj(Tf,Tg )≤βjdr(j)(f,g),∀f,g∈A.
Following the Contraction Mapping Principle, a lot of papers were devoted to ob-
taining ﬁxed point theorems in uniform spaces, amongst which we mention the work of
Gheorghiu [ 45], Gheorghiu and Rotaru [ 46]. In a recent paper, Vailakis and Martins-
da-Rocha [ 64] have established new results involving the local contractions in uniform
spaces.
Instead of considering an operator T:F→F, Vailakis and Martins-da-Rocha have
considered operators T:A→F, whereA⊂Fis a nonempty, τ-bounded, sequentially
τ-complete and T-invariant subset of X, and provided suﬃcient conditions to ensure
the existence of a ﬁxed point of TinA, as well as the existence and uniqueness of the
ﬁxed point of T.

6 1. INTRODUCTION
Definition 1.3.3.([64]) The subset Ais said to be τ-Hausdorﬀ if and only if for
each pairf,g∈A,f/negationslash=g, there exists j∈Jsuch thatdj(f,g)>0.
The subset A⊂Fis calledT-invariant if and only if T(A)⊂A, i.e.,T|A:A→A.
Definition 1.3.4.[64]LetFbe a uniform space and let D= (dj)j∈Jbe a family
of semimetrics deﬁned on F.
The sequence{xn}n∈N∗isτ−convergent tox∗ifdj(xn,x∗)→0asn→∞,∀j∈J.
The sequence{xn}n∈N∗is said to be dj-Cauchy if and only if for each j∈J,
dj(xn,xm)→0asn,m→∞.
The sequence{xn}n∈N∗is said to be τ−Cauchyif it isdj-Cauchy, for all j∈J.
The subset AofFis said to be sequentially τ-complete if every τ-Cauchy sequence in
Fconverges in Fwith respect to the τ-topology.
The diameter of the subset A⊂Fshall be considered: diamj(A)≡sup{dj(x,y) :x,y∈A}.
The subset A⊂Fis said to be τ-bounded if diamj(A)≡sup{dj(x,y) :x,y∈A}is
ﬁnite for every j∈J.
The main result in [ 64] is:
Theorem 1.3.5.[64]LetFbe a set. Consider a function r:J→Jand let
T:A→Abe a local contraction with regard to (D,r). Consider a nonempty, τ-
bounded, sequentially τ-complete, and T-invariant subset A⊂F.
E:(existence): If the condition
(1.3.1) ∀j∈J, limn→∞βjβr(j)···βrn+1(j)diamrn+1(j)(A) = 0
is satisﬁed, then the operator Tadmits a ﬁxed point f∗inA.
S:(existence and uniqueness)
Furthermore, if h∈Fsatisﬁes
(1.3.2) ∀j∈J, limn→∞βjβr(j)···βrn+1(j)drn+1(j)(h,A) = 0
then the sequence {Tnh}n∈Nis τ-convergent to f∗, implying that the ﬁxed
pointf∗is unique in A.
A more general global ﬁxed point result has been obtained more than forty years
ago by Gheorghiu [ 45], see also Gheorghiu and Rotaru [ 46]:
Theorem 1.3.6.([45]) LetXbe a uniform Hausdorﬀ space, sequentially complete,
and let (di)i∈Ibe a family of semidistances deﬁned on X. Letf:X→Xbe a mapping
for which there exist ϕ:I→I,q:I→R+such that
di(f(x),f(y))≤qidϕ(i)(x,y),∀x,y∈X,
and the series∞/summationdisplay
n=1qiqϕ(i)···qϕn(i)dϕn(i)(x,y)

4. MOTIVATION 7
is convergent for all i∈Iand for all x,y∈X.
Thenfhas a unique ﬁxed point.
Remark 1.3.7.IfTsatisﬁes condition (1.3.1), then the series is convergent.
The results obtained by Rincón-Zapatero and Rodrigues-Palmero [ 83] involving the
local contractions are applied to the study of various applications to economics. Also,
they consider metric spaces of functions with a metric constructed from a countable
family of seminorms. That family is chosen as the supremum of continuous functions
over the multitude of compact sets covering the set of consumption streams. The ﬁxed-
point results obtained by Martins da Rocha and Filipe Vailakis in [ 64] turned out to
be useful in solving recursive equations in economic dynamics. This results make it
easier the study of existence and uniqueness of solutions to recursive equations that
arise in economic dynamics.
4. Motivation
The present thesis is intended to unify two classes of contractive mappings that are
important in ﬁxed point theory:
1) the class of almost contractions, very extensively studied, presented in § 2;
2) the class of local contractions, with applications in economics, presented in 2.3.36.
Next, we present the starting point of our research work: these two types of con-
tractions mentioned before.
A. The wide range of applications of Banach Contraction Principle in nonlinear anal-
ysis has challenged researchers to obtain its conclusions under weaker assumptions
than (1.1.1), which do not force the continuity of the operator T.
The ﬁrst achievement in this respect has been stated by Kannan in 1968 [ 58], who
obtained a ﬁxed point theorem for discontinuous mappings. Chatterjea [ 36], Bian-
chini [29], Reich, Rus [ 85], Ćirić [37], Zamﬁrescu [ 106] and many other researchers
continued this direction of research, see Rhoades [ 80] for a classiﬁcation and com-
parison of various such contractive type mappings.
All the above quoted ﬁxed point theorems ensure, based on speciﬁc assumptions,
the following two conclusions for the contractive mapping T:X→X:
(a)Fix(T) =x, i.e.,Thas a unique ﬁxed point in X;
(b)limn→∞Tnx0=x, for anyx0∈X.
(hereTnstands for the nthiterate ofT).
More recently, Berinde [ 15] introduced a large class of contractive mappings, called
weak contractions [in [ 11]] and later named almost contractions (in [ 9] and after-
wards).
The class of almost contractions includes Banach contractions, Kannan contrac-
tions, Chatterjea contractions, Zamﬁrescu contractions, Reich-Rus contractions,

8 1. INTRODUCTION
Bianchini contractions and, partially, the so called quasi-contractions, due to Ćirić
[38]. But, unlikely the above mentioned classes of contractions, which admit a
unique ﬁxed point, an almost contraction may have two or more ﬁxed points by
simultaneously keeping almost all the other features of the Banach Contraction
Mapping Principle, including rate of convergence, error estimate, stability and so
on.
B. On the other hand, Rincón-Zapatero and Rodrigues-Palmero (2003, 2007) [ 83],
established another extension of the Banach Contraction Principle, with various
applications in economics: recursive utility is used in many contexts to model eco-
nomic agents’ preferences where current utility is expressed as a function (the ag-
gregator) of current consumption and the utility of future consumption. Later,
Martins da Rocha and Filipe Vailakis in [ 64] (2010) have developed the idea of
Rincón-Zapatero and Rodrigues-Palmero in [ 83]
The setting they are working is that of a set Fendowed with a family D=
(dj)j∈Jof semidistances deﬁned on F. They consider a weak topology τdeﬁned by
the familyD.
LetA⊂Fbe aτ-bounded sequentially τ-complete and T-invariant subset of F.
T:A→Ais called a local contraction with respect to (D,r), wherer:J→J, if
there exists βj∈[0,1)such that
dj(Tf,Tg )≤βjdr(j)(f,g),∀f,g∈A,
The ﬁxed point theorem of Martin da Rocha and Filipe Vailakis [ 64] essentially
states that, if Fisτ-Hausdorﬀ, and,∀j∈J,
limn→∞βjβr(j)···βrn+1(j)diamrn+1(j)(A) = 0,
thenThas a ﬁxed point f∗inA.
Moreover, if h∈Fsatisﬁes
∀j∈J, limn→∞βjβr(j)···βrn+1(j)drn+1(j)(h,A) = 0,
then the sequence {Tnh}n∈Nis τ-convergent to f∗
(heredj(h,A)≡inf{dj(h,g) :g∈A}, for everyj∈J).
The theoretical results in [ 64] were then applied to solve recursive equations in
economic dynamics with various applications in dynamic programming.
BothclassesareextensionsofthewellknownBanachcontractionmappings, introduced
by Stefan Banach in 1922 in his famous dissertation [ 6]. They represent the foundation
of metrical ﬁxed point theory, an extremely dynamic ﬁeld of research starting with
second half of the 20th century, see the monograph [ 86], for a selected list of reference
books.

5. STRUCTURE OF THE THESIS, MAIN RESULTS 9
The two essentially diﬀerent approaches presented in A and B, both emerging from
Banach Contraction Mapping Principle, give rise to a very interesting and challenging
problem: is it possible to unify almost contractions and local contractions to form
a common class of contractive mappings that keep most of the features of the two
sources?
The present thesis aims to answer this problem in the aﬃrmative. We present a
coherent theory of what we shall call local almost contractions (ALC-s), for which we
state and prove various ﬁxed point theorems, examples, particular cases and applica-
tions. We plan to cover most of the classes of mappings studied in ﬁxed point theory:
single-valued self mappings, multivalued non-self mappings, common ﬁxed points and
coincidence points of local almost contractions .
5. Structure of the Thesis, Main Results
The thesis is divided into seven chapters, closely linked to each other.
The1stchapter, Introduction, take stock a non-exhaustive list of concepts, notions
and basic results from the ﬁxed point theory in metric, respectively, pseudometric
spaces.
In the second chapter, we present the concept of almost local contractions (brieﬂy
ALC-s) in pseudometric spaces, we prove the existence, respectively, existence and
uniqueness theorems for the ﬁxed points of ALC-s. Further on, we present stability
and data dependence of the ﬁxed points. We introduce several classes of ALC-s, we
discuss about their approximate ﬁxed points, ALC-s in b-pseudometric spaces with a
detailed study of their ﬁxed points.
The personal contributions in the second chapter are:
Deﬁnitions 2.1.5, 2.1.6, 2.3.1, 2.3.8, 2.3.15, 2.3.17, 2.3.20, 2.3.22, 2.3.24, 2.3.26, 2.3.37,
2.3.50, 2.3.87, 2.3.89, 2.3.91, 2.3.93, 2.3.95,
Theorems 2.1.9, 2.1.12, 2.1.14, 2.2.3, 2.2.4, 2.3.3, 2.3.10, 2.3.13, 2.3.14, 2.3.18, 2.3.19,
2.3.21, 2.3.23, 2.3.27, 2.3.42, 2.3.44, 2.3.48, 2.3.51, 2.3.70, 2.3.71, 2.3.72, 2.3.73, 2.3.74,
2.3.76, 2.3.78, 2.3.79, 2.3.80, 2.3.81, 2.3.83, 2.3.98, 2.3.99, 2.3.100, 2.3.101, 2.3.102,
2.3.103, 2.3.104, 2.3.105, 2.3.106, 2.3.121, 2.3.122, 2.3.123, 2.3.124, 2.3.125, 2.3.126,
2.3.127, 2.3.128,
Propositions 2.3.25, 2.3.66
Corollaries 2.3.28, 2.3.29,
Lemmas2.3.55, 2.3.64, 2.3.67, 2.3.69, 2.3.109, 2.3.110, 2.3.111, 2.3.113, 2.3.114, 2.3.116,
2.3.117, 2.3.118, 2.3.119, 2.3.120,
Examples 2.1.3, 2.1.4, 2.2.5, 2.1.15, 2.2.6, 2.1.11, 2.2.7, 2.3.12, 2.3.16, 2.3.39, 2.3.41,
2.3.46, 2.3.77, 2.3.107,2.3.108.

10 1. INTRODUCTION
In Chapter 3, we study multivalued self ALC-s in pseudometric spaces, with several
ﬁxed point theorems and approximate ﬁxed points also.
The personal contributions in the third chapter are:
Deﬁnitions 3.1.11, 3.1.12,
Theorems 3.1.13, 3.1.15, 3.1.17, 3.1.18, 3.1.19, 3.1.20, 3.1.21, 3.1.23,
Example 3.1.24.
In Chapter 4, we introduce the non-self single valued ALC-s in pseudometric spaces.
We establish new ﬁxed point theorems, both existence and uniqueness types. This
chapter also contains the notion of α-graphic local contraction, then we study several
types of non-self ALC-s, in order to establish if they are α-graphic local contractions.
The personal contributions in the fourth chapter are:
Deﬁnitions 4.1.2, 4.1.9,
Theorems 4.1.5, 4.1.7, 4.1.10, 4.1.11, 4.1.13, 4.1.14,
Examples 4.1.15, 4.1.16, 4.1.17, 4.1.18, 4.1.19.
Chapter 5 investigates the non-self multivalued ALC-s in pseudometric spaces. We
state and prove two ﬁxed point theorems related to the above-mentioned operators.
The personal contributions in the 5thchapter are: Deﬁnition 3.2.6, Theorems 3.2.7,
3.2.10, and Corollary 3.2.8.
In Chapter 6, we summarise several known results on the complex valued b-metric
space. The main result of this chapter is represented by complex valued rational local
contractions, including the study of common ﬁxed points for two operators of the
mentioned type.
The personal contributions in the 6thchapter are: Deﬁnition 4.2.6, Theorems 4.2.7,
4.2.8, Corollaries 4.2.9, 4.2.10.
Chapter 7 focuses on the applicability of ALC-s in economy, by using them in dynamic
programming. First, we deﬁne the k-almost local contractions. Secondly, Bellman
operator and the same equation is introduced, having strong connections with the
ALC-s.
The personal contributions in the 7thchapter are:
Deﬁnition 5.3.1,
Propositions 5.3.2, 5.3.3,
Theorems 5.3.4, 5.3.5, 5.3.6.
6. Acknowledgments
The main results of the thesis were presented to the mathematical community as
follows:

6. ACKNOWLEDGMENTS 11
My published papers:
– Zakany, M., Fixed Point Theorems For Local Almost Contractions , Miskolc Mathe-
matical Notes, 18 (2017), No. 1, pp. 499-506; ISI journal
– Zakany, M., New classes of local almost contractions , Acta Universitatis Sapientiae,
Mathematica, 10 (2) (2018), pp. 378-394; ISI journal
– Zakany, M., On the continuity of almost local contractions , Creative Mathematics
and Informatics, 26 (2017), No. 2, pp. 241-246; ISI journal
Within the 17thSymposium of Symbolic and Numeric Algorithms for Scientiﬁc
ComputingSYNASCTimisoara, Romania, 2015, Ipresentedmyresearchresults"Fixed
Point Theorems for Local Almost Contractions".
Within the 18thSymposium of Symbolic and Numeric Algorithms for Scientiﬁc
Computing SYNASC Timisoara, Romania, 2016, I presented my new research results
"The Continuity of Almost Local Contractions".
Within the 19thSymposium of Symbolic and Numeric Algorithms for Scientiﬁc
Computing SYNASC Timisoara, Romania, 2017, I presented my research results "Mul-
tivalued Self Almost Local Contractions".
WithintheConferenceofMathematicsandInformaticswithapplications, organised
by Babes-Bolyai University, Cluj Napoca, Romania, 2016, I presented my research
results "Local Almost Contractions".
First, I would like express my sincerest thanks to my scientiﬁc advisor, Prof. Univ.
Dr. Vasile Berinde. I am extremely grateful for the knowledge and opportunities, for
his continued support throughout my Ph.D study and related research. In addition,
I would like to thank to all my colleges from the Doctoral School for all the help,
interesting discussions and thought provoking questions. My Ph.D study could not be
accomplished without the meticulous guidance and constant encouragement of prof.
Dr. Horvat-Marc Andrei.
I would like to express my deep gratitude to all my family and friends for your
amazing support and encouragement. A special thanks to my parents and my husband
who has motivated, guided and inspired me always.

CHAPTER 2
SINGLE VALUED SELF ALMOST LOCAL
CONTRACTIONS IN UNIFORM SPACES
1. Preliminaries
The purpose of this chapter is to combine the concepts of almost contraction and
local contraction, and thus build a ﬁxed point theory for what we shall call local almost
contractions.
Throughoutthischapter, XisaHausdorﬀtopologicalspacewithitstopologygenerated
by a family{dj}j∈Jof pseudometrics on X.
Definition 2.1.1.(see[108]) LetXbe a nonempty set.
A mapping d:X×X→R+is said to be a pseudometric if:
(1)d(x,x) = 0,∀x∈X;
(2)d(x,y) =d(y,x),∀x,y∈X;
(3)d(x,y)≤d(x,z) +d(z,y),∀x,y,z∈X.
Remark 2.1.2.Note thatx=y⇔d(x,y) = 0in the case of a metric, which
means that the distance between two diﬀerent elements could be zero for the pseudo-
metric, see Examples 2.1.3 and 2.1.4. Also, observe the diﬀerence from a pseudometric
and a semimetric (Deﬁnitions 1.3.1 and 2.1.1).
Example 2.1.3.[104]LetX= [0,n]×[0,n]⊂R2,n∈N∗.The diameter of the
subsetX= [0,n]×[0,n]⊂R2is given by the length of diagonal line square whose four
sides have length n, which isn√
2.
Consider the pseudometrics dj:X×X→R+
(2.1.1) dj/parenleftBig
(x1,y1),(x2,y2)/parenrightBig
=|y1−y2|·e−j,∀j∈J,∀(x1,y1),(x2,y2)∈X,
whereJis a subset of N.djis a pseudometric, but is not a metric, take for example:
dj((1,4),(2,4)) =|4−4|·e−j= 0, however (1,4)/negationslash= (2,4).
Example 2.1.4.([104]) LetX= [0,n]×[0,n]⊂R2,n∈N∗.The diameter of the
subsetX= [0,n]×[0,n]⊂R2is given by the length of diagonal line of the square
whose four sides have length n.
Consider the pseudometric:
(2.1.2) dj((x1,y1),(x2,y2)) =|x1−x2|·e−j,∀j∈J,∀(x1,y1),(x2,y2)∈X,
12

1. PRELIMINARIES 13
whereJis a subset of N. Similar to the previous example, djis a pseudometric, but is
not a metric, take for example:
dj((1,3),(1,5)) =|1−1|·e−j= 0, however (1,3)/negationslash= (1,5).
A family of pseudometrics introduces on the set Xa topologyτ, named uniform
space topology, in certain research papers (see [2],[35]).
Definition 2.1.5.([2]) LetXbe a uniform space and let D= (dj)j∈Jbe a family
of pseudometrics deﬁned on X.
We letτbe the weak topology on Xdeﬁned by the family D.
A sequence{xn}n∈N∗is said to be τ−Cauchyif it isdj-Cauchy,∀j∈J.
The subset AofXis said to be sequentially τ-complete if every τ-Cauchy sequence in
Xconverges in Xwith respect to the τ-topology.
The subset A⊂Xis said to be τ-bounded if diamj(A)≡sup{dj(x,y) :x,y∈A}is
ﬁnite for every j∈J.
Definition 2.1.6.([103]) LetXbe a uniform space and a subset A⊂X.
Let a function r:J→J. An operator T:A→Ais called almost local contraction
with regard to (D,r) if, for every j, there exist the constants θ∈(0,1)andL≥0such
that
(2.1.3) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈A.
Remark 2.1.7.1) By taking (X,d)metric space instead of the pseudometrics dj,
rthe identity function ( r(j) =j) and by considering dr(j)deﬁned onXin(2.1.3), we
can easily conclude that the almost contractions represent a particular case of almost
local contractions.
2) As the pseudometric possesses the property of symmetry, the almost local contraction
condition (2.1.3)implies the following condition:
(2.1.4) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,Ty ),∀x,y∈A.
Definition 2.1.8.[64]LetXbe a uniform space, and consider a nonempty subset
A⊂X. The space Xis calledτ-Hausdorﬀ if the following condition is valid: for each
pairx,y∈X,x/negationslash=y, there exists j∈Jsuch thatdj(x,y)>0.
IfAis a nonempty subset of X, then for each zinX, we let
dj(z,A)≡inf{dj(z,y) :y∈A}.
Theorem 2.1.9 is an existence ﬁxed point theorem for almost local contractions, as
they appear in [ 103].
Theorem 2.1.9.LetXbe a uniform space. Consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X. Take a function r:J→J

14 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
and letT:A→Abe an almost local contraction with regard to ( D,r), with the
constantsθ∈(0,1)andL≥0. If the condition
(2.1.5) ∀j∈J, limn→∞θn+1diamrn+1(j)(A) = 0
is satisﬁed, then the operator Tadmits a ﬁxed point x∗inA.
Proof:Letx0∈Abe arbitrary and let {xn}∞
n=0be the Picard iteration deﬁned by
xn+1=Txn, n∈N.
Takex:=xn−1,y:=xnin (2.1.3) to obtain
dj(Txn−1,Txn)≤θ·dr(j)(xn−1,xn),
which means
(2.1.6) dj(xn,xn+1)≤θ·dr(j)(xn−1,xn),∀j∈J,∀n≥1.
By using (2.1.6), we inductively obtain :
(2.1.7) dj(xn,xn+1)≤θn·drn(j)(x0,x1), n = 0,1,2,···
According to the triangle inequality , by (2.1.7) we get:
dj(xn,xn+p)≤θn(1 +θ+···+θp−1)drn(j)(x0,x1) =
=θn
1−θ(1−θp)·drn(j)(x0,x1), n,p∈N,p/negationslash= 0.
These relations show us that the sequence {xn}n∈Nisdj-Cauchy for each j∈J. As
the subsetAis assumed to be sequentially τ-complete, there exists x∗inAsuch that
{xn}n∈Nisτ-convergent to x∗. Besides, the sequence {xn}converges with respect to
the topology τtox∗, which implies
∀j∈J, dj(Tx∗,x∗) = limn→∞dj(Tx∗,Tn+1x).
The operator Tis an almost local contraction with regard to ( D,r), thus we have
∀j∈J, dj(Tx∗,x∗)≤θlimn→∞dr(j)(x∗,Tnx).
The convergence with respect to the τ-topology implies convergence for the pseudo-
metricdr(j), we obtain dj(Tx∗,x∗) = 0for everyj∈J.
This way, we prove that Tx∗=x∗, since we have a Hausdorﬀ space.
So, we prove the existence of the ﬁxed point for ALC-s.
Remark 2.1.10 .ForTverifying (2.1.3)withL= 0, by taking θ=βj, we can
easily conclude that an almost local contraction is a local contraction.
Further, for the case dj=d,∀j∈J, withd= metric on X, we obtain the well known
Banach contraction, with a unique ﬁxed point.

1. PRELIMINARIES 15
Example 2.1.11 .LetAbe the set of positive functions:
A={f|f: [0,∞)→[0,∞)},
which is a subset of the set of real functions F={f:R→R}.
Letdj(f,g) =|f(0)−g(0)|·e−j,∀f,g∈A,r(j) =j+ 4,∀j∈J,whereJis a subset
ofN.
Indeed,djis a pseudometric, but is not a metric, take for example dj(x,x2) = 0, but
x/negationslash=x2.
Consider the mapping Tf=|f|,∀f∈A, and apply condition (2.1.3)for ALC-s:
|f(0)−g(0)|·e−j≤θ·|f(0)−g(0)|·e−(j+4)+L·|g(0)−f(0)|·e−(j+4),
which is equivalent to: e4≤θ+L.
This inequality is true if j >0, θ =1
4∈(0,1), L = 80>0.
Hence,Tis an ALC with the monotony property assured, since
dr(j)(f,g)≤dj(f,g)⇔|f(0)−g(0)|·e−(j+4)≤|f(0)−g(0)|·e−j,
valid for all f,g∈Aand for every j∈J.However,Tis not a contraction, because the
contractive condition (2.2.11)leads us again to: 1≤θ, a contradiction.
The (2.1.5) condition for the existence of the ﬁxed point is valid:
∀j∈J, limn→∞θn+1diamrn+1(j)(A) = limn→∞/parenleftBigg1
4/parenrightBiggn+1
·sup/braceleftBig
|/parenleftBig
f(0)−g(0)/parenrightBig
e−j|/bracerightBig
= 0.
The mapping Thas an inﬁnite set of ﬁxed points: Fix(T) ={f∈A}=A, by taking:
|f(x)|=f(x),∀f∈A,x∈[0,∞).
The next theorem represents an existence and uniqueness theorem for the ALC-s.
Theorem 2.1.12 .([104]) Assume all the conditions of Theorem 2.1.9 are satisﬁed.
If, additionally:
(U) for every ﬁxed j∈Jthere exist two constants θ1∈(0,1)andL1≥0such that
θ1+L1<1and
(2.1.8) dj(Tx,Ty )≤θ1·dr(j)(x,y) +L1·dr(j)(y,Tx ),∀x,y∈A,
and
(2.1.9) limn→∞(θ1+L1)ndrn(j)(z,A) = 0,∀x,y∈A,
then the ﬁxed point x∗ofTis unique.

16 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Proof:Suppose, by contradiction, there are two diﬀerent ﬁxed points x∗andy∗of
T. Then for every ﬁxed j ∈J, we have:
0<dj(x∗,y∗) =dj(Tx∗,Ty∗)≤θ1dr(j)(x∗,y∗) +L1dr(j)(y∗,Tx∗) =
= (θ1+L1)·dr(j)(x∗,y∗)≤···≤ (θ1+L1)n·drn(j)(x∗,y∗)≤
≤(θ1+L1)n
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
/arrowsoutheast0ifn→∞·drn(j)(z,A).
Now, letting n→∞, we obtain a contradiction with condition (2.1.9), i.e., the ﬁxed
point is unique.
Remark 2.1.13 .The proof of Theorem 2.1.12 is quite similar to that of Vailakis
[64]from the local contractions, published in 2010. But the uniqueness of the ﬁxed point
in the case of almost local contractions can be proved without assuming the additional
condition (U), only with the monotony condition for the pseudometric, and uniqueness
condition from the almost contractions, as shown in the next theorem.
Theorem 2.1.14 .Assume all the conditions of Theorem 2.1.9 are satisﬁed. If we
add a monotony condition for the pseudometric, namely:
(2.1.10) dr(j)(f,g)≤dj(f,g),∀f,g∈A,∀j∈J,
and also we add the uniqueness condition (U’):
(2.1.11) dj(Tx,Ty )≤θu·dr(j)(x,y) +Lu·dr(j)(x,Tx ),∀x,y∈A,
with the constants θu∈(0,1)andLu≥0,then the ﬁxed point f∗ofTis unique.
Proof:Suppose, by contradiction, there are two distinct ﬁxed points f∗andg∗
ofT. Then, by using (2.1.10), and condition (2.1.11) for every ﬁxed j∈Jwith
f:=f∗,g:=g∗we get:
dj(f∗,g∗)≤θu·dr(j)(f∗,g∗) +Ludr(j)(f∗,Tf∗) =θu·dr(j)(f∗,g∗)≤θu·dj(f∗,g∗)
⇒dj(f∗,g∗)≤θu·dj(f∗,g∗)⇒(1−θu)·dj(f∗,g∗)≤0,
which is obviously a contradiction with dj(f∗,g∗)>0andθu∈(0,1).
So, we prove the uniqueness of the ﬁxed point.
Example 2.1.15 .LetX= [0,n]×[0,n]⊂R2,n∈N∗, T :X→X,
T(x,y) =

(x
2,y
2)if (x,y)/negationslash= (1,0)
(0,0)if (x,y) = (1,0)
The diameter of the subset X= [0,n]×[0,n]⊂R2is given by the length of diagonal
line of the square whose four sides have length n.
Consider the pseudometric:
(2.1.12) dj((x1,y1),(x2,y2)) =|x1−x2|·e−j,∀j∈J,∀(x1,y1),(x2,y2)∈X,

2. THEORETICAL BACKGROUND 17
where the family of indices Jis a subset ofN.djis a pseudometric, but is not a metric,
take for example:
dj((1,4),(1,6)) =|1−1|·e−j= 0, however (1,4)/negationslash= (1,6f)
In this case, consider the function r(j) =j+ 2. By applying the inequality (2.1.3)to
the mapping T, we get for all x= (x1,y1),y= (x2,y2)∈X
/vextendsingle/vextendsingle/vextendsinglex1
2−x2
2/vextendsingle/vextendsingle/vextendsingle·e−j≤θ·|x1−x2|·e−(j+2)+L·/vextendsingle/vextendsingle/vextendsinglex2−x1
2/vextendsingle/vextendsingle/vextendsingle·e−(j+2),
for allj∈J, which can be write as the equivalent form
|x1−x2|·e2≤2θ·|x1−x2|+L·|2×2−x1|,
The last inequality is true if we take θ=1
2∈(0,1),L= 10≥0. HenceTis an ALC,
with the monotony property assured. The uniqueness condition (2.1.11) becomes:
/vextendsingle/vextendsingle/vextendsinglex1
2−x2
2/vextendsingle/vextendsingle/vextendsingle·e−j≤θ·|x1−x2|·e−(j+2)+L·/vextendsingle/vextendsingle/vextendsinglex1−x1
2/vextendsingle/vextendsingle/vextendsingle·e−(j+2),
for allj∈J.From that, we obtain
/vextendsingle/vextendsingle/vextendsinglex1
2−x2
2/vextendsingle/vextendsingle/vextendsingle(e2−2θ)≤L·x1,∀x1,x2∈[0,n].
This inequality is valid for θ=1
3∈(0,1)andL= 3≥0.
The operator Tpossess the unique ﬁxed point: Fix(T) = (0,0).
Havinginview[ 72], weintroduce, similarlytothecaseofstrictalmostcontractions,
a new type of ALC-s, which is the strict ALC-s.
Definition 2.1.16 .LetXbe a uniform space and let D= (dj)j∈Jbe a family
of pseudometrics deﬁned on X. We letτbe the weak topology on Xdeﬁned by the
familyD. Consider a function r:J→Jand let a nonempty, τ-bounded, sequentially
τ-complete, and T-invariant subset A⊂X.
An operator T:A→Ais called strict almost local contraction if it satisﬁes both
conditions (2.1.5)and(2.1.9), with some real constants θ,θu∈(0,1)andL,Lu≥0,
respectively.
Remark 2.1.17 .The existence condition (2.1.5)and the uniqueness conditions
(2.1.9)are independent, according to Example 2.1.15.
Remark 2.1.18 .The strict almost contractions have a remarkable importance: they
always have a unique ﬁxed point, see [72].
2. Theoretical background
2.1. Stability and data dependence of the ﬁxed points for ALC-s
The aim of this section is to survey the most signiﬁcant contributions to the area of
stability and data dependence of the almost local contractions. It was ﬁrst introduced

18 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
by Harder and Hicks [ 50]. The key idea was to approximate the sequence {xn}∞
n=0
deﬁned by a given iterative method with a more practical sequence {yn}∞
n=0, whose
limit will properly approximate the ﬁxed point of the initial mapping.
Definition 2.2.1.[50]Let(X,d)be a metric space and T:X→Xa mapping,
x0∈Xand suppose that the sequence of successive approximations deﬁned by
xn+1=Txn, n∈Nwithx0=x,
converges to a ﬁxed point pofT.
Let{yn}∞
n=0be an arbitrary sequence in Xand denote:
(2.2.1) εn=d(yn+1,Tyn),n= 0,1,2,…
We say that the ﬁxed point iteration procedure is T-stable or stable with regard to Tif
and only if
(2.2.2) limn→∞εn= 0⇔limn→∞yn=p.
The following Lemma will be useful in future demonstrations.
Lemma 2.2.2.(see[19]) Consider{an}n≥0,{bn}n≥0two sequences of positive real
numbers and q∈(0,1)such that:
(i)an+1≤qan+bn,n≥0;
(ii)bn→0asn→∞.
Then:
limn→∞an= 0
It is our aim to prove that the Picard iteration is T-stable with respect to almost
local contractions.
Theorem 2.2.3.LetXbe a uniform space. Consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X. Take a function r:J→J
and letT:A→Abe an almost local contraction with regard to ( D,r), with the
constantsθu∈(0,1)andLu≥0. The pseudometric Tpossess the monotony property
, namely:
(2.2.3) dr(j)(x,y)≤dj(x,y),∀x,y∈A,∀j∈J.
If the uniqueness condition:
(2.2.4) dj(Tx,Ty )≤θu·dr(j)(x,y) +Lu·dr(j)(x,Tx ),∀x,y∈A,
is valid, then the ﬁxed point pofTis unique. Consider the Picard iteration
xn+1=Txn, n∈Nwithx0=x.
Then{xn}∞
n=0converges strongly to pand isT-stable, which means that: for {xn}given
by(2.2.1), the equivalence (2.2.2)holds.

2. THEORETICAL BACKGROUND 19
Proof:Let{xn}n≥0,{yn}n≥0two sequences of positive real numbers. By applying
triangle inequality and (2.2.4), after that we use the monotony of the pseudometric,
we obtain
dj(yn+1,p)≤dj(yn+1,Tyn) +dj(Tyn,p)≤
≤εn+θudr(j)(yn,p)≤θudj(yn,p) +εn.
First, assume that limn→∞εn= 0. Having in view that θu∈(0,1), by using Lemma 2.2.2,
we can conclude that limn→∞yn=p.
On the other hand, by using the deﬁnition of ALC, we can write
dj(xn+1,p) =dj(Txn,Tp)≤θudr(j)(xn,p)
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
≤dj(xn,p)+Ludr(j)(p,Tp )
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
=0.
It results
dj(xn+1,p)≤θudj(xn,p),
which means that limn→∞xn=p.
Conversely, assume that limn→∞yn=p,it follows that
εn=dj(yn+1,Tyn)≤dj(yn+1,p) +θudj(yn,p)→0
asn→∞.
2.2. Continuity of almost local contractions
ThissectioncanberegardedasanextensionofV.BerindeandM.Pacurar( 2015,[9])
analysis about the continuity of almost contractions in their ﬁxed points. The main
results are given by Theorem 2.2.4, which give us the answer about the continuity of
ALC-s in their ﬁxed points.
Theorem 2.2.4.LetXbe a uniform space and D= (dj)j∈Jbe a family of pseudo-
metrics deﬁned on X. Consider a function r:J→Jand take a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X. LetT:A→Abe an ALC with
regard to (D,r), satisfying condition (2.1.5). ThenTadmits a ﬁxed point. Moreover,
Tis continuous at f, for anyf∈Fix(T).
Proof:The mapping Tis an ALC, i.e., there exist the constants θ∈(0,1)and
someL≥0such that
(2.2.5) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈A
For any sequence {yn}∞
n=0inAconverging to f, we takey:=yn,x:=fin (2.2.5), and
we get
(2.2.6) dj(Tf,Tyn)≤θ·dr(j)(f,yn) +L·dr(j)(yn,Tf),n= 0,1,2,…

20 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
UsingTf=f, sincefis a ﬁxed point of T, we obtain:
(2.2.7) dj(Tyn,Tf)≤θ·dr(j)(f,yn) +L·dr(j)(yn,f),n= 0,1,2,…
Now by letting n→∞in (2.2.7), we get Tyn→Tf, which shows that Tis continuous
atf.
The ﬁxed point has been chosen arbitrarily, so the proof is complete.
Example 2.2.5.[104]LetX= [0,n]×[0,n]⊂R2,n∈N∗. The diameter of the
subsetX= [0,n]×[0,n]⊂R2is given by the length of diagonal line of the square
whose four sides have length n.
Consider the pseudometric:
(2.2.8) dj((x1,y1),(x2,y2)) =|x1−x2|·e−j,∀j∈J,∀(x1,y1),(x2,y2)∈X,
whereJis a subset of N.djis a pseudometric, but is not a metric, take for example:
dj((1,4),(1,3)) =|1−1|·e−j= 0, however (1,4)/negationslash= (1,3).
In this case, consider the function r(j) =j+ 1, wherej∈J.
ConsiderT:X→X,
T(x,y) =

(x,y
5)if (x,y)/negationslash= (1,1)
(0,0)if (x,y) = (1,1)
Tis not a contraction because the contractive condition:
(2.2.9) dj(Tx,Ty )≤θ·dj(x,y),
is not valid∀x,y∈X, and for any θ∈(0,1). Indeed, (2.2.9)is equivalent with:
|x1−x2|·e−j≤θ·|x1−x2|·e−j,∀j∈J
The last inequality leads us to 1≤θ, which is obviously false, considering θ∈(0,1).
However,Tbecomes an almost local contraction if:
|x1−x2|·e−j≤θ·|x1−x2|·e−(j+1)+L·|x2−x1|·e−(j+1),
which is equivalent to : e≤θ+L.
Forθ=1
3∈(0,1),L= 3≥0andj >0, the last inequality becomes true, i.e., Tis an
ALC with an inﬁnite set of ﬁxed points:
Fix(T) ={(x,0) :x∈[0,n]}⊂X.
Observe that the monotony of the almost local contraction is fulﬁlled, because
dr(j)(x,y)≤dj(x,y)⇔|x1−x2|·e−(j+1)≤|x1−x2|·e−j,
for allj∈Jand for all x,y∈X.In this case, we have:
∀j∈J, limn→∞θn+1diamrn+1(j)(X) = limn→∞(n+ 1)√
2·/parenleftBigg1
3/parenrightBiggn+1
= 0.

2. THEORETICAL BACKGROUND 21
The existence of the ﬁxed point is assured, according to condition (2.1.5)from Theorem
2.1.9.
Theorem 2.2.4 could be applied, because the continuity of Tin(0,0)∈Fix(T), but
discontinuity in (1,1), which is not a ﬁxed point of T.
Example 2.2.6.With the assumptions from Example 2.1.15 and the pseudometric
deﬁned by
(2.2.10) dj((x1,y1),(x2,y2)) =|x1−x2|·2−j,∀j∈J,∀(x1,y1),(x2,y2)∈X,
wherer(j) =j+ 3,∀j∈J, we get another example for ALC-s.
ConsiderT:X→X,
T(x,y) =

(x,y
2)if (x,y)/negationslash= (1,1)
(0,0)if (x,y) = (1,1)
Tis not a contraction because the contractive condition:
(2.2.11) dj(Tx,Ty )≤θ·dj(x,y),
is not valid∀x,y∈X, and for any θ∈(0,1). Indeed, (2.2.11)is equivalent with:
|x1−x2|·2−j≤θ·|x1−x2|·2−j,∀j∈J,∀(x1,y1),(x2,y2)∈X.
The last inequality leads us to 1≤θ, which is obviously false, considering θ∈(0,1).
However,Tbecomes an almost local contraction if:
|x1−x2|·2−j≤θ·|x1−x2|·2−(j+3)
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
dr(j)(x,y)+L·|x2−x1|·2−(j+3)
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
dr(j)(y,Tx ),
which is equivalent to : 8≤θ+L.
Forθ=1
5∈(0,1),L= 9≥0andj∈J, the last inequality becomes true, i.e., Tis an
ALC with an inﬁnite set of ﬁxed points: Fix(T) ={(x,0) :x∈[0,n]}.
In this case, we have:
∀j∈J, limn→∞θn+1diamrn+1(j)(A) = limn→∞/parenleftBigg1
5/parenrightBiggn+1
·n√
2 = 0.
The existence of the ﬁxed point is assured, according to condition (2.1.3)from Theorem
2.1.9.
Theorem 2.2.4 could be applied, because the continuity of Tin(0,0)∈Fix(T), but
discontinuity in (1,1), which is not a ﬁxed point of T.
Example 2.2.7.LetA= [0,n]×[0,n]⊂R2,n∈N∗, T :A→A,
T(x,y) =

(x
2,y
4)if (x,y)/negationslash= (1,0)
(0,0)if (x,y) = (1,0)

22 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
The diameter of the subset A= [0,n]×[0,n]⊂R2is given by the length of diagonal
line of the square whose four sides have length n, namelyn√
2
Consider the pseudometric:
(2.2.12) dj/parenleftBig
(x1,y1),(x2,y2)/parenrightBig
=|x1−x2|·e−2j,∀j∈J,∀(x1,y1),(x2,y2)∈A,
whereJis a subset of Q.djis a pseudometric, but is not a metric, take for example:
dj((1,5),(1,6)) =|1−1|·e−2j= 0, however (1,5)/negationslash= (1,6).
In this case, consider the function r(j) =j+1
2. By applying the inequality (2.1.3)to
the mapping T, we get for all x= (x1,y1),y= (x2,y2)∈A:
/vextendsingle/vextendsingle/vextendsinglex1
2−x2
2/vextendsingle/vextendsingle/vextendsingle·e−2j≤θ·|x1−x2|·e−2(j+1
2)+L·/vextendsingle/vextendsingle/vextendsinglex2−x1
2/vextendsingle/vextendsingle/vextendsingle·e−2(j+1
2),
for allj∈J, which can be put in the equivalent form
|x1−x2|·e≤2θ·|x1−x2|+L·|2×2−x1|,
The last inequality is true if we take θ=1
2∈(0,1),L= 4≥0. HenceTis an ALC,
with the unique ﬁxed point (0,0).
According to Theorem 2.2.4, Tis continuous at the ﬁxed point, at (0,0)∈Fix(T), but
is not continuous at (1,0)/∈Fix(T).
According to Deﬁnition 2.1.6, the ALC-s are deﬁned in a subset A⊂X. In the
caseA=X, then an almost local contraction becomes an usual almost contraction.
3. New classes of almost local contractions
In this section we present the extension for various type of operators on pseudo-
metric spaces.
a)Generalized ALC
Definition 2.3.1.LetFbe a uniform space and let a function r:J→J.Let
A⊂Fbe aτ-bounded sequentially τ-complete and T-invariant subset of F. A mapping
T:A→Ais called generalized almost local contraction if there exist a constant
θ∈(0,1)and someL≥0such that∀x,y∈A,∀j∈Jwe have:
dj(Tx,Ty )≤θ·dr(j)(x,y) +
+L·min{dr(j)(x,Tx ),dr(j)(y,Ty ),dr(j)(x,Ty ),dr(j)(y,Tx )}. (2.3.1)
Remark 2.3.2.It is obvious that any generalized ALC is an almost contraction,
by takingr(j) =j, therefore satisfy the inequality (1.2.2).
Theorem 2.3.3.Assume all conditions of Deﬁnition 2.3.1 are satisﬁed.
LetT:A→Abe a generalized ALC, i.e., a mapping satisfying (2.3.1), and also
verifying the condition (2.1.11)for the uniqueness of the ﬁxed point, with the constants
θu∈(0,1)andLu≥0. LetFix(T) ={f}. ThenTis continuous at f.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 23
Proof:SinceTis a generalized ALC, there exist a constant θu∈(0,1)and some
Lu≥0such that (2.1.11) is satisﬁed. We know by Theorem 2.1.14 that Thas a unique
ﬁxed point, say f.
Let{yn}∞
n=0be any sequence in Aconverging to f. Then by taking
y:=yn, x :=f
in the generalized almost local contraction condition (2.3.1), we get
(2.3.2) dj(Tf,Tyn)≤θu·dr(j)(f,yn),n= 0,1,2,···
sincefis a ﬁxed point for T, we have
min{dr(j)(x,Tx ),dr(j)(y,Ty ),dr(j)(x,Ty ),dr(j)(y,Tx )}=dr(j)(f,Tf ) = 0.
Now, by letting n→∞in (2.3.2), we get Tyn→Tfwhich shows that Tis continuous
atf.
b)Ćirić type strong almost local contraction
Definition 2.3.4.(see Berinde, [22],[21]) Let (X,d)be a complete metric space.
The mapping T:X→Xis called Ćirić almost contraction if there exist a constant
α∈[0,1)and someL≥0such that
(2.3.3) d(Tx,Ty )≤α·M1(x,y) +L·d(y,Tx ),for all x,y∈X,
where
M1(x,y) =max/braceleftBig
d(x,y),d(x,Tx ),d(y,Ty ),d(x,Ty ) +d(y,Tx )
2/bracerightBig
.
From the above deﬁnition the following question arises: is it possible to extend it
in the case of almost local contractions? First we need to remind two Lemmas of Ćirić
([39]), which will be essential in proving our main results.
Lemma 2.3.5.([39]) LetTbe a quasi-contraction on Xand letnbe any positive
integer. Then, for each x∈X, and all positive integers i,j,wherei,j∈{1,2,···n}
implies
d(Tix,Tjx)≤h·δ[O(x,n)],
where we denoted δ(A) = sup{d(a,b) :a,b∈A}for a subset A⊂X.
Remark 2.3.6.Observe that, by means of Lemma 2.3.5, for each n, there exists
k≤nsuch that
d(x,Tkx) =δ[O(x,n)].
Lemma 2.3.7.(see[39]) LetTbe a quasi-contraction on X.
Then the inequality
δ[O(x,n)]≤1
1−hd(x,Tkx)
holds for all x∈X.

24 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
The answer to the last question is aﬃrmative and is given by the next deﬁnition.
Definition 2.3.8.LetXbe a uniform space. Consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X. Take a function r:J→J.
The operator
T:A→Ais called Ćirić-type strong ALC with regard to ( D,r) if, for every j∈J,
there exist the constants θ∈(0,1)andL≥0such that
(2.3.4) dj(Tf,Tg )≤θ·M1r(j)(f,g) +L·dr(j)(g,Tf ),for all f,g∈A,
where
M1r(j)(f,g) = max/braceleftBig
dr(j)(f,g),dr(j)(f,Tf ),dr(j)(g,Tg ),dr(j)(f,Tg ) +dr(j)(g,Tf )
2/bracerightBig
.
Remark 2.3.9.Although this class is wider than the one of ALC-s, similar conclu-
sions can be stated, as it follows:
Theorem 2.3.10 .LetXbe a uniform space and let a function r:J→J.Let
A⊂Xbe aτ-bounded sequentially τ-complete and T-invariant subset of Xand let
T:A→Abe a Ćirić-type strong ALC relating to ( D,r) with the monotony property
(2.1.10)fulﬁlled. Then
(1)Thas a ﬁxed point, i.e., Fix(T) ={x∈X:Tx=x}/negationslash=φ;
(2) For any x0=x∈A, the Picard iteration {xn}∞
n=0converges to x∗∈Fix(T);
(3) The following a priori estimate is available:
(2.3.5) dj(xn,x∗)≤θn
(1−θ)2dj(x,Tx ), n = 1,2…
Proof:For the conclusion of the theorem, we have to prove that Thas at least a
ﬁxed point in the subset A⊂X. To this end, let x∈Abe arbitrary, and let {xn}∞
n=0
be the Picard iteration deﬁned by xn+1=Txn, n∈Nwithx0=x.
Takex:=xn−1,y:=xnin (2.3.4) to obtain
dj(xn,xn+1) =dj(Txn−1,Txn)≤θ·M1r(j)(xn−1,xn) +L·dr(j)(xn,Txn−1)
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
=0,
that is,
dj(xn,xn+1)≤θmax/braceleftBig
dr(j)(xn−1,xn),dr(j)(xn,xn+1),dr(j)(xn−1,xn+1) + 0
2/bracerightBig
,
having in view that dr(j)(xn,Txn−1) = 0.After using the triangle inequality, we obtain
dr(j)(xn−1,xn+1)≤dr(j)(xn−1,xn) +dr(j)(xn,xn+1).
At this point, we use the well known inequalityx+y
2≤max{x,y},∀x,y∈R,we
distinguish two cases:
(2.3.6) max/braceleftBig
dr(j)(xn−1,xn),dr(j)(xn,xn+1),dr(j)(xn−1,xn+1)
2/bracerightBig
=dr(j)(xn−1,xn)

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 25
or
(2.3.7) max/braceleftBig
dr(j)(xn−1,xn),dr(j)(xn,xn+1),dr(j)(xn−1,xn+1)
2/bracerightBig
=dr(j)(xn,xn+1).
The case (2.3.7) lead to the contradiction:
dj(xn,xn+1)≤θdr(j)(xn,xn+1)≤/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
(2.1.10)dj(xn,xn+1),
therefore (2.3.6) is valid.
Continuing in this way, for n≥1, by Lemma 2.3.5, we have
dj(Tnx,Tn+1x) =dj(TTn−1x,T2Tn−1x)≤θ·δ[O(Tn−1x,2)].
By using Remark 2.3.6, we can easily conclude: there exists a positive integer k1∈
{1,2}such that
δ[O(Tn−1x,2)] =dj(Tn−1x,Tk1Tn−1x),
and therefore
dj(xn,xn+1)≤θ·dj(Tn−1x,Tk1Tn−1x).
By using once again Lemma 2.3.5, we obtain, for n≥2,
dj(Tn−1x,Tk1Tn−1x) =dj(TTn−2x,Tk1+1Tn−2x)≤
≤θ·δ[O(Tn−2x,k 1+ 1)]≤θ·δ[O(Tn−2x,3)].
Continuing in this way, we get
dj(Tnx,Tn+1x)≤θ·δ[O(Tn−1x,2)]≤θ2·δ[O(Tn−2x,3)].
By applying repeatedly the last inequality, we get
(2.3.8) dj(Tnx,Tn+1x)≤θ·δ[O(Tn−1x,2)]≤···≤θn·δ[O(x,n+ 1)].
At this point, by Lemma 2.3.7, we obtain
δ[O(x,n+ 1)]≤δ[O(x,∞)]≤1
1−θdj(x,Tx ),
which by (2.3.8) yields
(2.3.9) dj(Tnx,Tn+1x)≤θn
1−θdj(x,Tx ).
The inequality (2.3.8) and the triangle inequality can be merged to obtain the following
estimate:
(2.3.10) dj(Tnx,Tn+px)≤θn
1−θ·1−θp
1−θdj(x,Tx ).
Let us remind the fact that 0< θ < 1, then, by using (2.3.10), we can conclude
that{xn}∞
n=0is a Cauchy sequence. As the subset Ais assumed to be sequentially

26 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
τ-complete, there exists x∗inAsuch that{xn}isτ-convergent to x∗. After simple
computations involving the triangular inequality and the Deﬁnition (2.3.4), we get
dj(x∗,Tx∗)≤dj(x∗,xn+1) +dj(xn+1,Tx∗) =
=dj(Tn+1x,x∗) +dj(Tn+1x,Tx∗)≤dj(Tn+1x,x∗) +
+θmax{dj(Tnx,u),dj(Tnx,Tn+1x),dj(x∗,Tx∗),dj(Tnx,Tx∗),dj(Tn+1x,x∗)}+
+L·dj(x∗,Txn)
Continuing in this way, we obtain
dj(x∗,Tx∗)≤dj(Tn+1x,x∗) +θ·[dj(Tnx,u) +dj(Tnx,Tn+1x) +
+dj(x∗,Tx∗) +dj(Tn+1x,x∗)] +L·dj(x∗,Txn).
These relations leads us to the following inequalities:
dj(x∗,Tx∗)≤1
1−θ[(1 +θ)dj(Tn+1x,x∗) +
+(θ+L)dj(x∗,Txn) +θdj(Tnx,Tn+1x)]. (2.3.11)
Lettingn→∞in (2.3.11) we obtain
dj(x∗,Tx∗) = 0,
which means that x∗is a ﬁxed point of T. The estimate (2.3.5) can be obtained from
(2.3.9) by letting p→∞.
This completes the proof.
Remark 2.3.11 .1) Theorem 2.3.10 represents a very important extension of Ba-
nach’s ﬁxed point theorem, Kannan’s ﬁxed point theorem, Chatterjea’s ﬁxed point theo-
rem, Zamﬁrescu’s ﬁxed point theorem, as well as of many other related results obtained
on the base of similar contractive conditions. These ﬁxed point theorems mentioned
before ensures the uniqueness of the ﬁxed point, but the Ćirić type ALC need not have
a unique ﬁxed point, according to Example 2.3.12.
2) Let us remind (see Rus, [88],[89]) that an operator T:X→Xis said to be a
weakly Picard operator (WPO) if the sequence {Tnx0}∞
n=0converges for all x0∈X
and the limits are ﬁxed point of T. The main merit of Theorem 2.3.10 is the very
large class of weakly Picard operators assured by using it. Obviously, the ﬁxed point
x∗attained by the Picard iteration depends on the initial guess x0∈X. However, the
error estimate (2.3.5)obtained in Theorem 2.3.10 is weaker as (1.1.3),(1.1.4)from the
Banach’s Contraction Principle or the estimate given in Theorem 2.3.10.
Example 2.3.12 .LetAbe the set of positive functions
A={f|f: [0,∞)→[0,∞)}, which is the subset of all real functions X={f:R→R},
A⊂X.
Consider the pseudometric dj(f,g) =|f(0)−g(0)|·j,∀j∈J;J⊂N,∀f,g∈A.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 27
Indeed,djis a pseudometric, but is not a metric, take for example dj(x3,x2) = 0, but
x3/negationslash=x2.
Consider the mapping Tf=|f|,∀f∈A,r(j) =j+ 1. Note that the monotony
condition (2.3.12)also holds. By using condition (2.1.3)for ALC-s, we obtain for all
j∈J:
|f(0)−g(0)|·j≤θ·|f(0)−g(0)|·(j+ 1) +L·|g(0)−f(0)|·(j+ 1),∀f,g∈A,
which is equivalent to: j≤(θ+L)(j+ 1),∀j∈J.
This inequality is true if θ=1
3∈(0,1), L = 1>0.
Hence,Tis an almost local contraction. However, Tis not a contraction, because the
contractive condition (2.2.11)leads us to the false: 1≤θ.
The mapTis Ćirić-type ALC, because
Mr(j)(f,g) =|f(0)−g(0)|·(j+ 1),∀f,g∈A,∀j∈J.
From(2.3.4)we have the equivalent form
|f(0)−g(0)|·j≤θ·|f(0)−g(0)|·(j+ 1) +L·|f(0)−f(0)|·(j+ 1),
Again, we obtain the inequality j≤(θ+L)(j+ 1), valid forθ=1
3∈(0,1),L= 1≥0.
The mapping Thas an inﬁnite set of ﬁxed points: Fix(T) ={f∈A}=A, by taking:
|f(x)|=f(x),∀f∈A,∀x∈[0,∞).
The uniqueness condition (2.3.13)is not valid, having in view the equivalent form:
|f(0)−g(0)|·j≤θ·|f(0)−g(0)|·(j+ 1) +L1·|f(0)−f(0)|·(j+ 1),
which leads us to the contradiction j≤θ(j+ 1), i.e., the mapping Tnot satisfy the
uniqueness condition (2.3.13).
The uniqueness of the ﬁxed point for a Ćirić type ALC can be assured by imposing
an additional contractive condition, according to the next theorem.
Theorem 2.3.13 .Under the assumptions of Theorem 2.3.10, let T:A→Abe a
Ćirić type strong almost local contraction with the additional inequality, which actually
means the monotony property for the pseudometric:
(2.3.12) dr(j)(f,g)≤dj(f,g),∀f,g∈A,∀j∈J.
If the mapping Tsatisﬁes the supplementary condition: there exist the constants
θ∈(0,1)and someL1≥0such that
(2.3.13) dj(Tf,Tg )≤θ·dr(j)(f,g) +L1·dr(j)(f,Tf ),for all f,g∈A,∀j∈J,
then
(1)Thas a unique ﬁxed point, i.e., Fix(T) ={f∗};

28 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
(2) The Picard iteration {xn}∞
n=0given byxn+1=Txn, n∈Nconverges to f∗,
for anyx0∈A;
(3) The a priori error estimate (2.3.5)holds;
(4) The rate of the convergence of the Picard iteration is given by
(2.3.14) dj(xn,f∗)≤θ·dr(j)(xn−1,f∗), n = 1,2,…,∀j∈J.
Proof:1) Suppose, by contradiction, there are two distinct ﬁxed points f∗and
g∗ofT. Then, by using (2.3.13), and condition (2.3.12) for every ﬁxed j∈Jwith
f:=f∗,g:=g∗we get:
dj(f∗,g∗)≤θ·dr(j)(f∗,g∗)≤θ·dj(f∗,g∗)⇔(1−θ)·dj(f∗,g∗)≤0,
which is obviously a contradiction with dj(f∗,g∗)>0. So, we prove the uniqueness of
the ﬁxed point.
The proof for 2) and 3) is quite similar to the proof from the Theorem 2.3.10.
4) At this point, letting g:=xn,f:=f∗in (2.3.13), it results the rate of convergence
given by (2.3.14). The proof is complete.
The contractive conditions (2.3.4) and (2.3.13) can be merged to maintain the
uniqueness of the ﬁxed point, stated by the next theorem.
Theorem 2.3.14 .Under the assumptions of deﬁnition 2.3.8, let T:A→Abe a
mapping for which there exist the constants θ∈(0,1)and someL≥0such that
for allf,g∈Aand∀j∈Jwe have:
dj(Tf,Tg )≤θ·Mr(j)(f,g) +
+L·min{dr(j)(f,Tf ),dr(j)(g,Tg ),dr(j)(f,Tg ),dr(j)(g,Tf )}, (2.3.15)
where
Mr(j)(f,g) = max{dr(j)(f,g),dr(j)(f,Tf ),dr(j)(g,Tg ),dr(j)(f,Tg ),dr(j)(g,Tf )}.
Then
(1)Thas a unique ﬁxed point, i.e., Fix(T) ={f∗};
(2) The Picard iteration {xn}∞
n=0given byxn+1=Txn, n∈Nconverges to f∗,
for anyx0∈A;
(3) The a priori error estimate (2.3.5)holds.
Particular case 1.(1) The famous Ćirić’ s ﬁxed point theorem for single
valued mappings given in [39]can be obtained from theorems 2.3.10, 2.3.14,
2.3.13 by taking L=L1= 0and considering rthe identity mapping: r(j) =j.
The Ćirić’ s contractive condition represents one of the most general metri-
cal condition that provide a unique ﬁxed point by means of Picard iteration.
Despite this observation, the contractive condition given for Ćirić-type ALC

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 29
(in(2.3.4)) admit a very high level of generalization. Note that the ﬁxed point
could be approximated by means of Picard iteration, just like in the case of
Ćirić’ s ﬁxed point theorem, although the uniqueness of the ﬁxed point is not
ensured by using (2.3.4).
(2) If the maximum from Theorem 2.3.14 becomes:
max/braceleftBig
dr(j)(f,g),dr(j)(f,Tf ),dr(j)(g,Tg ),dr(j)(f,Tg ),dr(j)(g,Tf )/bracerightBig
=dj(f,g),
for allf,g∈A, then we can easily obtain Theorem 2.1.9 from Theorem 2.3.10.
Also, by Theorem 2.3.13 we obtain Theorem 2.1.12 (see [103]).
After this presentation of the existence and the uniqueness theorems of the ﬁxed
points for the Ćirić-type ALC-s, it is natural to extend it to the Ćirić-type strict almost
local contractions.
Definition 2.3.15 .LetXbe a uniform space and D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X. We letτbe the weak topology on Xdeﬁned by the family
D. Consider a function r:J→Jand take a nonempty, τ-bounded, sequentially
τ-complete, and T-invariant subset A⊂X. LetT:A→Abe an ALC with regard
to (D,r), satisfying condition (2.1.5). Letr:J→J.The operator T:A→Ais
called Ćirić-type strict ALC relating to ( D,r) if it simultaneously satisﬁes conditions
(Ci−ALC )and (ALC−U), with some real constants θC∈(0,1),LC≥0and
θu∈(0,1),Lu≥0, respectively.
(Ci−ALC )dj(Tf,Tg )≤θC·Mr(j)(f,g) +LC·dr(j)(g,Tf ),for all f,g∈A,
for everyj∈J, where
Mr(j)(f,g) = max/braceleftBig
dr(j)(f,g),dr(j)(f,Tf ),dr(j)(g,Tg ),dr(j)(f,Tg ),dr(j)(g,Tf )/bracerightBig
.
(ALC−U)dj(Tf,Tg )≤θu·dr(j)(f,g) +Lu·dr(j)(f,Tf ),for all f,g∈A,∀j∈J,
We present an illustrative example ( 2.3.16) for our results: Ćirić’ type almost local
contractions, without having unique ﬁxed point.
Example 2.3.16 .By taking the mapping from Example 2.1.11, with a small mod-
iﬁcation, that is: let Xbe the set of positive functions
X={f|f: [0,∞)→[0,∞)},
which is a subset of the real functions F={f:R→R}.
Fixx0∈[0,∞)and take the pseudometric dj(f,g) =|f(x0)−g(x0)|·e−j,∀f,g∈
X,r(j) =j+ 2,∀j∈N.
We can conclude in the same manner that Tis also a Ćirić type ALC, i.e., it satisfy
the contractive condition (2.3.4).

30 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Indeed, we have Mr(j)(f,g) =|f(x0)−g(x0)|·e−(j+2). In this case, the condition (2.3.4)
becomes the contractive condition for almost local contractions (2.1.9).
By considering L= 0in the deﬁnition 2.3.8 of Ćirić-type ALC, we get a new type
of ALC, that is, the quasi-almost local contraction.
c)Quasi-Almost Local Contractions
Definition 2.3.17 .LetXbe a uniform space. Consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X. Take a function r:J→J.
The operator T:A→Ais called quasi-almost local contraction with regard to ( D,r)
if, for every j∈J, there exists the constant θ∈(0,1)such that
(2.3.16) dj(Tf,Tg )≤θ·Mr(j)(f,g),for all f,g∈A,
where
Mr(j)(f,g) = max{dr(j)(f,g),dr(j)(f,Tf ),dr(j)(g,Tg ),dr(j)(f,Tg ),dr(j)(g,Tf )}.
Theorem 2.3.18 .LetXbe a uniform space. Consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X.
Consider a function r:J→J, let a nonempty, τ-bounded, sequentially τ-complete,
andT-invariant subset A⊂Xand letT:A→Abe quasi-ALC with regard to ( D,r).
Then
(1)Thas a ﬁxed point , i.e., Fix(T) ={x∈X:Tx=x}/negationslash=φ;
(2) For any x0=x∈A, the Picard iteration {xn}∞
n=0converges to x∗∈Fix(T);
(3) The following a priori estimate is available:
(2.3.17) dj(xn,x∗)≤θn
(1−θ)2dj(x,Tx ), n = 1,2…
Proof:Obviously, we have to follow the steps from the proof of Theorem 2.3.10,
with the only diﬀerence that the constant L= 0, as in the case of quasi ALC-s.
The uniqueness of the ﬁxed point is also assured by imposing an additional condi-
tion, just like in the class of Ćirić-type ALC, as it follows.
Theorem 2.3.19 .With the assumptions of Theorem 2.3.10, let T:A→Abe a
quasi-almost local contraction with the additional inequality:
(2.3.18) dr(j)(f,g)≤dj(f,g),∀f,g∈A,∀j∈J.
If the mapping Tsatisﬁes the supplementary condition: there exist the constants
θ∈(0,1)such that
(2.3.19) dj(Tf,Tg )≤θ·dr(j)(f,g) +L1·dr(j)(f,Tf ),for all f,g∈A,∀j∈J,
then
(1)Thas a unique ﬁxed point, i.e., Fix(T) ={f∗};

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 31
(2) The Picard iteration {xn}∞
n=0given byxn+1=Txn, n∈Nconverges to f∗,
for anyx0∈A;
(3) The a priori error estimate (2.3.5)holds;
(4) The rate of the convergence of the Picard iteration is given by
(2.3.20) dj(xn,f∗)≤θ·dr(j)(xn−1,f∗), n = 1,2,…,∀j∈J.
d)Ćirić-Reich-Rus type almost local contraction
Definition 2.3.20 .LetXbe a uniform space. Consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X. Take a function r:J→J.
The operator
T:A→Ais called Ćirić-Reich-Rus type ALC regarding to ( D,r) if the mapping
T:A→Asatisfying the condition
(2.3.21) dj(Tf,Tg )≤δ·dr(j)(f,g) +L·[dr(j)(f,Tf ) +dr(j)(g,Tg )],
for allf,g∈A, whereδ,L∈R+andδ+ 2L<1.
Theorem 2.3.21 .If the pseudometric dsatisfy the condition:
dr(j)(f,g)≤dj(f,g),∀j∈J,∀f,g∈A,then any Ćirić-Reich-Rus type almost local
contraction, i.e., any mapping T:A→Asatisfying condition (2.3.21)with 0<L< 1
andδ+L
1−L∈(0,1), is an ALC.
Proof:Using condition (2.3.21) and the triangle inequality, we get
dj(Tf,Tg )≤δ·dr(j)(f,g) +L·[dr(j)(f,Tf ) +dr(j)(g,Tg )]≤
≤δ·dr(j)(f,g) +L·[dr(j)(g,Tf ) +dr(j)(Tf,Tg ) +dr(j)(f,g) +dr(j)(g,Tf )]
The condition for the pseudometric leads us to:
dj(f,g)>dr(j)(f,g),
dj(Tf,Tg )>dr(j)(Tf,Tg ),
dj(g,Tf )>dr(j)(g,Tf ).
At this point, we get after simple computations:
(2.3.22) (1−L)·dj(Tf,Tg )≤(δ+L)·dj(f,g) + 2L·dr(j)(g,Tf )
and which implies
(2.3.23) dj(Tf,Tg )≤δ+L
1−L·dj(f,g) +2L
1−L·dr(j)(g,Tf ),∀f,g∈A
Considerδ,L∈R+andδ+ 2L<1, the inequality (2.3.21) holds, with
δ+L
1−L∈(0,1)and2L
1−L≥0. Therefore, anyĆirić-Reich-Rustypealmostlocalcontraction
with the condition for the pseudometric, is an ALC.
e)Chatterjea-type ALC

32 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Definition 2.3.22 .LetXbe a uniform space. Consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X. Take a function r:J→J.
The operator
T:A→Ais called Chatterjea-type almost local contraction with regard to ( D,r) or
(δ,L)-Chatterjea contraction if there exists a constant 0≤c<1
2such that
(2.3.24) dj(Tf,Tg )≤c·[dr(j)(f,Tg ) +dr(j)(g,Tf )],∀f,g∈A.
Theorem 2.3.23 .If the pseudometric dsatisfy the monotony property:
(2.3.25) dr(j)(f,g)≤dj(f,g),∀j∈J,∀f,g∈A,
then any Chatterjea-type ALC, i.e., any mapping T:A→Asatisfying the condition
(2.3.24)is an almost local contraction.
Proof:Using condition (2.3.24) and the triangle inequality, we get
dj(Tf,Tg )≤c·[dr(j)(f,Tg ) +dr(j)(g,Tf )]≤
≤c[dr(j)(f,g) +dr(j)(g,Tf ) +dr(j)(Tf,Tg )] +c·dr(j)(g,Tf ).
The monotony of the pseudometric leads us to:
(2.3.26) dr(j)(g,Tf )≤dj(g,Tf ).
At this point, we get:
(2.3.27) dj(Tf,Tg )≤c[dr(j)(f,g) +dr(j)(g,Tf )] +c[dr(j)(g,Tf ) +dj(Tf,Tg )]
(1−c)dj(Tf,Tg )≤c·dr(j)(f,g) + 2c·dr(j)(g,Tf ),
and which implies
(2.3.28) dj(Tf,Tg )≤c
1−c·dj(f,g) +2c
1−c·dr(j)(g,Tf ),∀f,g∈A.
Consider 0≤c<1
2, the inequality (2.3.24) holds, with δ=c
1−candL=2c
1−c.
Therefore, any Chatterjea-type ALC with the monotony condition for the pseudomet-
ric, is an almost local contraction.
f)Zamﬁrescu-type ALC
Definition 2.3.24 .A mapT:A→Afor which at least one of the following is
true:
(z1)Tis an almost local contraction;
(z2)Tis a Kannan type almost local contraction;
(z3)Tis a Chatterjea type ALC,
is called Zamﬁrescu-type almost local contraction.
Now, it is natural to state the next result:
Proposition 2.3.25 .Any Zamﬁrescu-type ALC is an almost local contraction.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 33
g)Generalized Berinde-type ALC
Definition 2.3.26 .LetXbe a uniform space. Consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X. Take a function r:J→J.
The operator T:A→Ais called generalized Berinde-type ALC regarding to ( D,r) if
there exist a constant θ∈(0,1)and a function b:A→[0,∞)such that
(2.3.29) dj(Tx,Ty )≤θ·dr(j)(x,y) +b(y)·dr(j)(y,Tx ),∀x,y∈A,∀j∈J.
Theorem 2.3.27 .LetXbe a uniform space. Consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X. Take a function r:J→J.
LetTbe a mapping on the subset A. Consider a function b:A→[0,∞).Assume that
there exists θ∈(0,1)such that
(2.3.30)
(1 +θ)−1dj(x,Tx )≤dj(x,y)implies d j(Tx,Ty )≤θdr(j)(x,y) +b(y)dr(j)(y,Tx ),
for allx,y∈A. Then, for every x∈A, the sequence{Tnx}converges to a ﬁxed point
ofT.
Proof:Sinceθ∈(0,1), we have the inequality
(2.3.31) (1 +θ)−1dj(x,Tx )≤dj(x,Tx ),∀j∈J,
and we get
(2.3.32)dj(Tx,T2x)≤θdr(j)(x,Tx ) +b(Tx)dr(j)(Tx,Tx ) =θdr(j)(x,Tx ),∀x∈A.
Letu∈A, then from (2.3.32) we have
dj(Tnu,Tn+1u)≤θndrn(j)(u,Tu ),∀j∈J.
Ifdrn(j)(u,Tu )<C(constant ), then we obtain:
∞/summationdisplay
n=1dj(Tnu,Tn+1u)<∞.
Thus, the sequence {Tnu}isdj-Cauchy for each j. As the subset Ais assumed to
be sequentially τ-complete, there exists zinAsuch that (Tnx)n∈Nisτ-convergent to
z∈A.
Using (2.3.32), we can ﬁnd a subsequence {f(n)}of the sequence{n}such that
(1 +θ)−1dj(Tf(n)u,Tf(n)+1u)≤dj(Tf(n)u,z).
By (2.3.30), we have
dj(z,Tz ) = limn→∞dj(Tf(n)+1u,Tz )≤
≤limn→∞/parenleftBig
θdr(j)(Tf(n)u,z) +b(z)dr(j)(Tf(n)+1u,z)/parenrightBig
=
=θdr(j)(z,z) +b(z)dr(j)(z,z) = 0.
Therefore,zis a ﬁxed point of T.

34 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Corollary 2.3.28 .IfTis a generalized ALC on the subset A⊂X, then for every
x∈A, the sequence{Tnx}converges to a ﬁxed point of T.
Corollary 2.3.29 .LetTbe a generalized ALC on A⊂X. Assume that there
existθ∈(0,1)andB∈[0,∞)such that
(1 +θ)−1dj(x,Tx )≤dj(x,y)implies d j(Tx,Ty )≤θdr(j)(x,y) +Bdr(j)(Tx,y ),
for allx,y∈A. Then for every x∈A, the sequence{Tnx}converges to a ﬁxed point
ofT.
3.1. Almost local ϕ-contractions
We extend the ALC-s to the more general class of almost local ϕ-contractions. The
aim of this subsection is to study the properties and the existence of ﬁxed points of
this new type of ALC-s. First, let us recall some results and notions introduced by
Berinde in ([ 10]).
Definition 2.3.30 .[10]A mapϕ:R+→R+is called comparison function if it
satisﬁes:
(iϕ)ϕis monotone increasing, i.e., t1<t2⇒ϕ(t1)≤ϕ(t2);
(iiϕ)the sequence{ϕn(t)}∞
n=0converges to zero, for all t∈R+, whereϕnstands for
thenthiterate ofϕ.
Ifϕsatisﬁes (iϕ)and
(iiiϕ)the series
∞/summationdisplay
k=0ϕk(t)
converges for all t∈R+, thenϕis said to be a (c)-comparison function.
The following Lemma was proved in [ 19], it contains a few properties of (c)-
comparison functions.
Lemma 2.3.31 .[19]Ifϕ:R+→R+is ac-comparison function, then the following
hold:
(i)ϕis a comparison function;
(ii)ϕ(t)<t,for anyt∈R+;
(iii)ϕis continuous at zero;
(iv)the series∞/summationtext
k=0ϕk(t)converges for any t∈R+.
According to Berinde’s work (see [ 19]),ϕsatisﬁes (iiiϕ)if and only if there exist
0<c< 1and a convergent series of positive terms,∞/summationtext
n=0unsuch that
ϕk+1(t)≤c·ϕk(t) +uk,for allt∈R+andk≥k0(ﬁxed).

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 35
Also, it was proved: if ϕis a (c)- comparison function, then the sum of the comparison
series, that is,
(2.3.33) s(t) =∞/summationdisplay
k=0ϕk(t),t∈R+
is monotone increasing and continuous at zero, hence
(2.3.34) ϕk(t)→0ask→∞,∀t∈R+,
and that any (c)-comparison function is a comparison function.
The concept of (c)-comparison function was reformulated in [ 25] to that of b-
comparison function, as it follows:
Definition 2.3.32 .[25]Letb≥1be a real number. A map ϕ:R+→R+is called
b-comparison function if it satisﬁes:
(iϕ)ϕis increasing;
(iiϕ)there exist k0∈N,a∈(0,1)and a convergent series of non-negative terms
∞/summationtext
n=0vksuch that
(2.3.35) bk+1ϕk+1(t)≤abkϕk(t) +vk,
fork≥k0and anyt∈R+.
Remark 2.3.33 .Obviously, for b= 1, the concept of b-comparison function reduces
to that of (c)-comparison function.
Lemma 2.3.34 .[24]Ifϕ:R+→R+is ab-comparison function, then :
(i)the series∞/summationtext
k=0bkϕk(t)converges for any t∈R+;
(ii)The function sb:R+→R+deﬁned by
(2.3.36) sb(t) =∞/summationdisplay
k=0bkϕk(t),t∈R+
is increasing and continuous at zero.
Lemma 2.3.35 .(see[73]) Letϕ:R+→R+be ab-comparison function, with
constantb≥1andan∈R+,n∈Nsuch thatan→0asn→∞.
Then
(2.3.37)n/summationdisplay
k=0bn−kϕn−k(ak)→0asn→∞.
Almostϕ-contractions were ﬁrst introduced by V. Berinde in [ 20]. The aim of this
section is to extend almost ϕ-contractions to the more general case of almost local
ϕ-contractions.

36 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Definition 2.3.36 .(Berinde, [20]) Let (X,d)be a metric space. A self operator
T:X→Xis said to be an almost ϕ-contraction or (ϕ,L)-almost contraction, provided
that there exist a comparison function ϕand someL≥0, such that
(2.3.38) d(Tx,Ty )≤ϕ(d(x,y)) +L·d(y,Tx ),for allx,y∈X.
At this point, it is natural the extension of almost ϕ-contractions to the almost
localϕ-contractions..
Definition 2.3.37 .LetXbe a uniform space and D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X. We letτbe the weak topology on Xdeﬁned by the family
D. Take a nonempty, τ-bounded, sequentially τ-complete, and T-invariant subset A⊂
X. Letr:J→J, and letA⊂Xbe aτ-bounded sequentially τ-complete and T-
invariant subset of X. A mapping T:A→Ais called almost local ϕ-contraction if
there exist a comparison function ϕand someL≥0such that we have:
(2.3.39) dj(Tx,Ty )≤ϕ(dr(j)(x,y)) +L·dr(j)(y,Tx ),∀x,y∈A,∀j∈J.
Remark 2.3.38 .It is obvious that any almost local ϕ-contraction becomes an ALC
if we takeϕ(t) =θt,t∈R+and0<θ< 1.
Example 2.3.39 .Letϕ(t) =t
t+ 1, t∈R+, withr(j) =jand letTan almost
localϕ-contraction, i.e., a mapping which satisﬁes (2.3.39).
Thenϕis a nonlinear comparison function, but not veriﬁes the condition for the (c)-
comparison function. In this case, Tis an almost local ϕ-contraction without being an
ALC.
Remark 2.3.40 .1) Similar to the case of ALC-s, the fact that Tsatisﬁes(2.3.38),
for allx,y∈A, imply that the following dual inequality
(2.3.40) dj(Tx,Ty )≤ϕ(dr(j)(x,y)) +L·dr(j)(x,Ty ),∀x,y∈A,∀j∈J
obtained from (2.3.38)by replacing xwithyand vice versa, is also valid.
Hence, in order to prove that a mapping Tis an almost local ϕ-contraction, it is
necessary to check both inequalities (2.3.39)and(2.3.40).
2) The class of almost local ϕ-contractions includes a wide class of mappings, see for
Example 2.3.41, which contains a mapping with not one, but an inﬁnite number of ﬁxed
points.
Example 2.3.41 .Let[0,1]be the unit interval with the euclidean metric and the
operatorT: [0,1]→[0,1]the identity map, i.e., Tx=x, for allx∈[0,1]. By taking
ϕ(t) =a·t, t∈R,0<a< 1, θ =a, r (j) =jandL≥1−a, condition (2.1.7)
leads to
|x−y|≤a·|x−y|+L·|y−x|,

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 37
which is valid for all x,y∈[0,1]. Note that the mapping Thas an inﬁnite set of ﬁxed
points:
Fix(T) ={x∈[0,1] :Tx=x}= [0,1].
The next two theorems represent an existence theorem and, respectively, a unique-
ness theorem for the ﬁxed points of almost local ϕ-contractions.
Theorem 2.3.42 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, with the the monotony property fulﬁlled:
dr(j)(x,y)≤dj(x,y),∀j∈J,∀x,y∈A.
Consider a nonempty, τ-bounded, sequentially τ-complete, and T-invariant subset A⊂
X. We letτbe the weak topology on Xdeﬁned by the family D. Letr:J→Jand
T:A→Aan almost local ϕ-contraction with ϕa (c)-comparison function.Then
(1)Fix(T) ={x∈A:Tx=x}/negationslash=φ;
(2) For any x0∈A, the Picard iteration {xn}∞
n=0deﬁned byx0∈Aand
(2.3.41) xn+1=Txn, n = 0,1,2,…;
converges to a ﬁxed point x∗∈Fix(T)
(3) The a posteriori estimate
(2.3.42) dj(xn,x∗)≤s(dj(xn,xn+1)), n = 0,1,2…,∀x,y∈A,∀j∈J
holds, where s(t)is given by (2.3.33).
Proof:We shall prove that the set of ﬁxed points of Tis nonempty. Using the fact
thatTis an almost local ϕ-contraction, there exist a (c)-comparison function ϕand
someL≥0, such that
(2.3.43) dj(Tx,Ty )≤ϕ(dr(j)(x,y)) +L·dr(j)(y,Tx ),∀x,y∈A,∀j∈J
holds,∀x,y∈A. Letx0∈Abe arbitrary and{xn}∞
n=0be the Picard iteration deﬁned
by
xn+1=Txn, n∈N.
Takex:=xn−1,y:=xnin (2.3.43) to obtain
dj(Txn−1,Txn)≤ϕ(dr(j)(xn−1,xn)),
which yields
(2.3.44) dj(xn,xn+1)≤ϕ(dr(j)(xn−1,xn)),∀j∈J,∀n= 1,2,…
Sinceϕis increasing, the monotony of the pseudometric leads to:
(2.3.45) dj(xn+1,xn+2)≤ϕ(dr(j)(xn,xn+1))≤ϕ(dj(xn,xn+1)),∀x,y∈A,∀j∈J.

38 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
From that, we obtain by induction with related to n:
(2.3.46) dj(xn+k,xn+k+1)≤ϕk(dj(xn,xn+1)),∀x,y∈A,∀j∈J.
According to the triangle inequality, we get:
dj(xn,xn+p)≤dj(xn,xn+1) +dj(xn+1,xn+2) +…+dj(xn+p−1,xn+p)
≤r+ϕ(r) +…+ϕn+p−1(r),∀x,y∈A,∀j∈J, (2.3.47)
where we denoted r=dj(xn,xn+1). Again, by (2.3.45), we conclude
(2.3.48) dj(xn,xn+1)≤ϕn(dj(x0,x1)),n= 0,1,2,…
which, by property (iiϕ)from the deﬁnition (2.3.30) of a comparison function, implies
(2.3.49) limn→∞dj(xn,xn+1) = 0.
Having in view that ϕis positive, it is obvious that
(2.3.50) r+ϕ(r) +…+ϕn+p−1(r)<s(r),
wheres(t)is the sum of the series
s(t) =∞/summationdisplay
k=0ϕk(r),t∈R+.
Then by (2.3.50) and (2.3.47) we get
(2.3.51) dj(xn,xn+p)≤s(dj(xn,xn+1)),n∈N,p∈N,∀x,y∈A,∀j∈J.
Sincesis continuous at zero, (2.3.50) and (2.3.49) implies that the sequence {xn}n∈N
isdj-Cauchy for each j∈J. As the subset Ais assumed to be sequentially τ-complete,
there exists x∗inAsuch that the sequence {xn}isτ-convergent to x∗. We shall prove
thatx∗is a ﬁxed point of T. From the triangle inequality, we have:
dj(x∗,Tx∗)≤dj(x∗,xn+1) +dj(xn+1,Tx∗) =
=dj(xn+1,x∗) +dj(Txn,Tx∗),∀x,y∈A,∀j∈J.
By (2.1.7) we have
dj(Txn,Tx∗)≤ϕ(dr(j)(xn,x∗)) +L·dr(j)(x∗,Txn),∀x,y∈A,∀j∈J,
and hence
(2.3.52) dj(x∗,Tx∗)≤(1 +L)dr(j)(xn+1,x∗) +ϕ(dr(j)(xn,x∗)),
valid for all n= 0,1,2,…
Now letting n→∞in (2.3.52) and using the continuity of ϕat zero, we conclude
dj(x∗,Tx∗) = 0,
which means that x∗is a ﬁxed point of T, it follows that the Picard iteration converges
to a ﬁxed point x∗∈Fix(T).

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 39
The estimate (2.3.42) follows from (2.3.47) by taking p→∞.
This completes the proof.
Remark 2.3.43 .1) The a posteriori error estimates (2.3.42)and(2.3.48)leads us
to the a priori estimate for the Picard iteration {xn}∞
n=0.
2) By taking ϕ(t) =θ·t,t∈R+,0<θ< 1, from Theorem 2.3.42 we obtain the same
result for ALC-s, namely Theorem 2.1.9.
3) An almost local ϕ-contraction may have more than one ﬁxed point, as shown by
Example 2.3.41. In Theorem 2.3.42, the Picard iteration {xn}∞
n=0provide the ﬁxed
pointx∗, but it generally depends on the initial guess x0.
For the uniqueness of the ﬁxed point of Tit is necessary an additional condition, as
shown in Theorem 2.3.44.
Theorem 2.3.44 .LetXandTas in Theorem 2.3.42, the subset A⊂X. Assume
the monotony property for the pseudometric fulﬁlled:
dr(j)(x,y)≤dj(x,y),∀j∈J,∀x,y∈A.
SupposeTalso satisﬁes the following condition: there exist a comparison function Υ
and someL1>0such that
(2.3.53) dj(Tx,Ty )≤Υ(dr(j)(x,y)) +L1·dr(j)(x,Tx )
valid for all x,y∈A,∀j∈J.
Then
(1)Thas a unique ﬁxed point, i.e., Fix(T) ={x∗};
(2) The a posteriori error estimate
d(xn,x∗)≤s(d(xn,xn+1)), n = 0,1,2…
holds, where s(t)is given by (2.3.33);
(3) The rate of the convergence of the Picard iteration is given by
(2.3.54) dj(xn,x∗)≤Υ(dj(xn−1,x∗)), n = 1,2…
Proof:Suppose, by contradiction, there are two diﬀerent ﬁxed points x∗andy∗of
T. Then from (2.3.53), by taking x:=x∗andy:=y∗, we obtain
dj(x∗,y∗)≤Υ (dr(j)(x∗,y∗))
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
≤dj(x∗,y∗))≤Υ(dj(x∗,y∗)),∀x,y∈A,∀j∈J,
which by induction with regarding to n, yields
(2.3.55) dj(x∗,y∗)≤Υn(dj(x∗,y∗)),n= 1,2,…
Lettingn→∞in (2.3.55), we get
dj(x∗,y∗) = 0

40 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
which means that x∗=y∗, a contradiction.
In this way, we proved that the ﬁxed point is unique.
For the proof of (2.3.54), all we have to do is to change x:=x∗andy:=xnin the
inequality (2.3.53).
The proof is complete.
Remark 2.3.45 .1) Having in view the pairs of dual conditions (2.1.3)and(2.1.4),
respectively (2.3.39)and(2.3.40), condition (2.3.53)holds if and only if its dual
(2.3.56) dj(Tx,Ty )≤Υ(dr(j)(x,y)) +L1·dr(j)(y,Ty ),
is valid for all x,y∈A,∀j∈J.
2) Condition (2.3.53)is not necessary for the uniqueness of the ﬁxed point, according
to Example 2.3.46.
Example 2.3.46 .LetX= [1,n]×[1,n]⊂R2. Consider the pseudometric:
(2.3.57) dj((x1,y1),(x2,y2)) =|x1−x2|·e−j,∀j∈J,
whereJis a subset of N. Consider r(j) =j+ 1and the mapping T:X→X,
T(x,y) =

(x,−y)if (x,y)/negationslash= (1,1)
(0,0)if (x,y) = (1,1)
Let the (c)-comparison function: ϕ(t) =1
4t,ϕ:R+→R+,∀t∈R+.
Tis an almost local ϕ-contraction if:
|x1−x2|·e−j≤1
4·|x1−x2|·e−(j+1)+L·|x2−x1|·e−(j+1),
which is equivalent to : e≤1
4+L.
ForL= 3≥0, the last inequality becomes true, i.e., Tis an almost local ϕ-contraction
with an inﬁnite set of ﬁxed points: Fix(T) ={(x,0) :x∈[1,n]}.
Remark 2.3.47 .IfThas a unique ﬁxed point x∗and the Picard iteration {Tnx0}∞
n=0
converges to x∗for allx0∈A, then by Theorem 2.3.4, for any a∈(0,1), there exists
a pseudometric /rho1onAsuch that (X,/rho1)isτ-complete and Tis ana-contraction with
regard to the pseudometric /rho1.
Therefore, condition (2.3.53)could be reformulated using another pseudometric, which
leads us to a more general result.
Theorem 2.3.48 .LetXbe a uniform space, let A⊂Xbe aτ-bounded sequentially
τ-complete and T-invariant subset of Xandd,/rho1two pseudometrics on Asuch that
(X,d)isτ-complete.
LetT:A→Abe a self operator satisfying
(i)There exist a (c)-comparison function ϕandL≥0such that
dj(Tx,Ty )≤ϕ(dr(j)(x,y)) +L·dr(j)(y,Tx ),∀x,y∈A,∀j∈J;

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 41
(ii)There exist a comparison function ΥandL1≥0such that
/rho1j(Tx,Ty )≤Υ(/rho1r(j)(x,y)) +L1·ρr(j)(x,Tx ),∀x,y∈A,∀j∈J.
Then
(1)Thas a unique ﬁxed point, i.e., Fix(T) ={x∗};
(2) For any x0∈A, the Picard iteration {xn}∞
n=0deﬁned byx0∈Aand
xn+1=Txn, n = 0,1,2,…
converges to a ﬁxed point x∗∈Fix(T);
(3) The a posteriori error estimate
dr(j)(xn,x∗)≤s(dr(j)(xn,xn+1)), n = 0,1,2…,∀x,y∈A,∀j∈J
holds, where s(t)is given by (2.3.33);
(4) The rate of the convergence of the Picard iteration is given by
/rho1j(xn,x∗)≤Υ(/rho1j(xn−1,x∗)),∀x,y∈A,∀j∈J, n = 1,2…
Remark 2.3.49 .We obtain Theorem 2.3.44 as a particular case of Theorem 2.3.48
if we setdj≡/rho1j.
In order to extend the class of almost local ϕ-contractions, we try at ﬁrst with the
quasiϕ-contractions, as shown below.
Definition 2.3.50 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X. We letτbe the weak topology on Xdeﬁned by the family
D. Letr:J→J, and let the subset A⊂Xbe aτ-bounded sequentially τ-complete
andT-invariant subset of X. A mapping T:A→Ais called quasi ϕ-contraction if
there exists a comparison function ϕsuch that we have:
(2.3.58) dj(Tx,Ty )≤ϕ(dr(j)(x,y)),∀x,y∈A,∀j∈J.
Theorem 2.3.51 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X. We letA⊂Xa subset of X. We letτbe the weak
topology on Xdeﬁned by the family D. Letr:J→JandT:A→Abe a quasi
ϕ-contraction: a mapping which satisﬁes (2.3.58)withϕa comparison function. Then
Thas a unique ﬁxed point if there exists x0∈Asuch that the Picard iteration {xn}∞
n=0
deﬁned by
xn+1=Txn, n∈N
is bounded.

42 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Proof:If the sequence of the successive approximations {xn}∞
n=0is bounded, for a
certainx0∈A, it means that there exist a constant k>0and an element u∈Asuch
that
(2.3.59) dj(xn,u)≤k,∀n∈N;j∈J.
By choosing m,n∈N, we have from the triangle inequality:
dj(xn,xm)≤dj(xn,u) +dj(u,xm)≤2k.
After simple computations involving (2.3.58) and the deﬁnition of the Picard iteration,
we inductively obtain
dj(xn,xn+p) =dj(Txn−1,Txn+p−1)≤ϕ(dr(j)(xn−1,xn+p−1))≤
≤···≤ϕn−1(dr(j)(x1,xp+1))≤ϕn−1(2k), n,p∈N.
Note thatϕis a comparison function and according to (2.3.34), we conclude that the
sequence{xn}n∈Nisdj-Cauchy for each j∈J.
As the subset Ais assumed to be sequentially τ-complete, there exists x∗inAsuch
that{xn}n∈Nisτ-convergent to x∗. Then
limn→∞dj(xn,x∗) = 0.
By using again the triangle inequality, we obtain
0≤dj(Tx∗,x∗)≤dj(Tx∗,Txn) +dj(Txn,x∗)≤
≤ϕ/parenleftBig
dr(j)(x∗,xn)) +dj(xn+1,x∗)/parenrightBig
.
This means
dj(Tx∗,x∗) = 0,
because the comparison function ϕis continuous at zero. It follows that x∗is a ﬁxed
point ofT, hencex∗∈Fix(T).
The proof is complete.
Remark 2.3.52 .The converse of theorem 2.3.51 is also true:
If the mapping Thas a unique ﬁxed point, denoted by x∗, then the sequence of the
successive approximations is bounded. Indeed, by choosing x0=x∗, the sequence{xn}
is constant, therefore it is bounded.
3.2. (B)-almost local contractions
In this section our goal is to eliminate the inconvenience of having non-symmetric
relations for the deﬁnition of ALC-s, that is,
ALC 1, d j(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈A

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 43
and, respectively,
ALC 2, d j(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,Ty ),∀x,y∈A
under the assumptions of Deﬁnition 2.1.6 given for ALC-s.
Remark 2.3.53 .IfALC 1holds for some θ∈(0,1),L≥0for anyx,y∈A,A⊂X,
then its dual is valid for all x,y∈Aas well, and vice versa. That means, (ALC 1)and
(ALC 2)are equivalent, if they hold for any x,y∈A,A⊂X.
Remark 2.3.54 .A very important beneﬁt of this equivalence is represented by the
comfort of verifying only one of these conditions. However, in the case of xandyﬁxed,
it is compulsory to check both ALC 1andALC 2in order to clarify whether Tsatisﬁes
conditionALC 1for the particular values xandy, or not.
This inconvenient non-symmetry can be eliminate by introducing the following
results:
Lemma2.3.55 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of pseudo-
metrics deﬁned on X. Consider a nonempty, τ-bounded, sequentially τ-complete, and
T-invariant subset A⊂X. Consider a function r:J→J. An operator T:A→A
satisﬁes condition ALC 1with respect to (D,r)if and only if it satisﬁes condition
(m−ALC )dj(Tx,Ty )≤θ·dr(j)(x,y) +L·min{dr(j)(x,Ty ),dr(j)(y,Tx )},
for allx,y∈A, for allj∈J.
Proof:Without loss of generality, we can assume that L>0, asL= 0would lead
to the trivial case of Banach contractions.
”⇒”: In the beginning, suppose that Tsatisﬁes (ALC 1). By using Remark 2.3.53, it
also satisﬁes (ALC 2).
The inequality (ALC 1)becomes:
(2.3.60)1
L[dj(Tx,Ty )−θ·dr(j)(x,y)]≤dr(j)(y,Tx ),∀x,y∈A.
The other condition, ALC 2take the equivalent form:
(2.3.61)1
L[dj(Tx,Ty )−θ·dr(j)(x,y)]≤dr(j)(x,Ty ),∀x,y∈A.
The inequalities (2.3.60) and (2.3.61) can be merged to obtain
1
L[dj(Tx,Ty )−θ·dr(j)(x,y)]≤min{dr(j)(x,Ty ),dr(j)(y,Tx )},∀x,y∈A,
which is equivalent with (m−ALC ):
dj(Tx,Ty )≤θ·dr(j)(x,y) +L·min{dr(j)(x,Ty ),dr(j)(y,Tx )},
for anyx,y∈Aand∀j∈J.
”⇐” :Secondly, for the proof of the reciprocal assessment, suppose that Tsatisﬁes

44 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
(m−ALC ). Then, obviously it also fulﬁlls (ALC 1).
Hence, the conditions (ALC 1)and(m−ALC )are equivalent.
This completes the proof.
3.3. Approximate ﬁxed points
Theε-ﬁxed points of operators took a very constructive and practical approach
of ﬁxed point problems, since, in real situations, sometimes it is enough to obtain an
approximation of the solution. So, the existence of ﬁxed points is not strictly required,
but the proximity of ﬁxed points is of interest to researchers. This approximation is
also used when the conditions imposed for the existence of the ﬁxed points are too
strong.
Itisanaturalconsequencetointroducetheconceptsof ε-ﬁxed point (orapproximate
ﬁxed point ), which, in fact, represents a proximity ﬁxed point, and that of function
with the approximate ﬁxed point property and to establish qualitative and quantitative
theorems for various types of ALC-s.
In this section, the starting point is represented by the article of Tijs, Torre and
Branzei (see [ 101]) and also the paper of M. Berinde (see [ 8]).
Note that we consider operators on pseudometric spaces, not on metric or complete
metric spaces, the usual framework for ﬁxed point problems.
The following deﬁnition is very useful for the study of approximate ﬁxed points, it was
published in [ 53], probably attended by the work of Granas, Dugundji ([ 47]):
Definition 2.3.56 .([47]) Let (E,τ)be a topological space, αan open covering of
Eandf:E→Ean operator. Then x∈Eis calledα-ﬁxed point of fif there exist
U∈αsuch thatxandf(x)are inU.
The following deﬁnition was given by V.R.Klee [ 61] and it was mentioned also by
van der Walt [ 102]:
Definition 2.3.57 .[102]Let(E,τ)be a topological space, (X,d)be a metric space,
f:E→Xan operator, and ε>0. Thenfis calledε-continuous if each x∈Ehas a
neighborhood Usuch that
δ(f(U))≤ε,
whereδ(f(U))represents the diameter of the set f(U).
Definition 2.3.58 .(see[101]) Letf:X→X,ε> 0,×0∈X.
Then
(a)x0is anε-ﬁxed point (approximate ﬁxed point) of fif
d(f(x0),x0)<ε.
(b)fis calledε-operator if all x∈Xareε-ﬁxed points of f.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 45
Remark 2.3.59 .The set of all ε-ﬁxed points of f, for a given ε, will be denoted by
Fixε(f) ={x∈X|x is anε-ﬁxed point of f}.
Remark 2.3.60 .Hence, any ﬁxed point of fis also anε-ﬁxed point of f:
x∈Fix (f)⇒x∈Fixε(f),
but the converse is not always true, as shown in the example (Example 3.1.1in[72]).
Definition 2.3.61 .([101]) Letf:X→X. Thenfhas the approximate ﬁxed
point property if
∀ε>0,Fixε(f)/negationslash=?.
Remark 2.3.62 .The concept of asymptotically regular operator was ﬁrst introduced
in[33]in metric spaces, see also [60]and[88]. It seems to be very useful concept in
this section.
Lemma 2.3.63 .([8]) Let (X,d)be a metric space, f:X→Xsuch thatfis
asymptotic regular, i.e.,
d(fn(x),fn+1(x))→0asn→∞,∀x∈X.
Thenfhas the approximate ﬁxed point property.
Lemma 2.3.63 could be extended to a larger, more general space, such as the pseu-
dometric space, as in the following result:
Lemma 2.3.64 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, we choose a subset A⊂X. We letτbe the weak topology on Xdeﬁned
by the familyD. LetT:A→Aan asymptotic regular mapping. Then Thas the
approximate ﬁxed point property.
Proof:Fix an element x0∈A. SinceTis asymptotic regular, and having in view
the fact that the convergence with respect to the τ- topology implies convergence for
the pseudometric dr(j), we have
d(Tn(x),Tn+1(x))→0asn→∞⇔
∀ε>0,∃n0(ε)∈N∗such that∀n≥n0(ε),dj(Tn(x0),Tn+1(x0))<ε⇔
∀ε>0,∃n0(ε)∈N∗such that∀n≥n0(ε),dj(Tn(x0),T(Tn(x0)))<ε.
Denote:
y0=Tn(x0).
After simple computations, it follows that
∀ε>0,∃y0∈Asuch that dj(y0),T(y0))<ε.

46 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Thus, we have: for each ε>0, there exists an ε-ﬁxed point of Tin the subset A⊂X,
that is,y0.
So, we prove that Thas the approximate ﬁxed point property.
Remark 2.3.65 .There is an equivalence between the existence of ﬁxed points for a
given mapping and the approximate ﬁxed points of it, stated by the following result:
Proposition 2.3.66 .LetAbe a closed subset of a pseudometric space and
T:A→Aa compact map. Then Thas a ﬁxed point if and only if it has the ﬁxed
point property.
We will denote by δ(A)the diameter of the nonempty set A, namely:
δ(A) = sup{d(x,y)|x,y∈A}.
Lemma 2.3.67 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, we choose a subset A⊂X. We letτbe the weak topology on Xdeﬁned
by the familyD, letJa family of indices. Let T:A→Aan operator and ε > 0.
Suppose that:
(i)Fixε(T)/negationslash=?;
(ii)∀η>0,∃ϕ(η)>0such that
dj(x,y)−dj(T(x),T(y))≤η⇒dj(x,y)≤ϕ(η),∀x,y∈Fixε(T).
Then:
δ(Fε(T))≤ϕ(2ε).
Proof:Fixε>0andx,y∈Fε(T). By means of deﬁnition 2.3.58, we have:
dj(x,T(x))<ε,dj(y,T(y))<ε.
By using the triangle inequality, we obtain:
dj(x,y)≤dj(x,T(x)) +dj(T(x),T(y)) +dj(y,T(y))≤dj(T(x),T(y)) + 2ε.
This implies:
dj(x,y)−dj(T(x),T(y))≤2ε.
At this point, it follows from (ii):
dj(x,y)≤ϕ(2ε),
which means that
δ(Fixε(T))≤ϕ(2ε).
This completes the proof.
Remark 2.3.68 .According to Lemma 2.3.64, condition (i) from Lemma 2.3.67 can
be replaced by the asymptotic regularity condition.
Ergo, Lemma 2.3.67 can be reformulated as follows:

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 47
Lemma 2.3.69 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, we choose a subset A⊂X. We letτbe the weak topology
onXdeﬁned by the family D, letJa family of indices. Let T:A→Aan operator
such that for ε>0the following hold:
(i)
dj(Tn(x),Tn+1(x))→0asn→∞,∀x∈A.
(ii)∀η>0,∃ϕ(η)>0such that
dj(x,y)−dj(T(x),T(y))≤η⇒dj(x,y)≤ϕ(η),∀x,y∈Fixε(f).
Then:
δ(Fixε(T))≤ϕ(2ε).
A. Qualitative results for mappings in pseudometric spaces
Our ﬁrst goal is to state and prove, using Lemma 2.3.64, qualitative results for
a large class of operators deﬁned on pseudometric spaces. Most importantly, we are
interested in establishing conditions under which the approximate ﬁxed point property
is fulﬁlled for the mappings considered.
Theorem 2.3.70 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X. We letA⊂Xa subset of X. We letτbe the weak
topology on Xdeﬁned by the family D. Letr:J→J, withJa family of indices.
LetT:A→Aa quasi-almost local contraction, i.e., a mapping satisfying Deﬁnition
2.3.17.
Then:
∀ε>0,Fixε(T)/negationslash=?.
Proof:Takeε>0andx∈A.
dj(Tn(x),Tn+1(x)) =dj(T(Tn−1(x),T(Tn(x)))≤
≤θ·Mr(j)(Tn−1(x),Tn(x)))≤···≤θnMr(j)(x,T(x)),
where
Mr(j)(f,g) = max{dr(j)(f,g),dr(j)(f,Tf ),dr(j)(g,Tg ),dr(j)(f,Tg ),dr(j)(g,Tf )}.
Note thatθ∈(0,1), which implies θn→0asn→∞.
Hence, we have for all j∈J:
dj(Tn(x),Tn+1(x))→0,asn→∞,∀x∈A.
According to Lemma 2.3.64, it follows that Fixε(T)/negationslash=?,∀ε>0.

48 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Theorem 2.3.71 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, letA⊂Xa subset of X. We letτbe the weak topology
onXdeﬁned by the family D. Letr:J→J, withJa family of indices. Let
T:A→Aana-local contraction: a mapping for which there exists a∈(0,1)such
that:dj(Tx,Ty )≤a·dr(j)(x,y),for all x,y∈A.
Then:
∀ε>0,Fixε(T)/negationslash=?.
Proof:Letε>0andx∈A.
dj(Tn(x),Tn+1(x)) =dj(T(Tn−1(x),T(Tn(x)))≤
≤a·dr(j)(Tn−1(x),Tn(x))≤···≤an·dr(j)(x,T(x)).
Note thata∈(0,1), which implies that we have for all j∈J:
dj(Tn(x),Tn+1(x))→0,asn→∞,∀x∈A.
According to Lemma 2.3.64, it follows that Fixε(T)/negationslash=?,∀ε>0.
This completes the proof.
Theorem 2.3.72 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, letA⊂Xa subset of X. We letτbe the weak topology
onXdeﬁned by the family D. Letr:J→J, withJa family of indices. Let T:A→A
a Kannan-type ALC: a mapping for which there exists a∈[0,1
2)such that
(2.3.62) dj(Tx,Ty )≤a·[dr(j)(x,T(x)) +dr(j)(y,T(y))],for all x,y∈A,
The pseudometric satisﬁes the monotony property:
dr(j)(x,y)≤dj(x,y),∀j∈J,∀x,y∈A,∀j∈J,
Then:
∀ε>0,Fixε(T)/negationslash=?.
Proof:Letε > 0andx∈A. After following similar computations as in the
previous theorem, and by using the monotony of the pseudometric (2.3.12), we obtain:
dj(Tn(x),Tn+1(x)) =dj(T(Tn−1(x),T(Tn(x)))≤
≤a·[dr(j)(Tn−1(x),T(Tn−1(x))) +dr(j)(Tn(x),T(Tn(x)))] =
=a·dr(j)(Tn−1(x),Tn(x)) +a·dr(j)(Tn(x),Tn+1(x))≤
≤a·dr(j)(Tn−1(x),Tn(x)) +a·dj(Tn(x),Tn+1(x)),
which implies
(1−a)dj(Tn(x),Tn+1(x))≤a·dr(j)(Tn−1(x),Tn(x))⇒
dj(Tn(x),Tn+1(x))≤a
1−adr(j)(Tn−1(x),Tn(x))≤···≤/parenleftBiga
1−a/parenrightBigndr(j)(x,T(x)).

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 49
Having in view that a∈[0,1
2), this meansa
1−a∈[0,1), hence,/parenleftBig
a
1−a/parenrightBign→0asn→∞.
It results that we have for all j∈J:
dj(Tn(x),Tn+1(x))→0,asn→∞,∀x∈A.
According to Lemma 2.3.64, it follows that Fixε(T)/negationslash=?,∀ε>0.
This completes the proof.
Theorem 2.3.73 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X. We letA⊂Xa subset of X. We letτbe the weak
topology on Xdeﬁned by the family D. Letr:J→J, withJa family of indices.
LetT:A→Aa Chatterjea-type ALC: a mapping satisfying the condition: there exists
a constant 0≤c<1
2such that
dj(Tf,Tg )≤c[dr(j)(f,Tg ) +dr(j)(g,Tf )],∀f,g∈A.
If the monotony is valid for the pseudometric, namely (2.3.12),
then:
∀ε>0,Fixε(T)/negationslash=?.
Proof:Letε>0andx∈A.
After simple computations, and by using the monotony of the pseudometric (2.3.12),
we obtain:
dj(Tn(x),Tn+1(x)) =dj(T(Tn−1(x),T(Tn(x)))≤
≤c·[dr(j)(Tn−1(x),T(Tn(x))) +dr(j)(Tn(x),T(Tn−1(x)))] =
=c·[dr(j)(Tn−1(x),Tn+1(x)) +dr(j)(Tn(x),Tn(x))] =
=c·dr(j)(Tn−1(x),Tn+1(x)).
By using the triangle inequality, the previous inequality and the monotony property
for the pseudometric, we get:
dr(j)(Tn−1(x),Tn+1(x))≤dr(j)(Tn−1(x),Tn(x)) +dr(j)(Tn(x),Tn+1(x))≤
≤dr(j)(Tn−1(x),Tn(x)) +dj(Tn(x),Tn+1(x))⇒
⇒(1−c)dj(Tn(x),Tn+1(x))≤c·dr(j)(Tn−1(x),Tn(x)),
which is equivalent with:
dj(Tn(x),Tn+1(x))≤c
1−cdr(j)(Tn−1(x),Tn(x))≤···≤
≤/parenleftBigc
1−c/parenrightBigndr(j)(x,T(x)).
Remind that c∈[0,1
2), soc
1−c∈[0,1), which means:/parenleftBig
c
1−c/parenrightBign→0asn→∞.
We obtain, for all j∈J:
dj(Tn(x),Tn+1(x))→0,asn→∞,∀x∈A.

50 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
According to Lemma 2.3.64, it follows that Fixε(T)/negationslash=?,∀ε>0.
The proof is complete.
Theorem 2.3.74 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, letA⊂Xa subset of X. We letτbe the weak topology
onXdeﬁned by the family D. Letr:J→J, withJa family of indices.
LetT:A→Aa Zamﬁrescu-type ALC: a mapping satisfying conditions: ∃a,k,c∈
R,a∈[0,1),k∈/bracketleftBig
0,1
2/parenrightBig
,c∈/bracketleftBig
0,1
2/parenrightBig
such that∀x,y∈A,at least one of the following is
true:
(i)dj(T(x),T(y))≤a·dr(j)(x,y);
(ii)dj(T(x),T(y))≤k·[dr(j)(x,T(x)) +dr(j)(y,T(y))];
(iii)dj(T(x),T(y))≤c·[dr(j)(x,T(y)) +dr(j)(y,T(x))].
Assume the monotony property (2.3.12)assured for the pseudometric.
Then:
∀ε>0,Fixε(T)/negationslash=?.
Proof:Our ﬁrst goal is to merge the independent conditions (i), (ii) and (iii)
into a single one for an easier proof of the approximate ﬁxed point property for the
Zamﬁrescu-type almost local contractions.
Letx,y∈A.
a) Assuming (ii) valid, we obtain by using the monotony property for the pseudometric:
dj(T(x),T(y))≤k·[dr(j)(x,T(x)) +dr(j)(y,T(y))]≤
≤k·dr(j)(x,T(x)) +k[dr(j)(y,x) +dr(j)(x,T(x)) +dr(j)(T(x),T(y))] =
= 2k·dr(j)(x,T(x)) +k·dr(j)(x,y) +k·dr(j)(T(x),T(y))≤
≤2k·dr(j)(x,T(x)) +k·dr(j)(x,y) +k·dj(T(x),T(y))
This implies
(2.3.63) dj(T(x),T(y))≤2k
1−kdr(j)(x,T(x)) +k
1−kdr(j)(x,y).
b) Supposing (iii) holds, we get:
dj(T(x),T(y))≤c[dr(j)(x,T(y)) +dr(j)(y,T(x))]≤
≤c[dr(j)(x,y) +dr(j)(y,T(y))] +c[dr(j)(y,T(y)) +dr(j)(T(y),T(x))] =
=c·dr(j)(T(x),T(y)) + 2c·dr(j)(y,T(y)) +c·dr(j)(x,y)≤
≤c·dj(T(x),T(y)) + 2c·dr(j)(y,T(y)) +c·dr(j)(x,y)
By dividing with 1−c, we have that:
(2.3.64) dj(T(x),T(y))≤2c
1−cdr(j)(y,T(y)) +c
1−cdr(j)(x,y).

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 51
Continuing in this way, we obtain a similar inequality:
dj(Tx,Ty )≤c[dr(j)(x,T(y)) +dr(j)(y,T(x))]≤
≤c[dr(j)(x,T(x)) +dr(j)(T(x),T(y))] +c[dr(j)(y,x) +dr(j)(x,T(x))] =
=c·dr(j)(T(x),T(y)) + 2c·dr(j)(x,T(x)) +c·dr(j)(x,y)≤
≤c·dj(T(x),T(y)) + 2c·dr(j)(x,T(x)) +c·dr(j)(x,y),
which implies that
(2.3.65) dj(T(x),T(y))≤2c
1−cdr(j)(x,T(x)) +c
1−cdr(j)(x,y).
Now, having in view (i), (2.3.63), (2.3.64), and (2.3.65); for a more concentrated ap-
proach, we denote:
ρ= max/braceleftBig
a,k
1−k,c
1−c/bracerightBig
.
Obviously, we have ρ∈[0,1].
The mapping Tsatisfying at least one of the inequalities (i), (ii), (iii), we conclude:
(2.3.66) dj(T(x),T(y))≤2ρ·dr(j)(x,T(x)) +ρ·dr(j)(x,y),
and
(2.3.67) dj(T(x),T(y))≤2ρ·dr(j)(y,T(y)) +ρ·dr(j)(x,y).
Using conditions (2.3.66) and (2.3.67), we have for a given x∈X:
dj(Tn(x),Tn+1(x)) =dj(T(Tn−1(x),T(Tn(x)))≤
≤2ρ·dr(j)(Tn−1(x),T(Tn−1(x))) +ρ·dr(j)(Tn−1(x),Tn(x)) =
= 3ρ·dr(j)(Tn−1(x),Tn(x))≤···≤ (3ρ)n·dr(j)(x,T(x)).
We obtain, for all j∈J:
dj(Tn(x),Tn+1(x))→0,asn→∞,∀x∈A.
According to Lemma 2.3.64, it follows that Fε(T)/negationslash=?,∀ε>0.
Now, the proof is complete.
Remark 2.3.75 .Anya-almost local contraction, Kannan-type ALC and Chatterjea-
type ALC is also a Zamﬁrescu-type almost local contraction, this is the reason why
Theorems 2.3.71, 2.3.72 and 2.3.73 are actually contained in Theorem 2.3.74.
Theorem 2.3.76 .Under the assumptions of Theorem 2.3.74, let T:A→Aan
ALC: a mapping satisfying condition (2.1.3)from Deﬁnition 2.1.6 .
Then:
∀ε>0,Fixε(T)/negationslash=?.

52 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Proof:Letε>0andx∈A.
By using the deﬁnition of the almost local contractions, we are able to calculate the
pseudodistances :
dj(Tn(x),Tn+1(x)) =dj(T(Tn−1(x),T(Tn(x)))≤
≤θ·dr(j)(Tn−1(x),Tn(x)) +Ldr(j)(Tn(x),T(Tn−1(x))) =
=θ·dr(j)(Tn−1(x),Tn(x))≤···≤θn·dr(j)(x,T(x)).
Let us remind that θ∈[0,1), which implies that we have for all j∈J:
dj(Tn(x),Tn+1(x))→0,asn→∞,∀x∈A.
According to Lemma 2.3.64, it follows that Fixε(T)/negationslash=?,∀ε>0.
This completes the proof.
Example 2.3.77 .LetX= [0,1]with the pseudometric:
dj(x,y) =|x−y|·e−j,∀j∈J,∀x,y∈X,
whereJis a subset of Nand letT: [0,1]→{0,1}be deﬁned by
T(x) =

0ifx∈[0,1)
1ifx= 1.
Consider the function r(j) =j+ 1,∀j∈J, whereJis a subset of N.
Then:
(1)Fix(T) ={0,1};
(2)Fixε(T) = [0,ε·ej], for anyε>0and somej∈J.
The mapping Tis almost local contraction because:
– the ALC condition for every x,y∈[0,1),x/negationslash=ybecomes:
|0−0|·e−j≤(θ·|x−y|+L·|y−0|)·e−(j+1),
which is obviously true;
– the ALC condition for every x∈[0,1),y= 1becomes:
|1−0|·e−j≤(θ·|x−1|+L·|1−0|)·e−(j+1),
which leads us to the conclusion
e≤θ·|x−1|+L.
The last inequality holds for some θ∈(0,1)and for some L≥3.
Conclusion 1) is obvious, since:
– forx∈[0,1), the ﬁxed point condition T(x) =xleads us to x= 0;
– forx= 1,T(x) =xleads us to x= 1.
2) Letε>0,j∈Jﬁxed and choose xanε-ﬁxed point of T, which means
(2.3.68) |x−T(x)|·e−j≤ε.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 53
According to the deﬁnition of T, we have two cases:
a) Ifx∈[0,1),in this case we have T(x) = 0,which implies from (2.3.68):
|x−0|·e−j≤ε⇔x≤ε·ej⇔x∈[0,ε·ej].
b) Ifx= 1, by using (2.3.68), we get
|1−1|≤ε⇔ε≥0.
The conditions from Theorem 2.3.76 are fulﬁlled, therefore can be applied. Thus, we
have:Fixε(T) = [0,ε·ej], for anyε>0.
B. Quantitative results for mappings in pseudometric spaces In the sequel,
our main goal is to establish some quantitative results regarding the studied operators,
using Lemma 2.3.67.
Theorem 2.3.78 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, letA⊂Xa subset of X. We letτbe the weak topology
onXdeﬁned by the family D. Letr:J→J, withJa family of indices. Let
T:A→Aana-local contraction and assume the monotony property (2.3.12)assured
for the pseudometric.
Then:
δ(Fixε(T))≤2ε
1−a,∀ε>0.
Proof:Letε>0. Condition (i) from Lemma 2.3.67 was proved in Theorem 2.3.71.
It remain to show that (ii) holds for a-contractions.
Fixγ >0andx,y∈Fixε(T). We make the assumption
dj(x,y)−dj(T(x),T(y))≤γ.
In order to obtain the conclusion of the Lemma, we will prove that there exists an
ϕ(γ)>0such thatdj(x,y)≤ϕ(γ). It results, after using the monotony property:
dj(x,y)≤dj(T(x),T(y)) +γ≤a·dr(j)(x,y) +γ≤a·dj(x,y) +γ.
Now, we can conclude that
(1−a)dj(x,y)≤γ,
which implies dj(x,y)≤γ
1−a.
Ergo, we have that: ∀γ >0,∃ϕ(γ) =γ
1−asuch that
dj(x,y)−dj(T(x),T(y))≤γ⇒dj(x,y)≤ϕ(γ).
Now, by Lemma 2.3.67. Thus, we have:
δ(Fε(T))≤ϕ(2ε),∀ε>0.

54 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
The last inequality actually means that
δ(Fixε(T))≤2ε
1−a,∀ε>0.
The proof is complete.
Theorem 2.3.79 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, letA⊂Xa subset of X. We letτbe the weak topology
onXdeﬁned by the family D. Letr:J→J, withJa family of indices. Let
T:A→Aa Kannan-type ALC and assume the monotony property (2.3.12)assured
for the pseudometric.
Then:
δ(Fixε(T))≤2ε(1 +a),∀ε>0.
Proof:Letε>0. Again, condition (i) from Lemma 2.3.67 is satisﬁed, as one can
see in the proof of Theorem 2.3.71.
It remain to show that (ii) holds for Kannan-type ALC-s.
Fixγ >0andx,y∈Fixε(T). We make the assumption
dj(x,y)−dj(T(x),T(y))≤γ.
Asx,yare approximate ﬁxed point of T, this means :
dj(x,T(x))<ε
which implies
dj(x,y)≤γ+dj(T(x),T(y))≤γ+a[dr(j)(x,T(x)) +dr(j)(y,T(y))] =
=γ+adj(x,T(x)) +adj(y,T(y))
⇒dj(x,y)≤2aε+γ.
Therefore,∀γ >0,∃ϕ(γ) =γ+ 2aε> 0such that
dj(x,y)−dj(T(x),T(y))≤γ⇒dj(x,y)≤γ.
Now, by Lemma 2.3.67 it results that
δ(Fixε(T))≤ϕ(2ε),∀ε>0.
The last inequality leads us to the conclusion:
δ(Fixε(T))≤2ε(1 +a),∀ε>0.
The proof is complete.
Theorem 2.3.80 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, letA⊂Xa subset of X. We letτbe the weak topology
onXdeﬁned by the family D. Letr:J→J, withJa family of indices. Let

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 55
T:A→Aa Chatterjea-type ALC and assume the monotony property (2.3.12)assured
for the pseudometric.
Then:
δ(Fixε(T))≤2ε(1 +a)
1−2a,∀ε>0.
Proof:Letε >0. Again, condition (i) from Lemma 2.3.67 is satisﬁed, according
to the proof of Theorem 2.3.71.
It remain to show that (ii) holds for Chatterjea-type ALC-s.
Letγ >0andx,y∈Fixε(T). Assume that
dj(x,y)−dj(T(x),T(y))≤γ.
Then we have
dj(x,y)≤γ+dj(T(x),T(y))≤γ+adr(j)(x,T(y)) +adr(j)(y,T(x))≤
≤γ+adj(x,T(y)) +adj(y,T(x))≤γ+a[dj(x,y) +dj(y,T(y))] +
+a[dj(y,x) +dj(x,T(x))].
Butx,yare approximate ﬁxed points of T, it results that
dj(x,y)≤2adj(x,y) + 2εa+γ
⇒(1−2a)dj(x,y)≤2εa+γ⇒
⇒dj(x,y)≤γ+ 2εa
1−2a.
Hence∀γ >0,∃ϕ(γ) =γ+2εa
1−2asuch that
dj(x,y)−dj(T(x),T(y))≤γ⇒dj(x,y)≤ϕ(γ).
Again, by Lemma 2.3.67 it results that
δ(Fixε(T))≤ϕ(2ε),∀ε>0.
The last inequality leads us to the conclusion:
δ(Fixε(T))≤2ε(1 +a)
1−2a,∀ε>0.
This completes the proof.
Theorem 2.3.81 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, letA⊂Xa subset ofX. We letτbe the weak topology on
Xdeﬁned by the family D. Letr:J→J, withJa family of indices. Let T:A→A
a Zamﬁrescu-type ALC and assume the monotony property (2.3.12)assured for the
pseudometric.
Then:
δ(Fixε(T))≤2ε(1 +ρ)
1−ρ,∀ε>0.

56 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Proof:Letε >0. Again, condition (i) from Lemma 2.3.67 is satisﬁed, as we can
see in the proof of Theorem 2.3.71.
It remain to show that condition (ii) from Lemma 2.3.67 holds for Zamﬁrescu-type
ALC-s.
Letγ >0andx,y∈Fε(T). Assume that
dj(x,y)−dj(T(x),T(y))≤γ.
Then we have
dj(x,y)≤γ+dj(T(x),T(y))≤γ+ 2ρ·dr(j)(x,T(x)) +ρ·dr(j)(x,y)≤
≤γ+ 2ρ·dj(x,T(x)) +ρ·dj(x,y)⇒
⇒(1−ρ)dj(x,y)≤2ρε+γ⇒
⇒dj(x,y)≤γ+ 2ρε
1−ρ.
Therefore, for all γ >0,∃ϕ(γ) =γ+2ρε
1−ρ>0such that
dj(x,y)−dj(T(x),T(y))≤γ⇒dj(x,y)≤γ.
Again, by Lemma 2.3.67 it follows that
δ(Fixε(T))≤ϕ(2ε),∀ε>0.
The last inequality leads us to the desired conclusion:
δ(Fixε(T))≤2ε(1 +ρ)
1−ρ,∀ε>0.
This completes the proof.
Remark 2.3.82 .The case of almost local contractions is very similar to the Zamﬁrescu-
type ALC-s, but we need an additional condition, that is, θ+L<1, in order to get the
result, as we can see in the next theorem.
Theorem 2.3.83 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X, letAa subset of X. We letτbe the weak topology
onXdeﬁned by the family D. Letr:J→J, withJa family of indices. Let
T:A→Aan ALC with θ+L < 1.Assume the monotony property (2.3.12)assured
for the pseudometric.
Then:
δ(Fixε(T))≤2 +L
1−θ−L·ε,∀ε>0.
Proof:Letε>0. Again, condition (i) from Lemma 2.3.67 holds.
It remain to show that (ii) holds for ALC-s.
Letγ >0andx,y∈Fixε(T). Assume that
dj(x,y)−dj(T(x),T(y))≤γ.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 57
By using the triangle inequality and the monotony property for the pseudometric, we
get
dj(x,y)≤γ+dj(T(x),T(y))≤γ+θ·dr(j)(x,y) +L·dr(j)(y,T(x))≤
≤γ+θ·dr(j)(x,y) +L·dr(j)(x,y) +L·dr(j)(x,T(x))≤
≤γ+θ·dj(x,y) +L·dj(x,y) +L·dj(x,T(x))≤
≤(θ+L)dj(x,y) +Lε+γ
Thus, we have:
(1−θ−L)dj(x,y)≤Lε+γ⇒dj(x,y)≤Lε+γ
1−θ−L.
Therefore, for all γ >0,∃ϕ(γ) =Lε+γ
1−θ−Lsuch that
dj(x,y)−dj(T(x),T(y))≤γ⇒dj(x,y)≤ϕ(γ).
Again, by Lemma 2.3.67 it follows that
δ(Fixε(T))≤ϕ(2ε),∀ε>0.
The last inequality leads us to the conclusion:
δ(Fixε(T))≤2 +L
1−θ−L·ε,∀ε>0.
Now, the proof is complete.
3.4. Almost local contractions in b-pseudometric spaces
Inthissubsection,thenotionofalmostlocalcontractionina b-pseudometricspaceis
considered. Inthisframeworksomenewﬁxedpointresultsaregiven. Alargenumberof
generalizations for the concept of metric space were given by several authors, the most
important of them: [ 90], [34], [5], [40], and recent works, amongst which we mention
[31], [32]. The concept of b-metric space was introduced by Czerwik in [ 40]. Since
then several publications were studied the ﬁxed point of single valued and multivalued
operators in b-metric spaces (see [ 24], [97], [40]). The starting point of this part was
the book of M. Păcurar [ 72], who take an in depth look into the question of ﬁxed points
inb-metric spaces, but only for the almost contractions.
Definition 2.3.84 .(see[90]) LetXbe a nonempty set.
A mapping db:X×X→R+is called b-metric if the following hold:
(1)db(x,y) = 0if and only if x=y;
(2)db(x,y) =db(y,x),∀x,y∈X;
(3)db(x,z)≤b·[db(x,y) +db(y,z)],∀x,y,z∈X,
whereb≥1is a given real number
A nonempty set Xendowed with a b-metricdb:X×X→R+is called b-metric space.

58 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Example 2.3.85 .(see[5]) Letp∈(0,1)and letlpbe the space of all real sequences
{xn}n≥0⊂Rsuch that
∞/summationdisplay
n=1|xn|p<∞.
Letd:lp×lp→R+deﬁned by
d(x,y) =/parenleftBig∞/summationdisplay
n=1|xn−yn|p/parenrightBig1
p,
for anyx={xn}n≥0,y={yn}n≥0.Thendis ab-metric onlpwith constant b= 21
p>1,
hence (lp,d)is a b-metric space.
Remark 2.3.86 .In the sequel, our goal is to extend the b-metrics to the more
general case of b-pseudometrics, as it follows:
Definition 2.3.87 .The mapping db:X×X→R+is said to be a b-pseudometric
if:
(1)db(x,y) =db(y,x),∀x,y∈X;
(2)db(x,z)≤b·[db(x,y) +db(y,z)],∀x,y,z∈X,
whereb≥1is a given real number;
(3)db(x,x) = 0,∀x∈X.
Remark 2.3.88 .Obviously, a metric space is a b-metric space with b= 1.
However, a b-metric on Xneed not be a metric on the set X.
In the beginning of this subsection, we will present a set of concepts which help us
to establish a theory of some ﬁxed point theorems, related to a various type of ALC-s
inb-pseudometric spaces.
The following concepts are given in metric spaces, they appear originally in [ 91], but
we will present them in the framework of a b-pseudometric space.
Definition 2.3.89 .Let(X,d)be ab-pseudometric space and f:X→Xa
(weakly)Picard operator. Then fis said to be a good (weakly) Picard operator if
/summationdisplay
n∈Nd(fn(x),fn+1(x))<∞,
for anyx∈X.
Remark 2.3.90 .For the particular case b= 1, we get from Deﬁnition2.3.89 the
well-known deﬁnition of good (weakly) Picard operator in a metric space.
Definition 2.3.91 .Let(X,d)be ab-pseudometric space and f:X→Xa
(weakly)Picard operator. Then fis said to be a special (weakly) Picard operator if
/summationdisplay
n∈Nd(fn(x),f∞(x))<∞,
for anyx∈X.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 59
Remark 2.3.92 .For the particular case b= 1, from Deﬁnition 2.3.91 we get the
well-known deﬁnition of a special (weakly) Picard operator in a metric space.
The property of well-posedness for a ﬁxed point problem for an operator proposed
in [42] was studied also in [ 75], [92] and [65]. In ab-metric space was introduced in
[72] by M.Păcurar, we introduce it in a b-pseudometric space as follows:
Definition 2.3.93 .Let(X,d)be ab-pseudometric space and f:X→Xa Picard
operator with Ff={x∗}. Suppose there exist zn∈X,n∈Nsuch that
d(zn,f(zn))→0asn→∞.
If this implies
zn→x∗asn→∞,
then we say that the ﬁxed point problem for the operator fis well posed.
Remark 2.3.94 .For the particular case b= 1, from Deﬁnition2.3.93 we obtain the
well-known deﬁnition of a well posed ﬁxed point problem in a metric space.
Definition 2.3.95 .Let(X,d)be ab-pseudometric space and f:X→Xan
operator. Suppose there exist zn∈X,n∈Nsuch that
d(zn+1,f(zn))→0asn→∞.
If there exists x∈Xsuch that
d(zn,fn(x))→0asn→∞,
then we say that the operator fhas the the limit shadowing property.
Remark 2.3.96 .For the particular case b= 1, from Deﬁnition 2.3.95 we obtain the
usual deﬁnition of limit shadowing property for an operator on a metric, respectively
b-metric space.
Example 2.3.97 .(see[91]) Let (X,d)be a complete metric space and f:X→X
a Banach contraction with α∈[0,1). Thenfis a good Picard operator, the ﬁxed point
problem for the operator fis well posed, fhas the limit shadowing property.
3.5.ϕ-contractions in b-pseudometric spaces
The type of ϕ-contractions have been studied in pseudometric spaces (see section
3.1 in chapter 2). In the sequel, we intend to analyze them in the more general case of
b-pseudometric spaces, as it follows:
Theorem 2.3.98 .Let(X,d)be ab-pseudometric space, ϕ:R+→R+a compar-
ison function, consider a function r:J→J, also consider a nonempty, τ-bounded,

60 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
sequentially τ-complete, and T-invariant subset A⊂X.
Letf:X→Xbe a quasiϕ-contraction, as in (2.3.58).
Thenfhas a unique ﬁxed point if and only if there exists x0∈Xsuch that the
Picard iteration{xn}n≥0given by (1.1.2)is bounded.
Proof:a)” =⇒”Iffhas a unique ﬁxed point x∗, then by choosing x0=x∗, the
sequence of successive approximations is bounded, being constant.
b)”⇐= ”Assume (xn)is bounded for a certain x0∈X. It results there exist a constant
c>0and an element y∈Xsuch that
dj(xn,y)≤c,∀n∈N,∀j∈J.
Form,n∈N, by using the deﬁnition of the b-pseudometric, we get
dj(xn,xm)≤b[dj(xn,y) +dj(y,xm)]≤2b·c.
Note thatϕis monotone increasing, by using the condition for the quasi ϕ-contraction,
we obtain
dj(xn,xn+p) =dj(f(xn−1),f(xn+p−1))mon.
≤ϕ(dr(j)(xn−1,xn+p−1))≤
≤ ···≤ϕn−1(dr(j)(x1,xp+1))≤ϕn−1(2b·c), n,p∈N,
for allj∈J.
These relations show us that the sequence {xn}n∈Nisdj-Cauchy for each j∈J.
As the subset Ais assumed to be sequentially τ-complete, there exists x∗inAsuch
that{fnx}n∈Nisτ-convergent to x∗. Besides, the sequence {fnx}n∈Nconverges with
respect to the topology τtox∗, which implies
x∗= limn→∞xn⇒limn→∞dj(xn,x∗) = 0.
By using the deﬁnition of a quasi ϕ-contraction, we obtain:
0≤dj(f(x∗),x∗))≤b·[dj(f(x∗),f(xn)) +dj(f(xn),x∗)]≤
≤b·[ϕ(dr(j)(x∗,xn)) +dj(xn+1,x∗)].
Having in view that ϕis continuous at zero, we deduce: dj(f(x∗),x∗) = 0.
This actually means that x∗is a ﬁxed point of the operator f. The uniqueness of the
ﬁxed point is based on reductio ad absurdum method.
The proof is complete.
At this point (see [ 24]), ﬁxd≥1a real number and let ϕ:R+→R+be a
comparison function for which there exist a series of positive terms∞/summationtext
n=0vn, which is
convergent and a number 0≤α<1such that
(2.3.69) bk+1ϕk+1(t)≤α·bk·ϕk(t) +vk,∀t∈R,

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 61
for eachk≥N, withNﬁxed.
By using (2.3.69), the series
(2.3.70)∞/summationdisplay
k=0bkϕk(t)
converges for each t∈R+and its sum, denoted by sb(t)is monotone increasing and
continuous at zero.
Theorem 2.3.99 .Let(X,d)be ab-pseudometric space, ϕ:R+→R+a comparison
function, consider a function r:J→J, consider a nonempty, τ-bounded, sequentially
τ-complete, and T-invariant subset A⊂X.
Letf:X→Xa quasiϕ-contraction, with ϕsatisfying condition (2.3.69).
Ifx0∈Xis chosen such that the sequence of successive approximations is bounded and
Fix(f) ={x∗}, then we have
(2.3.71) dj(xn,x∗)≤b·sb(dj(xn,xn+1)),n≥0,
wheresb(t)is the sum of the series (2.3.70).
Proof:By means of contraction condition (2.3.58), we obtain for n≥1,
dj(xn,xn+1) =dj(f(xn−1),f(xn))≤ϕ(dr(j)(xn−1,xn)).
Forn≥2, we can write
(2.3.72) dj(xn−1,xn)≤ϕ(dr(j)(xn−2,xn−1)).
Having in view the monotony of ϕ, it results
dj(xn,xn+p)≤b·[dj(xn,xn+1) +dj(xn+1,xn+p)]≤
≤b·dj(xn,xn+1) +b2[dj(xn+1,xn+2) +dj(xn+2,xn+p)]≤
≤ ···≤b·dj(xn,xn+1) +b2dj(xn+1,xn+2) +···+bpdj(xn+p−1,xn+p),
which yields:
(2.3.73) dj(xn,xn+p)(2.3.72)
≤bp−1/summationdisplay
k=0bkϕk(dr(j)(xn,xn+1))
Then, by taking p→∞in (2.3.73), we get
(2.3.74) dj(xn,x∗)≤b·sb(dj(xn,xn+1)),n≥0.
The proof is complete.
Theorem 2.3.100 .Let(X,d)be ab-pseudometric space with b≥1.
The monotony property is valid with respect to the b-pseudometric, let ϕ:R+→R+a
b-comparison function, consider a function r:J→J, consider a nonempty, τ-bounded,
sequentially τ-complete, and T-invariant subset A⊂X.

62 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Letf:X→Xa quasiϕ-contraction, with ϕsatisfying condition (2.3.69).
Then:
(1)fis a Picard operator;
(2) the following a priori and a posteriori estimates are available:
dj(xn,x∗)≤b·sb(ϕn(dr(j)(x0,x1))),n≥0, (2.3.75)
dj(xn,x∗)≤b·sb(dj(xn,xn+1)),n≥0, (2.3.76)
wheresb(t)is the sum of the series (2.3.70).
(3) for any x∈X:
(2.3.77) dj(x,x∗)≤b·sb(dj(x,f(x))).
Proof:1) Fixx0∈X, then consider the sequence of successive approximations
xn=f(xn−1), we get:
dj(xn,xn+1) =dj(f(xn−1),f(xn))≤ϕ(dr(j)(xn−1,xn)),∀n≥1.
We inductively obtain the following inequality:
(2.3.78) dj(xn,xn+1)≤ϕn(dr(j)(x0,x1)).
Having in view that djis ab-pseudometric, for n≥0,p≥1, we get:
(2.3.79)dj(xn,xn+p)≤bdj(xn,xn+1) +b2dj(xn+1,xn+2) +···+bpdj(xn+p−1,xn+p).
Accordingly (2.3.78), we obtain:
(2.3.80)
dj(xn,xn+p)≤bϕn(dr(j)(x0,x1)) +b2ϕn+1(dr(j)(x0,x1) +···+bpϕn+p−1(dr(j)(x0,x1)),
which yields
(2.3.81)dj(xn,xn+p)≤1
bn−1[bnϕn(dr(j)(x0,x1)) +···+bn+p−1ϕn+p−1(dr(j)(x0,x1))].
Denote
(2.3.82) Sn=n/summationdisplay
k=0bkϕk(dr(j)(x0,x1),n≥1.
The inequality (2.3.81) becomes:
(2.3.83) dj(xn,xn+p)≤1
bn−1[Sn+p−1−Sn−1], n≥1,p≥1.
But the series
∞/summationdisplay
k=0bkϕk(dr(j)(x0,x1))
is convergent, hence there is
(2.3.84) S= limn→∞Sn∈R+.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 63
Having in view that b≥1, by (2.3.83) we conclude that the sequence {xn}n≥0is
τ-Cauchy, which yields: there exists x∗∈Xsuch that
x∗= limn→∞xn.
Next, we have to prove that x∗is a ﬁxed point of f. After simple computations, we
get:
(2.3.85)dj(xn+1,f(x∗)) =dj(f(xn),f(x∗))≤ϕ(dr(j)(xn,x∗))≤ϕ(dj(xn,x∗)),n≥0.
Remind that ϕis continuous at zero, and by taking n→∞we obtain:
dj(x∗,f(x∗)) = 0,
which actually means that x∗is a ﬁxed point for f. Suppose by contradiction there
exists another ﬁxed point y∗∈X, diﬀerent from x∗.
By using the deﬁnition of a quasi ϕ-contraction, Lemma 2.3.31 and the monotony
(2.3.12) of the pseudometric, we have:
dj(x∗,y∗) =dj(f(x∗),f(y∗))≤ϕ(dr(j)(x∗,y∗))<dr(j)(x∗,y∗)≤dj(x∗,y∗).
The above inequality is obviously a contradiction. Ergo, we have proven the uniqueness
of the ﬁxed point, which means that fis a Picard operator.
2) By using (2.3.80), it results that
dj(xn,xn+p)≤b[ϕ0(ϕn(dr(j)(x0,x1))) +bϕ(ϕn(dr(j)(x0,x1))) +
+···+bp−1ϕp−1(ϕn(dr(j)(x0,x1)))], (2.3.86)
wheren≥0,p≥1.At this point, we let p→∞in (2.3.86), and we get the ﬁrst
required estimate:
dj(xn,x∗)≤b·sb(ϕn(dr(j)(x0,x1))),n≥0.
Secondly, for n≥1,k≥0we obtain
dj(xn+k,xn+k+1) =dj(f(xn+k−1),f(xn+k))≤ϕ(dr(j)(xn+k−1,xn+k)).
We obtain by induction the following inequality:
(2.3.87) dj(xn+k,xn+k+1)≤ϕk(dr(j)(xn,xn+1)),n≥1,k≥0.
Now, by replacing (2.3.87) in (2.3.79), we get
dj(xn,xn+p)≤b[dj(xn,xn+1) +bϕ(dr(j)(xn,xn+1)) +
+···+bp−1ϕp−1(dr(j)(xn,xn+1))], (2.3.88)
forn≥0,p≥1.After using the monotony property for the pseudometric, we get:
dj(xn,xn+p)≤b[dj(xn,xn+1) +bϕ(dj(xn,xn+1)) +
+···+bp−1ϕp−1(dj(xn,xn+1))]. (2.3.89)

64 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Lettingp→∞in (2.3.89) we obtain the second required estimate:
dj(xn,x∗)≤b·sb(dj(xn,xn+1)),n≥0.
3) Lettingxn=x, the a posteriori estimate (2.3.100) becomes for an arbitrary x∈X:
dj(x,x∗)≤b·sb(dj(x,f(x))).
This completes the proof.
Inthesequel, ourmaingoalistoverifyifaquasi ϕ-contractionfulﬁllstheproperties
mentioned in the beginning of this subsection.
Theorem 2.3.101 .Letf:A→Abe as in Theorem 2.3.100.
Thenfis a good Picard operator.
Proof:Choosex0∈X.By using (2.3.80) from the proof of Theorem 2.3.100, we
get:
(2.3.90) dj(fn(x0),fn+1(x0)) =dj(xn,xn+1)≤ϕn(dr(j)(x0,x1)),n≥0,
which is also valid for the case b= 1.
Next, we use the deﬁnition of a b-quasimetric and we obtain:
∞/summationdisplay
n=0dj(fn(x0),fn+1(x0))≤bnϕn(dr(j)(x0,x1)) =sb(dr(j)(x0,x1)).
Finally, according to Lemma 2.3.34 and the inequality/summationtext
n∈Nd(fn(x),fn+1(x))<∞, we
conclude that fis a good Picard operator.
This completes the proof.
Theorem 2.3.102 .Letf:A→Abe as in Theorem 2.3.100. Then the ﬁxed point
problem for fis well posed.
Proof:Let the sequence{zn}n∈N⊂Xsuch that
(2.3.91) dj(zn,f(zn))→0asn→∞.
By (2.3.77), if we choose x=zn,n∈N, we get:
(2.3.92) dj(zn,x∗)≤b·sb(dj(zn,f(zn))),n∈N.
Remind that sbis continuous at zero, according to Lemma 2.3.34. Letting n→∞in
(2.3.92), and combining it with (2.3.91), we obtain:
dj(zn,x∗)→0,n→∞,
which means that the ﬁxed point problem for fis well posed.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 65
Theorem 2.3.103 .Letf:A→Abe as in Theorem 2.3.100, with the monotony
property for the pseudometric. If ϕsatisﬁes:
(2.3.93) ϕ(a1t1+a2t2)≤a1ϕ(t1) +a2ϕ(t2),
for anya1,a2,t1,t2∈R+, thenfhas the limit shadowing property.
Proof:Let the sequence{zn}n∈N⊂Xsuch that
(2.3.94) dj(zn+1,f(zn))→0asn→∞.
Forn≥0we obtain:
(2.3.95) dj(zn+1,x∗)≤b·dj(zn+1,f(zn)) +b·dj(f(zn),f(x∗)).n∈N.
Having in view that fis aϕ-contraction, we obtain:
(2.3.96) dj(zn+1,x∗)≤b·dj(zn+1,f(zn)) +bϕ(dr(j)(zn,x∗)),n∈N.
The monotony of the pseudometric leads to:
(2.3.97) dj(zn+1,x∗)≤b·dj(zn+1,f(zn)) +bϕ(dj(zn,x∗)),n∈N.
After simple computations, we get :
dj(zn,x∗)≤b·dj(zn,f(zn−1)) +bϕ(dj(zn−1,x∗)),n≥1,
which replaced in (2.3.97), by using (2.3.93) yields
dj(zn+1,x∗)≤b·dj(zn+1,f(zn)) +b2ϕ(dj(zn,f(zn−1))) +b2ϕ2(dj(zn−1,x∗)),n≥1.
At this point, we inductively obtain:
dj(zn+1,x∗)≤b·dj(zn+1,f(zn)) +b2ϕ(dj(zn,f(zn−1))) +
+···+bn+1ϕn(dj(z1,f(z0))) +bn+2ϕn+1(dj(z0,x∗)).
Thus, we have:
dj(zn+1,x∗)≤bn/summationdisplay
k=0bkϕk(dj(zn−k+1,f(zn−k))) +
+bn+2ϕn+1(dj(z0,x∗)). (2.3.98)
Consider Lemma 2.3.35 with the substitution: an=dj(zn+1,f(zn)), it results that
(2.3.99)n/summationdisplay
k=0bkϕk(dj(zn−k+1,f(zn−k)))→0asn→∞.
We must discuss two diﬀerent cases:
Case I: Ifz0=x∗, this means bnϕn(dj(z0,x∗)) = 0.Case II: Ifz0/negationslash=x∗, we have that
bnϕn(dj(z0,x∗))→0asn→∞,according to Lemma 2.3.34. At this point, we let
n→∞in (2.3.98) and it follows that
(2.3.100) dj(zn+1,x∗)→0,n→∞.

66 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
By using Theorem 2.3.100, we conclude that for any x∈Athe Picard iteration
{fn(x)}n≥0converges to x∗.This means that the following inequalities holds:
(2.3.101) dj(zn+1,fn(x))≤dj(zn+1,x∗) +dj(x∗,fn(x)),n≥0
for somex∈X. Taking the limit n→∞in (2.3.101), we have that
dj(zn+1,fn(x))→0,asn→∞,
which actually means that fhas the limit shadowing property.
The next Theorem states and proves the data dependence of the ﬁxed point for
quasiϕ-contractions on b-pseudometric spaces with ϕab-comparison function:
Theorem 2.3.104 .Letf:A→Abe as in Theorem 2.3.100 with the ﬁxed point x∗
andg:A→Asuch that:
(1)ghas at least one ﬁxed point, denoted by y∗∈Fg;
(2) there exists η>0such that
(2.3.102) dj(f(x),g(x))≤η,for anyx∈X.
Then
dj(x∗,y∗)≤b·sb(η),
whereFf={x∗}andsbwas introduced in Lemma 2.3.34.
Proof:From 2.3.77 in Theorem 2.3.100, by using the substitution x:=y∗, we
obtain:
dj(x∗,y∗)≤b·sb(dj(y∗,f(y∗))) =b·sb(dj(g(y∗),f(y∗))).
We use again Lemma 2.3.34, it follows from that: sbis increasing, which by (2.3.102)
yields
dj(x∗,y∗)≤b·sb(η).
This completes the proof.
3.6. Fixed points of almost local contractions in b-pseudometric spaces
Theorem 2.3.105 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
b-pseudometrics deﬁned on X, with constant b≥1. We choose a subset A⊂Xand we
letτbe the weak topology on Xdeﬁned by the family D. Letf:A→Abe an almost
local contraction with constants θ∈[0,1
b)andL≥0.
Assume a monotony property valid for the ALC f:
(2.3.103) dr(j)(f,g)≤dj(f,g),∀f,g∈A,∀j∈J.
Then:
(i)fis a weakly Picard operator;

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 67
(ii) If theb-pseudometric is continuous, then for any x∈Athe following error esti-
mates hold:
dj(fn(x),f∞(x))≤bθn
1−bθdj(x,f(x)),n≥1; (2.3.104)
dj(fn(x),f∞(x))≤bθ
1−bθdj(fn−1(x),fn(x)),n≥1; (2.3.105)
Proof:(i) In the beginning, we will prove that the operator fhas at least one ﬁxed
point inX, i.e., the set of ﬁxed points is nonempty. To this end, we let x0∈Aand
{xn}n≥0be the Picard iteration which starts from x0. By using the deﬁnition of the
Picard iteration and the ALC-s, we get:
dj(xn,xn+1) =dj(f(xn−1),f(xn))≤θ·dr(j)(xn−1,xn) +Ldr(j)(xn,xn),
for alln∈N. We obtain by induction with respect to n:
(2.3.106) dj(xn,xn+1)≤θn·dr(j)(x0,x1), n = 1,2,···
Forn≥0,p≥1we can write:
dj(xn,xn+p)≤b[dj(xn,xn+1) +dj(xn+1,xn+p)] =
=bdj(xn,xn+1) +bdj(xn+1,xn+p)≤
≤bdj(xn,xn+1) +b2[dj(xn+1,xn+2) +dj(xn+2,xn+p))]≤···≤
≤bdj(xn,xn+1) +b2[dj(xn+1,xn+2) +···+bpdj(xn+p−1,xn+p).
By (2.3.106) it follows that:
dj(xn,xn+p)≤bθndr(j)(x0,x1) +b2θn+1dr(j)(x0,x1) +···+bpθn+p−1dr(j)(x0,x1) =
=bθndr(j)(x0,x1)[1 +bθ+ (bθ)2+···+ (bθ)p−1] =
=b·1−(bθ)p
1−bθdr(j)(x0,x1)·θn, (2.3.107)
forn≥0,p≥1.
Remind that θ∈[0,1
b), withb≥1, it is obvious that 0≤bθ < 1, which yields from
(2.3.107) the conclusion that {xn}n≥0is adj−Cauchy sequence in the b-pseudometric
space. This means that it is convergent with its limit denoted by
(2.3.108) x∗= limn→∞xn.
Applying the deﬁnition of the b-pseudometric, we get:
dj(x∗,f(x∗))≤b[dj(x∗,f(xn)) +dj(f(xn),f(x∗))].

68 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
After using the deﬁnition of an ALC and the monotony property (2.3.103), we obtain
from the last inequality:
dj(x∗,f(x∗))≤bdj(x∗,f(xn)) +bθdr(j)(xn,x∗) +bLdr(j)(x∗,f(xn))≤
≤bdj(x∗,f(xn)) +bθdj(xn,x∗) +bLdj(x∗,f(xn)) =
=b(1 +L)dj(x∗,xn+1) +bθdj(xn,x∗).
Having in view (2.3.108) and letting n→∞, it results that
dj(x∗,f(x∗)) = 0,
which means that x∗is a ﬁxed point of f. Hencefis a weakly Picard operator.
(ii)Inthesequel, remindthatthe b-pseudometriciscontinuousand 0≤bθ< 1. Letting
p→∞in (2.3.107), we get the a priori error estimate (2.3.104). By using the induction
in (2.3.106), we obtain:
(2.3.109) dj(xn+k,xn+k+1)≤θk+1·dr(j)(xn−1,xn),
for anyn,k∈N,n≥1. From that, we can write:
dj(xn,xn+p)≤bdj(xn,xn+1) +b2dj(xn+1,xn+2) +···+bpdj(xn+p−1,xn+p)≤
≤bθdj(xn−1,xn) + (bθ)2dj(xn−1,xn) +···+ (bθ)pdj(xn−1,xn) =
=bθ·1−(bθ)p
1−bθ·dj(xn−1,xn).
Again, by the continuity of the b-pseudometric and having in view that 0≤bθ< 1, by
lettingp→∞in the last inequality, it results the a posteriori estimate, namely the
second relation in (2.3.106).
The proof is complete.
In the sequel, our main goal is to extend the ALC-s to the case of strict almost
contractions, as a result of which we get an existence and uniqueness theorem:
Theorem 2.3.106 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
b- pseudometrics deﬁned on X, with constant b≥1. Assume the monotony property
for theb-pseudometric (2.3.103). We choose again a subset A⊂Xand we let τbe
the weak topology on Xdeﬁned by the family D. Letf:A→Aa strict ALC with
constantsθ∈[0,1
b)andL≥0, andθu∈[0,1
b),Lu≥0.
Assume a uniqueness condition for the mapping f(see[103]), that is:
(2.3.110) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,T(x)),∀x,y∈A.
Then:
(i)fis a Picard operator;

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 69
(ii) If theb-pseudometric is continuous, then the following error estimates hold:
(2.3.111) dj(xn,x∗)≤bθn
1−bθdj(x0,x1),n≥1,
(2.3.112) dj(xn,x∗)≤bθ
1−bθdj(xn−1,xn),n≥1;
(iii) Assume the continuity of the b-pseudometric. The rate of convergence of the
Picard iteration is given by
(2.3.113) dj(xn,x∗)≤θudj(xn−1,x∗),n≥1,
whereFf={x∗}.
Proof:(i) The ﬁrst part of the conclusion of the theorem, namely the existence
of the ﬁxed point is assured by Theorem 2.3.105. In order to prove the uniqueness of
the ﬁxed point, suppose that fhas two diﬀerent ﬁxed points x∗,y∗∈A. Then, using
the monotony condition (2.1.10) for the b-pseudometric and the uniqueness condition
(2.3.110), we can write that:
dj(f(x∗),f(y∗))≤θudr(j)(x∗,y∗) +Ludr(j)(x∗,f(x∗))≤
≤θudj(x∗,y∗) +Ludj(x∗,f(x∗)),
which means that dj(x∗,y∗)≤θudj(x∗,y∗).
As0≤θu<1, we get the obvious contradiction dj(x∗,y∗)<dj(x∗,y∗).
It results that Ff={x∗},hencefis a Picard operator.
(ii) The a priori and a posteriori estimates (2.3.111), (2.3.112) follows by Theorem
2.3.105. (iii) From (2.3.110) we obtain:
dj(f(x∗),f(xn−1))≤θudr(j)(x∗,xn−1) +Ludr(j)(x∗,f(x∗))
≤θudj(x∗,xn−1) +Ludj(x∗,f(x∗)),
which means:
dj(xn,x∗)≤θudj(xn−1,x∗),n≥1.

We present two examples of ALC-s in b-pseudometric spaces: Examples 2.3.107 and
2.3.108.
Example 2.3.107 .LetX={−1,0,1}×{− 1,0,1}⊂R2.
Consider the b-pseudometric: dj((x1,y1),(x2,y2)) = 0, ifx1=x2;
dj((x1,y1),(x2,y2)) =e−j, if|x1−x2|=1;dj((x1,y1),(x2,y2)) =b·e−j, if|x1−x2|= 2,
whereb≥2andJis a subset of Q.
djis a pseudometric, but is not a metric, take for example:
dj((1,−1),(1,0)) = 0, however (1,−1)/negationslash= (1,0).

70 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
In this case, consider the function r(j) =j+1
3, wherej∈J.
ConsiderT:X→X,
T(x,y) =

(x,−y)if (x,y)/negationslash= (1,1)
(0,0)if (x,y) = (1,1)
Tis not a contraction because the contractive condition:
(2.3.114) dj(Tx,Ty )≤θ·dj(x,y),
is not valid∀x,y∈X, and forθ∈[0,1). Indeed, (2.3.114) is equivalent with:
|x1−x2|·e−j≤θ·|x1−x2|·e−j,∀j∈J.
The last inequality leads us to 1≤θ, obviously false, considering θ∈(0,1).
However,Tbecomes an almost local contraction if:
|x1−x2|·e−j≤θ·|x1−x2|·e−(j+1
3)+L·|x2−x1|·e−(j+1
3),
which is equivalent to : e1
3≤θ+L.
Forθ=1
4∈(0,1),L= 2≥0andj∈J, the last inequality becomes true, i.e., Tis an
almost local b-contraction with three ﬁxed points:
Fix(T) ={(−1,0),(0,0),(1,0)}.
Example 2.3.108 .LetX={0,1,2}⊂R. Deﬁne the b-pseudometric: dj(x,y) = 0,
ifx=y,dj(x,y) =e−j, if|x−y|= 1,dj(x,y) =b·e−j, if|x−y|= 2,where
b≥2andJis a subset of Q. Consider the function r(j) =j+1
2, for every j∈J.
Consider the mapping T:X→X,
T(x,y) =

(x
2,y
2)if (x,y)/negationslash= (1,0)
(0,0)if (x,y) = (1,0)
By applying the inequality (2.1.3)to the mapping T, we get for all
x= (x1,y1),y= (x2,y2)∈X
/vextendsingle/vextendsingle/vextendsinglex1
2−x2
2/vextendsingle/vextendsingle/vextendsingle·e−j≤θ·|x1−x2|·e−(j+1
2)+L·/vextendsingle/vextendsingle/vextendsinglex2−x1
2/vextendsingle/vextendsingle/vextendsingle·e−(j+1
2),
for allj∈J, which can be write as the equivalent form
|x1−x2|·e1
2≤2θ·|x1−x2|+L·|2×2−x1|.
The last inequality is true if we take θ=1
2∈(0,1),L= 4≥0.
HenceTis an almost local b-contraction, with the unique ﬁxed point (0,0).
Next, we shall make a comparison to other type of contractive conditions in
b-pseudometric spaces.
Lemma 2.3.109 .In ab-pseudometric space with the monotony property (2.3.103),
any Kannan-type ALC with constant k∈[0,1
2b)is an ALC with θ=kb
1−kbandL=2kb
1−kb.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 71
Proof:Supposef:A→Ais a Kannan-type ALC with constant k∈[0,1
2b).
This means:
(2.3.115) dj(f(x),f(y))≤k[dr(j)(x,f(x)) +dr(j)(y,f(y))],∀x,y∈A.
Applyingthedeﬁnitionofthe b-pseudometricandthemonotonyproperty, wecanwrite:
dj(f(x),f(y))≤kbdr(j)(x,y) +kbdr(j)(y,f(x)) +kbdr(j)(y,f(x)) +kbdr(j)(f(x),f(y))≤
≤kbdj(x,y) +kbdj(y,f(x)) +kbdj(y,f(x)) +kbdj(f(x),f(y)),
for allx,y∈A.
After simple computations we get:
dj(f(x),f(y))≤kb
1−kbdj(x,y) +2kb
1−kbdj(y,f(x)),
for anyx,y∈A. The last inequality shows that fis an ALC, i.e., it satisﬁes (2.1.3)
withθ=kb
1−kb∈[0,1)andL=2kb
1−kb≥0.
This completes the proof of the Lemma.
Lemma 2.3.110 .In ab-pseudometric space with the monotony property (2.3.103),
any Kannan-type ALC with constant k∈[0,1
b(b+1))is a strict almost local contraction
withθ=kb
1−kbandL=2kb
1−kband, respectively θu=kb2
1−kbandLu=k(1+b2)
1−kb.
Proof:Having in view that b≥1, the condition k∈[0,1
b(b+1))impliesk<1
2b. This
means that the conclusion of Lemma 2.3.109 is valid. Furthermore, from (2.3.115), it
results:
dj(f(x),f(y))≤kdr(j)(x,f(x)) +kbdr(j)(f(y),f(x)) +kbdr(j)(f(x),y)≤
≤kdj(x,f(x)) +kbdj(f(y),f(x)) +kb2dj(f(x),x) +kb2dj(x,y).
From that, we have:
dj(f(x),f(y))≤kb2
1−kbdj(x,y) +k(b2+ 1)
1−kbdj(x,f(x)),
for anyx,y∈A, i.e.,fis a strict ALC with θu=kb2
1−kbandLu=k(1+b2)
1−kb.
The following two Lemmas refers to the Ćirić-Reich-Rus ALC-s.
Lemma 2.3.111 .In ab-pseudometric space with the monotony property (2.3.103),
any Ćirić-Reich-Rus-type ALC with constants α,β∈R+such thatα+ 2bβ < 1, is an
ALC withθ=α+bβ
1−bβandL=2bβ
1−bβ.
Proof:Denote byf:A→Aa Ćirić-Reich-Rus-type ALC, where Ais a subset of
theb-pseudometric space X. Letα,β∈R+,α+ 2bβ < 1, such that
(2.3.116) dj(f(x),f(y))≤αdr(j)(x,y) +β[dr(j)(x,f(x)) +dr(j)(y,f(y))],

72 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
for anyx,y∈A.
At this point, after using the monotony property (2.3.103), we can write:
dj(f(x),f(y))≤αdj(x,y) +bβdj(x,y) +bβdj(y,f(x)) +
+bβdj(y,f(x)) +bβdj(f(x),f(y)),
which implies
dj(f(x),f(y))≤α+bβ
1−bβdj(x,y) +2bβ
1−bβdj(y,f(x)),
for everyx,y∈A, i.e.,fsatisﬁes (2.1.3) with θ=α+bβ
1−bβ∈[0,1)andL=2bβ
1−bβ≥0.
This completes the proof.
Remark 2.3.112 .Forb= 1in Lemma 2.3.111, the values from Theorem 2.3.21
are obtained.
Lemma 2.3.113 .In ab-pseudometric space with the monotony property (2.3.103),
any Ćirić-Reich-Rus -type ALC with constants α,β∈R+such thatα+b(b+ 1)β <1
is a strict almost local contraction with θ=α+bβ
1−bβandL=2bβ
1−bβ≥0, and respectively,
θu=α+b2β
1−bβandLu=β(b2+1)
1−bβ.
Proof:Asb≥1, assumption α+b(b+ 1)β <1impliesα+ 2bβ < 1, therefore the
conclusions of Lemma 2.3.111 holds. Furthermore, according to (2.3.116), we obtain:
dj(f(x),f(y))≤αdr(j)(x,y) +βdr(j)(x,f(x)) +βdr(j)(f(y),y)≤
≤αdj(x,y) +βdj(x,f(x)) +bβdj(f(y),f(x)) +b2βdj(f(x),x) +b2βdj(x,y),
therefore
dj(f(x),f(y))≤α+b2β
1−bβdj(x,y) +β(b2+ 1)
1−bβdj(x,f(x)),
for everyx,y∈A, i.e.,fsatisﬁes (2.3.110) with θu=α+b2β
1−bβ∈[0,1)andLu=β(b2+1)
1−bβ,
which means that fis a strict ALC.
Lemma 2.3.114 .In ab-pseudometric space with the monotony property (2.3.103),
any Chatterjea-type ALC with constant c∈[0,1
b(b+1))is an almost local contraction
withθ=cb2
1−cbandL=c(b2+1)
1−cb.
Proof:LetAa subset of the b-pseudometric space Xand letf:A→Aa
Chatterjea-type ALC with c∈[0,1
b(b+1))such that
(2.3.117) dj(f(x),f(y))≤c[dr(j)(x,f(y)) +dr(j)(y,f(x))],for anyx,y∈A.
From that, we can write:
dj(f(x),f(y))≤cdj(f(y),x) +cdj(y,f(x))≤
≤cbdj(f(y),f(x)) +cbdj(f(x),x) +cdj(y,f(x))≤
≤cbdj(f(x),f(y)) +cb2dj(x,y) +cb2dj(y,f(x)) +cdj(y,f(x)),

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 73
therefore
dj(f(x),f(y))≤cb2
1−cbdj(x,y) +c(b2+ 1)
1−cbdj(y,f(x)),
for anyx,y∈A, that is,fsatisﬁes (2.1.3) with θ=cb2
1−cb∈[0,1)andL=c(b2+1)
1−cb≥0.
Now, the proof is complete.
Remark 2.3.115 .Forb= 1in Lemma 2.3.114, we obtain the conclusion of Theo-
rem 2.3.23.
Lemma 2.3.116 .In ab-pseudometric space with the monotony property (2.3.103),
any Chatterjea-type ALC with constant c∈[0,1
b(b+1))is a strict almost local contraction
withθ=cb2
1−cbandL=c(b2+1)
1−cband, respectively, θu=cb
1−cbandLu=2cb
1−cb.
Proof:The conclusions of Lemma 2.3.114 holds and from (2.3.117) we have that:
dj(f(x),f(y))≤cbdr(j)(x,f(y)) +cbdr(j)(f(x),f(y)) +cbdr(j)(y,x) +cbdr(j)(x,f(x)).
After applying the monotony property for the pseudometric, we get:
dj(f(x),f(y))≤cbdj(x,f(y)) +cbdj(f(x),f(y)) +cbdj(y,x) +cbdj(x,f(x)),
therefore
dj(f(x),f(y))≤cb
1−cbdj(x,y) +2cb
1−cbdj(x,f(x)),
for anyx,y∈A, that is,fsatisﬁes (2.3.110) with θu=cb
1−cbandLu=2cb
1−cb≥0.
Having in view that c∈[0,1
b(b+1))andb≥1, hence, it results that c<1
2b, which means
θu∈[0,1).
In the sequel, it is natural to study the Zamﬁrescu-type ALC-s in this new space
setting.
Lemma 2.3.117 .In ab-pseudometric space, any Zamﬁrescu-type ALC with con-
stantsα∈[0,1),k∈[0,1
2b)andc∈[0,1
b(b+1))is an ALC with θ= max/braceleftBig
α,bk
1−bk,b2c
1−bc/bracerightBig
andL= max/braceleftBig
2bk
1−bk,(b2+1)c
1−bc/bracerightBig
.
Lemma 2.3.118 .In ab-pseudometric space, any Zamﬁrescu-type ALC with con-
stantsα∈[0,1),k∈[0,1
b(b+1))andc∈[0,1
2b)satisﬁes the inequality (2.3.110) with
θu= max/braceleftBig
α,b2k
1−bk,bc
1−bc/bracerightBig
andLu= max/braceleftBig(b2+1)k
1−bk,2bc
1−bc/bracerightBig
.
Lemma 2.3.119 .In ab-pseudometric space, any Zamﬁrescu-type ALC with con-
stantsα∈[0,1),k∈[0,1
b(b+1))andc∈[0,1
b(b+1))is a strict ALC with θ= max/braceleftBig
α,bk
1−bk,b2c
1−bc/bracerightBig
andL= max/braceleftBig
2bk
1−bk,(b2+1)c
1−bc/bracerightBig
and, respectively, θu= max/braceleftBig
α,b2k
1−bk,bc
1−bc/bracerightBig
andLu=
max/braceleftBig(b2+1)k
1−bk,2bc
1−bc/bracerightBig
.
Our next goal is to study the case of quasi- almost local contractions.

74 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Lemma 2.3.120 .In ab-pseudometric space, any quasi-ALC with constant
h∈[0,1
b(b+1))is an ALC with θ=b2h
1−bhandL=b2h
1−bh.
Proof:Letf:A→Abe a quasi-ALC with constant h∈[0,1
b(b+1))
such that
(2.3.118)
dj(f(x),f(y))≤hmax{dr(j)(x,y),dr(j)(x,f(x)),dr(j)(y,f(y)),dr(j)(x,f(y)),dr(j)(y,f(x))}.
for allx,y∈A. Denote
Mr(j)(x,y) = max{dr(j)(x,y),dr(j)(x,f(x)),dr(j)(y,f(y)),dr(j)(x,f(y)),dr(j)(y,f(x))}.
and we distinguish ﬁve diﬀerent cases:
I.Mr(j)(x,y) =dj(x,y). We can write
dj(f(x),f(y))≤hdr(j)(x,y).
II.Mr(j)(x,y) =dj(x,f(x)).Then
dj(f(x),f(y))≤hdr(j)(x,f(x))≤hbdr(j)(x,y) +hbdr(j)(y,f(x)),
therefore (2.1.3) is satisﬁed with θ=hb∈[0,1)andL=hb≥0.
Remind that ALC-s are not symmetric operators, so we have to check both conditions
(2.1.3) and (2.1.4) in order to prove that an operator fis an ALC. Then, for any
x,y∈Awe can write:
dj(f(x),f(y))≤hdr(j)(x,f(x))≤hbdr(j)(x,f(y)) +hbdr(j)(y,f(x)).
Thus, we have:
dj(f(x),f(y))≤hb
1−hbdr(j)(x,f(y)),
which means that (2.1.4) is satisﬁed with θ= 0andL=hb
1−hb.
From that, the almost local contraction condition (2.1.3) is veriﬁed with:
θ= max{hb,0}=hbandL= max{hb,hb
1−hb}=hb
1−hb.
III. IfMr(j)(x,y) =dj(y,f(y)),in a similar manner to case II., it results that (2.1.3) is
fulﬁlled with θ=hbandL=hb
1−hb.
IV. IfMr(j)(x,y) =dj(x,f(y)),then we can write
dj(f(x),f(y))≤hdr(j)(x,f(y)).
This means that (2.1.4) holds with θ= 0andL=h.
By using the b-pseudometric property, it results
dj(f(x),f(y))≤hdr(j)(x,f(y))≤
≤bhdr(j)(f(y),f(x)) +b2hdr(j)(f(x),y) +b2hdr(j)(y,x),
therefore
dj(f(x),f(y))≤b2h
1−hbdr(j)(x,y) +b2h
1−hbdr(j)(y,f(x)).

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 75
From that, we may state that for the previously chosen xandy, the ALC (2.1.3)
condition holds with θ= max{b2h
1−bh,0}=b2h
1−bh∈[0,1)and
L= max{h,b2h
1−bh}=b2h
1−bh.
V.Mr(j)(x,y) =dj(y,f(x)).This case is quite similar to case IV.
These ﬁve cases leads us to the conclusion that for any x,y∈A, the ALC condition
(2.1.3) is fulﬁlled with
θ= max/braceleftBig
h,hb,b2h
1−bh/bracerightBig
=b2h
1−bh∈[0,1)
and
L= max/braceleftBig
0,bh
1−bh,b2h
1−bh/bracerightBig
=b2h
1−bh≥0.
The proof is complete.
In the sequel, we will return to the class of strict ALC-s.
Theorem 2.3.121 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of b-
pseudometrics deﬁned on X, with constant b≥1. We choose again a subset A⊂X
and we letτbe the weak topology on Xdeﬁned by the family D. Letf:A→Aa strict
ALC with constants θ∈[0,1
b)andL≥0,θu∈[0,1
b)andLu≥0.
Thenfis a good Picard operator.
Proof:In the proof of Theorem, we have from (2.3.23) the following relation:
(2.3.119) dj(xn,xn+1)≤θn·dr(j)(x0,x1), n = 1,2,···
If we letx=x0∈A, we obtain:
/summationdisplay
n≥0dj(fn(x),fn+1(x)) =/summationdisplay
n≥0dj(xn,xn+1) = limn→∞n/summationdisplay
k=0dj(xk,xk+1)≤
≤limn→∞(1 +θ+···+θn)dj(x0,x1) = limn→∞1−θn+1
1−θdj(x0,x1).
From that, by using θ∈[0,1), we obtain
/summationdisplay
n≥0dj(fn(x),fn+1(x))≤1
1−θdj(x0,x1).
Therefore,fis a good Picard operator, according to Deﬁnition 2.3.89.
Theorem 2.3.122 .Letf:A→Aan operator as appears in Theorem 2.3.106.
Thenfis a special Picard operator.
Proof:We showed in Theorem 2.3.106 the following error estimate:
dj(xn,x∗)≤bθn
1−bθdj(x0,x1),n≥1.

76 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
We letx=x0∈Aand we obtain
/summationdisplay
n≥0dj(fn(x),x∗) = limn→∞n/summationdisplay
k=0dj(xk,x∗) =
= limn→∞n/summationdisplay
k=0bθk
1−bθdj(x0,x1) =bdj(x0,x1)
1−bθlimn→∞n/summationdisplay
k=0θk.
It results
/summationdisplay
n≥0dj(fn(x),x∗) =b
(1−bθ)(1−θ)dj(x0,x1)<∞,
which means that fis a special Picard operator, having in view Deﬁnition 2.3.91. This
completes the proof.
Theorem 2.3.123 .Letf:A→Aan operator as appears in Theorem 2.3.106.
Then the ﬁxed point problem is well posed.
Proof:Chooseyn∈A,n∈Nsuch that
(2.3.120) dj(yn,f(yn))→0asn→∞.
According to Theorem 2.3.106, the operator fadmit a unique ﬁxed point, denoted by
x∗∈A.
Sincefis an ALC and having in view the deﬁnition of a b-pseudometric, we obtain:
dj(yn,x∗)≤bdj(yn,f(yn)) +bdj(f(yn),x∗) =
=bdj(yn,f(yn)) +bdj(f(x∗),f(yn))≤
≤bdj(yn,f(yn)) +bθudj(x∗,yn) +bLudj(x∗,f(x∗)).
From that, we conclude:
dj(yn,x∗)≤b
1−bθudj(yn,f(yn)),n≥0.
Applying (2.3.120), we have:
yn→x∗,asn→∞,
which means that the ﬁxed point problem is well posed.
Theorem 2.3.124 .Letf:A→Aan operator as appears in Theorem 2.3.106.
Thenfhas the limit shadowing property.
Proof:Choose the point yn∈A,n∈Nsuch that
(2.3.121) dj(yn+1,f(yn))→0asn→∞
According to Theorem 2.3.106, the operator fhas a unique ﬁxed point, denoted by
x∗∈A,this is actually the limit of the Picard iteration deﬁned by
xn+1=Txn, n∈N,

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 77
for anyx0∈A.By means of the b-pseudometric deﬁnition, we can write:
dj(yn,x∗)≤bdj(yn,f(yn−1)) +bdj(f(yn−1),x∗) =
=bdj(yn,f(yn−1)) +bdj(f(x∗),f(yn−1).
By using the uniqueness condition for ALC-s, we obtain:
dj(yn,x∗)≤bdj(yn,f(yn−1)) +bθudr(j)(yn−1,x∗) +bLudr(j)(x∗,f(x∗)).
Butx∗is a ﬁxed point of the operator f. Thus, we have:
dj(yn,x∗)≤bdj(yn,f(yn−1)) +bθudr(j)(yn−1,x∗).
Take the following substitutions: an=dj(yn,x∗),q=bθu,bn=bdj(yn+1,f(yn)).Note
thatbθu∈[0,1), we get from (2.3.121) and by using Lemma 2.2.2:
(2.3.122) dj(yn,x∗)→0,asn→∞.
Note thatx∗∈Ais the limit of the Picard iteration:
(2.3.123) x∗= limn→∞fn(x0),
for somex0∈A, thus we obtain that
dj(yn,fn(x0))≤bdj(yn,x∗) +bdj(x∗,fn(x0)).
By using (2.3.122) and (2.3.123), the last inequality leads us to:
dj(yn,fn(x0))→0,asn→∞,
which means, according to Deﬁnition 2.3.95, that fhas the limit shadowing property.
This completes the proof.
In the sequel, we propose to study the data dependence of the ﬁxed point for the
class of strict almost local contractions.
Theorem 2.3.125 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
b-pseudometrics deﬁned on X, with constant b≥1. We choose again a subset A⊂X
and we let τbe the weak topology on Xdeﬁned by the family D. Letf:A→Aan
operator as appears in Theorem 2.3.106 and g:A→Aa mapping satisfying:
(i)ghas at least one ﬁxed point, denoted by x∗
g∈Fix(g);
(ii) there exists η>0such that
(2.3.124) dj(f(x),g(x))≤η,
for anyx∈A.
Then
dj(x∗
f,x∗
g)≤bη
1−bθu,
wherex∗
fis the unique ﬁxed point of the operator f.

78 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS IN UNIFORM SPACES
Proof:Theorem 2.3.106 assures a unique ﬁxed point for the operator f, sayx∗
f.
Using theb-pseudometric property, we get:
dj(x∗
f,x∗
g) =dj(f(x∗
f),g(x∗
g))≤bdj(f(x∗
f),f(x∗
g)) +bdj(f(x∗
g),g(x∗
g)).
ByusingtheuniquenessconditionforALC-s,the(2.3.124)inequality,andthemonotony
property (2.1.10), it results that
dj(x∗
f,x∗
g)≤bθudr(j)(x∗
f,x∗
g) +bLudr(j)(x∗
f,f(x∗
f)) +bη≤
≤bθudj(x∗
f,x∗
g) +bLudj(x∗
f,f(x∗
f)) +bη.
In Theorem 2.3.106 we have the condition 1−bθu>0, it follows that:
dj(x∗
f,x∗
g)≤bη
1−bθu.
Now the proof is complete.
Theorem 2.3.126 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
b-pseudometrics deﬁned on X, with constant b≥1. Let theA⊂Xsubset and we let
τbe the weak topology on Xdeﬁned by the family D. Letf:A→Aan operator as
appears in Theorem 2.3.106 and fn:A→Aa mapping satisfying:
(i) the operator fnhas at least a ﬁxed point x∗
n∈Fix(fn),n∈N;
(ii)fnconverges uniformly to fasn→∞.
Thenxn→x∗asn→∞, whereFix(f) ={x∗}.
Proof:By using Theorem 2.3.106, the operator fhas a unique ﬁxed point, denoted
byx∗∈A.Having in view that fnconverges uniformly to fasn→∞,there exists
ηn∈R+,n∈Nsuch thatηn→0asn→∞and
dj(fn(x),f(x))≤ηn,
for anyx∈A.According to Theorem 2.3.125, it results that
dj(x∗
n,x∗)≤ηn·b
1−bθu,n∈N,
for each mappings fandfn.
From that, by taking n→∞, we obtain ηn→0, therefore the operator fhas unique
ﬁxed point. Now the proof is complete.
Theorem 2.3.127 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of
b-pseudometrics deﬁned on X, with constant b≥1. Let theA⊂Xsubset and we let
τbe the weak topology on Xdeﬁned by the family D. Letf:A→Aan operator as
appears in Theorem 2.3.106 and fn:A→A,n∈Na mapping satisfying:
(i) the operators fn,n∈Nare strict almost contractions with constants a∈[0,1
b),
K≥0andau∈[0,1
b),Ku≥0, respectively;
(ii)fn→fasn→∞.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 79
Thenx∗
n→x∗asn→∞, whereFix(fn) ={x∗
n},n∈NandFix(f) ={x∗}.
Proof:According to Theorem 2.3.106, the operators fn,n∈Nandfhave a unique
ﬁxed point, thus we will denote the sets of their ﬁxed points by Fix(f) ={x∗}and
Fix(fn) ={x∗
n}, respectively. Applying the deﬁnition of the b-pseudometric and the
monotony property for the b-pseudometric, we may write for any n∈N:
dj(x∗
n,x∗) =dj(fn(x∗)
n,f(x∗))≤bdj(fn(x∗
n),fn(x∗)) +bdj(fn(x∗),f(x∗))≤
≤bθudr(j)(x∗
n,x∗) +bLudr(j)(x∗
n,fn(x∗
n)) +bdj(fn(x∗),f(x∗))≤
≤bθudj(x∗
n,x∗) +bLudj(x∗
n,fn(x∗
n)) +bdj(fn(x∗),f(x∗)) =
=bθudj(x∗
n,x∗) +bdj(fn(x∗),f(x∗)).
From that, we conclude:
dj(x∗
n,x∗)≤b
1−bθudj(fn(x∗),f(x∗)).
By using condition ii), letting n→∞, it results that:
dj(x∗
n,x∗)→0,asn→∞.
The proof is complete.
At the end of this subsection we present a Maia type result related to strict ALC-s
onb-pseudometric spaces.
Theorem 2.3.128 .LetXbe a uniform space, we let τbe the weak topology on X
deﬁned by the family D. Consider dandρtwob- pseudometrics deﬁned on X, with
constantb≥1. Let the subset A⊂Xandf:A→Aan operator such that:
(i)dj(x,y)≤ρj(x,y), for anyx,y∈A;
(ii) the subset A⊂Xisτ-complete;
(iii)fis continuous with respect to d;
(iv)f: (X,ρ)→(X,ρ)is a strict ALC with constants θ∈[0,1
b),L≥0andθu∈
[0,1
b),Lu≥0, respectively.
Then:
(1)Fix(f) ={x∗};
(2) the Picard iteration {xn}n≥0converges to x∗for anyx0∈A.
Proof:Letx0∈A.According to condition iv), it results that {fn(x0)}n∈Nisτ-
Cauchy in (X,ρ), by applying the proof of Theorem 2.3.105. By using condition i), it
results that it is Cauchy sequence in (X,d), as well.
At this point, by ii) and iii) it is obvious that it converges in (X,d)to the unique ﬁxed
point off, denoted by x∗.
Now the proof is complete.

CHAPTER 3
MULTIVALUED ALMOST LOCAL
CONTRACTIONS
1. Multivalued self almost local contractions
The notion of multivalued contraction was ﬁrst introduced by Nadler in [ 68].
Let(X,d)be a metric space, we will denote the family of all nonempty bounded
and closed subsets of XwithCB(X).
ForA,B⊂X, we consider
D(x,A) =inf{d(x,y) :y∈A},the distance between xandA,
D(A,B) =inf{d(a,b) :a∈A,b∈B},the distance between AandB,
δ(A,B) =sup{d(a,b) :a∈A,b∈B},the diameter of AandB,
H(A,B) =max{sup{D(a,B) :a∈A},sup{D(b,A) :b∈B}},thePompeiu-Hausdorﬀ
metric onCB(X)induced by d.
We know thatCB(X)is a metric space with the Pompeiu-Hausdorﬀ distance func-
tionH. It is also known, that if (X,d)is a complete metric space then ( CB(X),H) is
a complete metric space, too. (Rus [ 88])
LetP(X)be the family of all nonempty subsets of Xand letT:X→P (X)be a
multivalued mapping. An element x∈Xwithx∈T(x)is called a ﬁxed point of T.
We will denote by Fix(T)the set of all ﬁxed points of T, i.e.,
Fix(T) ={x∈X:x∈T(x)}.
Definition 3.1.1.(Nadler [68]) Letf:X→Xbe a single-valued map and
T:X→CB (X)be a multivalued map .
(i) A point x∈Xis a ﬁxed point of f(resp.T) ifx=fx(resp.x∈Tx).
The set of all ﬁxed point of f(resp.T) is denoted by Fix(f), (resp.Fix(T)).
(ii) A point x∈Xis a coincidence point of fandTiffx∈Tx.
The set of all coincidence points of fandTwill be denoted by C(f,T).
(iii) A point x∈Xis a common ﬁxed point of fandTifx=fx∈Tx.
The set of all common ﬁxed points of fandTis denoted by F(f,T).
The following lemma could be found in Rus [ 88], it is useful for the next theorem.
Lemma 3.1.2.(see[68]) Let (X,d)be a metric space, let A,B⊂Xandq>1.
Then, for every a∈A, there exists b∈Bsuch that
(3.1.1) d(a,b)≤qH(A,B).
80

1. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS 81
Definition 3.1.3.(see[16]) Let (X,d)be a metric space and T:X→P (X)be a
multivalued operator. Tis said to be a multivalued almost contraction or a multivalued
(θ,L)-almost contraction if there exist two constants θ∈(0,1),L≥0such that
(3.1.2) H(Tx,Ty )≤θ·d(x,y) +L·D(y,Tx ),∀x,y∈X.
Remark 3.1.4.Because of the symmetry of the distance dandH, the almost
contraction condition (3.1.2)includes the following dual one:
(3.1.3) H(Tx,Ty )≤θ·d(x,y) +L·D(x,Ty ),∀x,y∈X.
Obviously, to prove the almost contractiveness of T, it is necessary to check both (3.1.2)
and(3.1.3).
Theorem 3.1.5.(Berinde V., Berinde M. [16]) Let (X,d)be a metric space and
T:X→P (X)be a (θ,L)- contraction. Then
(1)Fix(T)/negationslash=φ
(2) for any x0∈X, there exists an orbit {xn}∞
n=0ofTat the point x0that converges
to a ﬁxed point uofT, for which the following estimates hold:
(3.1.4) d(xn,u)≤hn
1−hd(x0,x1), n = 0,1,2…
(3.1.5) d(xn,u)≤h
1−hd(xn−1,xn), n = 1,2…
for a certain constant h<1.
With the assumptions from the deﬁnition of almost local contractions (see 2.1.6
and 2.1.5) and denote:
Dj(A,B) =inf{dj(a,b) :a∈A,b∈B},
δj(A,B) =sup{dj(a,b) :a∈A,b∈B},
Hj(A,B) =max{sup{Dj(a,B) :a∈A},sup{Dj(b,A) :b∈B}},
the Pompeiu-Hausdorﬀ metric on CB(X)induced by dj.
Remark 3.1.6.From the deﬁnition of Dj, we have the following result:
ifDj(a,B) = 0, this implies that a∈B.
Definition 3.1.7.(see[9]) A multivalued map T:X→C(X)is said to be
continuous at the point pif
limn→∞d(xn,p) = 0implies limn→∞H(Txn,Tp) = 0.
Observe that in the work of Rhoades (see [ 81]) instead of H, the author used the
functionalD. The following concept was published by Rus ([ 93]), respectively Rus et
al. ([94]) and we need them to study the ﬁxed points of multivalued ALC-s.

82 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
Definition 3.1.8.Let(X,d)be a metric space and T:X→P (X)be a multivalued
operator.Tis said to be a multivalued weakly Picard (brieﬂy MWP) operator if and
only if for each x∈Xand anyy∈T(x), there exists a sequence {xn}∞
n=0such that
(i)x0=x,x 1=y;
(ii)xn+1∈T(xn)for alln=o,1,2,···;
(iii) the sequence {xn}∞
n=0is convergent and its limit is a ﬁxed point of T.
Remark 3.1.9.A sequence{xn}∞
n=0satisfying conditions i) and ii) will be called a
sequence of successive approximations of T, starting from (x,y), or a Picard iteration
associated to T, or a (Picard) orbit of Tat the initial value x0.
A few examples of MWP operators will be presented from [94].
Approximate ﬁxed points of multivalued almost local contractions
Definition 3.1.10 .(see[7]) A multivalued mapping T:X→P (X)is said to have
the approximate ﬁxed point property provided
(3.1.6) inf
x∈Xd(x,Tx ) = 0,
or, equivalently, for any ε>0, there exists z∈Xsuch that
(3.1.7) d(z,Tz )≤ε,
or, equivalently, for any ε>0, there exists xε∈Xsuch that
(3.1.8) T(xε)∩B(xε,ε)/negationslash=φ,
whereB(x,r)denotes a closed ball of radius rcentered at x.
Main results
The purpose of the present chapter is to present the author’s main contributions to the
case of multivalued ALC-s, starting from the multivalued almost contractions. To this
end, we will improve a lot of corresponding results in literature [ 41], [44], [56], [57],
[62], [67], [49], [71], [76] and many others.
Definition 3.1.11 .Letr:J→J, letS⊂Xbe aτ-bounded sequentially τ-
complete and T-invariant subset of X. An operator T:S→P (S)is called a multival-
ued almost local contraction (ALC) related to ( D,r) if, for every j∈J, there exist the
constantsθ∈(0,1)andL≥0such that
(3.1.9) Hj(Tx,Ty )≤θ·dr(j)(x,y) +L·Dr(j)(y,Tx ),∀x,y∈S,∀j∈J.
The following deﬁnition represents an extension of generalized almost contractions,
concept introduced by Berinde V. and Berinde M. in [ 16] in the more general case of
generalized ALC-s.

1. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS 83
Definition 3.1.12 .Letr:J→J, letS⊂Xbe aτ-bounded sequentially τ-
complete and T-invariant subset of X. An operator T:S→P (S)is called a generalized
multivalued (α,L)almost local contraction with respect ( D,r) if, for every j∈J, there
exists a function α: [0,∞)→[0,1)satisfying lim supr→t+α(r)<1for everyt∈[0,∞)
such that
Hj(Tx,Ty )≤α(dr(j)(x,y))dr(j)(x,y) +L·min{dr(j)(x,Tx ),
dr(j)(y,Ty ),dr(j)(x,Ty ),dr(j)(y,Tx )},∀x,y∈S,∀j∈J. (3.1.10)
Theorem 3.1.13 .Letr:J→J, letS⊂Xbe aτ-bounded sequentially τ-complete
andT-invariant subset of X.
Consider the operator T:S→ P (S)a generalized multivalued (α,L)almost local
contraction related to ( D,r).
ThenThas at least one ﬁxed point.
Proof:The proof is very similar to that of Theorem 3.1.5 by replacing the term
θdr(j)(x,y)d(x,y)with the expression α(dr(j)(x,y))dr(j)(x,y), whereα: [0,∞)→[0,1)
satisfying conditions: lim supr→t+α(r)<1for everyt∈[0,∞)such that (3.1.10) holds.
Therefore we do not present the proof of this Theorem.
In order to introduce a ﬁxed point theorem related to these type of multivalued
ALC-s, we state and prove the following Lemma.
Lemma 3.1.14 .Letr:J→J,S⊂Xbe aτ-bounded sequentially τ-complete and
T-invariant subset of X.
Consider the mapping T:S→C(S). Then, for every x∈Swithdj(x,Tx )>0and
anyb∈(0,1),there exists y∈Tx,y/negationslash=x, such that
bdr(j)(x,y)≤dj(x,Tx ),∀j∈J.
Proof:Having in view that Txis nonempty and closed, the inequality dj(x,Tx )>
0implies that there exists y∈Tx,y/negationslash=x. From the deﬁnition of dj(x,Tx )we know
that, for every ε>0,there exists y∈Txsuch that
dr(j)(x,y)≤dj(x,Tx ) +ε.
At this point, by considering ε=/parenleftBig
1
b−1/parenrightBig
dj(x,Tx )>0, we obtain the conclusion of
this Lemma.
Theorem 3.1.15 .Letr:J→J, letS⊂Xbe aτ-bounded sequentially τ-complete
andT-invariant subset of X.
Consider the operator T:S→C (S)and assume that the following conditions are
fulﬁlled:
(i) the map f:X→R+f(x) =dr(j)(x,Tx ),x∈Sis lower semi-continuous;

84 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
(ii) there exist L≥0,b∈(0,1)andϕ: (0,∞)→[0,b)such that for all t∈(0,∞),
(3.1.11) lim sup
r→t+ϕ(r)<b,
and for all x∈S,∃y∈Ix
bsuch that
dj(y,Ty )≤α(dr(j)(x,y))dr(j)(x,y) +L·min{dr(j)(x,Tx ),
dr(j)(y,Ty ),dr(j)(x,Ty ),dr(j)(y,Tx )},∀j∈J. (3.1.12)
ThenThas a ﬁxed point.
Proof:If there exists x∈Ssuch thatdr(j)(x,Tx ) = 0, this means x∈Tx, which
meansxis a ﬁxed point of T.
Having in view that the range of Tis closed, for each b∈(0,1)and anyx∈X,
satisfyingdr(j)(x,Tx )>0, it results by Lemma 3.1.14 that there exists y∈Txsuch
thaty∈Ix
b,i.e.,
(3.1.13) bdr(j)(x,y)≤dj(x,Tx ).
Further, suppose that we have y∈Ix
b,y/negationslash=x,otherwisey=x∈Txis actually a ﬁxed
point ofT, which would complete the proof.
Takex1∈Sarbitrary but ﬁxed with dr(j)(x1,Tx 1)>0.Combining (3.1.13) and
assumption (ii), there exists x2∈Tx1,x2/negationslash=x1,satisfying:
(3.1.14) bdr(j)(x1,x2)≤dj(x1,Tx 1).
The last inequality can be write, using (3.1.11) in the equivalent form:
(3.1.15) dj(x2,Tx 2)≤ϕ/parenleftBig
dr(j)(x1,x2))dr(j)(x1,x2),ϕ(dr(j)(x1,x2)/parenrightBig
<b,
because ofdr(j)(x2,Tx 1) = 0.
At this point, (3.1.14) and (3.1.15) can be merged as:
dj(x1,Tx 1)−dj(x2,Tx 2)≥bdr(j)(x1,x2)−ϕ(dr(j)(x1,x2))·dr(j)(x1,x2) =
= [b−ϕ(dr(j)(x1,x2))](dr(j)(x1,x2)>0.
For the obtained x2, by continuing the construction of the sequence {xn}, we claim
there exists x3∈Tx2,x3/negationslash=x2,satisfying
(3.1.16) bdr(j)(x2,x3)≤dj(x2,Tx 2),
such that
(3.1.17) dj(x3,Tx 3)≤ϕ/parenleftBig
dr(j)(x2,x3))dr(j)(x2,x3),ϕ(dr(j)(x2,x3)/parenrightBig
<b.
Again, by (3.1.16) and (3.1.17) we get
dj(x2,Tx 2)−dj(x3,Tx 3)≥bdr(j)(x2,x3)−ϕ(dr(j)(x2,x3))·dr(j)(x2,x3) =
= [b−ϕ(dr(j)(x2,x3))]·dr(j)(x2,x3)>0.

1. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS 85
Thus, we obtain
dr(j)(x2,x3)≤1
bdj(x2,Tx 2)≤1
bϕ(dr(j)(x1,x2))·dr(j)(x1,x2)<dr(j)(x1,x2).
By induction with related to n>1, we conclude there exist xn+1∈Txn,xn/negationslash=xn+ 1,
such that
(3.1.18) bdr(j)(xn,xn+1)≤dj(xn,Txn)
and, similar to (3.1.17), satisfying
(3.1.19)dj(xn+1,Txn+1)≤ϕ/parenleftBig
dr(j)(xn,xn+1))dr(j)(xn,xn+1),ϕ(dr(j)(xn,xn+1)/parenrightBig
<b.
By using (3.1.18) and (3.1.19), we have
dj(xn,Txn)−dj(xn+1,Txn+1)≥bdr(j)(xn,xn+1)−
−ϕ(dr(j)(xn,xn+1))·dr(j)(xn,xn+1) =
= [b−ϕ(dr(j)(xn,xn+1))]·dr(j)(xn,xn+1)>0, (3.1.20)
and also, we get:
(3.1.21) dr(j)(xn,xn+1)<dr(j)(xn,xn−1).
It is obvious that {dj(xn,Txn)}n∈Nand{dr(j)(xn,xn+1)}n∈Nare both decreasing se-
quences of positive numbers, which means they are convergent.
By applying (3.1.10), it results that there exists s∈[0,b)such that
(3.1.22) ϕ(dr(j)(xn,xn+1))<b0,∀n>n 0.
Denotea=b−b0,we get from (3.1.20) the following inequality:
(3.1.23) dj(xn,Txn)−dj(xn+1,Txn+1)≥a/parenleftBig
dr(j)(xn,xn+1)/parenrightBig
,∀n>n 0.
At this point, we use (3.1.18), (3.1.19), (3.1.22) and we obtain for all n>n 0:
dj(xn+1,Txn+1)≤ϕ(dr(j)(xn,xn+1))≤ϕ(dr(j)(xn,xn+1))
b·dr(j)(xn,Txn)≤
≤…≤ϕ(dr(j)(xn,xn+1))·…·ϕ(dr(j)(x1,x2))
bn·dr(j)(x1,Tx 1) =
=ϕ(dr(j)(xn,xn+1))·…·ϕ(dr(j)(xn0+1,xn0+2))
bn−n0·
·ϕ(dr(j)(xn0,xn0+1))·…·ϕ(dr(j)(x1,x2))
bn0·dr(j)(x1,Tx 1)<
</parenleftBiggb0
b/parenrightBiggn−n0
·ϕ(dr(j)(xn0,xn0+1))·…·ϕ(dr(j)(x1,x2))
bn0·dr(j)(x1,Tx 1).
Note thatb0<b, therefore we have:
limn→∞/parenleftBiggb0
b/parenrightBiggn−n0
= 0,

86 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
thus, it follows from the previous inequalities:
limn→∞dj(xn,Txn) = 0.
We claim that{xn}is a Cauchy sequence. In order to show that, we use the triangle
inequality and (3.1.23), for n,p∈N,n,p>n 0and we get:
dr(j)(xn,xn+p)≤n+p−1/summationdisplay
j=ndr(j)(xj,xj+1)≤n+p−1/summationdisplay
j=ndr(j)(xj,xj+1)−dr(j)(xj+1,xj+2) =
=1
a/parenleftBig
dr(j)(xn,xn+1)−dr(j)(xn+p,xn+p)/parenrightBig
, (3.1.24)
which means that the sequence {xn}n∈Nisdj-Cauchy for each j∈J. As the subset S⊂
Xis assumed to be sequentially τ-complete, there exists x∗inSsuch that{Tnx}n∈N
isτ-convergent to x∗. Besides, the sequence {Tnx}n∈Nconverges with respect to the
topologyτtox∗, which implies
0≤dj(x∗,Tx∗)≤lim infn→∞dj(xn,Txn) = limn→∞dj(xn,Txn) = 0.
Using the fact that Tx∗is closed, we obtain x∗∈Tx∗,i.e.,x∗is a ﬁxed point of T.
Now the proof is complete.
The following Lemma is useful for the proof of future theorems.
Lemma 3.1.16 .LetSbe a subset of Xand letD= (dj)j∈Jbe a family of pseudo-
metrics deﬁned on X. We letτbe the weak topology on Xdeﬁned by the family D.
LetA,B⊂Sandq>1.
Then, for every j∈Janda∈A, there exists b∈Bsuch that
(3.1.25) dj(a,b)≤qHj(A,B).
Proof:IfHj(A,B) = 0, then for every a∈A, we have:
Hj(A,B)≥Dj(a,B)⇒Dj(a,B) = 0.
From that, we conclude: there exists b∈Bsuch thatdj(a,b) = 0.
The inequality (3.1.25) is valid, i.e., 0≤0.
IfHj(A,B)>0, then let us denote
(3.1.26) ε= (h−1−1)H(A,B)>0.
Using the deﬁnition of Hj(A,B)andDj(a,B), we conclude that for any ε >0there
existsb∈Bsuch that
(3.1.27) dj(a,b)≤qDj(a,B) +ε≤Hj(A,B) +ε.
Combining (2.3.107) and (3.1.27), we get the conclusion of the lemma.
Theorem 3.1.17 .With the assumptions of Deﬁnition 3.1.11, let T:S→P (S)be
a multivalued ALC. Then we have:

1. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS 87
(1)Fix(T)/negationslash=φ;
(2) for any x0∈S, there exists an orbit {xn}∞
n=0ofTat the point x0that converges
to a ﬁxed point uofT, for which the following estimates hold:
(3.1.28) dj(xn,u)≤hn
1−hdj(x0,x1), n = 0,1,2…
(3.1.29) dj(xn,u)≤h
1−hdj(xn−1,xn), n = 1,2…
for a certain constant h<1.
Proof:We consider q >1, letx0∈Xandx1∈Tx0. IfHj(Tx0,Tx 1) = 0, that
means from the deﬁnition of DjandHj:
(3.1.30) 0 =Hj(Tx0,Tx 1)≥Dj(x1,Tx 1),
and that is possible only if Dj(x1,Tx 1) = 0, from here, we conclude x1∈Tx1, which
leads us to the conclusion Fix(T)/negationslash=φ.
LetHj(Tx0,Tx 1)/negationslash= 0. According to Lemma 3.1.16, there exists x2∈Tx1such that
(3.1.31) dj(x1,x2)≤qHj(Tx0,Tx 1).
By (3.1.9), we have
dj(x1,x2)≤q[θ·dr(j)(x0,x1) +L·Dr(j)(x1,Tx 0)] =qθ·dr(j)(x0,x1),
sincex1∈Tx0,Dr(j)(x1,Tx 0) = 0.
We takeq>1such that
h=qθ< 1,
and we obtain dj(x1,x2)<h·dj(x0,x1).
IfHj(Tx1,Tx 2) = 0thenDj(x2,Tx 2) = 0, that means x2∈Tx2using Remark 3.1.6.
LetHj(Tx1,Tx 2)/negationslash= 0. Again, using Lemma 3.1.16, there exists x3∈Tx2such that
(3.1.32) dj(x2,x3)≤qh·dj(x1,x2),∀j∈J.
We obtain an orbit {xn}∞
n=0ofTat the point x0satisfying
(3.1.33) dj(xn,xn+1)≤h·dj(xn−1,xn),∀j∈J, n = 1,2,…
By (3.1.33), we inductively obtain
(3.1.34) dj(xn,xn+1)≤hndj(x0,x1),∀j∈J,
and, respectively,
(3.1.35) dj(xn+k,xn+k+1)≤hk+1dj(xn−1,xn), k∈N.
Using the inequality (3.1.34), we obtain
(3.1.36) dj(xn,xn+p)≤hn(1−hp)
1−hdj(x0,x1), n,p∈N.

88 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
Recall 0< h < 1, conditions (3.1.35), (3.1.36) show us that the sequence {xn}n∈Nis
dj-Cauchy for each j, which shows that {xn}∞
n=0is a Cauchy sequence. That means
{xn}∞
n=0is convergent with the limit u:
(3.1.37) u= limn→∞xn.
After simple computations, we get for all j∈J:
Dr(j)(u,Tu )≤Dr(j)(u,xn+1) +Dr(j)(xn+1,Tu)≤dr(j)(u,xn+1) +Hr(j)(Txn,Tu),
which, by (3.1.9) yields
(3.1.38) Dr(j)(u,Tu )≤dr(j)(u,xn+1) +θdr(j)(xn,u) +L·Dr(j)(u,Txn),∀j∈J.
Lettingn→∞and using the fact that xn+1∈Txnimplies by (3.1.37),
Dr(j)(u,Txn)→0, asn→∞. We get
Dr(j)(u,Tu ) = 0.
SinceTuis closed, this implies u∈Tu.
We letp→∞in (3.1.36) to obtain (3.1.28). Using (3.1.35), we get
(3.1.39) d(xn,xn+p)≤h(1−hp)
1−hd(xn−1,xn), p∈N,n≥1,
and letting p→∞in (3.1.39), we obtain (3.1.29).
The proof is complete.
The next Theorem shows that any multivalued ALC is continuous at the ﬁxed
point.
Theorem 3.1.18 .With the assumptions of Deﬁnition 3.1.11, let T:S→P (S)be
a multivalued ALC: a mapping for which there exist the constants
θ∈(0,1)andL≥0such that, for every j∈J, the next inequality is valid:
(3.1.40) Hj(Tx,Ty )≤θ·dr(j)(x,y) +L·Dr(j)(y,Tx ),∀x,y∈S.
ThenFix(T)/negationslash=φand for any p∈Fix(T),Tis continuous at p.
Proof:The existence of the ﬁxed points follows by Theorem 3.1.17.
It remains to prove the continuity of the mapping Tatp. To this end, let {yn}∞
n=0be
any sequence in the subset Sconverging to the ﬁxed point p. Then by taking y:=yn
andx:=pin the multivalued ALC condition (3.1.40), we get
(3.1.41) dj(Tp,Tyn)≤δ·dr(j)(p,yn) +L·Dr(j)(yn,Tp),n= 0,1,2,…
Using the deﬁnition of Dr(j)(yn,Tp), we know that is the smallest distance between
ynand any element from Tp, take for example p∈Tp. Now, we have the following
inequalities:
Dr(j)(yn,Tp)≤Dj(yn,Tp)≤dj(yn,p).

1. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS 89
By replacing Dr(j)(yn,Tp)from (3.1.41) with dj(yn,p), we get:
(3.1.42) dj(Tyn,Tp)≤(δ+L)·dj(yn,p),n= 0,1,2,…
Now, by letting n→∞in (3.1.42), we get Tyn→Tpasn→∞.
Thus,Tis continuous at p.
The following Theorem proves that any generalized strict multivalued ALC is con-
tinuous at the ﬁxed point.
Theorem 3.1.19 .With the assumptions of Deﬁnition 3.1.11, let T:S→CB (S)
be a multivalued (θ,L)-strict ALC: a mapping for which there exist θ∈[0,1)and some
L≥0such that∀x,y∈S,
Hj(Tx,Ty )≤θdr(j)(x,y) +Lmin{dr(j)(x,Tx ),dr(j)(y,Ty ),dr(j)(x,Ty ),dr(j)(y,Tx )},
for allj∈J.
ThenFix(T)/negationslash=φand for any p∈Fix(T),Tis continuous at p.
Proof:The proof for the existence of the ﬁxed point is very similar to the proof of
Theorem 3.1.17, with minor diﬀerences, but will be presented below:
We consider q >1, letx0∈Xandx1∈Tx0. IfHj(Tx0,Tx 1) = 0, that means from
the deﬁnition of DjandHj:
(3.1.43) 0 =Hj(Tx0,Tx 1)≥Dj(x1,Tx 1),
and that is possible only if Dj(x1,Tx 1) = 0. Thus, we obtain x1∈Tx1, which leads us
to the conclusion Fix(T)/negationslash=φ.
LetHj(Tx0,Tx 1)/negationslash= 0. According to Lemma 3.1.16, there exists x2∈Tx1such that
(3.1.44) dj(x1,x2)≤qHj(Tx0,Tx 1).
By (3.1.19) we have
dj(x1,x2)≤q[θ·dr(j)(x0,x1) +L·Dr(j)(x1,Tx 0)] =qθ·dr(j)(x0,x1),
since we have:
min{dr(j)(x0,Tx 0),dr(j)(x1,Tx 1),dr(j)(x0,Tx 1),dr(j)(x1,Tx 0)}=dr(j)(x1,Tx 0) = 0.
We takeq>1such that
h=qθ< 1,
and we obtain dj(x1,x2)<h·dj(x0,x1).
IfHj(Tx1,Tx 2) = 0, thenTx1=Tx2. Thus,x2∈Tx2, by using Remark 3.1.6.
LetHj(Tx1,Tx 2)/negationslash= 0. Again, using Lemma 3.1.16, there exists x3∈Tx2such that
(3.1.45) dj(x2,x3)≤qh·dj(x1,x2),∀j∈J.

90 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
We obtain an orbit {xn}∞
n=0ofTat the point x0satisfying
(3.1.46) dj(xn,xn+1)≤h·dj(xn−1,xn),∀j∈J, n = 1,2,…
By (3.1.46), we inductively obtain
(3.1.47) dj(xn,xn+1)≤hndj(x0,x1),∀j∈J,
and, respectively,
(3.1.48) dj(xn+k,xn+k+1)≤hk+1dj(xn−1,xn), k∈N.
Using the inequality (3.1.47), we obtain
(3.1.49) dj(xn,xn+p)≤hn(1−hp)
1−hdj(x0,x1), n,p∈N
Recall 0< h < 1, conditions (3.1.48),(3.1.49) show us that the sequence {xn}n∈Nis
dj-Cauchy for each j, which shows that {xn}∞
n=0is a Cauchy sequence. That means
{xn}∞
n=0is convergent with the limit u:
(3.1.50) u= limn→∞xn.
We get for all j∈J:
Dr(j)(u,Tu )≤Dr(j)(u,xn+1) +Dr(j)(xn+1,Tu)≤dr(j)(u,xn+1) +Hr(j)(Txn,Tu),
which, by (3.1.19), yields
(3.1.51) Dr(j)(u,Tu )≤dr(j)(u,xn+1) +θdr(j)(xn,u) +L·Dr(j)(u,Txn),∀j∈J.
Lettingn→∞and using the fact that xn+1∈Txnimplies by (3.1.50),
Dr(j)(u,Txn)→0, asn→∞. We get
Dr(j)(u,Tu ) = 0.
SinceTuis closed, this implies u∈Tu.
Let{yn}∞
n=0be any sequence in the subset Sconverging to p. Applying the condition
(3.1.19), by taking y:=ynandx:=p, we get:
dj(Tp,Tyn)≤θdr(j)(p,yn),n= 0,1,2,···,
having in view that:
min{dr(j)(p,Tp ),dr(j)(yn,Tyn),dr(j)(p,Tyn),dr(j)(yn,Tp)}=dr(j)(p,Tp ) = 0.
The last inequality can be write as:
(3.1.52) dj(Tyn,Tp)≤θdr(j)(yn,p),n= 0,1,2,···,
If we letn→∞in (3.1.52), we obtain Tyn→Tp, which means that Tis continuous
atp.
Now, the proof is complete.

1. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS 91
Theorem 3.1.19 can be extended to the more general case of generalized almost
local contractions, as it follows:
Theorem 3.1.20 .With the assumptions of Deﬁnition 3.1.11, let T:S→CB (S)
be a generalized multivalued (θ,L)-almost local contraction: a mapping for which there
existθ∈[0,1)and someL≥0such that∀x,y∈Sand∀j∈J:
(3.1.53) Hj(Tx,Ty )≤θdr(j)(x,y) +Ldr(j)(y,Tx ).
ThenFix(T)/negationslash=φand for any p∈Fix(T),Tis continuous at p.
Proof:The existence of the ﬁxed point follows from Theorem 3.1.17.
Let{yn}∞
n=0be any sequence in the subset Sconverging to p. By taking y:=ynand
x:=pin the generalized ALC (3.1.53) condition, we get:
dj(Tp,Tyn)≤θdr(j)(yn,p) +Ldr(j)(yn,Tp),n= 0,1,2,···.
Note that: Tp=p,which implies:
(3.1.54) dj(Tyn,Tp)≤(θ+L)·dr(j)(yn,p),n= 0,1,2,···,
If we letn→∞in (3.1.54), we obtain Tyn→Tpasn→∞, which means that Tis
continuous at p.
Next, we will prove that every generalized multivalued ALC has the approximate
ﬁxed point property.
Theorem 3.1.21 .Every generalized multivalued (α,L)-almost local contraction has
the approximate ﬁxed point property.
Proof:LetSbe a subset of Xand letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX. We letτbe the weak topology on Xdeﬁned by the family D.
LetT:S→Cl(S)a generalized multivalued almost local contraction, where Cl(S)
denote the families of all nonempty closed subsets of S. Take the following sequences:
xn∈Sandyn∈T(xn)be such that
(3.1.55) limn→∞dr(j)(xn,yn) = inf
x∈Sdj(x,Tx ),∀j∈J.
Without loss of generality, we may assume that the sequence {α(dr(j)(xn,yn))}n∈Nis
convergent. By using (3.1.55) and the deﬁnition 3.1.12, it follows that:
inf
x∈Sdj(x,Tx )≤inf{dj(y,Ty ) :y∈/uniondisplay
x∈STx}= inf
x∈Sinf
y∈Txdj(y,Ty )≤inf
x∈Sinf
y∈TxHr(j)(Tx,Ty )≤
≤inf
x∈Sinf
y∈Tx[α(dr(j)(x,y))dr(j)(x,y) +L·dr(j)(y,Tx )] =
= inf
x∈Sinf
y∈Tx[α(dr(j)(x,y))dr(j)(x,y)]≤
≤inf
n∈N[α(dr(j)(xn,yn))dr(j)(xn,yn)]≤limn→∞α(dr(j)(xn,yn)) limn→∞dr(j)(xn,yn)≤
≤ lim sup
r→(infx∈Sdj(x,Tx ))+α(r)·inf
x∈Sdj(x,Tx ).

92 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
Having in view that lim supr→(infx∈Sdj(x,Tx ))+α(r)<1, we obtain
inf
x∈Sdj(x,Tx ) = 0.
The proof is complete.
Remark 3.1.22 .Theorem 3.1.21 is valid also for multivalued (θ,L)-almost local
contractions, as shown in the next Theorem.
Theorem 3.1.23 .Every multivalued (θ,L)-almost local contraction has the approx-
imate ﬁxed point property, where θ∈[0,1)andL≥0.
Proof:The proof is very similar to that of Theorem 3.1.21, all we have to do is
just replace the term α(dr(j)(x,y))dr(j)(x,y)withθdr(j)(x,y),therefore we do not prove
it here.
Example 3.1.24 .By considering the multivalued map T: [0,1]→Cl([0,1])deﬁned
byTx= [0,1]for allx∈[0,1].We have:Hj(Tx,Ty ) = 0,∀x,y∈[0,1]⊂[0,∞)and
consider the application r:J→J,J=Nthe identity map, i.e., r(j) =j,∀j∈J. We
can write:Hj(Tx,Ty )≤1
3·dr(j)(x,y) + 0·Dr(j)(y,Tx ),∀x,y∈[0,1],∀j∈J.
ThereforeTis a multivalued ALC with θ=1
3∈[0,1),L= 0≥0.The set of ﬁxed points
is:Fix(T) = [0,1],hencex∈Txis valid for every x∈[0,1].Also,Tis continuous at
every ﬁxed point of T, thus Theorem 3.1.18 is valid.
Example 3.1.25 .(see[68]) Let (X,d)be a complete metric space and
T:X→CB (X)be a multivalued α-contraction (0< α < 1). ThenTis an MWP
operator.
Example 3.1.26 .(see Reich [76],[77],[79]) Let (X,d)be a complete metric space
andT:X→CB (X)be a multivalued operator for which there exist α,β,γ∈R+,α+
β <1such that
(1)H(Tx,Ty )≤θ·d(x,y) +L·D(y,Tx ),∀x∈Xand∀y∈Tx,
(2)Tis a closed multivalued mapping.
ThenTis an MWP operator.
2. Non-self multivalued almost local contractions
In [16], M. Berinde and V. Berinde introduce the non-self multivalued almost con-
tractions.
Definition 3.2.1.[16]The notations are the same as in the case of self multivalued
ALC-s. Let (X,d)be a metric space and Ka nonempty subset of X.
A mapT:K→CB(X)is called a multivalued non-self almost contraction if there
exist two constants δ∈(0,1)andL≥0such that
(3.2.1) H(Tx,Ty )≤δ·d(x,y) +L·D(y,Tx ),∀x,y∈K.

2. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS 93
Definition 3.2.2.[16]A metric space (X,d)is called convex if for each x,y∈X
withx/negationslash=ythere exists x∈X,x/negationslash=z/negationslash=ysuch that
(3.2.2) d(x,y) =d(x,z) +d(z,y).
Remark 3.2.3.The convex metric space is quite similar to the metric space of
hyperbolic type includes all normed linear spaces and all spaces with hyperbolic metric.
Note that, in a convex metric space each two points are the endpoints of at least one
metric segment. (see [3])
Proposition 3.2.4.(Assad and Kirk [3]) LetKbe a closed subset of a closed and
convex metric space X. Ifx∈Kandy /∈K, then there exists a point z∈∂Ksuch
that
(3.2.3) d(x,y) =d(x,z) +d(z,y)
Lemma 3.2.5.(see[3]) Let (X,d)be a metric space and A,B∈CB(X). Ifx∈A,
then for each positive number α, there exists y∈B, such that
(3.2.4) d(x,y)≤H(A,B) +α.
Fundamental results
The main result of this section is represented by two ﬁxed point theorems for
multivalued non-self almost local contractions.
Definition 3.2.6.LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X. We letτbe the weak topology on X, which is convex
space, deﬁned by the family D, letKa nonempty closed subset of X. Letr:J→J,
whereJdenote a family of indices.
An operator T:K→CB(X)is called a multivalued almost local contraction (ALC)
with respect (D,r) if, for every j∈J, there exist the constants θ∈(0,1)andL≥0
such that
(3.2.5) Hj(Tx,Ty )≤θ·dr(j)(x,y) +L·Dr(j)(y,Tx ),∀x,y∈K.
Theorem 3.2.7.Under the assumptions of deﬁnition 3.2.6, suppose that
T:K→CB(X)is a multivalued almost local contraction, that is,
(3.2.6) Hj(Tx,Ty )≤θ·dr(j)(x,y) +L·Dr(j)(y,Tx ),∀x,y∈K,
withθ∈(0,1)andL≥0such thatθ(1 +L)<1, and, also, assume that
Dr(j)(y,Tx )≤Dj(y,Tx )for eachj∈J,∀x,y∈K.
IfTsatisﬁes Rothe’s type condition, that is, x∈∂K=⇒Tx⊂K, thenThas a ﬁxed
point inK, which means: there exists z∈Ksuch thatz∈Tz.

94 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
Proof:Consider two sequences {xn}and{yn}by constructing them as it follows:
Letx0∈Kandy1∈Tx0. Ify1∈K, letx1=y1. Ify1/∈K, then there exists x1∈∂K
such that
(3.2.7) dj(x0,x1) +dj(x1,y1) =dj(x0,y1).
We havex1∈Kand using Lemma 3.2.5 with α=θ, we choose y2∈Tx1such that
(3.2.8) dj(y1,y2)≤Hj(Tx0,Tx 1) +θ.
Again, ify2∈K, putx2=y2. Ify2/∈K, then there exists x2∈∂Ksuch that
(3.2.9) dj(x1,x2) +dj(x2,y2) =dj(x1,y2).
Therefore,x2∈K, and by Lemma 3.2.5 and α=θ2, we can choose y3∈Tx2such that
(3.2.10) dj(y2,y3)≤Hj(Tx1,Tx 2) +θ2.
Continuing in this way, we construct two sequences {xn}and{yn}such that
(i)yn+1∈Txn,
(ii)dj(yn,yn+1)≤Hj(Txn−1,Txn) +θn, with
(iii)yn∈K⇒yn=xn,
(iv)yn/negationslash=xnwhenyn/∈K, and thenxn∈∂Ksatisfying the condition
(3.2.11) dj(xn−1,xn) +dj(xn,yn) =dj(xn−1,yn).
Our next goal is to prove that {xn}is a Cauchy sequence. For the simplicity, denote:
P={xi∈{xn}:xi=yi},
Q={xi∈{xn}:xi/negationslash=yi}.
It is obvious that if xn∈Q, then both xn−1andxn+1belong to the set P. We
distinguish three possibilities as it follows:
Case 1. Ifxn,xn+1∈P, having in view the deﬁnition of the set P, we haveyn=xn
andyn+1=xn+1. We get
dj(xn,xn+1) =dj(yn,yn+1)
≤Hj(Txn−1,Txn) +θn
≤θ·dj(xn−1,xn) +L·Dr(j)(xn,Txn−1) +θn
≤θ·dj(xn−1,xn) +θn,
becauseyn∈Txn−1.
Case 2. Ifxn∈P,xn+1∈Q.

2. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS 95
In this case we have yn=xnbutyn+1/negationslash=xn+1. We conclude from here:
dj(xn,xn+1)≤dj(xn,xn+1) +dj(xn+1,yn+1)
=dj(xn,yn+1)
=dj(yn,yn+1)
≤Hj(Txn−1,Txn) +θn
≤θ·dj(xn−1,xn) +L·Dr(j)(xn,Txn−1) +θn
≤θ·dj(xn−1,xn) +θn.
Case 3. Ifxn∈Q,xn+1∈P, thenyn/negationslash=xn,yn+1=xn+1,yn−1=xn−1andyn∈Txn−1.
After simple computations, we get
dj(xn,xn+1) =dj(xn,yn+1)
≤dj(xn,yn) +dj(yn,yn+1)
≤dj(xn,yn) +Hj(Txn−1,Txn) +θn
≤dj(xn,yn) +θ·dj(xn−1,xn) +L·Dr(j)(xn,Txn−1) +θn.
Sinceθ<1, we have
dj(xn,xn+1)≤dj(xn,yn) +dj(xn−1,xn) +L·Dr(j)(xn,Txn−1) +θn
=dj(xn−1,yn) +L·Dr(j)(xn,Txn−1) +θn
≤dj(xn−1,yn) +L·Dj(xn,Txn−1) +θn
≤dj(xn−1,yn) +L·dj(xn,yn) +θn
=dj(xn−1,yn) +L·dj(xn−1,yn)−L·dj(xn−1,xn) +θn
≤(1 +L)dj(yn−1,yn) +θn
≤(1 +L)Hj(Txn−2,Txn−1) + (1 +L)θn−1+θn
≤(1 +L)θ·dj(xn−2,xn−1) + (1 +L)LDr(j)(xn−1,Txn−2)
+ (1 +L)θn−1+θn
≤(1 +L)θ·dj(xn−2,xn−1) + (1 +L)θn−1+θn.
Having in view the condition h=θ(1 +L)<1, we conclude that
(3.2.12) dj(xn,xn+1)<h·dj(xn−2,xn−1) +h·θn−2+θn.
By combining all three cases, we get
(3.2.13) dj(xn,xn+1)≤αdj(xn−1,xn) +αn,
or the other possibility:
(3.2.14) αdj(xn−2,xn−1) +αn−1+αn,

96 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
where
(3.2.15) α= max{θ,h}=h.
We inductively obtain:
(3.2.16) dj(xn,xn+1)≤hn−1
2·ω+hn
2,
where
(3.2.17) ω= max{dj(x0,x1),dj(x1,x2)}.
By takingn>m, we have
dj(xn,xm)≤dj(xn,yn−1) +dj(xn−1,yn−2)
+···+dj(xm−1,xm)
≤(hn−1
2+hn−2
2+…hm−1
2)ω
+αn
2·n+αn−2
2·(n−1) +···+αm
2·m.
These relations show us that the sequence {xn}n∈Nisdj-Cauchy for each j∈J. As
the subsetAis assumed to be sequentially τ-complete, there exists zinKsuch that
(3.2.18) z= limn→∞xn.
As we constructed the sequence {xn}n∈N, there is a subsequence {xp}such that
(3.2.19) yp=xp∈Txp−1.
In what follows, we propose to prove that z∈Tz. In fact, by (i), xp∈Txp−1. Since
xp→zasp→∞, we have
(3.2.20) D(z,Txp−1)→0,asq→∞.
After simple computations, we get
Dj(z,Tz )≤dj(z,xp) +dj(xp,Tz)
≤dj(z,xp) +Hj(Txp−1,Tz)
≤dj(z,xp) +θ·dj(xp−1,z) +L·Dr(j)(z,Txp−1)
≤dj(z,xp) +θ·dj(xp−1,z) +L·Dj(z,Txp−1).
Now, if we let q→∞, implies that Dj(z,Tz ) = 0. Thus, we get z∈Tz.
The proof is complete.
Note that, by Theorem 3.2.7 we obtain a ﬁxed point theorem for multivalued non-self
almost contractions stated by Maryam A. Alghamdi, Vasile Berinde, and Naseer
Shahzad [1] as a particular case by letting r(j) =j,θ=δ, and (X,d)be a complete
convex metric space.

2. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS 97
Corollary 3.2.8.Under the assumptions of deﬁnition 3.2.6, suppose that
T:K→CB(X)is a multivalued contraction, that is,
(3.2.21) Hj(Tx,Ty )≤θ·dr(j)(x,y),∀x,y∈K,
withθ∈(0,1). Further, we assume that
Dr(j)(y,Tx )≤Dj(y,Tx ),
for eachj∈J,∀x,y∈K.
IfTsatisﬁes Rothe’s type condition, that is, x∈∂K=⇒Tx⊂K, thenThas a ﬁxed
point inK, which means: there exists z∈Ksuch thatz∈Tz.
Remark 3.2.9.Corollary 3.2.8 represents a particular case of Theorem 3.2.7 by taking
L= 0, therefore we skip over the proof.
Theorem 3.2.10 .LetXbe a uniform space and let D= (dj)j∈Jbe a family of pseudo-
metrics deﬁned on X, where (X,d)is convex pseudometric space. We let τbe the weak
topology on X, deﬁned by the family D, with the monotony property (2.3.103) assured.
LetKa nonempty closed subset of X. Taker:J→J, whereJdenote a family of
indices.
The operator T:K→CB(X)satisfy the following contractive condition:
for everyx,y∈K,
Hj(Tx,Ty )≤α·dr(j)(x,y) +β·max{Dr(j)(x,Tx ),Dr(j)(y,Ty )}+
+γ[Dr(j)(x,Ty ) +Dr(j)(y,Tx )],∀j∈J (3.2.22)
withα,β,γ≥0such thatp:=/parenleftBig
1+α+γ
1−β−γ/parenrightBig/parenleftBig
α+β+γ
1−γ/parenrightBig
<1.
IfTsatisﬁes Rothe’s boundary condition, that is, x∈∂K=⇒Tx⊂K, thenThas a
ﬁxed point in K, which means: there exists z∈Ksuch thatz∈Tz.
Proof:As in the proof of the previous Theorem, consider two sequences {xn}and
{yn}by constructing them step by step as it follows:
Letx0∈Kandy1∈Tx0. Ify1∈K, letx1=y1. Ify1/∈K, then there exists x1∈∂K
such that
(3.2.23) dj(x0,x1) +dj(x1,y1) =dj(x0,y1).
Takey2∈Tx1and, according to Lemma 3.2.5, we have:
(3.2.24) dj(x1,y2)≤Hj(Tx0,Tx 1) + (1−β−γ)ε,
whereε=p.Again, ify2∈K, putx2=y2. Ify2/∈K, then there exists x2∈∂Ksuch
that
(3.2.25) dj(x1,x2) +dj(x2,y2) =dj(x1,y2).
Continuing in this way, we introduce two sequences {xn}and{yn}such that

98 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
(i)yn+1∈Txn,
(ii)dj(yn,yn+1)≤Hj(Txn−1,Txn) + (1−β−γ)εn,
(iii)xn+1=yn+1ifyn+1∈K,
(iv) ifyn+1/∈K, thenxn+1shall satisfy the condition
(3.2.26) dj(xn,xn+1) +dj(xn+1,yn+1) =dj(xn,yn+1).
Our next goal is to prove that {xn}is a Cauchy sequence. Denote:
P={xi∈{xn}:xi=yi},
Q={xi∈{xn}:xi/negationslash=yi}.
We distinguish three cases as it follows:
Case 1. Ifxn,xn+1∈P, having in view (3.2.22), we get:
dj(xn,xn+1)≤Hj(Txn−1,Txn) + (1−β−γ)εn≤
≤α·dr(j)(xn−1,xn) +β·max{Dr(j)(xn−1,Txn−1),Dr(j)(xn,Txn)}+
+γ[Dr(j)(xn−1,Txn) +Dr(j)(xn,Txn−1)] + (1−β−γ)εn≤
≤α·dr(j)(xn−1,xn) +β·max{dr(j)(xn−1,xn),dr(j)(xn,xn+1)}+
+γdr(j)(xn−1,xn+1) + (1−β−γ)εn≤
≤max{(α+β+γ)dr(j)(xn−1,xn) + (1−β−γ)εn
1−γ,
,(α+γ)dr(j)(xn−1,xn) + (1−β−γ)εn
1−β−γ}≤
≤max/braceleftBigg(α+β+γ)
1−γ,α+γ
1−β−γ/bracerightBigg
dr(j)(xn−1,xn) +εn=
=k·dr(j)(xn−1,xn) +εn,
where we denote k:=(α+β+γ)
1−γ.
Case 2. Ifxn∈P,xn+1∈Q, from (3.2.22), we have:
dj(xn,xn+1)≤dj(xn,xn+1) +dj(xn+1,yn+1) =
=dj(xn,yn+1) =
=dj(yn,yn+1)≤
≤Hj(Txn−1,Txn) + (1−β−γ)εn≤
≤α·dr(j)(xn−1,xn) +β·max{Dr(j)(xn−1,Txn−1),Dr(j)(xn,Txn)}+
+γ[Dr(j)(xn−1,Txn) +Dr(j)(xn,Txn−1)] + (1−β−γ)εn≤
≤α·dr(j)(xn−1,xn) +β·max{dr(j)(xn−1,xn),dr(j)(xn,yn+1)}+
+γdr(j)(xn−1,yn+1) + (1−β−γ)εn.

2. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS 99
We observe that: if in the coeﬃcient of β, the maximum is dr(j)(xn−1,xn), and if the
monotony condition holds, then we can write:
dj(xn,yn+1)≤(α+β+γ)dj(xn−1,xn) + (1−β−γ)εn
1−γ.
On the other hand, if in the coeﬃcient of β, the maximum is dr(j)(xn,yn+1), and if the
monotony condition holds, then we get:
dj(xn,yn+1)≤(α+γ)dj(xn−1,xn) + (1−β−γ)εn
1−β−γ.
After analyzing all cases, we conclude:
(3.2.27) dj(xn,xn+1)≤k·dr(j)(xn−1,xn) +εn,∀j∈J.
Case 3. Ifxn∈Q,xn+1∈P, thenyn/negationslash=xn,yn+1=xn+1,yn−1=xn−1andyn∈Txn−1.
After using (3.2.22), by applying case 2, we get:
dj(xn,xn+1) =dj(xn,yn+1)≤
≤dj(xn,yn) +dj(yn,xn+1)≤
≤dj(xn,yn) +Hj(Txn−1,Txn) + (1−β−γ)εn≤
≤dj(xn−1,yn) +α·dj(xn−1,xn) +β·max{Dr(j)(xn−1,Txn−1),Dr(j)(xn,Txn)}+
+γ[Dr(j)(xn−1,Txn) +Dr(j)(xn,Txn−1)] + (1−β−γ)εn≤
≤dj(xn−1,yn) +α·dj(xn−1,xn) +β·max{dr(j)(xn−1,yn),dr(j)(xn,xn+1)}+
+γ[dr(j)(xn−1,xn+1) +dr(j)(xn,yn)] + (1−β−γ)εn.
According to the triangle inequality, and the deﬁnition of sets P,Q, we can write:
dj(xn−1,xn+1) +dj(xn,yn)≤dj(xn−1,xn) +dj(xn,xn+1) +dj(xn,yn) =
=dj(xn−1,yn) +dj(xn,xn+1).
Therefore
dj(xn,xn+1)≤max/braceleftBig(1 +α+β+γ)dr(j)(xn−1,yn) + (1−β−γ)εn
1−γ,
(1 +α+γ)dr(j)(xn−1,yn) + (1−β−γ)εn
1−β−γ/bracerightBig
≤
≤max/braceleftBigg(1 +α+β+γ)
1−γ,(1 +α+γ)
1−β−γ/bracerightBigg
dr(j)(xn−1,yn) +εn≤
≤1 +α+γ
1−β−γdr(j)(xn−1,yn) +εn≤
≤p·dj(xn−2,xn−1) +(1 +α+γ)εn−1
1−β−γ+εn.
After that step, it results by induction with respect to n, that:
dj(x2n,x2n+1)≤pn/parenleftBigg
δ+3n
1−β−γ/parenrightBigg
,

100 3. MULTIVALUED ALMOST LOCAL CONTRACTIONS
and also results the following inequality:
dj(x2n+1,x2n+2)≤p2n+1
2/parenleftBigg
δ+3n+ 1
1−β−γ/parenrightBigg
.
The last two inequalities let us to conclude that for any m>n,
dj(xm,xn)≤m−1/summationdisplay
i=ndj(xi,xi+1)≤δm−1/summationdisplay
i=npi
2+1
1−β−γm−1/summationdisplay
i=npi
2(3i+ 1),
which means that (xn)n∈Nisdj-Cauchy for each j∈J.
As the subset Kis assumed to be sequentially τ-complete, hence convergent with the
limitz. But for the sequence {xn)n∈Nwe can always choose a subsequence {xnk}n∈N
such thatxnk=ynk.
From that, it follows that:
Dj(xnk,Tz)≤Hj(Txnk−1,Tz)≤
≤αdr(j)(xnk−1,z) +βmax{Dr(j)(xnk−1,Txnk−1),Dr(j)(z,Tz )}+
+γ[Dr(j)(xnk−1,Tz) +Dr(j)(z,Txnk−1)]≤
≤αdr(j)(xnk−1,z) +βmax{dr(j)(xnk−1,xnk),Dr(j)(z,Tz )}+
+γ[Dr(j)(xnk−1,Tz) +dr(j)(z,xnk)].
Now, if we let k→∞,we obtainDj(z,Tz )≤(β+γ)Dj(z,Tz ),i.e. ,z∈Tz.Thus,T
has the ﬁxed point z.
The proof is complete.
Example 3.2.11 .(see[74]) LetX= [0,1], and the mapping ddd:X×X→Cdeﬁned by
ddd(x,y) =|x−y|2+i|x−y|2for allx,y∈X. Then (X,ddd)is a complex valued b-metric
space withs= 2.

CHAPTER 4
NON-SELF SINGLE VALUED ALMOST LOCAL
CONTRACTIONS
1. Introduction
In this chapter, the notion of non-self single valued ALC-s in a pseudometric space is
considered. In this framework some new ﬁxed point results are given.
LetXbe a uniform space and let D= (dj)j∈Jbe a family of pseudometrics deﬁned
onX. We letτbe the weak topology on Xdeﬁned by the family D.
LetKa nonempty closed subset of XandT:K→Xa non-self single valued ALC,
that is a mapping satisfying (2.1.3).
Ifx∈Kis verifying Tx /∈K, then we can always choose an y∈∂K(the boundary of
K) such that y= (1−λ)x+λ·Tx, (0<λ< 1), which actually means that
(4.1.1) d(x,Tx ) =d(x,y) +d(y,Tx ),y∈∂K.
In general, the set Yof pointsysatisfying condition (4.1.1) above may contain more
than one element.
We shall need the following concept from [ 17]:
Definition 4.1.1.([17]) LetXbe a uniform space and let Ka nonempty closed subset
ofX,T:K→Xa non-self mapping. Let x∈KwithTx /∈Kand lety∈∂Kbe the
corresponding elements given by (4.1.1). If, for any such elements x, we have
(4.1.2) d(y,Ty )≤d(x,Tx ),
for at least one of the corresponding y∈Y, then we say that Thas property (M).
Next, we leave the usual metric space to the pseudometric space, by giving the new
deﬁnition of property (M).
Definition 4.1.2.LetXbe a uniform space and let Ka nonempty closed subset of
X,T:K→Xa non-self mapping. Denote by D= (dj)j∈Ja family of pseudometrics
deﬁned onX. We shall consider τbe the weak topology on Xdeﬁned by the family D,
letJa family of indices and a function r:J→J.
Letxj∈Kfor a certain j∈JwithTxj/∈Kand letyj∈∂Kbe the corresponding
elements given by
(4.1.3) dj(x,Tx ) =dj(x,y) +dj(y,Tx ),yj∈∂K.
101

102 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
If, for the aforementioned elements xj, we have
(4.1.4) dj(y,Ty )≤dj(x,Tx ),
for at least one of the corresponding yj∈Y, then we say that Thas property (M).
Very recently, Rus, I.A. and Serban, M.A published their work (see [ 95]), regarding
the non-self operators.
Definition 4.1.3.(see[95]) Let (X,d)be a metric space, Y∈Pcl(X)andf:Y→X
be a continuous non-self operator. The maximal displacement functional corresponding
tofrepresents the functional Ef:P(Y)→R+∪{+∞}deﬁned by
(4.1.5) Ef(A) := sup{d(x,f(x))|x∈A},
whereP(X) ={Y⊂X|Y/negationslash=φ}.
We have that:
(i)A,B∈P(Y),A⊂BimpliesEf(A)≤Ef(B),
(ii)Ef(A) =Ef(A), for allA∈P(Y).
Definition 4.1.4.(see[95]) An operator f:Y→Xis calledα-graphic contraction
if0≤α<1andx∈Y,f(x)∈Yimply
(4.1.6) d(f2(x),f(x))≤αd(x,f(x)).
Main results
The next Theorem states and proves the existence of the ﬁxed point for non-self
single valued ALC.
Theorem 4.1.5.LetXbe a uniform space and let Ka nonempty closed subset of X,
T:K→Xa non-self ALC, that is, a mapping for which there exist the constants
θ∈(0,1)andL≥0such that
(4.1.7) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈K.
Assume that conditions from Theorem 2.1.9 are veriﬁed, also assume again that the
monotony property for the pseudometric is valid.
IfThas property (M) and satisﬁes Rothe’s boundary condition
(4.1.8) T(∂K)⊂K,
thenThas a ﬁxed point in K.
Proof:IfT(K)⊂K, thenTis in fact a self mapping on the closed set Kand
the conclusion follows by Theorem 2.1.9 for X=K. It is natural to consider the case
T(K)/negationslash⊂K. Letx0∈∂K. Using (4.1.8) we know that Tx0∈K. Deﬁne a sequence
{xn}n∈Nin the following way:

1. INTRODUCTION 103
Letx1=Tx0. IfTx1∈K, setx2=Tx1. IfTx1/negationslash∈K, we can choose an element x2on
the segment x1,Tx 1which belong to ∂K, that is,
x2= (1−λ)x1+λTx 1(0<λ< 1).
The terms of the sequence {xn}deﬁned this way are satisfying one of the following
properties:
(i)xn=Txn−1, ifTxn−1∈K,
(ii)xn= (1−λ)xn−1+λTxn−1∈∂K(0<λ< 1), ifTxn−1/negationslash∈K.
By introducing the following notations, we shall simplify our proof:
P={xk∈{xn}:xk=Txk−1},
Q={xk∈{xn}:xk/negationslash=Txk−1}.
It is obvious that {xn}⊂Kand also, if xk∈Q, then both xk−1andxk+1belong to the
setP. Moreover, according to (4.1.8), we cannot have two consecutive terms of {xn}
in the setQ(but it is possible to have two consecutive terms of {xn}in the setP).
We shall prove that {xn}is aτ-Cauchy sequence.
To prove this, we must discuss three diﬀerent cases:
Case I.xn,xn+1∈P,
Case II.xn∈P,xn+1∈Q,
Case III.xn∈Q,xn+1∈P.
Case I.xn,xn+1∈P. Having in view the deﬁnition of the set P, we havexn=Txn−1
andxn+1=Txn. By (2.1.3) we get
dj(xn+1,xn) =dj(Txn,Txn−1)≤θ·dr(j)(xn,xn−1) +L·dr(j)(xn,Txn−1),
which means
(4.1.9) dj(xn+1,xn)≤θ·dr(j)(xn,xn−1),
sincexn=Txn−1.
Case II.xn∈P,xn+1∈Q.
In this case we have xn=Txn−1butxn+1/negationslash=Txnand
dj(xn,xn+1) +dj(xn+1,Txn) =dj(xn,Txn).
That means
dj(xn,xn+1)≤dj(xn,Txn) =dj(Txn−1,Txn),
and by using (2.1.3), we get
dj(xn,xn+1)≤θ·dr(j)(xn,xn−1) +L·dr(j)(xn,Txn−1) =θ·dr(j)(xn,xn−1),
which yields again inequality (4.1.9).
Case III.xn∈Q,xn+1∈P.

104 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
In this case, we have xn−1∈P. The mapping Thas property (M), this means: there
exists an indice j∈Jsuch that
(4.1.10) dj(xn,xn+1) =dj(xn,Txn)≤dj(xn−1,Txn−1).
Sincexn−1∈P, we havexn−1=Txn−2and by (2.1.3) we get
dj(Txn−2,Txn−1)≤θ·dr(j)(xn−2,xn−1) +L·dr(j)(xn−1,Txn−2) =θ·dr(j)(xn−2,xn−1).
From that, by combining the last inequality with (4.1.10), we obtain
(4.1.11) dj(xn,xn+1)≤θ·dr(j)(xn−2,xn−1).
At this point, after analyzing all three cases, and using (4.1.9), (4.1.11), and the
monotony property for the pseudometric, it follows that the sequence {xn}verify the
inequality:
dj(xn,xn+1)≤θ·max{dr(j)(xn−2,xn−1),dr(j)(xn−1,xn)}≤
≤θ·max{dj(xn−2,xn−1),dj(xn−1,xn)}, (4.1.12)
for alln≥2. Now, by induction for n≥2from (4.1.12) one obtains
dj(xn,xn+1)≤θ[n/2]·max{dj(x0,x1),dj(x1,x2)},
where [n/2]denotes the greatest integer not exceeding n/2.
Moreover, for m>n>N ,
dj(xn,xm)≤∞/summationdisplay
i=Ndj(xi,xi−1)≤2·θ[N/2]
1−θmax{dj(x0,x1),dj(x1,x2)}.
The last inequality shows that {xn}isτ-Cauchy sequence.
Note that{xn}⊂KandKis closed, which means that {xn}converges to some point
inK.
Denote
(4.1.13) x∗= limn→∞xn,
and let{xnk}⊂P, be an inﬁnite subsequence of {xn}denoted for simplicity also by
{xn}. It is clear that such a subsequence always exists.
Using the triangle inequality and the deﬁnition of P, we get:
dj(x∗,Tx∗)≤dj(x∗,xn+1) +dj(xn+1,Tx∗) =dj(xn+1,x∗) +dj(Txn,Tx∗).
Using (2.1.3), we obtain
dj(Txn,Tx∗)≤θ·dr(j)(xn,x∗) +L·dr(j)(x∗,Txn),
and hence
(4.1.14) dj(x∗,Tx∗)≤dj(xn+1,x∗) +θ·dr(j)(xn,x∗) +L·dr(j)(x∗,Txn)

1. INTRODUCTION 105
for alln≥0. Lettingn→∞in (4.1.14), we get the ﬁnal conclusion for our proof, i.e.,
dj(x∗,Tx∗) = 0
which shows that x∗is a ﬁxed point of T.
Remark 4.1.6.A mapping Tsatisfying (2.1.3), i.e.,Tis a non-self ALC, the mapping
Tmay be discontinuous (see Example 4.1.16), however Tis continuous at the ﬁxed
point. For the argumentation, if {yn}is a sequence in K, convergent to x∗=Tx∗,
then by(2.1.3)we get
dj(Tyn,x∗) =dj(Tx∗,Tyn)≤θ·dr(j)(x∗,yn) +L·dr(j)(yn,Tx∗).
From that inequality, by letting n→∞, we obtain the continuity of Tat the ﬁxed point
x∗, that is:dj(Tyn,x∗)→0asn→∞, which means Tyn→x∗.
The proof is complete.
Example 4.1.16 contains a non-self mapping with property (M), but discontinuous,
without being an almost local contraction. The following Theorem assures the
uniqueness of the ﬁxed point for non-self ALC.
Theorem 4.1.7.LetXbe a uniform space and let Ka nonempty closed subset of X,
T:K→Xa non-self ALC, that is, a mapping for which there exist the constants
θ∈(0,1)andL1≥0such that
(4.1.15) dj(Tx,Ty )≤θ·dr(j)(x,y) +L1·dr(j)(y,Tx ),∀x,y∈K.
Assume that an additional condition holds:
(U) for every ﬁxed j∈Jthere exists:
(4.1.16) limn→∞(θ+L)ndiamrn(j)(z,A) = 0,∀x,y∈X.
IfThas property (M) and satisﬁes Rothe’s boundary condition:
(4.1.17) T(∂K)⊂K,
thenThas a unique ﬁxed point in K
Remark 4.1.8.The proof is quite similar to the case of single valued self almost local
contractions (see Theorem 2.1.12).
Starting from the work of Rus-Șerban [ 95], our main aim is to extend these notions in
the more general case of non-self single valued almost local contractions.
Definition 4.1.9.LetXbe a uniform space and let D= (dj)j∈Jbe a family of
pseudometrics deﬁned on X. We letτbe the weak topology on Xdeﬁned by the family
D.

106 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
LetKa nonempty closed subset of X. An operator T:K→Xis calledα-graphic
local contraction if 0≤α<1andx∈K,f(x)∈Kimply
(4.1.18) dj(T2x,Tx )≤αdr(j)(x,Tx ),∀j∈J.
In the sequel, we study several types of non-self ALC-s, in order to establish if they
areα-graphic local contractions.
Theorem 4.1.10 .Use the assumptions of Deﬁnition 4.1.9, and assume the monotony
property (2.1.10)valid for the pseudometric. Let K∈Pcl(X)andT:K→Xa non-
self Ćirić-Reich-Rus operator, then Tis a non-self α-graphic local contraction with
α=δ+L
1−L.
Proof:Letx∈Ksuch thatTx∈K. Note that Tis a Ćirić-Reich-Rus type almost
local contraction, that is,
(4.1.19) dj(Tx,Ty )≤δ·dr(j)(x,y) +L·[dr(j)(x,Tx ) +dr(j)(y,Ty )],∀j∈J,
for allx,y∈K, whereδ,L∈R +andδ+ 2L < 1. If we replace y:=Txin the last
inequality, after applying the monotony property (2.1.10) for the pseudometric, we can
write:
dj(Tx,T2x)≤δ·dr(j)(x,Tx ) +L·[dr(j)(x,Tx ) +dr(j)(Tx,T2x)]≤
≤δ·dr(j)(x,Tx ) +L·[dr(j)(x,Tx ) +dj(Tx,T2x)],
which implies:
dj(Tx,T2x)≤δ+L
1−L·dr(j)(x,Tx ),∀j∈J.
The proof is complete.
Theorem 4.1.11 .Use the assumptions of Deﬁnition 4.1.9, and assume the monotony
property (2.1.10)valid for the pseudometric. Let K∈Pcl(X)and
T:K→Xa non-self almost local contraction, then Tis a non-self α-graphic local
contraction with α=θ.
Proof:Letx∈Ksuch thatTx∈K. We use the deﬁnition of an ALC, that is,
dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈K,∀j∈J,
withθ∈[0,1)andL≥0.Again, we apply the same substitution, y:=Txand we
obtain:
dj(Tx,T2x)≤θ·dr(j)(x,Tx ) +L·dr(j)(Tx,Tx ),∀x,y∈K,∀j∈J.
From that, we conclude:
dj(Tx,T2x)≤θ·dr(j)(x,Tx ),∀x,y∈K,∀j∈J,
which shows that Tis a non-self α-graphic local contraction with α=θ.
This completes the proof.

1. INTRODUCTION 107
Remark 4.1.12 .A non-self generalized ALC is also a non-self α-graphic local contrac-
tion withα=θ. Indeed, in the deﬁnition of a generalized ALC, that is, a mapping
satisfying
dj(Tx,Ty )≤θ·dr(j)(x,y) +
+L·min{dr(j)(x,Tx ),dr(j)(y,Ty ),dr(j)(x,Ty ),dr(j)(y,Tx )},
if we repeat the substitution y:=Tx, we obtain exactly the same inequality as in the
case of non-self ALC:
dj(Tx,T2x)≤θ·dr(j)(x,Tx ),∀x,y∈K,∀j∈J.
Thus,Tis a non-self α-graphic local contraction with α=θ.
Theorem 4.1.13 .A non-self generalized Berinde-type ALC is also a non-self α-graphic
local contraction with α=θ. Under the assumptions of Deﬁnition 4.1.9, the operator
T:K→Xis called generalized Berinde type ALC with respect ( D,r) if there exist a
constantθ∈[0,1)and a function bfrom the subset Kinto[0,∞)such that
dj(Tx,Ty )≤θ·dr(j)(x,y) +b(y)·dr(j)(y,Tx ),∀x,y∈K,∀j∈J.
After applying the well-known substitution y:=Tx, we get:
dj(Tx,T2x)≤θ·dr(j)(x,Tx ) +b(y)·dr(j)(Tx,Tx ),∀x,y∈K,∀j∈J.
From that, we can write:
dj(Tx,T2x)≤θ·dr(j)(x,Tx ),∀x,y∈K,∀j∈J,
which shows that Tis a non-self α-graphic local contraction with α=θ.
This completes the proof.
Theorem 4.1.14 .Use the assumptions of Deﬁnition 4.1.9, and assume the monotony
property (2.1.10)valid for the pseudometric. Let K∈Pcl(X)and
T:K→Xa non-self Chatterjea-type ALC, then Tis a non-self α-graphic local
contraction with α=c
1−c.
Proof:Remindthatanon-selfChatterjea-typeALCwithregardto( D,r)or(δ,L)-
Chatterjea contraction is a mapping T:K→Xfor which there exists a constant
0≤c<1
2such that
(4.1.20) dj(Tx,Ty )≤c·[dr(j)(x,Ty ) +dr(j)(y,Tx )],∀x,y∈K.

108 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
Letx∈Ksuch thatTx∈K. We use (4.1.20), and after applying the substitution
y:=Tx, by using the triangle inequality, also the monotony property, we get:
dj(Tx,T2x)≤c·[dr(j)(x,T2x) +dr(j)(Tx,Tx )] =
=c·[dr(j)(x,Tx ) +dr(j)(Tx,T2x)]≤
≤c·[dr(j)(x,Tx ) +dj(Tx,T2x)].
From that, we conclude:
dj(Tx,T2x)≤c
1−cdr(j)(x,Tx ),∀j∈J,
which means that Tis a non-self α-graphic local contraction with α=c
1−c.
Example 4.1.15 .ConsiderX= [1,2]∪{4}with the usual norm, let
K={1,2,4}⊂Xand take the non-self mapping T:K→X,deﬁned by
T(x,y) =

0if x∈{1,2}
1
3if x= 4
The only value x∈KwithTx /∈Kis represented by x= 4and its corresponding set
isY={2}, having in view that (4.1.1)need to be fulﬁlled. We have:
d(y,Ty ) =d(2,0) =|2−0|= 2,
d(x,Tx ) =d/parenleftbigg
4,1
3/parenrightbigg
=/vextendsingle/vextendsingle/vextendsingle/vextendsingle4−1
3/vextendsingle/vextendsingle/vextendsingle/vextendsingle=11
3.
Obviously, (4.1.2)holds, therefore Thas property (M).
Example 4.1.16 .LetXbe the set of real numbers with the usual metric, K= [0,1]
and letT:K→Xbe deﬁned by Tx=−1
10, ifx=9
10, andTx=x
x+1, ifx/negationslash=9
10. We
choose the identity function r(j) =j.
Tsatisﬁes condition (4.1.2), becauseThas property (M), Tis discontinuous at9
10, the
unique ﬁxed point of Tis0, andTis continuous at 0.Thas property (M). Indeed, if
x=9
10∈K,Tx=−1
10/negationslash∈K, then using the condition (4.1.2)we have/vextendsingle/vextendsingle/vextendsingley
y+1/vextendsingle/vextendsingle/vextendsingle≤1. This
is valid for both y∈{0,1}, soy∈∂K. However, Tnot satisfy the ALC condition,
take for example x/negationslash=9
10andy=x
x+1in(2.1.3)to get, for any x>0,
dj(Tx,Ty ) =dj/parenleftBigx
x+ 1/parenrightBig
=/vextendsingle/vextendsingle/vextendsinglex2
(x+ 1)(2x+ 1)/vextendsingle/vextendsingle/vextendsingle;dj(x,y) =/vextendsingle/vextendsingle/vextendsinglex2
(x+ 1)/vextendsingle/vextendsingle/vextendsingle;dj(y,Tx ) = 0.
By replacing these distances in (2.1.3), we get the equivalent form:1
2x+1≤θ < 1,
x>0.
If we take now x→0in the last double inequality, we obtain a contradiction:
1≤θ<1.

2. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS 109
Example 4.1.17 .LetX= [0,1]∪{2}be endowed with the usual norm and let K=
{0,1,2}. Consider a self-mapping T:K→Xdeﬁned byTx= 0, ifx∈{0,1}and
T2 =1
2. The only element x∈KwithTx/negationslash∈Kisx= 2and the corresponding set is
Y={1}, and since
d(y,Ty ) =d(1,T1) =|1−0|<|2−0.5|=d(2,T2) =d(x,Tx ).
Hence, property (M) obviously holds.
Example 4.1.18 .Iff:Y→Xisα-contraction then fisα-graphic contraction.
Example 4.1.19 .Iff:Y→Xisα-Kannan operator,i.e., 0≤α<1
2and
d(f(x),f(y))≤α[d(x,f(x)) +d(y,f(y))],∀x,y∈Y,
thenfisα
1−α-graphic contraction.
2. Complex valued b-metric spaces and rational local contractions
The aim of this section is to extend the framework to the more generous set of
complex numbers, providing new results.
The notion of b-metric space was ﬁrst introduce by Bakhtin [ 5] which represents a
more general form of metric space. Furthermore, the complex valued metric space has
been introduced by Azam et al. [ 4].
Definition 4.2.1.[5]LetCbe the set of complex numbers and z1,z2∈C.
Deﬁne a partial order -onC, as follows:
z1-z2if and only if Re(z1)≤Re(z2),Im(z1)≤Im(z2).
Thusz1-z2if one of the following holds:
(1)Re(z1) =Re(z2)andIm(z1) =Im(z2),
(2)Re(z1)<Re (z2)andIm(z1) =Im(z2),
(3)Re(z1) =Re(z2)andIm(z1)<Im (z2),
(4)Re(z1)<Re (z2)andIm(z1)<Im (z2).
We will consider z1z2ifz1/negationslash=z2and one of (2), (3), (4) is satisﬁed.
This means
(i)0-z1z2implies|z1|<|z2|,
(ii)z1-z2andz2≺z3implyz1≺z3,
(iii) 0-z1-z2implies|z1|≤|z2|,
(iv) ifa,b∈R,0≤a≤bandz1-z2, thenaz1-bz2for allz1,z2∈C.
The next deﬁnition was introduced by Rao et al.[ 74], will be very useful for the study
of rational local contractions in complex valued b-metric spaces.

110 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
Definition 4.2.2.[74]LetXbe a uniform space and let s≥1be a given real number.
A functionddd:X×X→Cis called a complex valued b-metric on Xif for allx,y,z∈X
the following conditions are satisﬁed:
(i)0-ddd(x,y)andddd(x,y) = 0if and only if x=y
(ii)ddd(x,y) =ddd(y,x),
(iii)ddd(x,y)-s[ddd(x,z) +ddd(z,y)].
The pair (X,ddd)is called a complex valued b-metric space.
In the sequel, we state the deﬁnition of the convergent and Cauchy sequence in a
complex valued b-metric space, which shall be helpful in this section.
Definition 4.2.3.(see[74]) Let (X,ddd)be a complex valued b-metric space and let
{xn}be a sequence in Xand assume x∈X.
(i) If for every c∈C, with 0≺rthere isN∈Nsuch that for all n>N,ddd(xn,x)≺c,
then{xn}is said to be convergent, {xn}converges to x, andxis the limit point
of{xn}. We denote this by limn→∞xn=xor{xn}→xasn→∞
(ii) If for every c∈C, with 0≺rthere isN∈Nsuch that for all n > N,
ddd(xn,xn+m)≺c, wherem∈N, then{xn}is said to be Cauchy sequence.
(iii) If every Cauchy sequence in Xis convergent, then (X,ddd)is called a complete
complex valued b-metric space.
Lemma 4.2.4.(see[74]) Let (X,ddd)be a complex valued b-metric space and let {xn}be
a sequence in X. Then{xn}converges to xif and only if|ddd(xn,x)|→0asn→∞.
Lemma 4.2.5.(see[74]) Let (X,ddd)be a complex valued b-metric space and let {xn}be
a sequence in X. Then{xn}is a Cauchy sequence if and only if |ddd(xn,xn+m)|→0as
n→∞, wherem∈N.
Fundamental results
The starting point of this subsection is represented by complex valued rational local
contractions, which turn to be a very generous concept because generalizations are
available.
Definition 4.2.6.LetXbe a uniform space and let D= (dddj)j∈Jbe a family of complex
valued b-metric on X. Let (X,ddd)be a complete complex valued b-metric space with the
coeﬃcients≥1. The mapping T:X→Xis called complex valued rational local
contraction if the following inequality
(4.2.1) dddj(Tx,Ty )-αdddj(x,y) +βdddr(j)(x,Tx )dddr(j)(y,Ty )
dddr(j)(x,Ty ) +dddr(j)(y,Tx ) +dddr(j)(x,y)
is valid for all x,y∈Xsuch thatx/negationslash=y,dddr(j)(x,Ty ) +dddr(j)(y,Tx ) +dddr(j)(x,y)/negationslash= 0,
whereα,βrepresent non-negative real numbers.

2. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS 111
Next, we extend the study of ﬁxed point theorems in complex valued b-metric spaces
to the more general case of ﬁxed points for rational local contractions.
Theorem 4.2.7.Let(X,ddd)be a complete complex valued b-metric space with the co-
eﬃcients≥1and let two mappings S,T:X→Xsatisfying
(4.2.2) dddj(Sx,Ty )-αdddr(j)(x,y) +βdddr(j)(x,Sx )dddr(j)(y,Ty )
dddr(j)(x,Ty ) +dddr(j)(y,Sx ) +dddr(j)(x,y)
for allx,y∈Xsuch thatx/negationslash=y, whereα,βare non-negative reals satisfying the
conditionα+sβ < 1ordddj(Sx,Ty ) = 0ifdddr(j)(x,Ty ) +dddr(j)(y,Sx ) +dddr(j)(x,y) = 0.
ThenSandThave a unique common ﬁxed point.
Proof:For any arbitrary x0∈X, deﬁne a sequence {xn}inXsuch that
x2n+1=Sx2n, (4.2.3)
x2n+2=Tx2n+1, (4.2.4)
forn= 0,1,2,3,…
Our ﬁrst goal is to prove that the sequence {xn}is Cauchy sequence.
Takex=x2nandy=x2n+1in (4.2.2). Thus, we obtain
dddj(x2n+1,x2n+2) =
=dddj(Sx2n,Tx 2n+1)-αdddr(j)(x2n,x2n+1) +
+βdddr(j)(x2n,Sx 2n)dddr(j)(x2n+1,Tx 2n+1)
dddr(j)(x2n,Tx 2n+1) +dddr(j)(x2n+1,Sx 2n) +dddr(j)(x2n,x2n+1)=
=αdddr(j)(x2n,x2n+1) + (4.2.5)
+βdddr(j)(x2n,Sx 2n)dddr(j)(x2n+1,x2n+2)
dddr(j)(x2n,x2n+2) +dddr(j)(x2n+1,x2n+1) +dddr(j)(x2n,x2n+1)
We obtain
|dddj(x2n+1,x2n+2)|≤
≤α|dddr(j)(x2n,x2n+1)|+ (4.2.6)
+β|dddr(j)(x2n,x2n+1)|·|dddr(j)(x2n+1,x2n+2)|
|dddr(j)(x2n,x2n+2)|+|dddr(j)(x2n,x2n+1)|.
Due to triangular inequality, we have
(4.2.7)|dddj(x2n+1,x2n+2)|≤|dddj(x2n+1,x2n)|+|dddj(x2n,x2n+2)|,
which yields
|dddj(x2n+1,x2n+2)| ≤α·|dddr(j)(x2n,x2n+1)|+
+β·|dddr(j)(x2n,x2n+1)|=
= (α+β)·|dddr(j)(x2n,x2n+1)|.

112 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
So, we get
(4.2.8) |dddj(x2n+1,x2n+2)|≤(α+β)·|dddr(j)(x2n,x2n+1)|.
In a similar way, we obtain
(4.2.9) |dddj(x2n+2,x2n+3)|≤(α+β)·|dddr(j)(x2n+1,x2n+2)|.
Having in view that α+sβ < 1ands≥1, we getα+β <1. Denoteδ=α+β <1,
we obtain for all n≥0 :
|dddj(x2n+1,x2n+2)| ≤δ·|dddr(j)(x2n,x2n+1)|+
≤δ2·|dddr(j)(x2n−1,x2n)|≤···≤ (4.2.10)
≤δ2n+1·|dddr(j)(x0,x1)|.
After simple computations we get
|dddj(xn+1,xn+2)| ≤δ·|dddr(j)(xn,xn+1)|≤δ2·|dddr(j)(xn−1,xn)|≤
≤ ···≤δn+1·|dddr(j)(x0,x1)|. (4.2.11)
Continuing in this way, for any n>m, andm,n∈N, we have
|dddj(xn,xm)| ≤s|dddr(j)(xn,xn+1)|+s|dddr(j)(xn+1,xm)|≤
≤s|dddr(j)(xn,xn+1)|+s2|dddr(j)(xn+1,xn+2)|+
+s2|dddr(j)(xn+1,xn+2)|≤ (4.2.12)
≤s|dddr(j)(xn,xn+1)|+s2|dddr(j)(xn+1,xm)|+
+s3|dddr(j)(xn+2,xn+3)|+s3|dddr(j)(xn+3,xm)|≤
≤ ···≤
≤s|dddr(j)(xn,xn+1)|+s2|dddr(j)(xn+1,xm)|+
+s3|dddr(j)(xn+2,xn+3)|+···+
+sm−n−2|dddr(j)(xm−3,xm−2)|+
+sm−n−1|dddr(j)(xm−2,xm−1)|+
+sm−n|dddr(j)(xm−1,xm)|.

2. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS 113
By using (4.2.11), we obtain
|dddj(xn,xm)| ≤sδn|dddr(j)(x0,x1)|+s2δn+1|dddr(j)(x0,x1)|+
+s3δn+2|dddr(j)(x0,x1)|+···+
+sm−n−2δm−3|dddr(j)(x0,x1)|+ (4.2.13)
+sm−n−1δm−2|dddr(j)(x0,x1)|+
+sm−nδm−1|dddr(j)(x0,x1)|=
=m−n/summationdisplay
i=1siδi+n−1|dddr(j)(x0,x1)|.
Therefore,
|dddj(xn,xm)| ≤m−n/summationdisplay
i=1si+n−1δi+n−1|dddr(j)(x0,x1)|=
=m−1/summationdisplay
t=nstδt|dddr(j)(x0,x1)|. (4.2.14)
Thus, we obtain:
|dddj(xn,xm)|≤∞/summationdisplay
t=n(sδ)t·|dddr(j)(x0,x1)|
From that, we conclude
(4.2.15) |dddj(xn,xm)|≤(sδ)n
1−sδ|dddr(j)(x0,x1)|→0.
asm,n→∞.
Thus, we prove that {xn}is a Cauchy sequence in X. SinceXis complete, there exists
u∈Xsuch thatxn→uasn→∞.
Next, we show that the mapping Shas the ﬁxed point u. To this end, assume the
contrary, that is,
(4.2.16) |dddj(u,Su )|=|z|>0.
According to the triangular inequality and (4.2.2), we get
z=dddj(u,Su )-sdddj(u,x 2n+2) +sdddj(x2n+2,Su) =
=sdddj(u,x 2n+2) +sdddj(Tx2n+1,Su)-
-sdddj(u,x 2n+2) +sαdddj(u,x 2n+1) +
+sβdddr(j)(u,Su )dddr(j)(x2n+1,Tx 2n+1)
dddr(j)(u,Tx 2n+1) +dddr(j)(x2n+1,Su) +dddr(j)(u,x 2n+1)=
=sdddj(u,x 2n+2) +sαdddj(u,x 2n+1) +
+sβdddr(j)(u,Su )dddr(j)(x2n+1,x2n+2)
dddr(j)(u,x 2n+2) +dddr(j)(x2n+1,Su) +dddr(j)(u,x 2n+1).

114 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
Thus, we obtain
|z|=|dddj(u,Su )|≤
≤s|dddr(j)(u,x 2n+2)|+sα·|dddr(j)(u,x 2n+1)|+ (4.2.17)
+sβ[|dddr(j)(u,Su )|·|dddr(j)(x2n+1,x2n+2)|]
|dddr(j)(u,x 2n+2)|+|dddr(j)(x2n+1,Su)|+|dddr(j)(u,x 2n+1)|.
Taking the limit of (4.2.17) as n→∞, we conclude that
|z|=|dddj(u,Su )|≤0,
which is a contradiction with (4.2.16). So, |z|= 0, henceSu=u.Proceeding in a
similar way, we obtain Tu=u.
Our goal is to show that SandThave the unique common ﬁxed point u. To prove
this, assume that u∗is another ﬁxed point of SandT. Then
dddj(u,u∗) =dddj(Su,Tu∗)-
-αdddr(j)(u,u∗) + (4.2.18)
+β[dddr(j)(u,Su )dddr(j)(u∗,Tu∗)]
dddr(j)(u,Tu∗) +dddr(j)(u∗,Su) +dddr(j)(u,u∗).
From that, we conclude
|dddj(u,u∗)| ≤
≤α|dddr(j)(u,u∗)|+ (4.2.19)
+β[|dddr(j)(u,Su )|·|dddr(j)(u∗,Tu∗)|]
|dddr(j)(u,Tu∗)|+|dddr(j)(u∗,Su)|+|dddr(j)(u,u∗)|-
-α|dddr(j)(u,u∗)|,
which means u=u∗. Therefore, the uniqueness of the ﬁxed point is proved.
In the second case:
(4.2.20) dddr(j)(x,Ty ) +dddr(j)(y,Sx ) +dddr(j)(x,y) = 0,
putx=x2nandy=x2n+1in the last condition and we get:
(4.2.21) dddr(j)(x2n,Tx 2n+1) +dddr(j)(x2n+1,Sx 2n) +dddr(j)(x2n,x2n+1) = 0,
for anyn= 0,1,2,….
We obtain dddr(j)(Sx2n,Tx 2n+1) = 0, which yields x2n=Sx2n=x2n+1=Tx2n+1=
x2n+2. Thus, we have x2n+1=Sx2n=x2n. From that, we conclude: there exist
K1andl1such thatK1=Sl1=l1, whereK1=x2n+1andl1=x2n. Continuing
in this way, there exist K2andl2such thatK2=Tl2=l2, whereK2=x2n+2and
l2=x2n+1. Asdddr(j)(l1,Tl2) +dddr(j)(l2,Sl1) +dddr(j)(l1,l2) = 0(according to (4.2.20)),
impliesdddr(j)(Sl1,Tl2) = 0. Thus,K1=Sl1=l1=Tl2=K2. So, we get from here
K1=Sl1=l1=SK 1. In a quite similar way, we can have K2=TK 2. ButK1=K2
impliesSK 1=KT 1=K1thereforeK1=K2is common ﬁxed point of SandT.

2. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS 115
For the proof of the uniqueness of common ﬁxed point, assume that K∗
1inXis another
common ﬁxed point of SandT. That means SK∗
1=TK∗
1=K∗
1.
From the condition (4.2.20), we have dddr(j)(K1,TK∗
1)+dddr(j)(K∗
1,SK 1)+dddr(j)(K1,K∗
1) =
0, thereforedddr(j)(K1,K∗
1) =dddr(j)(SK 1,TK∗
1) = 0, and this means that K1=K∗
1.
This completes the proof of the theorem.
Theorem 4.2.8.Let(X,ddd)be a complete complex valued b-metric space with the co-
eﬃcients≥1and let two mappings S,T:X→Xsatisfying
dddj(Sx,Ty )-αdddr(j)(x,y) +β[ddd2
r(j)(x,Ty ) +ddd2
r(j)(y,Sx )]
dddr(j)(x,Ty ) +dddr(j)(y,Sx )+
+γ[dddr(j)(x,Sx ) +dddr(j)(y,Ty )], (4.2.22)
for allx,y∈Xsuch thatx/negationslash=y, whereα,β,γare non-negative reals satisfying the
conditionα+ 2sβ+ 2γ <1ordddj(Sx,Ty ) = 0ifdddr(j)(x,Ty ) +dddr(j)(y,Sx ) = 0.
ThenSandThave a unique common ﬁxed point.
Proof:For any arbitrary x0∈X, deﬁne a sequence {xn}inXsuch that
x2n+1=Sx2n, (4.2.23)
x2n+2=Tx2n+1, (4.2.24)
forn= 0,1,2,3,…
Our ﬁrst goal is to prove that the sequence {xn}is Cauchy sequence.
Takex=x2nandy=x2n+1in (4.2.22). We have
dddj(x2n+1,x2n+2) =
=dddj(Sx2n,Tx 2n+1)-αdddr(j)(x2n,x2n+1) +
+β·[ddd2
r(j)(x2n,Tx 2n+1) +ddd2
r(j)(x2n+1,Sx 2n)]
dddr(j)(x2n,Tx 2n+1) +dddr(j)(x2n+1,Sx 2n)+
+γ[dddr(j)(x2n,Sx 2n) +dddr(j)(x2n+1,Tx 2n+1)] =
=αdddr(j)(x2n,x2n+1) +
+β·[ddd2
r(j)(x2n,x2n+2) +ddd2
r(j)(x2n+1,x2n+1)]
dddr(j)(x2n,x2n+2) +dddr(j)(x2n+1,x2n+1)+
+γ·[dddr(j)(x2n,x2n+1) +dddr(j)(x2n+1,x2n+2)].
From here, by using the modulus, we obtain:
|dddj(x2n+1,x2n+2)|≤
≤α·|dddr(j)(x2n,x2n+1)|+
+β·|ddd2
r(j)(x2n,x2n+2)|
|dddr(j)(x2n,x2n+2)|+
+γ·[|dddr(j)(x2n,x2n+1)|+|dddr(j)(x2n+1,x2n+2)|].

118 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
Thus,{xn}is a Cauchy sequence in X. SinceXis complete, there exists some u∈X
such thatxn→uasn→∞.
Next, we prove that uis ﬁxed point of S. To this end, assume not, then there exists
z∈Xsuch that
(4.2.34) dddj(u,Su ) =|z|>0.
According to the triangular inequality and (4.2.2), we get
z=dddj(u,Su )-sdddr(j)(u,x 2n+2) +sdddr(j)(x2n+2,Su) =
=sdddr(j)(u,x 2n+2) +sdddr(j)(Su,x 2n+1)-
-sdddr(j)(u,x 2n+2) +sαdddr(j)(u,x 2n+1) +
+sβ[ddd2
r(j)(u,Tx 2n+1) +ddd2
r(j)(x2n+1,Su)]
dddr(j)(u,Tx 2n+1) +dddr(j)(x2n+1,Su)+
+sγ[dddr(j)(u,Su ) +dddr(j)(x2n+1,Tx 2n+1)] = (4.2.35)
=sdddr(j)(u,x 2n+2) +sα·dddr(j)(u,x 2n+1) +
+sβ[ddd2
r(j)(u,x 2n+2) +ddd2
r(j)(x2n+1,Su)]
dddr(j)(u,x 2n+2) +dddr(j)(x2n+1,Su)+
+sγ[z+dddr(j)(x2n+1,x2n+2)].
The last inequality implies that
|z|=|dddj(u,Su )|≤
≤s|dddr(j)(u,x 2n+2)|+sα·|dddr(j)(u,x 2n+1)|+ (4.2.36)
+sβ[|ddd2
r(j)(u,x 2n+2)|+|ddd2
r(j)(x2n+1,Su)|]
|dddr(j)(u,x 2n+2)|+|dddr(j)(x2n+1,Su)|+
+sγ[|z|+|dddr(j)(x2n+1,x2n+2)|].
Taking the limit of (4.2.36) as n→∞, we conclude that
|z|=|dddj(u,Su )|≤0,
and this is a contradiction with (4.2.34). So |z|= 0, henceSu=u.The proof for the
fact thatuis ﬁxed point for the mapping Tis quite similar, so Tu=u.
Now, all we have to do is to show that SandThave the unique common ﬁxed point
u. To show this, assume that u∗is another ﬁxed point of SandT. Then
dddj(u,u∗) =dddj(Su,Tu∗)-
-αdddr(j)(u,u∗) + (4.2.37)
+β[ddd2
r(j)(u,Tu∗) +ddd2
r(j)(u∗,Su)]
dddr(j)(u,Tu∗) +dddr(j)(u∗,Su)+
+γ[dddr(j)(u,Su ) +dddr(j)(u∗,Tu∗)]≺
≺dddr(j)(u,u∗),

2. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS 119
which is a contradiction. So u=u∗, and we prove the uniqueness of the common ﬁxed
point inX.
For the second case dddj(Sx,Ty ) = 0ifdddr(j)(x,Ty )+dddr(j)(y,Sx ) = 0, the proof of unique
commonﬁxedpointisquitesimilartothatappearinginTheorem4.2.7. Thiscompletes
the proof of the theorem.
As a natural consequence of theorem 4.2.8, two corollaries could be stated as it
follows:
Corollary 4.2.9.Let(X,ddd)be a complete complex valued b-metric space with the
coeﬃcients≥1and letT:X→Xbe a mapping satisfying
dddj(Tx,Ty )-αdddr(j)(x,y) +β[ddd2
r(j)(x,Ty ) +ddd2
r(j)(y,Tx )]
dddr(j)(x,Ty ) +dddr(j)(y,Tx )+
+γ[dddr(j)(x,Tx ) +dddr(j)(y,Ty )], (4.2.38)
for allx,y∈Xsuch thatx/negationslash=y, whereα,β,γare non-negative reals satisfying the
conditionα+ 2sβ+ 2γ <1ordddj(Tx,Ty ) = 0ifdddr(j)(x,Ty ) +dddr(j)(y,Tx ) = 0.
ThenThas a unique ﬁxed point.
Proof:The proof arise from Theorem 4.2.8, by taking S=T.
Corollary 4.2.10 .Let(X,ddd)be a complete complex valued b-metric space with the
coeﬃcients≥1and letT:X→Xbe a mapping satisfying (for some ﬁxed n):
dddj(Tnx,Tny)-αdddr(j)(x,y) +
+β[ddd2
r(j)(x,Tny) +ddd2
r(j)(y,Tnx)]
dddr(j)(x,Tny) +dddr(j)(y,Tnx)+ (4.2.39)
+γ[dddr(j)(x,Tnx) +dddr(j)(y,Tny)],
for allx,y∈Xsuch thatx/negationslash=y, whereα,β,γare non-negative reals satisfying the
conditionα+ 2sβ+ 2γ <1ordddj(Tnx,Tny) = 0ifdddr(j)(x,Tny) +dddr(j)(y,Tnx) = 0.
ThenThas a unique ﬁxed point.
Proof:By using Corollary 4.2.8, one obtain that u∈Xsuch thatTnu=u. To
show the uniqueness of the ﬁxed point, we use the following relations:
dddj(Tu,u ) =dddj(TTnu,Tnu) =dddj(TnTu,Tnu)-
-αdddr(j)(Tu,u ) +
+β[ddd2
r(j)(Tu,Tnu) +ddd2
r(j)(u,TnTu)]
dddr(j)(Tu,Tnu) +dddr(j)(u,TnTu)+ (4.2.40)
+γ[dddr(j)(Tu,TnTu) +dddr(j)(u,Tnu)] =
= (α+ 2β)dddj(Tu,u ).

120 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
By taking modulus of (4.2.40) and using α+ 2β <1, we get
|dddj(Tu,u )|≤(α+ 2β)|dddj(Tu,u )|<|dddj(Tu,u )|,
which is obviously a contradiction. Thus, Tu=u.
HenceTu=Tnu=u. Therefore, the uniqueness of the ﬁxed point is proved.
This completes the proof.

CHAPTER 5
Application of almost local contractions in dynamic
programming
1. Introduction
The aim of this chapter is to illustrate the use of local almost contractions to solve
the Bellman equation. k-almost local contractions
The theory of dynamic programming (DP) with uncountable state space begins with
the essential work of Blackwell (see [ 30]). His results were extended in a large
number of directions, with important applications in economy, operations and
engineering. In fact, the theory of stochastic optimal growth lies in the framework of
dynamic programming.
Throughout the ﬁrst part of this chapter we recalled some important concepts and
results from Rincon-Zapatero [ 82] and Matkowski-Nowak [ 66].
Xwill denote a topological space such that
(5.1.1) X=∞/uniondisplay
j=1Kj,
with{Kj}an increasing sequence of compact subsets of X. Suppose that
X=∞/uniondisplay
j=1Int(Kj).
Denote byC(X)the set of all real-valued, continuous functions over X. We will
introduce a countable family of pseudometrics, as follows:
(5.1.2) dj(f,g) := max
x∈Kj|f(x)−g(x)|, j∈N.
Let the operator d, deﬁned by
(5.1.3) d(f,g) =∞/summationdisplay
j=12−jdj(f,g)
1 +dj(f,g),∀f,g∈C(X),
which is obviously a complete metric on C(X).
Definition 5.1.1.[82]Takek∈{0,1}, an operator T:C(X)→C(X)is called
k-Local Contraction (k-LC) relative to the subset A⊆C(X), if and only if
dj(Tf,Tg )≤βjdj+k(f,g),∀j∈N,
where 0≤βj<1for everyj∈N.
IfA=C(X), we simply call Tak-LC.
121

122 5. APPLICATION OF ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING
In the sequel, 0denotes the function ϕsuch thatϕ(x) = 0for allx∈X.
Definition 5.1.2.[82]The setA⊆C(X)is said to be bounded if there exist a sequence
{mj},mj<∞such thatdj(φ,0)≤mj, for allφ∈A, and∀j∈N.
Note that: if the set Acontains an unbounded function φ, then the sequence {mj}
also need to be unbounded.
Next, we introduce a seminorm on F(X), as follows from the work of Matkowski [ 66]:
LetF(X)be a vector space of functions ϕ:X→R. The mentioned seminorm on
F(X)is deﬁned by
||ϕ||j:= sup
x∈Kj|ϕ(x)|, ϕ∈F(X).
Suppose that the set of all operators ϕ∈F(X)endowed with the norm /bardbl·/bardbljis a
Banach space. If c>1andm={mj}is an increasing unbounded sequence of
positive real numbers, denote Fm(X)for the set of all operators ϕ∈F(X)such that
(5.1.4)∞/summationdisplay
j=1/bardblϕ/bardblj
mjcj<∞.
Note that the operator /bardbl·/bardbl:Fm(X)→Rdeﬁned by
(5.1.5) /bardblϕ/bardbl:=∞/summationdisplay
j=1/bardblϕ/bardblj
mjcj
is a complete norm on Fm(X), which means that (Fm(X),/bardbl·/bardbl)is a Banach space.
Denote by
Fmb(X) :={ϕ∈F(X) :/bardblϕ/bardblj≤mj,for allj∈N}
a closed subset of Fm(X).
Definition 5.1.3.[66]Takek∈{0,1}. An operator T:Fm(X)→F(X)is called
k-Local Contraction (k-LC) relative to the subset A⊆Fm(X), if and only if there exists
a coeﬃcient β∈[0,1)such that
/bardblTϕ−Tψ/bardblj≤β/bardblϕ−ψ/bardblj+k∀,ϕ,ψ∈A,j∈N.
Remark 5.1.4.It is obvious from the deﬁnition that a 0−LCis also a 1−LC.
Useful development in dynamic programming
Before proceeding further, we present some preliminary concepts from [ 66] and [63].
Denote by (X,Σ)a measurable space, Ya separable metric space. The mapping
B:X→Sis said to be (weakly) measurable if and only if
B−1(D) :={x∈X:B(x)∩D/negationslash= Φ}∈Σ,
for an open set D⊂Y.In this context, Srepresents the family of nonempty subsets
ofY. In the sequel, let Xbe a metric space and let Ea set-valued mapping. Then E

1. INTRODUCTION 123
is called continuous if E−1(D)is closed for every closed set D⊂Yand open for each
setD⊂Y.Note that every measurable mapping Bwith nonempty compact values
B(x)for allx∈Xhaving a measurable selector, see [ 63].
LetBa measurable compact set-valued mapping and denote
(5.1.6) C:={(x,a) :x∈X,a∈E(x)}.
According to Himmelberg’s work ([ 52]),Cis a measurable subset of X×Yequipped
with the product σ-algebra.
Proposition 5.1.5.(see[66]) Takeh:C→Ra measurable function such that
a→h(x,a)is continuous on B(x)for everyx∈X. Then
h∗(x) := max
a∈B(x)h(x,a)
is measurable and, also, we can ﬁnd a measurable operator g∗:X→Ysuch that
g∗(x)∈arg max
a∈B(x)h(x,a)
for allx∈X.
We shall brieﬂy go over the most important concepts of dynamic programming,
following [ 66]: Thediscrete-time Markov decision process include the objects:
X,Y,{A(x)}x∈X,u,q,βsuch that:
M1:Xrepresents the state space endowed with a σ-algebra Σ.
M2:Yis calledthe space of actions of the decision maker , which is actually a separable
metric space. For every x∈X, the compact subset A(x)⊂Yindicates the set
of all actions available in state x∈X.The mapping Cis deﬁned according to
(5.1.6).
M3:u:C→Rrepresents the (product) measurable instantaneous return function.
M4:qindicates a transition probability from CtoX, termed the law of motion among
states.
M5: The coeﬃcient β∈(0,1)represents the discount factor.
The sequence π={πt}is apolicy, whereπtis a measurable mapping which realize a
connection between an action at∈A(st)and any admissible history of the process st.
The set of all policies will be denoted by Π. Our work is focused on non-randomized
policies, because they are adequate to study the discounted models. We deﬁne the
expected discounted return over an inﬁnite future as:
(5.1.7) J(x,Π) :=Eπ
x/parenleftBig∞/summationdisplay
t=1βt−1u(xt,at)/parenrightBig
,
for each initial state x1=xand any policy π∈Π. In that equality Eπ
xindicates the
expectation operator regarding to the unique conditional probability measure Pπ
x
deﬁned byπand the transition probability qstated by the Ionescu Tulcea Theorem

124 5. APPLICATION OF ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING
(see [69]). In the sequel, we assume that the expected returns (5.1.7) are well-deﬁned.
Next, we add some regularity assumptions regarding the return and transition
probability functions.
R1: LetXbe a metric space and Kja strictly increasing family of compact sets such
that
(5.1.8) X=∞/uniondisplay
j=1Int(Kj).
Denote by Cc(X)the space of all continuous functions on Xwith compact sup-
ports. Assume that:
a) the set-valued mapping x→A(x)is continuous,
b) the return function uis continuous,
c) the correlation
(x,a)→/integraldisplay
Xv(y)q(dy|x,a)
is also continuous on the set C, for anyv∈Cc(X).
In the case of Xnot being a topological space, we use another regularity
condition:
R2: For every x∈X,any measurable set D⊂X, the functions a→u(x,a)and
a→q(D|x,a)are continuous on A(x).
If these two regularity conditions are fulﬁlled, we deﬁne
(5.1.9) uj(x) := max
a∈A(x)|u(x,a)|, x∈Kj;rj:= sup
x∈Kjuj(x).
Consider the sequences {mj}and{Kj}mentioned before. Suppose that (5.1.1) is
valid. For new results, we we need some new assumptions, as follows:
P1: We have the identity: q(Kj|x,a) = 1, for eachj∈Nandx∈Kj,a∈A(x).
P2: Suppose that there exists c>1such that
(5.1.10) ζ:=θc·sup/braceleftBigmj+1
mj,j∈N/bracerightBig
, ξ :=Lc·sup/braceleftBigmj+1
mj,j∈N/bracerightBig
, ζ,ξ < 1.
Furthermore, there exists a function g∈Mm(X)such that|uj(x)|≤g(x),for
eachj∈Nandx∈Kj. IfXis a metric space, then g∈Cm(X). Moreover, if
x∈Kj,a∈A(x),j∈N, one obtain q(Kj+1|x,a) = 1.
Remark 5.1.6.Actually, (5.1.10)leads to the conclusion:
∞/summationdisplay
t=1(θc)tmt<∞
The basis for the theory of discounted Markov decision processes is represented by the
Bellman functional equation. Its form appears in various papers. In this subsection,
use the Bellman equation stated by Matkowski and Nowak [ 66], as follows:
(5.1.11) Lv(x,a) :=u(x,a) +β/integraldisplay
Xv(y)q(dy|x,a),(x,a)∈C.

1. INTRODUCTION 125
for the integrable function v:X→R.
The Bellman equation can be put in the form
(5.1.12) v∗(x) = max
a∈A(x)Lv∗(x,a), x∈X.
Dynamic programming is probably the most important tool in economic analysis.
The purpose of this section is to study dynamic programming (DP) problems setting
as reduced form models. Starting from the recent work of Martin-da-Rocha and
Vailakis (2010) (see [ 64]) and also Rincon-Zapatero, Rodriguez-Palmero (2003) (see
[82]), it is our aim to develop their results and to obtain new ones.
First, we need some concepts and notations used in DP.
Definition 5.1.7.(see[82]) The dynamic optimization problem consists in solving the
following maximization problem:
v∗(x0) = max
{xt+1}∞
t=0∞/summationdisplay
t=0βtU(xt,xt+1)
xt+1∈Γ(xt) (t= 0,1,2,…) (5.1.13)
x0∈Xﬁxed,
whereXis a subset ofR,U:Graph (Γ)→Rrepresents the return function, β∈(0,1)
is the discounting factor, Γ :X→2Xis the technological correspondence giving the
set of admissible actions from any x∈X,v∗is the value function, and v∗(x0)is the
optimal value as a function of the initial condition x0.
At this point, consider the space Z=X×X×···, then we deﬁne the operator
Π :X→Zby
Π(x0) =/braceleftBig
˜x= (xt) = (x0,x1,···)∈Z|xt+1∈Γ(xt),t= 0,1,···/bracerightBig
, x 0∈X.
For every ˜x∈Π(x0), let
S(˜x) =∞/summationdisplay
t=0βtU(xt,xt+1)
be the total discounted returns. The following assumptions are frequently used in
dynamic programming:
(DP1) Γis nonempty, continuous and compact valued.
(DP2)U:Graph (Γ)→Ris continuous.
Definition 5.1.8.(The Bellman operator, see [82]) LetVbe the set of functions
fromXto[−∞,∞). The Bellman operator BonVis deﬁned by
(5.1.14)Bf(x) = max
y∈Γ(x){U(x,y) +βf(y)}, x∈X,f∈V.

126 5. APPLICATION OF ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING
The Bellman equation is: Bf=f, which is very similar to the ﬁxed point condition.
The Bellman operator has various properties in the literature, such as: monotony and
discounting (see [ 55], [98]).
In the sequel, we set as the real metric dRthe Euclidean distance: dR(x,y) =|x−y|.
By using this metric (norm) we obtain a family of semidistances (seminorms) {dj}on
C(X), deﬁned as
(5.1.15) dj(f,g) = max
x∈Kj|f(x)−g(x)|=/bardblf−g/bardblKj (dj(f,0) =/bardblf/bardblKj)
Consider the same metric d, as in the previous subsection:
d(f,g) =∞/summationdisplay
j=12−j/bardblf−g/bardblKj
1 +/bardblf−g/bardblKj.
The following Theorem was proved by Rincon-Zapatero and Rodriguez-Palmero in
[82]:
Theorem 5.1.9.([82]) LetBbe a Bellman operator satisfying (DP1) and (DP2) so
that there exists a countable increasing sequence {Kj}of nonempty and compact subsets
ofXwithX=/uniontext
jKjsatisfying Γ(Kj)⊆Kjfor allj∈N.
Then the following hold:
(a) The Bellman equation has a unique solution ˆfonC(X). Furthermore, ˆfsatisﬁes
/bardblˆf/bardblKj≤/bardblΨ/bardblKj
1−βj∈N.
(b) The value function v∗is continuous and coincides with the ﬁxed point ˆf.
(c) For any f∈C(X),Bnf→v∗asn→∞.
2. Examples
Example 5.2.1.([26]) (Discounted DP Problems)
Consider an α-discounted total cost DP problem. We have
H(x,u,J ) =E{g(x,u,w ) +αJ(f(x,u,w ))},
whereα∈(0,1), the function gis uniformly bounded and represents cost per stage, w
is random with distribution, depending on (x,u).
The equation J=TJ, i.e.,
J(x) = inf
u∈U(x)H(x,u,J ) = inf
u∈U(x)E{g(x,u,w ) +αJ(f(x,u,w ))}
is actually Bellmann’s equation with the unique solution J∗. The mapping Hcould be
used in diﬀerent forms, such as:
H(x,u,J ) = min/bracketleftBig
V(x),E{g(x,u,w ) +αJ(f(x,u,w ))}/bracketrightBig
,

3. MAIN RESULTS 127
or
H(x,u,J ) =E{g(x,u,w ) +αmin/bracketleftBig
V(f(x,u,w )),J(f(x,u,w ))/bracketrightBig
}],
whereVis a known function with the property V(x)≥J∗(x)for allx∈X.If the
solutionJ∗is not aﬀected by using diﬀerent variants Vfor the mapping H, we will
apply policy iteration algorythms for a more favorably computation of the value x.
Example 5.2.2.([26])(Discounted Semi-Markov Problems)
Withx,y,uas in the previous example, take the mapping
H(x,u,J ) =G(x,u) +n/summationdisplay
y=1mxy(u)J(y),
where the function Grepresents cost per stage, and mxy(u)are non-negative numbers
with
n/summationdisplay
y=1mxy<1,∀x∈X,u∈U(x).
The equation J=TJis Bellmann’s equation for a continuous-time semi-Markov deci-
sion problem, after it is converted into an equivalent discrete-time problem.
Example 5.2.3.([26])(Minimax Problems)
Consider a minimax version of Example 5.2.1, where an antagonistic player chooses v
from a setV(x,u). Consider the mapping
H(x,u,J ) = sup
v∈V(x,u)/bracketleftBig
g(x,u,v ) +αJ(f(x,u,v ))/bracketrightBig
.
Again, the equation J=TJis Bellmann’s equation for an inﬁnite horizon minimax
DP problem. A generalization is represented by the mapping
H(x,u,J ) = sup
v∈V(x,u)E/braceleftBig
g(x,u,v,w ) +αJ(f(x,u,v,w ))/bracerightBig
,
wherewis random with given distribution, and the expected value is with respect to
that distribution. This form could be found in zero-sum sequential games.
3. Main results
Next, we develop the concept of k−LC-s in the more general case of k-almost local
contractions (k-ALC).
Definition 5.3.1.An operator T:A→Ais calledk-almost local contraction (k-
ALC), with k∈{0,1}, relative to the subset A⊆Fm(X), if and only if there exist the
constantsθj,Lj∈[0,1)such that
(5.3.1)/bardblTϕ−Tψ/bardblj≤θj/bardblϕ−ψ/bardblj+k+Lj/bardblψ−Tϕ/bardblj+k∀,ϕ,ψ∈A,j∈N.

128 5. APPLICATION OF ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING
In the following, our main goal is to ﬁnd the conditions of existence for the ﬁxed
point onFm(X)for a 0−ALC, although the operator Tneed not be a contraction on
the metric generated by the family of seminorms. Proposition 5.3.2 below proves that
a0−ALCis an almost local contraction relative to a subset of Fm(X).
In the sequel, 0denotes the function ψsuch thatψ(x) = 0for allx∈Fm(X).
Proposition 5.3.2.a) Let the subset A⊆Fm(X)andT:A→Aan operator.
IfTis a0−ALC, then for each ϕ,ψ∈Athere exist the constants θ,L∈[0,1), such
that
(5.3.2)/bardblTϕ−Tψ/bardbl≤θ/bardblϕ−ψ/bardbl+L/bardblψ−Tϕ/bardbl ∀,ϕ,ψ∈A,j∈N.
IfT0∈Fm(X), thenTmapsFm(X)into itself. Furthermore, Tadmit a ﬁxed point
ϕ∗∈Fm(X).
b) If we add the conditions
/bardblT0/bardblj≤(1−θ−L)·m,∀j∈N
θ+L < 1, (5.3.3)
thenTmapsFmb(X)into itself, i.e., T:Fmb(X)→Fmb(X).
Proof:a) Chooseϕ,ψ∈A.For the proof of (5.3.2), apply the deﬁnition (5.1.5)
and (5.1.4), we obtain:
/bardblTϕ−Tψ/bardbl=∞/summationdisplay
j=1/bardblTϕ−Tψ/bardblj
mjcj≤∞/summationdisplay
j=1θ/bardblϕ−ψ/bardblj+L/bardblψ−Tϕ/bardblj
mjcj=
=∞/summationdisplay
j=1θ/bardblϕ−ψ/bardblj
mjcj+∞/summationdisplay
j=1L/bardblψ−Tϕ/bardblj
mjcj=
=θ/bardblϕ−ψ/bardbl+L/bardblψ−Tϕ/bardbl,
for allj∈N.
Observe that, for all ϕ∈Fm(X), we can write:
/bardblTϕ/bardbl=/bardblTϕ−T0+T0/bardbl≤/bardblTϕ−T0/bardbl+/bardblT0/bardbl≤
≤θ/bardblϕ−0/bardbl+L/bardbl0−Tϕ/bardbl+/bardblT0/bardbl=
=θ/bardblϕ/bardbl+L/bardblTϕ/bardbl+/bardblT0/bardbl.
After rearranging the terms, we obtain:
(1−L)/bardblTϕ/bardbl≤θ/bardblϕ/bardbl+/bardblT0/bardbl.
From that, by dividing with 1−L, we get
/bardblTϕ/bardbl≤θ
1−L/vextenddouble/vextenddouble/vextenddoubleϕ/vextenddouble/vextenddouble/vextenddouble+1
1−L/vextenddouble/vextenddouble/vextenddoubleT0/vextenddouble/vextenddouble/vextenddouble,

3. MAIN RESULTS 129
forθ,L∈[0,1). Having in view that ϕ∈Fm(X)and alsoT0∈Fm(X), it results that
TmapsFm(X)into itself, because the seminorm is ﬁnite in Fm(X),which yields:
/bardblTϕ/bardbl<∞.
b) For the desired conclusion, suppose that for every j∈Nand∀ϕ∈Fmb(X), condi-
tions (5.3.3) are veriﬁed.
/bardblTϕ/bardblj≤ /bardblTϕ−T0/bardblj+/bardblT0/bardblj≤θ/bardblϕ−0/bardblj+L/bardbl0−Tϕ/bardblj+/bardblT0/bardblj=
=θ/bardblϕ/bardblj+L/bardblTϕ/bardblj+/bardblT0/bardblj.
Once again, after rearranging the terms in a proper order, we can write:
(1−L)·/bardblTϕ/bardblj≤θ·mj+ (1−L−θ)·mj.
By using suppositions (5.3.3), we can write:
(1−L)·/bardblTϕ/bardblj≤(1−L)·mj.
Note thatL∈[0,1), we have from the last inequality:
/bardblTϕ/bardblj≤mj,
which means: T:Fmb(X)→Fmb(X).The existence of a ﬁxed point is guaranteed by
the existence of ﬁxed points for almost local contractions. The proof is complete.
Proposition 5.3.3.Let the subset A⊆Fm(X)andT:A→Aa1−ALC. Denote
(5.3.4) ζ:=θc·sup/braceleftBigmj+1
mj,j∈N/bracerightBig
,ξ:=Lc·sup/braceleftBigmj+1
mj,j∈N/bracerightBig
,
thenT:A→Ais an ALC, having the contraction coeﬃcients ζ,ξ < 1and with the
ﬁxed pointϕ∗∈A.
Proof:Chooseϕ,ψ∈A. Then we have:
/bardblTϕ−Tψ/bardbl=∞/summationdisplay
j=1c−j/bardblTϕ−Tψ/bardblj
mj≤∞/summationdisplay
j=1θc−j/bardblϕ−ψ/bardblj+1
mj+∞/summationdisplay
j=1Lc−j/bardblψ−Tϕ/bardblj+1
mj=
=∞/summationdisplay
j=1/parenleftBig
θcmj+1
mj/parenrightBig
c−j−1/bardblϕ−ψ/bardblj+1
mj+1+∞/summationdisplay
j=1/parenleftBig
Lcmj+1
mj/parenrightBig
c−j−1/bardblψ−Tϕ/bardblj+1
mj+1≤
≤ζ∞/summationdisplay
j=1/bardblϕ−ψ/bardblj+1
mj+1cj+1+ξ∞/summationdisplay
j=1/bardblψ−Tϕ/bardblj+1
mj+1cj+1=ζ/bardblϕ−ψ/bardbl+ξ/bardblψ−Tϕ/bardbl.
We prove that Tis an almost local contraction with the ﬁxed point
ϕ∗∈A.
Our main goal is to state and prove an existence theorem of a solution to the Bellman
equation in the space Cm(X), whenXis a metric space, or in Mm(X)if we want to
study a more general case.

130 5. APPLICATION OF ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING
Theorem 5.3.4.a) We accept condition P1. If condition R1 (or R2 with rj<∞for
everyj∈N) is valid, then there exists an increasing unbounded sequence m={mj}
and a unique mapping v∗∈Cm(X).
b) In the same conditions as in a), but in the space Mm(X), there exists an increasing
unbounded sequence m={mj}and a unique mapping v∗∈Mm(X).
Proof:a) If we make assumption R1, then Berge’s theorem (1963, [ 28]) assures
the continuity of every function uion the compact set Kj.Thus,rj<∞,∀j∈N.That
means: we can select an increasing unbounded sequence m={mj}such thatmj≥rj.
Choose the closed subset Cm(X)of the Banach space Cm(X).We shall introduce the
operatorTas follows:
Tv(x) := max
a∈A(x)/parenleftBig
(1−θ)u(x,a) +θ/integraldisplay
Xv(y)·q(dy|x,a) + (5.3.5)
+L/integraldisplay
X(v−Tv)(y)·q(dy|x,a)/parenrightBig
,
withv∈Cmb(X),x∈X.Note that (5.3.5) take the form of an integral equation.
DenoteH(y) := (v−Tv)(y), then (5.3.5) becomes:
Tv(x) := max
a∈A(x)/parenleftBig
(1−θ)u(x,a) +θ/integraldisplay
Xv(y)·q(dy|x,a) + (5.3.6)
+L/integraldisplay
XH(y)·q(dy|x,a)/parenrightBig
.
Again, by the maximum theorem of Berge ([ 28]),Tvis continuous on each set Kj.
According to (5.1.8), we have that the mapping Tvis continuous also on the set X.
Observe that the mapping T:Cmb(X)→X. Thus,Tis a non-self mapping, which
will be studied in chapter 5. In the light of these considerations, we can easily conclude
thatTis a Ćirić-Reich-Rus- type 0-ALC. Indeed, by using our assumption on q(from
the condition P2), after simple computations, we get:
/bardblTv−Tw/bardblj≤θ·/bardblv−w/bardblj+L·/parenleftBig
/bardblv−Tv/bardblj+/bardblw−Tw/bardblj/parenrightBig
for eachj∈Nand∀v,w∈Cmb(X). Since we prove that any Ciric-Reich-Rus type
ALC is an ALC (theorem 2.3.21) and by using Proposition 5.3.2, it means that our
mappingTis a 0-almost local contraction with unique ﬁxed point w∗∈Cmb(X)such
thatTw∗=w∗.Takev∗=w∗
1−θ∈Cm(X), which is a solution for the Bellman equation.
b)IfwemakeassumptionR2and rj<∞foreveryj∈N,theproofisverysimilartothe
case (a), by applying theorem 5.3.2. Obviously, the new ﬁxed point is v∗∈Mm(X).
Theorem 5.3.5.Suppose condition P2 satisﬁed. If condition R1 is valid, then the
Bellman equation has the unique solution v∗∈Cm(X).
Proof:If we make assumptions P2 and R1, then the operator Tfrom (5.3.5) will
be deﬁned for any v∈Cm(X),hence/bardblv/bardbl<∞.Denote
ρ:=/bardblg/bardbl, u∗(x) := max
a∈A(x)|u(x,a)|.

3. MAIN RESULTS 131
The operator Tis deﬁned on the closed ball Bρ:={v∈Cm(X) :/bardblv/bardbl≤ρ}∈Cm(X).
That means: u∗∈Bρ. By applying the maximum theorem of Berge, the operator Tv
is continuous, for every v∈Bρ.For ﬁnalize the proof, we need another notation:
(5.3.7) µ(x) := max
a∈A(x)/vextendsingle/vextendsingle/vextendsingle/integraldisplay
Xv(y)·q(dy|x,a)/vextendsingle/vextendsingle/vextendsingle, x∈X.
Obviously, the mapping µis continuous. By condition P2, one conclude that
/bardblµ/bardblj≤/bardblv/bardblj+1for eachj∈Nand forx∈Kj.We obtain:
/bardblµ/bardbl=inf/summationdisplay
j=1/bardblµ/bardblj
mjcj≤inf/summationdisplay
j=1/bardblv/bardblj+1
mjcj=inf/summationdisplay
j=1/bardblv/bardblj+1
mj+1cj+1·mj+1cj+1
mjcj≤
≤ /bardblv/bardbl·1
θ·θc·mj+1
mj=/bardblv/bardblζ
θ≤ρ
θ.
Again, the operator Tseems to be a non-self mapping, similar to the previous theorem.
In the light of the last inequalities regarding µ, we obtain that Tis a Ciric-Reich-Rus-
type 1-ALC. Indeed:
/bardblTv−Tw/bardblj≤θ·/bardblv−w/bardblj+1+L·/parenleftBig
/bardblv−Tv/bardblj+1+/bardblw−Tw/bardblj+1/parenrightBig
.
By applying Proposition 5.3.3, and by using the previous result that any Ćirić-Reich-
Rus type ALC is an ALC (theorem 2.3.21), leads us to the conclusion that the mapping
Tis a 1-almost local contraction with unique ﬁxed point w∗∈Cm(X)such that
Tw∗=w∗.Takev∗=w∗
1−θ∈Cm(X), which is a solution for the Bellman equation.
Starting from theorem 5.1.9, the other main result of this subsection is presented
below:
Theorem 5.3.6.LetBbe a Bellman operator satisfying (DP1) and (DP2). Then
(a)/bardblBf−f/bardbl≤/bardblf−Bg/bardbl+β·/bardblf−g/bardbl, for allf,g∈C(X);
(b)/bardblBf−g/bardbl≤/bardblBg−g/bardbl+β·/bardblf−g/bardbl, for allf,g∈C(X);
(c)/bardblBf−f/bardbl≤/bardblBg−g/bardbl+ (β+ 1)/bardblf−g/bardbl, for allf,g∈C(X);
Proof:(a) For the required inequality, consider:
Bf(x) = max[ U(x,y) +βf(y)]≤
≤max[U(x,y) +βg(y) +βmax|f(y)−g(y)|].
After simple computations, we get:
|g(x)−Bf(x)|=|Bf(x)−g(x)|≤
≤/vextendsingle/vextendsingle/vextendsingleBg(x) +βmax|f(y)−g(y)|−g(x)/vextendsingle/vextendsingle/vextendsingle,
for allf,g∈C(X). Hence, we can write: /bardblBf−f/bardbl≤/bardblf−Bg/bardbl+β·/bardblf−g/bardbl, for all
f,g∈C(X).

132 5. APPLICATION OF ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING
(b) The conclusion is immediate from the above inequalities:
/bardblBf−g/bardbl≤/bardblBg−g/bardbl+β·/bardblf−g/bardbl,∀f,g∈C(X).
(c) We know from the proof of theorem 5.1.9 ([ 82]) that:/bardblBf−Bg/bardbl≤β/bardblf−g/bardbl, which
implies
Bf≤Bg+f−f+g−g+β/bardblf−g/bardbl.
This inequality could be reformulated as:
Bf−f≤Bg−g+β/bardblf−g/bardbl+/bardblf−g/bardbl.
Therefore, we have /bardblBf−f/bardbl≤/bardblBg−g/bardbl+ (β+ 1)/bardblf−g/bardbl, for allf,g∈C(X).
This completes the proof.
Remark 5.3.7.By introducing a new operator T:=B−I(Ibeing the identity
function), we can reformulate (c) from the previous theorem:
(5.3.8) /bardblTf/bardbl≤/bardblTg/bardbl+ (β+ 1)/bardblf−g/bardbl,∀f,g∈C(X).
Next, we extend the classical contractions in the more general case of almost local
contractions. To this end, we start from the papers of Bertsekas (2012)(see [ 26], [27]).
In the beginning, we need some concepts from Bertsekas ([ 26]):
Take two sets: XandU, which shall be considered the set of ”states” and,
respectively, the set of ”controls”. For every x∈X, letU(x)⊂Ube a nonempty
subset of controls that are feasible at state x. In dynamic programming context we
use a function µ:X→U, whereµ(x)∈U(x), for allx∈X, which will be a ”policy”.
Let us consider the set of all policies M, and letR(X)be the set of real-valued
functionsJ:X→R. LetH:X×U×R(X)→Rbe a given mapping. We also
introduce the mapping Tdeﬁned by
(5.3.9) (TJ)(x) = inf
u∈U(x)H(x,u,J ),∀x∈X.
The mapping Tsatisﬁes the condition (TJ)(x)>−∞for allx∈X, which means
thatTmapsR(X)intoR(X). For each policy µ∈M, we introduce another
mappingTµ:R(X)→R(X)as follows:
(5.3.10) (TµJ)(x) =H(x,µ(x),J),∀x∈X.
The purpose of this construction is to ﬁnd a function J∗∈R(X)such that
(5.3.11) J∗(x) = inf
u∈U(x)H(x,u,J∗),∀x∈X,
which actually means that we want to ﬁnd a ﬁxed point of T. The second goal is to
obtain a policy µ∗such thatTµ∗J∗=TJ∗.
In his research paper, Bertsekas ([ 26]) introduced the weighted sup-norm, in the
following manner:

3. MAIN RESULTS 133
Take a function v:X→R, with positive values:
v(x)>0,∀x∈X,
letB(X)the space of real-valued functions JonXsuch thatJ(x)
v(x)is bounded as x
ranges over X, consider the weighted sup-norm on B(X):
(5.3.12) /bardblJ/bardbl= sup
x∈X|J(x)|
v(x).
At the end of this chapter, we give a few examples, according to Bertsekas [ 26]. In
his research paper, Bertsekas worked with contractions:
Definition 5.3.8.([26]) For allJ∈B(X)andµ∈M, the functions TµJandTJ
belong toB(X). Moreover, for some α∈(0,1), we have
(5.3.13) /bardblTµJ−TµJ/prime/bardbl≤α/bardblJ−J/prime/bardbl,∀J,J/prime∈B(X),µ∈M.
Condition (5.3.13) becomes:
|H(x,u,J )−H(x,u,J/prime)|
v(x)≤α/bardblJ−J/prime/bardbl,∀x∈X,u∈U(x),∀J,J/prime∈B(X).
We can easily observe that condition (5.3.13) leads us to
(5.3.14) /bardblTJ−TJ/prime/bardbl≤α/bardblJ−J/prime/bardbl,∀J,J/prime∈B(X).
Note that the contraction assumption (5.3.13) is valid for the mapping Hin examples
5.2.1, 5.2.2, 5.2.3. with v(x)≡1, which means that vrepresents the unit function.
In the following, we extend the model of Bertsekas, the strong correlation with
dynamic programming justiﬁes the introduction of a new type of contraction, namely
the weighted sup-norm almost local contractions.
Definition 5.3.9.Consider the functions u,v:X→R, with positive values:
v(x)>0,u(x)>0,∀x∈X,
and take the weighted sup-norm on B(X)deﬁned by (5.3.12). For allJ∈B(X)and
µ∈M, the functions TµJandTJbelong toB(X). The mappings T,Tµ:B(J)→B(J)
are called almost local contractions in dynamic programming, shortly: ALC-DP if, for
every policy µ∈M, there exist the constants α∈(0,1)andL≥0such that:
(5.3.15)/bardblTµJ−TµJ/prime/bardblv≤α/bardblJ−J/prime/bardblu+L/bardblJ/prime−TµJ/bardblu,∀J,J/prime∈B(X),µ∈M.
(5.3.16)/bardblTJ−TJ/prime/bardblv≤α/bardblJ−J/prime/bardblu+L/bardblJ/prime−TJ/bardblu,∀J,J/prime∈B(X),
where
(5.3.17) /bardblJ/bardblv= sup
x∈X|J(x)|
v(x),/bardblJ/bardblu= sup
x∈X|J(x)|
u(x).
Remark 5.3.10 .It can be easily seen that the functions TandTµrepresent in fact two
almost local contractions.

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