Technical University of Cluj-Napoca [632188]

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Technical University of Cluj-Napoca [632188]

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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
Scientiﬁc Advisor: Ph.D. Student: [anonimizat]. Univ. Dr. Vasile Berinde Zakany Monika
Baia Mare
2018

Contents
Chapter 1. Introduction……………………………………………….. 1
1. Overview of the thesis…………………………………………….. 1
2. Strict contractions, Picard operators………………………………… 3
3. Almost contractions………………………………………………. 4
4. Local contractions………………………………………………… 7
Chapter 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS…. 9
1. Introduction …………………………………………………….. 9
2. Stability and data dependence of the ﬁxed points for almost local
contractions……………………………………………………… 16
3. Continuity of almost local contractions……………………………… 17
4. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS …………… 20
5. ALMOST LOCAL ϕ-CONTRACTIONS ……………………………. 31
6. (B)-almost local contractions………………………………………. 39
7. APPROXIMATE FIXED POINTS…………………………………. 41
8. Almost local contractions in b-pseudometric spaces…………………… 54
8.1.ϕ-contractions in b-pseudometric spaces………………………….. 56
8.2. Fixed points of almost local contractions in b-pseudometric spaces ….. 63
Chapter 3. Almost local contractions in dynamic programming applications …. 75
1.k-almost local contractions………………………………………… 75
2. Useful development in dynamic programming………………………… 79
Chapter 4. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS…… 88
1. Preliminaries…………………………………………………….. 88
2. Approximate ﬁxed points of multivalued almost local contractions……… 99
Chapter5. NON-SELFSINGLEVALUEDALMOSTLOCALCONTRACTIONS102
Chapter 6. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS110
1. NON-SELF MULTIVALUED ALMOST CONTRACTIONS…………… 110
2. FIXED POINT THEOREMS FOR NON-SELF MULTIVALUED ALC…. 110
Chapter 7. COMPLEX VALUED b-METRIC SPACES AND RATIONAL
LOCAL CONTRACTIONS …………………………………. 119
1. Complex valued b-metric…………………………………………..119
1

2 CONTENTS
2. Fixed point theorems for rational local contractions ………………….. 120
Bibliography ………………………………………………………….. 131

CHAPTER 1
Introduction
1. Overview of the thesis
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 ( or weak
contractions) and 2) the class of local contractions.
Both classes are extensions of the well known class of Banach contraction mappings,
which were introduced by Stefan Banach in 1922 in his famous dissertation [ 5] and
which are at the foundation of metrical ﬁxed point theory, an extremely dynamic ﬁeld
of research starting with second half of the XXth century, see the monographs [Rus],
for a selected list of reference books.
Banach contraction mapping, originally formulated by Banach in the setting of a
complete normed space (what we call now a Banach space), states that a contraction
mappingT:X→Xdeﬁned on a complete metric space (X,d)has a unique ﬁxed
pointxinXwhich can be obtained as the limit of any Picard iteration, i.e.
limn→∞Tnx0=x,
for anyx0∈X.
Banach mapping principle has important applications in nonlinear analysis, being
themostusedtoolinobtainingexistenceanduniquenessresultsfornonlinearfunctional
questions.
However, any Banach contraction is continuous on X, due to the contraction condition
itself:
(1.1.1) d(Tx,Ty )≤α·d(x,y),∀x,y∈X
whereα∈(0,1)is the contraction coeﬃcient.
A. The wide range of applications of Banach contraction principle in nonlinear analy-
sis have challenged researchers to obtain its conclusions unther weaker assumptions
than (1.1.1), which do not force the continuity of the operator T.
The ﬁrst achievement in this respect has been reported by Kannan in 1968 [ 53],
who obtained a ﬁxed point theorem for discontinuous mappings. Chatterjea [ 32],
Bianchini [ 26], Reich, Rus [ 79], Ćirić [33], Zamﬁrescu [ 100] and many other re-
searchers continued this direction of research, see Rhoades for a classiﬁcation and
comparison of various such contractive type mappings.
1

2 1. INTRODUCTION
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 nth iterate of T).
More recently, Berinde [ 13] introduced a large class of contractive mappings, called
weak contractions [in [ 9]] and later almost contractions (in [ 8] and afterwards).
The class of almost contractions includes Banach contractions, Kannan contrac-
tions, Chatterjea contractions, Zamﬁrescu contractions, Reich-Rus contractions,
Bianchini contractions and, partially, the so called quasi-contractions, due to Ćirić
[34]. But, unlikely the above mentioned classes of contractions, which possess a
unique ﬁxed point, an almost contraction may have two or more ﬁxed points by
simultaneously keeping almost all other features of the Banach contraction mapping
principle, including rate of convergence, error estimate, stability and so on.
B. On the other hand, Martin da Rocha and Filipe Vailakis [ 59], in a paper with
applications in economics, established another extension of the Banach contraction
principle. The setting they are working is that of a set Fendowed with a family
D= (dj)j∈Jof semidistances deﬁned on F. They consider on Fweak topology σ
deﬁned by the family D.
LetA⊂Fbe aσ-bounded sequencially σ-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 [ 59] essentially
states that, if Fisσ-Hausdorﬀ, and, for ∀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 σ- convergent to f∗
(heredj(h,A)≡inf{dj(h,g) :g∈A}, for everyj∈J).
The theoretical results in [ 59] were then applied to solve recursive equations in
economic dynamics with many applications in dynamic programming.
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

2. STRICT CONTRACTIONS, PICARD OPERATORS 3
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 shall present
a coherent theory of what we shall call local almost contractions (ALC), for which we
shall present various ﬁxed point theorems, examples, applications and particular cases.
We plan to cover most of the classes of mappings studied in ﬁxed point theory: single-
valued self mappings; multi-valued nonself mappings; common ﬁxed points and coin-
cidence points of local almost contractions etc.
2. Strict contractions, Picard operators
The Banach’s contraction principle represent probably the most important tool in
nonlinear analysis. It was ﬁrst established in 1922 by Stefan Banach. In a complete
metric space it is given by the next theorem.
Theorem 1.2.1.(see[5]) Let (X,d)be a complete metric space and T:X→X
be a map satisfying
(1.2.1) 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
(1.2.2) xn+1=Txn, n = 0,1,2···
converges to p, for anyx0∈X.
(iii) The following a priori and a posteriori error estimates hold:
(1.2.3) d(xn,p)≤an
1−ad(x0,x1),n= 0,1,2…
(1.2.4) d(xn,p)≤a
1−ad(xn−1,xn),n= 1,2…
(iv) The rate of convergence of Picard iteration is given by
(1.2.5) 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 [ 80] ,[82].
A mapping satisfying (1.1.1) is usually called a strict contraction or an a-contraction.
Hence, in essence, Theorem 1.2.1 shows that any contraction is a Picard operator.
A mapping satisfying condition (1.1.1) is always continuous; that fact lead researchers
to look up for discontinuous classes of such kind of mappings for which conclusions of
Theorem 1.2.1 still holds.
In 1968, R. Kannan [ 53] found a positive answer to this problem by proving a ﬁxed

4 1. INTRODUCTION
theorem that extends Theorem 1.2.1 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
(1.2.6) d(Tx,Ty )≤b[d(x,Tx ) +d(y,Ty )],∀x,y∈X.
3. Almost contractions
Definition 1.3.1.(see[13]) 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
(1.3.1) d(Tx,Ty )≤δ·d(x,y) +L·d(y,Tx ),∀x,y∈X.
Remark 1.3.2.The term of almost contraction is the same as weak contraction,
and it was ﬁrst introduced by V. Berinde in [16].
Remark 1.3.3.A strict contraction satisﬁes (1.3.1), withδ=aandL= 0, there-
fore it is an almost contraction with a unique ﬁxed point.
Other examples of almost contractions are given in [ 11], [12], [9], [16]. There are
manyotherexamplesofcontractiveconditionswhichimpliesthealmostcontractiveness
condition, see for example Taskovic [ 94], Rus [81].
We present an existence theorem 1.3.4, then an existence and uniqueness theorem
1.3.5, as they are presented in [ 16]. Their main merit is that they extend Banach’s con-
traction principle, Kannan, Chatterjea and Zamﬁrescu’s ﬁxed point theorem ([ 100]).
These results provide a method for approximating the ﬁxed point, for which both a
priori and a posteriori error estimates are available.
Theorem 1.3.4.([16]) 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=0given byxn+1=Txnconverges
to somex∗∈Fix(T);
(3) The following estimates
(1.3.2) d(xn,x∗)≤δn
1−δd(x0,x1), n = 0,1,2…
(1.3.3) d(xn,x∗)≤δ
1−δd(xn−1,xn), n = 1,2…
hold, where δis the constant appearing in (1.3.1).
Theorem 1.3.5.([16]) 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
(1.3.4) d(Tx,Ty )≤θ·d(x,y) +L1·d(x,Tx ),∀x,y∈X

3. ALMOST CONTRACTIONS 5
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
(1.3.5) 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 (1.3.1).
Definition 1.3.6.Let(X,d)be a complete metric space. A mapping T:X→X
satisfying the condition: there exists 0≤b<1/2such that
(1.3.6) d(Tx,Ty )≤b[d(x,Tx ) +d(y,Ty )],∀x,y∈X
is called Kannan mapping.
Definition 1.3.7.([32]) 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.
Definition 1.3.8.Let(X,d)be a complete metric space. The mapping T:X→X
satisfying the contractive condition:
there exists 0≤h<1
2such that
(1.3.7)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.
Remark 1.3.9.(1) A large number of examples of almost contractions were given
by V. Berinde in [16]. For example it was proved that:
– any Zamﬁrescu mapping from Theorem Z in [100]is an almost contraction;
– any quasi-contraction with 0<h<1
2is an almost contraction;
– any Kannan mapping (in [53]) is the same kind of almost contraction
(2) The almost contractive condition (1.3.1)has been used in a large class of applica-
tions, see for example Taskovic [94], Rus[81]for some of them.
(3) The ﬁxed point x∗attained by the Picard iteration depends on the initial guess.

6 1. INTRODUCTION
A mapT:X→Xis called weakly Picard operator (see [81],[83]) if the sequence
{Tnx0}∞
n=0converges for all the initial points x0∈Xand the limits are ﬁxed points
ofT. Therefore, the class of weak contractions provides a large class of weakly Picard
operators.
(4) Condition (1.3.1)implies another concept: the Banach orbital condition
d(Tx,T2x)≤a·d(x,Tx ),∀x∈X
studied by various authors in the context of ﬁxed point theorems, see for example Hicks
and Rhoades [46], Ivanov [49], Rus[80]and Taskovic [94].
The next Theorem shows that an almost contraction is continuous at any ﬁxed
point of it, according to [ 8].
Theorem 1.3.10 .([8]) Let (X,d)be a complete metric space and T:X→Xbe
an almost contraction. Then Tis continuous at p, for anyp∈Fix(T).
Definition 1.3.11 .(see[93]) 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
functionbfromXinto[0,∞)such that
(1.3.8) d(Tx,Ty )≤r·d(x,y) +b(y)·d(y,Tx ),∀x,y∈X
Definition 1.3.12 .([35]) Let (X,d)be a metric space. Any mapping T:X→X
is called Ćirić-Reich-Rus contraction if it is satisﬁed the condition:
(1.3.9) d(Tx,Ty )≤α·d(x,y) +β·[d(x,Tx ) +d(y,Ty )],∀x,y∈X,
whereα,β∈R +andα+ 2β <1
Theorem 1.3.13 .([32]) Any mapping Tsatisfying the Chatterjea contractive con-
dition, i.e.: there exists 0≤c<1
2such that
d(Tx,Ty )≤c[d(x,Ty ) +d(y,Tx )],∀x,y∈X,
is a weak contraction.
A kind of dual of Kannan mapping is due to Chatterjea [ 32]. The new contractive
condition is similar to (1.3.6): there exists 0≤c<1
2such that
(1.3.10) d(Tx,Ty )≤c[d(x,Ty ) +d(y,Tx )],∀x,y∈X,
Example 1.3.14 .(see[8]) LetT: [0,1]→[0,1]a mapping given by Tx=2
3for
x∈[0,1), andT1 = 0. ThenThas the following properties:
1)Tsatisﬁes (1.3.7)withh∈[2
3,1), i.e.,Tis a quasi-contraction;
2)Tsatisﬁes (1.3.1), withδ≥2
3andL≥0, i.e.,Tis also almost contraction;
3)Thas a unique ﬁxed point, x∗=2
3.
4)Tis not continuous on [0,1]but it is continuous at x∗=2
3.

4. LOCAL CONTRACTIONS 7
4. Local contractions
The concept of local contraction was ﬁrst introduced by Martins da Rocha and
Filipe Vailakis in [ 59] (2010), where they studied the existence and uniqueness of ﬁxed
points for the local contractions.
Definition 1.4.1.(see[101],[102]) The function d:X×X→R+is said to be
semimetric on Xif:
(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 that case.
Inordertoavoidtheconfusionbetweentheweaktopologyandtheoriginaltopology
onX, theoriginaltopologyon Xisoftencalledthestrongtopologyon X. Observethat
the diﬀerence between a semimetric and a semidistance is that the ﬁrst one represent
a functiond:X×X→R+, meanwhile the second one signiﬁes the numerical value of
that function.
Definition 1.4.2.(see[59]) LetFbe a set and letD= (dj)j∈Jbe a family of
semimetrics deﬁned on F. We letσbe the weak topology on Fdeﬁned by the family
D. DenoteJa family of indices (which frequently can be considered as a subset of N)
and letrbe a function from JtoJ.
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 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.
In a recent paper, Vailakis and Martins-da-Rocha [ 59] have established a local
variant of the contraction mapping principle in uniform spaces originally proved by
Gheorghiu [ 41], Gheorghiu and Rotaru [ 42].
Instead of considering an operator T:X→Xlike in [59] , Vailakis and Martins-da-
Rocha have considered operators T:A→X, whereA⊂Xis a nonempty, σ-bounded,
sequencially σ-completeand T-invariant subsetof X, and provided suﬃcientconditions
toensuretheexistenceofaﬁxedpointof TinA, aswellastheexistenceanduniqueness
of the ﬁxed point of T.
Definition 1.4.3.([59]) The subset Ais considered σ- Hausdorﬀ if and only if
for each pair f,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.

8 1. INTRODUCTION
Definition 1.4.4.[59]LetXbe a set and letD= (dj)j∈Jbe a family of semimet-
rics deﬁned on X. We shall consider the weak topology on Xdenoted by σ, deﬁned by
the familyD.
The sequence (xn)n∈N∗isσ−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 σ−Cauchyif it isdj-Cauchy, for all j∈J.
The subset AofXis said to be sequencially σ-complete if every σ-Cauchy sequence in
Xconverges in Xfor theσ-topology.
The diameter of the subset A⊂Xshall be considered: diamj(A)≡sup{dj(x,y) :x,y∈A}.
The subset A⊂Xis said to be σ-bounded if diamj(A)≡sup{dj(x,y) :x,y∈A}is
ﬁnite for every j∈J.
The main results in [ 59] are:
Theorem 1.4.5.[59]LetFbe a set. Consider a function r:J→Jand let
T:A→Abe a local contraction with regard to (D,r). Consider a nonempty, σ-
bounded, sequentially σ- complete, and T- invariant subset A⊂F.
E:(existence): If the condition
(1.4.1) ∀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)
Moreover, if h∈Asatisﬁes
(1.4.2) ∀j∈J, limn→∞βjβr(j)···βrn+1(j)drn+1(j)(h,A) = 0
then the sequence (Tnh)n∈Nis σ- convergent to f∗.
A more general global ﬁxed point result has been obtained more than fourty years
ago by Gheorghiu [ 41], Gheorghiu and Rotaru [ 42]:
Theorem 1.4.6.([41]) LetXbe a uniform Hausdorﬀ space, sequencially complete,
and let (di)i∈Ibe a family of semidistances deﬁned on X. Letf:X→Xbe a mapping
for which there exists 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)
is convergent for all i∈Iand for all x,y∈X.
Thenfhas a unique ﬁxed point.
Remark 1.4.7.IfTsatisﬁes condition (1.4.1), then the series is convergent.

CHAPTER 2
SINGLE VALUED SELF ALMOST LOCAL
CONTRACTIONS
1. Introduction
Theaimofthispaperistocombinetheconceptsofalmost contraction ,andlocal contraction ,
and thus build a ﬁxed point theory for what we shall call local almost contractions.
Throughout this chapter, Xis a Hausdorﬀ topological space with his topology gener-
ated by a family{dj}j∈Jof pseudometrics on X.
Definition 2.1.1.(see[102]) The mapping d:X×X→R+is said to be
a pseudometric if:
(1)d(x,y) =d(y,x),∀x,y∈X;
(2)d(x,y)≤d(x,z) +d(z,y),∀x,y,z∈X;
(3)d(x,x) = 0,∀x∈X.
Remark 2.1.2.Note thatx=y⇔d(x,y) = 0in the metric case, which means
that the distance between two diﬀerent elements could be zero for the pseudometric,
see Example 2.1.4 and 2.1.5. Also, observe the diﬀerence from a pseudometric and a
semimetric (deﬁnition 1.4.1 and 2.1.1).
A family of pseudometrics introduces on the set Xthe topology τ, named uniform space
topology, in certain research papers.
Example 2.1.3.([102]) LetXbe the set of all real numbers and take the mapping
d:X×X→R+such that
d(x,y) = 0,∀x,y∈X.
Observe that dis not a metric, but it is a pseudometric, because condition (3)from
deﬁnition 2.1.1 is not veriﬁed.
Example 2.1.4.([98]) LetX= [0,n]×[0,n]⊂R2,n∈N∗.The diameter of the
subsetX= [0,n]×[0,n]⊂R2is given by the diagonal line of the square whose four
sides have length n.
We shall use the pseudometric:
(2.1.1) dj((x1,y1),(x2,y2)) =|y1−y2|·e−j,∀j∈J,
9

10 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
whereJis a subset of N. This is a pseudometric, but not a metric, take for example:
dj((1,4),(2,4)) =|4−4|·e−j= 0, however (1,4)/negationslash= (2,4).
Example 2.1.5.([98]) LetX= [0,n]×[0,n]⊂R2,n∈N∗.The diameter of the
subsetX= [0,n]×[0,n]⊂R2is given by the diagonal line of the square whose four
sides have length n.
We shall use the pseudometric:
(2.1.2) dj((x1,y1),(x2,y2)) =|x1−x2|·e−j,∀j∈J,
whereJis a subset of N. Similar to the previous example, this is a pseudometric, but
not a metric, take for example:
dj((1,3),(1,5)) =|1−1|·e−j= 0, however (1,3)/negationslash= (1,5)
Definition 2.1.6.([97]) LetXbe a set and letD= (dj)j∈Jbe a family of pseu-
dometrics 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 sequencially τ-complete if every τ-Cauchy sequence in
Xconverges in Xfor theτ-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 2.1.7.([97])LetXbe a set and a subset A⊂X. Letrbe a function
fromJtoJ. An operator T:A→Ais called an 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 2.1.8.1)The almost contractions represent a particular case of almost
local contractions, by taking (X,d)metric space instead of the pseudometrics djand
dr(j)deﬁned onX. Also, to obtain the almost contractions, we take in (2.1.3)forrthe
identity function, so we have r(j) =j.
2) The pseudometric possess the property of symmetry, that is the reason why the
almost local contraction condition (2.1.3)includes a second condition:
(2.1.4) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,Ty ),∀x,y∈A
Definition 2.1.9.The spaceXisτ- Hausdorﬀ if the following condition is valid:
for each pair x,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}.

1. INTRODUCTION 11
Theorem 2.1.10 is an existence ﬁxed point theorem for almost local contractions,
as they appear in [ 97].
Theorem 2.1.10 .Consider a function r:J→Jand letT:A→Abe an almost
local contraction with regard to ( D,r). Consider a nonempty, τ- bounded, sequentially
τ- complete, and T- invariant subset A⊂X. 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{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 yields
(2.1.6) dj(xn,xn+1)≤θ·dr(j)(xn−1,xn),∀j∈J
Using (2.1.3), we obtain by induction regarding to n:
(2.1.7) dj(xn,xn+1)≤θn·dr(j)(x0,x1), n = 0,1,2,···
According to the triangle rule, by (2.1.7) we get:
dj(xn,xn+p)≤θn(1 +θ+···+θp−1)dr(j)(x0,x1) =
=θn
1−θ(1−θp)·dr(j)(x0,x1), n,p∈N,p/negationslash= 0
These relations show us that the sequence (xn)n∈Nisdj- Cauchy for each j∈J. The
subsetAis assumed to be sequentially τ-complete, there exists x∗inAsuch that
(Tnx)n∈Nisτ- convergent to x∗. Besides, the sequence (Tnx)n∈Nconverges for the
topologyσtox∗, which implies
∀j∈J, dj(Tx∗,x∗) = limn→∞dj(Tx∗,Tn+1x).
Recall that the operator Tis an almost local contraction with regarding to ( D,r). From
that, we have
∀j∈J, dj(Tx∗,x∗)≤θlimn→∞dr(j)(x∗,Tnx).
The convergence for the τ- topology implies convergence for the pseudometric dr(j), we
obtaindj(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 almost local contractions.

12 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Remark 2.1.11 .ForTveriﬁes(2.1.3)withL= 0, andr:J→Jthe identity
function, we ﬁnd Theorem 1.4.5 by taking θ=βj.
Further, for the case dj=d,∀j∈J, withd= metric on X, we obtain the well known
Banach contraction, with his unique ﬁxed point.
Example 2.1.12 .([98])LetX= [0,n]×[0,n]⊂R2,n∈N∗. The diameter of the
subsetX= [0,n]×[0,n]⊂R2is given by the diagonal line of the square whose four
sides have length n.
We shall use the pseudometric:
(2.1.8) dj((x1,y1),(x2,y2)) =|x1−x2|·e−j,∀j∈J,
whereJis a subset of N. This is a pseudometric, but 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, we shall use the function r(j) =j
2, wherej∈J.
Considering T: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:
(2.1.9) dj(Tx,Ty )≤θ·dj(x,y),
is not valid∀x,y∈X, and for any θ∈[0,1). Indeed, (2.1.9)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
2+L·|x2−x1|·e−j
2
which is equivalent to : e−j
2≤θ+L.
Forθ=1
3∈[0,1),L= 2≥0andj >0, the last inequality becomes true, i.e. Tis
an almost local contraction with an inﬁnity number of ﬁxed points:
FixT ={(x,0) :x∈R}.
In this case, we have:
∀j∈J, limn→∞θn+1diamrn+1(j)(A) = limn→∞/parenleftBigg1
3/parenrightBiggn+1
·(n−1)2= 0
This way, the existence of the ﬁxed point is assured, according to condition (2.1.3)from
Theorem 2.1.10.

1. INTRODUCTION 13
Remark 2.1.13 .In theorems 2.1.10, 2.1.16 the coeﬃcient of contraction θ∈(0,1)
is constant, but local contractions have a coeﬃcient of contraction θj∈[0,1)which
depends on j∈J. Our ﬁrst goal is to extend the local almost contractions to the most
general case of θj∈(0,1).
In the sequel, we extend Deﬁnition 2.1.7 to the case of almost local contractions
with variable coeﬃcient of contraction.
Definition 2.1.14 .([97]) Letrbe a function from JtoJ. An operator T:A→A
is called almost local contraction with regard to ( D,r)or(θj,Lj)- contraction, if there
exist a constant θj∈(0,1)and someLj≥0such that
(2.1.10) dj(Tx,Ty )≤θj·dr(j)(x,y) +Lj·dr(j)(y,Tx ),∀x,y∈A
Theorem 2.1.15 .([97]) With the assumptions of Theorem 2.1.10, if we modify the
condition (2.1.5)with the following one:
(2.1.11) ∀j∈J, limn→∞θjθr(j)···θrn(j)diamrn+1(j)(A) = 0,
then the operator Tadmits a ﬁxed point x∗inA.
Proof:Letxan element in A. From the deﬁnition of almost local contraction T,
for every pair of integers q>n> 0, we have
dj(Tqx,Tnx)≤θjdr(j)(Tq−1x,Tn−1x) +Ljdr(j)(Tn−1x,Tqx)≤
≤θj[θr(j)dr2(j)(Tq−2x,Tn−2x) +Ljdr2(j)(Tn−2x,Tq−1x)] +
+Lj[θr(j)dr2(j)(Tn−2x,Tq−1x) +Lr(j)dr2(j)(Tq−1x,Tn−1x)] =
=θj[θr(j)dr2(j)(Tq−2x,Tn−2x) +Lj(θj+θr(j))dr2(j)(Tn−2x,Tq−1x)] +
+LjLr(j)dr2(j)(Tq−1x,Tn−1x)≤···≤
≤θjθr(j)···θrn−1(j)drn(j)(Tq−nx,x) +···+LjLr(j)···Lrn−1(j)drn(j)(Tq−nx,x)
SinceAisT-invariant,Tq−ngbelongs toA, which yields
dj(Tqx,Tnx)≤θjθr(j)···θrn−1(j)diamrn(j)(A).
That means: the sequence (Tnx)n∈Nisdj-Cauchy for each j. The subset Ais assumed
to be sequentially τ-complete, there exists x∗inAsuch that (Tnx)n∈Nisτ- convergent
tox∗. Besides, the sequence (Tnx)n∈Nconverges for the topology τtox∗, which implies
∀j∈J, dj(Tx∗,x∗) = limn→∞dj(Tx∗,Tn+1x).
Recall that the operator Tis an almost local contraction with regarding to ( D,r).
From that, we have
∀j∈J, dj(Tx∗,x∗)≤θjlimn→∞dr(j)(x∗,Tnx) +Ljdr(j)(Tnx,Tx∗).

14 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
The convergence for the τ- topology implies convergence for the pseudodistance dr(j),
we obtaindj(Tx∗,x∗) = 0for everyj∈J. Therefore, Tx∗=x∗. The existence of the
ﬁxed point is proved.
The next Theorem represent an existence and uniqueness theorem for the almost
local contractions with constant coeﬃcient of contraction.
Theorem 2.1.16 .([98]) If to the conditions of Theorem 2.1.10, we add:
(U) for every ﬁxed j∈Jthere exists:
(2.1.12) limn→∞(θu+Lu)ndrn(j)(z,A) = 0,∀x,y∈A
then the ﬁxed point x∗ofTis unique.
Proof:Suppose, by contradiction, there are two diﬀerent ﬁxed points x∗andy∗of
T. Then for every ﬁxed j ∈Jwe have:
0<dj(x∗,y∗) =dj(Tx∗,Ty∗)≤θudr(j)(x∗,y∗) +Ludr(j)(y∗,Tx∗) =
= (θu+Lu)·dr(j)(x∗,y∗)≤···≤ (θu+Lu)n·drn(j)(x∗,y∗)≤
≤(θu+Lu)ndrn(j)(z,A)
Now, letting n→∞, we obtained a contradiction with condition (2.1.12), i.e., the
ﬁxed point is unique.
Remark 2.1.17 .The proof of Theorem 2.1.16 is quite similar to that of Vailakis
([59]) 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 addi-
tional condition (U), only with the monotonicity condition for the pseudometric, and
uniqueness condition from the almost contractions, as shown in the next Theorem.
Theorem 2.1.18 .If to the conditions of Theorem 2.1.10, we add a monotonicity
condition for the pseudometric, namely:
(2.1.13) dr(j)(f,g)≤dj(f,g),∀f,g∈A,∀j∈J
and also we add the uniqueness condition:
(2.1.14) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,T(x)),∀x,y∈A,
then the ﬁxed point f∗ofTis unique.
Proof:Suppose, by contradiction, there are two distinct ﬁxed points f∗andg∗
ofT. Then, by using (2.1.13), and condition (2.1.14) for every ﬁxed j∈Jwith
f:=f∗,g:=g∗we get:
dj(f∗,g∗)≤θ·dr(j)(f∗,g∗) +Ldr(j)(f∗,T(f∗)) =θ·dr(j)(f∗,g∗)≤θ·dj(f∗,g∗)
⇒dj(f∗,g∗)≤θ·dj(f∗,g∗)⇒(1−θ)·dj(f∗,g∗)≤0,

1. INTRODUCTION 15
which is obviously a contradiction with dj(f∗,g∗)>0.
So, we prove the uniqueness of the ﬁxed point.
Remark 2.1.19 .The transition between these two types of contractions (the ALC-s
and the local contractions) is well illustrated by Theorem 2.1.18, since the uniqueness
condition borrowed from any of them assure unique ﬁxed point for their combination.
Having in view the work of M.Păcurar ([ 67]), we introduce, similarly to the case
of strict almost contractions, a new type of ALC-s, which is the strict almost local
contractions.
Definition 2.1.20 .([67]) LetXbe a set and let 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 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(2.1.12), with some real constants θ∈[0,1),L≥0andθu∈
[0,1),Lu≥0, respectively.
Remark 2.1.21 .The existence (2.1.5)and, respectively, uniqueness (2.1.12)con-
ditions are independent, according to the following example.
Example 2.1.22 .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 diagonal line of the
square whose four sides have length n.
We shall use the pseudometric:
(2.1.15) dj((x1,y1),(x2,y2)) =|x1−x2|·e−j,∀j∈J,
where the family of indices Jis a subset ofN. This is a pseudometric, but 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, we shall use the function r(j) =j
2. By applying the inequality (2.1.3)to
our 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|·e−j
2≤2θ·|x1−x2|+L·|2×2−x1|,
The last inequality became true if we take θ=1
2∈(0,1),L= 4≥0. HenceTis an
almost local contraction, with the unique ﬁxed point (0,0).

16 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Condition (2.1.5)is valid for θ=1
2∈[0,1),L= 4≥0.
On the other hand, condition (2.1.12)is veriﬁed for θu=1
4∈[0,1),Lu=1
3≥0.
Remark 2.1.23 .The strict almost contractions have a notorious importance: they
always have a unique ﬁxed point.
2. Stability and data dependence of the ﬁxed points for almost local
contractions
The stability of ﬁxed point iteration procedure was ﬁrst introduced by Berinde in
[16]. The starting point of this new concept was to approximate the sequence {xn}∞
n=0
deﬁned by a given iterative method with a more practical sequence {yn}∞
n=0, which
limit will properly approximate the ﬁxed point of the initial mapping.
Definition 2.2.1.([16]) 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 we shall use the following notations:
(2.2.1) εn=d(yn+1,Tyn),n= 0,1,2,…
We shall say that the ﬁxed point iteration procedure is T- stable or stable with regard
toTif and only if
(2.2.2) limn→∞εn= 0⇔limn→∞yn=p.
The following Lemma shall be useful in future demonstrations.
Lemma 2.2.2.(see[17]) 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, in certain conditions.
Theorem 2.2.3.In addition to the conditions of Theorem 2.1.10, we add a mono-
tonicity property for the pseudometric, namely:
(2.2.3) dr(j)(f,g)≤dj(f,g),∀f,g∈A,∀j∈J

3. CONTINUITY OF ALMOST LOCAL CONTRACTIONS 17
and also we add the uniqueness condition:
(2.2.4) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,T(x)),∀x,y∈A,
then the ﬁxed point x∗ofTis unique. Consider the Picard iteration
xn+1=Txn, n∈Nwithx0=x.
Then{xn}∞
n=0converges strongly to x∗and isT- stable, which means that: for {xn}
given by (2.2.1), the equivalence (2.2.2)holds.
Proof:By applying triangle rule and (2.2.4), then the monotonicity of the pseu-
dometric, we obtain
dj(yn+1,x∗)≤dj(yn+1,Tyn) +dj(Tyn,x∗)≤
≤εn+θdr(j)(yn,x∗)≤θdj(yn,x∗) +εn
First, assume that limn→∞εn= 0. Having in view that θ∈[0,1), by using Lemma 2.2.2,
we can conclude that limn→∞yn=x∗.
On the other hand, by applying the deﬁnition of the ALC, we can write
dj(xn+1,x∗)≤θdj(xn,x∗),
which means limn→∞xn=x∗.
Conversely, assume that limn→∞yn=x∗,it follows that
εn=dj(yn+1,Tyn)≤dj(yn+1,x∗) +θdj(yn,x∗)→0
asn→∞.
3. Continuity of almost local contractions
ThissectioncanberegardedasanextensionofV.BerindeandM.Pacurar( 2015,[8])
analysis about the continuity of almost contractions in their ﬁxed points. The main
results are given by Theorem 2.3.1, which give us the answer about the continuity of
local almost contractions in their ﬁxed points.
Theorem 2.3.1.LetXbe a set andD= (dj)j∈Jbe a family of pseudometrics
deﬁned on X; letT:X→Xbe an almost local contraction satisfying condition
(2.1.5), thenTadmits a ﬁxed point.
MoreoverTis continuous at f, for anyf∈Fix(T).
Proof:The mapping Tis an almost local contraction, i.e. there exist the constants
θ∈(0,1)and someL≥0
(2.3.1) dj(Tx,Ty )≤θ·dr(j)j(x,y) +L·dr(j)(y,Tx ),∀x,y∈X

18 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
For any sequence {yn}∞
n=0inXconverging to f, we takey:=yn,x:=fin (2.3.1), and
we get
(2.3.2) dj(Tf,Tyn)≤θ·dr(j)(f,yn) +L·dr(j)(yn,Tf),n= 0,1,2,…
UsingTf=f, sincefis a ﬁxed point of T, we obtain:
(2.3.3) dj(Tyn,Tf)≤θ·dr(j)(f,yn) +L·dr(j)(yn,f),n= 0,1,2,…
Now by letting n→∞in (2.3.3) we get Tyn→Tf, which shows that Tis continuous
atf.
The ﬁxed point has been chosen arbitrarily, so the proof is complete.
According to Deﬁnition 2.1.7, the almost local contractions are deﬁned in a subset
A⊂X. In the case A=X, then an almost local contraction is actually an usual
almost contraction.
Example 2.3.2.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 diagonal line of the
square whose four sides have length n.
We shall use the pseudometric:
(2.3.4) dj/parenleftBig
(x1,y1),(x2,y2)/parenrightBig
=|x1−x2|·e−j,∀j∈J,
whereJis a subset of N. This is a pseudometric, but 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, we shall use the function r(j) =j
2. By applying the inequality (2.1.3)to
our 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|·e−j
2≤2θ·|x1−x2|+L·|2×2−x1|,
The last inequality became true if we take θ=1
2∈(0,1),L= 4≥0. HenceTis an
almost local contraction, with the unique ﬁxed point (0,0).
According to Theorem 2.3.1, Tis continuous in the ﬁxed point, at (0,0)∈Fix(T), but
is not continuous at (1,0)/∈Fix(T).
Example 2.3.3.With the assumptions in Example 2.1.22 and the pseudometric
deﬁned by (2.3.4)wherej∈J, andr(j) =j
2,we get another example for almost local

3. CONTINUITY OF ALMOST LOCAL CONTRACTIONS 19
contractions.
Considering T: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:
(2.3.5) dj(Tx,Ty )≤θ·dj(x,y),
is not valid∀x,y∈X, and for any θ∈(0,1). Indeed, (2.3.5)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
2+L·|x2−x1|·e−j
2
which is equivalent to : e−j
2≤θ+L.
Forθ=1
3∈[0,1),L= 2≥0andj∈J, the last inequality becomes true, i.e. Tis
an almost local contraction with many ﬁxed points: FixT ={(x,0) :x∈R}
In this case, we have:
∀j∈J, limn→∞θn+1diamrn+1(j)(A) = limn→∞/parenleftBigg1
3/parenrightBiggn+1
·(n−1)2= 0
This way, the existence of the ﬁxed point is assured, according to condition (2.1.3)from
Theorem 2.1.10
Theorem 2.3.1 is again valid, because the continuity of Tin(0,0)∈Fix(T), but
discontinuity in (1,1), which is not a ﬁxed point of T.
Example 2.3.4.LetXbe the set of positive functions:
X={f|f: [0,∞)→[0,∞)},
which is a subset of the real functions F={f:R→R}.
Letdj(f,g) =|f(0)−g(0)|·e−j,∀f,g∈X,r(j) =j
2,∀j∈J
Indeed,djis a pseudometric, but not a metric, take for example dj(x,x2) = 0, but
x/negationslash=x2
Considering the mapping Tf=|f|,∀f∈X, and using condition (2.1.3)for almost
local contractions:
|f(0)−g(0)|·e−j≤θ·|f(0)−g(0)|·e−j
2+L·|g(0)−f(0)|·e−j
2
which is equivalent to: e−j/2≤θ+L
This inequality becames true if j >0, θ =1
4∈(0,1), L = 3>0.
Hence,Tis an ALC. However, Tis not a contraction, because the contractive condition

20 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
(2.3.5)leads us again to the false assumption: 1≤θ. The mapping Thas inﬁnite
number of ﬁxed points: FixT ={f∈X}=X, by taking:
|f(x)|=f(x),∀f∈X,x∈[0,∞)
4. 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 2.4.1.Letrbe a function from JtoJ. LetA⊂Fbe aτ-bounded
sequencially 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≥0
such that∀x,y∈X,∀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 )} (2.4.1)
Remark 2.4.2.It is obvious that any generalized almost local contraction is an
almost contraction, i.e., it does satisfy the inequality (1.3.1).
Theorem 2.4.3.LetT:A→Abe a generalized almost local contraction, i.e., a
mapping satisfying (2.4.1), and also verifying the condition (2.5.8)for the unicity of
ﬁxed point. Let Fix(T) ={f}. ThenTis continuous at f.
Proof:SinceTis a generalized almost local contraction, there exist a constant
θ∈(0,1)and someL≥0such that (2.4.1) is satisﬁed. We know by Theorem (2.1.16)
thatThas a unique ﬁxed point, say f.
Let{yn}∞
n=0be any sequence in Xconverging to f. Then by taking
y:=yn, x :=f
in the generalized almost local contraction condition (2.4.1), we get
(2.4.2) dj(Tf,Tyn)≤θ·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 (2.4.2), we get Tyn→Tfwhich shows that Tis continuous
atf.
b)Ćirić type almost local contraction

4. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 21
Definition 2.4.4.(see Berinde, [19]) 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
(2.4.3) d(Tx,Ty )≤α·M(x,y) +L·d(y,Tx ),for all x,y∈X,
where
M(x,y) =max{d(x,y),d(x,Tx ),d(y,Ty ),d(x,Ty ),d(y,Tx )}.
From the above deﬁnition the following question arises: it is possible to expand it
to the case of almost local contractions? The answer is aﬃrmative and is given by the
next deﬁnition. But ﬁrst we need to remind the Lemma of Ćirić ([ 35]), which will be
essential in proving our main results.
Lemma 2.4.5.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 2.4.6.Observe that, by means of Lemma 2.4.5, for each n, there exist
k≤nsuch that
d(x,Tkx) =δ[O(x,n)].
Lemma 2.4.7.(see[35]) LetTbe a quasi-contraction on X.
Then the inequality
δ[O(x,n)]≤1
1−hd(x,Tkx)
holds for all x∈X.
Definition 2.4.8.Under the assumptions of deﬁnition 2.1.7, the operator
T:A→Ais called Ćirić-type almost local contraction with regard to ( D,r) if, for
everyj∈J, there exist the constants θ∈[0,1)andL≥0such that
(2.4.4) dj(Tf,Tg )≤θ·Mr(j)(f,g) +L·dr(j)(g,Tf ),for all f,g∈A,
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
.
Remark 2.4.9.Although this class is more wide than the one of almost local con-
tractions, similar conclusions can be stated as in the case of almost local contractions,
as it follows:
Theorem 2.4.10 .Consider a function r:J→J, let a nonempty, τ- bounded,
sequentially τ- complete, and T- invariant subset A⊂Xand letT:A→Abe Ćirić-
type almost local contraction relating to ( D,r). Then

22 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
(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:
(2.4.5) 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 (2.4.4) to obtain
dj(xn,xn+1) =dj(Txn−1,Txn)≤θ·Mr(j)(xn−1,xn),
sincedj(xn,Txn−1) =dj(Txn−1,Txn−1) = 0. Continuing in this manner, for n≥1, by
Lemma 2.4.5 we have
dj(Tnx,Tn+1x) =dj(TTn−1x,T2Tn−1x)≤θ·δ[O(Tn−1x,2)].
ByusingRemark2.4.6, wecaneasilyconclude: thereexistapositiveinteger 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 2.4.5, 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 manner, 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
(2.4.6) dj(Tnx,Tn+1x)≤θ·δ[O(Tn−1x,2)]≤···≤θn·δ[O(x,n+ 1)].
At this point, by Lemma 2.4.7, we obtain
δ[O(x,n+ 1)]≤δ[O(x,∞)]≤1
1−θdj(x,Tx ),
which by (2.4.6) yields
(2.4.7) dj(Tnx,Tn+1x)≤θn
1−θdj(x,Tx ).
The inequality (2.4.6) and the triangle inequality can be merged to obtain the following
estimate
(2.4.8) dj(Tnx,Tn+px)≤θn
1−θ·1−θp
1−θdj(x,Tx ).

4. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 23
Let us remind the fact that 0≤θ≤1, then, by using (2.4.8), we can conclude that
{xn}∞
n=0is a Cauchy sequence. The subset Ais assumed to be sequentially τ-complete,
there exists x∗inAsuch that{xn}isτ- convergent to x∗. After simple computations
involving the triangular inequality and the Deﬁnition (2.4.4), we get
dj(x∗,Tx∗)≤dj(x∗,xn+1) +dj(xn+1,Tx∗) =
=dj(Tn+1x,x∗) +dj(Tnx,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 manner, 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)]. (2.4.9)
Lettingn→∞in (2.4.9) we obtain
dj(x∗,Tx∗) = 0,
which means that x∗is a ﬁxed point of T. The estimate (2.4.5) can be obtained from
(2.4.7) by letting p→∞.
This completes the proof.
Remark 2.4.11 .1) Theorem 2.4.10 represent 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 almost local contrac-
tion need not have a unique ﬁxed point, according to Example 1.2.3 below.
2) Let us remind (see Rus, [82],[83]) 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 2.4.10 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 (2.4.5)obtained in Theorem 2.4.10 is weaker as (1.2.3),(1.2.4)from the
Banach’s contraction principle or the estimate given in Theorem 2.4.11.

24 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
The uniqueness of the ﬁxed point of a Ćirić type almost local contraction can
be assured by imposing an additional contractive condition, quite similar to (2.4.4),
according to the next theorem.
Theorem 2.4.12 .With the assumptions of Theorem 2.4.10, let T:A→Abe a
Ćirić type almost local contraction with the additional inequality, which actually means
the monotonicity of the pseudometric:
(2.4.10) 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
(2.4.11) 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∗};
(2) The Picard iteration {xn}∞
n=0given byxn+1=Txn, n∈Nconverges to f∗,
for anyx0∈A;
(3) The a priori error estimate (2.4.5)holds;
(4) The rate of the convergence of the Picard iteration is given by
(2.4.12) 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 (2.4.11), and condition (2.4.10) 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 2.4.10.
4) At this point, letting g:=xn,f:=f∗in (2.4.11), it results the rate of convergence
given by (2.4.12). The proof is complete.
The contractive conditions (2.4.4) and (2.4.11) can be merged to maintain the
unicity of the ﬁxed point, stated by the next theorem.
Theorem 2.4.13 .Under the assumptions of deﬁnition 2.4.8, let T:A→Abe a
mapping for which there exist the constants θ∈[0,1)and someL≥0such that
for allf,g∈Aand∀j∈J
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 )}, (2.4.13)

4. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 25
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 (2.4.5)holds.
Particular case 1.(1) The famous Ćirić’ s ﬁxed point theorem for single
valued mappings given in [35]can be obtained from Theorems 2.4.10, 2.4.13,
2.4.12 by taking L=L1= 0and considering rthe identity mapping: r(j) =j.
The Ćirić’ s contractive condition represent 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 almost
local contraction (in (2.4.4)) possess a very high level of generalisation. 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 (2.4.4).
(2) If the maximum from Theorem 2.4.13 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 2.1.10 from Theorem
2.4.10. Also, by Theorem 2.4.12 we obtain Theorem 2.1.16 (see Zakany, [97]).
After this presentation of the existence and the uniqueness theorems of the ﬁxed
points for the Ćirić-type almost local contractions, it is natural to extend it to the
Ćirić-type strict almost local contractions.
Definition 2.4.14 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned on X. In order to underline the local character of these type of contractions,
we letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned by the family
D. Letrbe a function from JtoJ. The operator T:A→Ais called Ćirić-type
strict almost local contraction 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
.

26 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
(ALC−U)dj(Tf,Tg )≤θu·dr(j)(f,g) +Lu·dr(j)(f,Tf ),for all f,g∈A,∀j∈J,
We end with a few examples that have an illustrative role. They presents Ćirić’
type almost local contractions, without having unique ﬁxed point.
Example 2.4.15 .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.
We shall use the pseudometric dj(f,g) =|f(0)−g(0)|·j,∀j∈J;J⊂N,∀f,g∈A.
Indeed,djis a pseudometric, but not a metric, take for example dj(x3,x2) = 0, but
x3/negationslash=x2
Considering the mapping Tf=|f|,∀f∈A,r(j) =j+ 1. Note that the restrictive
condition (2.4.10)is also veriﬁed. By using condition (2.1.3)for almost local contrac-
tions:
|f(0)−g(0)|·j≤θ·|f(0)−g(0)|·(j+ 1) +L·|g(0)−f(0)|·(j+ 1)
which is equivalent to: j≤(θ+L)(j+ 1)
This inequality becames true if j >1, θ =1
5∈(0,1), L = 3>0,andj
j−1∈(1,2).
Hence,Tis an almost local contraction. However, Tis not a contraction, because the
contractive condition (2.3.5)leads us to the false assumption: 1≤θ.
The mapTis Ćirić-type almost local contraction, because
Mr(j)(f,g) =|f(0)−g(0)|·(j−1),
and from (2.4.4)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 get the inequality j≤(θ+L)(j−1). The mapping Thas inﬁnite number of
ﬁxed points: FixT ={f∈A}=A, by taking:
|f(x)|=f(x),∀f∈A, x∈[0,∞).
The uniqueness condition (2.4.11)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 (2.4.11).
In fact, not even (2.4.13)is satisﬁed, by computing Mr(j)(f,g) =|f(0)−g(0)|·(j−1)
andmin{dr(j)(f,Tf ),dr(j)(g,Tg ),dr(j)(f,Tg ),dr(j)(g,Tf )}=|f(0)−g(0)|·(j−1)(since
j >1). By replacing these values in (2.4.13), we get
|f(0)−g(0)|·j≤θ·|f(0)−g(0)|·(j−1) +L·|f(0)−f(0)|·(j−1),
which also lead to the previous contradiction.

4. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 27
Example 2.4.16 .By taking the mapping from Example 2.3.4, with a small modi-
ﬁcation, which 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}.
Letdj(f,g) =|f(x0)−g(x0)|·ej,∀f,g∈X,r(j) =j
2,∀j∈Z.
We can conclude in the same manner that Tis also a Ćirić type almost local contraction,
i.e., it satisfy the contractive condition (2.4.4).
Indeed, we have Mr(j)(f,g) =|f(x0)−g(x0)|·ej
2. This way, the condition (2.4.4)
became the contractive condition for almost local contractions ( (2.1.10))
By considering L= 0in the deﬁnition 2.4.8 of Ćirić-type almost local contraction,
we get a new type of ALC, that is the quasi-almost local contraction.
c)Quasi-almost local contractions
Definition 2.4.17 .Under the assumptions of deﬁnition 2.1.7, the operator
T:A→Ais called quasi-almost local contraction with regard to ( D,r) if, for every
j∈J, there exist the constant θ∈[0,1)such that
(2.4.14) 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 2.4.18 .Consider a function r:J→J, let a nonempty, τ- bounded,
sequentially τ- complete, and T- invariant subset A⊂Xand letT:A→Abe quasi-
almost local contraction 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:
(2.4.15) 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 2.4.10,
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 almost local contraction, as it follows.
Theorem 2.4.19 .With the assumptions of Theorem 2.4.10, let T:A→Abe a
quasi-almost local contraction with the additional inequality:
(2.4.16) dr(j)(f,g)≤dj(f,g),∀f,g∈A,∀j∈J.

28 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
If the mapping Tsatisﬁes the supplementary condition: there exist the constants
θ∈[0,1)such that
(2.4.17) 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∗};
(2) The Picard iteration {xn}∞
n=0given byxn+1=Txn, n∈Nconverges to f∗,
for anyx0∈A;
(3) The a priori error estimate (2.4.5)holds;
(4) The rate of the convergence of the Picard iteration is given by
(2.4.18) dj(xn,f∗)≤θ·dr(j)(xn−1,f∗), n = 1,2,…,∀j∈J
d)Ćirić-Reich-Rus type almost local contraction
Definition 2.4.20 .Under the assumptions of deﬁnition 2.1.7, the operator
T:A→Ais called Ćirić-Reich-Rus type almost local contraction regarding to ( D,r)
if the mapping T:A→Asatisfying the condition
(2.4.19) 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 2.4.21 .If the pseudometric dsatisfy the condition:
dr(j)(f,g)<dj(f,g),∀j∈J,∀f,g∈Athen any Ćirić- Reich- Rus type almost local
contraction, i.e. any mapping T:A→Asatisfying the condition (2.4.19)withL/negationslash= 1
is an almost local contraction.
Proof:Using condition (2.4.19) and the triangle rule, 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 )
From this point, we get after simple computations:
(2.4.20) (1−L)·dj(Tf,Tg )≤(δ+L)·dj(f,g) + 2L·dr(j)(g,Tf )
and which implies
(2.4.21) dj(Tf,Tg )≤δ+L
1−L·dj(f,g) +2L
1−L·dr(j)(g,Tf ),∀f,g∈A

4. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 29
Considering δ,L∈R +andδ+ 2L<1, the inequality (2.4.19) holds, with
δ+L
1−L∈(0,1)and2L
1−L≥0. Therefore, anyĆirić-Reich-Rustypealmostlocalcontraction
with the condition for the pseudometric, is an almost local contraction.
e)Chatterjea type ALC
Definition 2.4.22 .Under the assumptions of deﬁnition 2.1.7, the operator
T:A→Ais called Chatterjea type almost local contraction with regard to ( D,r) or
(δ,L)- Chatterjea contraction if there exist a constant 0≤c<1
2such that
(2.4.22) dj(Tf,Tg )≤c·[dr(j)(f,Tg ) +dr(j)(g,Tf )],∀f,g∈A.
Theorem 2.4.23 .If the pseudometric dsatisfy the monotonicity property:
(2.4.23) dr(j)(f,g)≤dj(f,g),∀j∈J,∀f,g∈A,∀j∈J,
then any Chatterjea type almost local contraction, i.e., any mapping T:A→Asatis-
fying the condition (2.4.22)is an almost local contraction.
Proof:Using condition (2.4.22) and the triangle rule, 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 condition for the pseudometric leads us to:
(2.4.24) dj(g,Tf )<dr(j)(g,Tf )
From this point, we get after simple computations:
(2.4.25) 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
(2.4.26) dj(Tf,Tg )≤c
1−c·dj(f,g) +2c
1−c·dr(j)(g,Tf ),∀f,g∈A
Considering 0≤c <1
2, the inequality (2.4.22) holds, with δ=b
1−bandL=2b
1−b.
Therefore, any Chatterjea type almost local contraction with the condition for the
pseudometric, is an almost local contraction.
f)Zamﬁrescu type ALC
Definition 2.4.24 .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 almost local contraction,
is called Zamﬁrescu-type almost local contraction.

30 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Now, it is natural to state the next result:
Proposition 2.4.25 .Any Zamﬁrescu type almost local contraction is an almost
local contraction.
g)Generalized Berinde type ALC
Definition 2.4.26 .Under the assumptions of deﬁnition 2.1.7, the operator
T:A→Ais called generalized Berinde type ALC regarding to ( D,r) if there exist a
constantθ∈[0,1)and a function bfrom the subset Ainto[0,∞)such that
(2.4.27) dj(Tx,Ty )≤θ·dr(j)(x,y) +b(y)·dr(j)(y,Tx ),∀x,y∈A,∀j∈J.
Theorem 2.4.27 .We shall use assumptions from Deﬁnition 2.1.7.
LetTbe a mapping on the subset A. Consider a function bfromAinto[0,∞). Assume
that there exists θ∈[0,1)such that
(2.4.28)
(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,{Tnx}converges to a ﬁxed point of T.
Proof:Sinceθ∈[0,1), we have the inequality
(2.4.29) (1 +θ)−1dj(x,Tx )≤dj(x,Tx ),∀j∈J,
and we get
(2.4.30)dj(Tx,T2x)≤θdr(j)(x,Tx ) +b(Tx)dr(j)(Tx,Tx ) =θdr(j)(x,Tx ),∀x∈A
Letu∈A, then from (2.4.30) we have
dj(Tnu,Tn+1u)≤θndr(j)(u,Tu ),∀j∈J,
and hence∞/summationdisplay
n=1dj(Tnu,Tn+1u)<∞.
Thus, the sequence {Tnu}isdj- Cauchy for each j. The subset Ais assumed to be
sequentially τ-complete, there exists zinAsuch that (Tnx)n∈Nisτ- convergent to
z∈A.
Using (2.4.30), 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 (2.4.28), we have
dj(z,Tz ) = limn→∞dj(Tf(n)+1u,Tz )≤
≤limn→∞(θdr(j)(Tf(n)u,z) +b(z)dr(j)(Tf(n)+1u,z)) =
=θdr(j)(z,z) +b(z)dr(j)(z,z) = 0.
Therefore,zis a ﬁxed point of T.

5. ALMOST LOCAL ϕ-CONTRACTIONS 31
Corollary 2.4.28 .IfTis a generalized ALC on the subset A, then for every
x∈A,A⊂X, the sequence{Tnx}converges to a ﬁxed point of T.
Corollary 2.4.29 .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, A⊂X, the sequence{Tnx}converges to a
ﬁxed point of T.
5. ALMOST LOCAL ϕ-CONTRACTIONS
The class of ϕ-contractions represent a generalisation of strict contractions, which
is why we extend the ALC-s to the more general class of almost local ϕ-contractions.
The aim of this section is to study the properties and the ﬁxed points of this new type
of ALC-s. First, let us remind a few concepts from Rus ([ 81]) and Berinde ([ 10]).
Definition 2.5.1.[81]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.
According to Berinde’s work (see [ 17]),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).
Also, it was proved that if ϕis a (c)- comparison function, then the sum of the com-
parison series, which is,
(2.5.1) s(t) =∞/summationdisplay
k=0ϕk(t),t∈R+
is monotone increasing and continuous at zero,hence
(2.5.2) ϕk(t)→0ask→∞,∀t∈R+,
and that any (c)- comparison function is a comparison function.
The concept of c- comparison function was reformulated in [ 22] to that of b-
comparison function, as it follows:

32 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Definition 2.5.2.Letb≥1be a real number. A map ϕ:R+→R+is called
b-comparison function if it satisﬁes:
(iϕ)ϕis increasing;
(iiϕ)there existk0∈N,a∈(0,1)and a convergent series of nonnegative terms∞/summationtext
n=0vk
such that
(2.5.3) bk+1ϕk+1(t)≤abkϕk(t) +vk
fork≥k0and anyt∈R+
Remark 2.5.3.Obviously, for b= 1the concept of b-comparison function became
to that ofc-comparison function.
The following Lemma was proved in [ 17], it contain a few properties regarding
c-comparison functions.
Lemma 2.5.4.Ifϕ:R+→R+is a (c)-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+.
The following Lemma was proved in [ 21] and [22]:
Lemma 2.5.5.Ifϕ:R+→R+is a(b)-comparison function, then :
(i)the series∞/summationtext
k=0bkϕk(t)converges for any t∈R+;
(ii)The function sb:R+→R+deﬁned by
(2.5.4) sb(t) =∞/summationdisplay
k=0bkϕk(t),t∈R+
is increasing and continuous at zero.
Lemma 2.5.6.(see[68]) Ifϕ:R+→R+is a(b)-comparison function, with con-
stantb≥1andan∈R+,n∈Nsuch thatan→0asn→∞.
Then
(2.5.5)n/summationdisplay
k=0bn−kϕn−k(ak)→0asn→∞.
Weakϕ- contractions was ﬁrst introduced by V. Berinde in [ 18]. The aim of this
section is to extend weak ϕ- contractions to the more general case of weak ALC- s.
Definition 2.5.7.(Berinde, [18]) Let (X,d)be a metric space. A self operator
T:X→Xis said to be a weak ϕ- contraction or (ϕ,L)- weak contraction, provided
that there exist a comparison function ϕand someL≥0, such that
(2.5.6) d(Tx,Ty )≤ϕ(d(x,y)) +L·d(y,Tx ),for allx,y∈X.

5. ALMOST LOCAL ϕ-CONTRACTIONS 33
From this point,it is natural the extension of weak ϕ- contractions to the weak
ALC- s.
Definition 2.5.8.LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX. We letτbe the weak topology on Xdeﬁned by the family D. Letrbe a
function from JtoJ, and letA⊂Xbe aτ-bounded sequencially 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:
(2.5.7) dj(Tx,Ty )≤ϕ(dr(j)(x,y)) +L·dr(j)(y,Tx ),∀x,y∈A,∀j∈J.
Remark 2.5.9.It is obvious that any almost local ϕ- contraction became an almost
local contraction if we take ϕ(t) =θt,t∈R+and0<θ< 1.
There exist almost local ϕ- contractions which appear not to be almost local contractions
with relating to the same pseudometric, according to Example 2.5.10
Example 2.5.10 .Letϕ(t) =t
t+ 1, t∈R+, withr(j) =jand letTan almost
localϕ- contraction, i.e. a mapping which satisﬁes (2.5.7).
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 2.5.11 .1) Similar to the case of ALC-s, if Tsatisﬁes(2.5.6),
for allx,y∈A, does imply that the following dual inequality
(2.5.8) dj(Tx,Ty )≤ϕ(dr(j)(x,y)) +L·dr(j)(x,Ty ),∀x,y∈A,∀j∈J
obtained from (2.5.6)by replacing xwithyand viceversa, is also valid.
Clearly, in order to prove that a mapping Tis an almost local ϕ- contraction, it is
necessary to check both (2.5.7)and(2.5.8)inequalities.
2) The class of almost local ϕ- contractions includes a very large type of mappings, see
for example 2.5.12, which contain a mapping with not one, but inﬁnite number of ﬁxed
points.
Example 2.5.12 .Let[0,1]be the unit interval with the euclidian 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|,
which is valid for all x,y∈[0,1].
Note that the set of ﬁxed points, F(T) ={x∈[0,1] :Tx=x}= [0,1], has an
inﬁnite number of elements.

34 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
The next two theorems represent an existence theorem and, respectively, a uniqueness
theorem for the ﬁxed points of almost local ϕ- contractions.
Theorem 2.5.13 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned on X. In order to underline the local character of these type of contractions,
we letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned by the family
D. Letrbe a function from JtoJandT:A→Aan almost local ϕ- contraction with
ϕa (c)- comparison function.Then
(1)F(T) ={x∈A:Tx=x}/negationslash=φ;
(2) For any x0∈A, the Picard iteration {xn}∞
n=0deﬁned byx0∈Aand
(2.5.9) xn+1=Txn, n = 0,1,2,…
converges to a ﬁxed point x∗∈Fix(T)
(3) The a posteriori estimate
(2.5.10) dj(xn,x∗)≤s(dj(xn,xn+1)), n = 0,1,2…,∀x,y∈A,∀j∈J
holds, where s(t)is given by (2.5.1).
Proof:We shall prove that the set of ﬁxed points of Tis nonempty, which means
thatThas at least one ﬁxed point in the subset A. Using the fact that Tis an an
almost local ϕ- contraction, there exist a (c)- comparison function ϕand someL≥0,
such that
(2.5.11) 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 (2.5.11) to obtain
dj(Txn−1,Txn)≤ϕ(dr(j)(xn−1,xn))
which yields
(2.5.12) dj(xn,xn+1)≤ϕ(dr(j)(xn−1,xn)),∀j∈J,∀n= 1,2,…
Sinceϕis non decreasing, by (2.5.12) we have
(2.5.13) dj(xn+1,xn+2)≤ϕ(dr(j)(xn,xn+1)),∀x,y∈A,∀j∈J.
From that relation, we obtain by induction with related to n:
(2.5.14) dj(xn+k,xn+k+1)≤ϕk(dr(j)(xn,xn+1)),∀x,y∈A,∀j∈J.

5. ALMOST LOCAL ϕ-CONTRACTIONS 35
According to the triangle rule, 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 (2.5.15)
where we denoted r=dj(xn,xn+1). Again, by (2.5.13) we conclude
(2.5.16) dj(xn,xn+1)≤ϕn(dr(j)(x0,x1)),n= 0,1,2,…
which, by property (iiϕ)from the deﬁnition (2.5.1) of a comparison function implies
(2.5.17) limn→∞dj(xn,xn+1) = 0.
Having in view that ϕis positive, it is clear that
(2.5.18) 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 (2.5.18) and (2.5.15) we get
(2.5.19) dj(xn,xn+p)≤s(dr(j)(xn,xn+1)),n∈N,p∈N,∀x,y∈A,∀j∈J.
Sincesis continuous at zero, (2.5.18) and (2.5.17) implies that the sequence (xn)n∈N
isdj- Cauchy for each j∈J. 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
(2.5.20) 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 (2.5.20) 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).
The estimate (2.5.10) follows from (2.5.15) by taking p→∞.
This completes the proof.

36 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Remark 2.5.14 .1)The a posteriori error estimates (2.5.10)and(2.5.16)leads us
to the a priori estimate for the Picard iteration {xn}∞
n=0.
2)An almost local ϕ- contraction could be discontinuous, as shown by Example 2.5.15.
3) By taking ϕ(t) =θ·t,t∈R+,0<θ< 1, from Theorem 2.5.13 we obtain the same
result for almost local contractions, i.e. Theorem 2.1.10.
Example 2.5.15 .
Remark 2.5.16 .An almost local ϕ- contraction may have more than one ﬁxed
point, as shown by Example 2.5.12. In Theorem 2.5.13, the Picard iteration {xn}∞
n=0
provide the ﬁxed point x∗, 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 2.5.17.
Theorem 2.5.17 .LetXandTas in Theorem 2.5.13, the subset A⊂X. Suppose
Talso satisﬁes the following condition: there exist a comparison function Υand some
L1>0such that
(2.5.21) dj(Tx,Ty )≤Υ(dr(j)(x,y)) +L1·dr(j)(x,Tx )
is valid for all x,y∈A,∀x,y∈A,∀j∈J.
Then
(1)Thas a unique ﬁxed point, i.e. F(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 (2.5.1);
(3) The rate of the convergence of the Picard iteration is given by
(2.5.22) 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 (2.5.21), by taking x:=x∗andy:=y∗, we obtain
dj(x∗,y∗)≤Υdj(x∗,y∗),∀x,y∈A,∀j∈J,
which by induction with regarding to n, yields
(2.5.23) dj(x∗,y∗)≤Υn·dj(x∗,y∗),n= 1,2,…
Lettingn→∞in (2.5.23), we get
dj(x∗,y∗) = 0
which means that x∗=y∗, a contradiction.
In this manner,we proove that the ﬁxed point is unique.
For the proof of (2.5.22), all we have to do is to change x:=x∗andy:=xnin the

5. ALMOST LOCAL ϕ-CONTRACTIONS 37
inequality (2.5.21).
The proof is complete.
Remark 2.5.18 .1) Having in view the pairs of dual conditions (2.1.3)and(2.1.4),
(2.5.7)and(2.5.8), condition (2.5.21)holds if and only if its dual
(2.5.24) dj(Tx,Ty )≤Υ(dr(j)(x,y)) +L1·dr(j)(y,Ty ),
is valid for all x,y∈A,∀j∈J.
2) Condition (2.5.21)is not necessary for the unicity of the ﬁxed point, according to
Example 2.5.19.
Example 2.5.19 .LetX= [1,n]×[1,n]⊂R2. We shall use the pseudometric:
(2.5.25) dj((x1,y1),(x2,y2)) =|x1−x2|·ej,∀j∈Q.
We shall use r(j) =j
2and 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+.
Tis an almost local ϕ-contraction if:
|x1−x2|·ej≤1
4·|x1−x2|·ej
2+L·|x2−x1|·ej
2
which is equivalent to : ej
2≤1
4+L.
ForL= 3≥0andj <0, the last inequality becomes true, i.e. Tis an almost local
ϕ-contraction with many ﬁxed points: FixT ={(x,0) :x∈R}
Remark 2.5.20 .IfThas a unique ﬁxed point x∗and the Picard iteration {Tnx0}∞
n=0
converges to x∗for allx0∈A, then by Theorem 2.4.4, for any a∈(0,1), there exist a
pseudometric /rho1onAsuch that (X,/rho1)isτ- complete and Tis ana- contraction with
regard to the pseudometric /rho1.
Therefore, condition (2.5.21)could be reformulated using another pseudometric, which
leads us to a more general result.
Theorem 2.5.21 .LetXbe a nonempty set, let A⊂Xbe aτ-bounded sequencially
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 exists a c-comparison function ϕandL≥0such that
dj(Tx,Ty )≤ϕ(dr(j)(x,y)) +L·dr(j)(y,Tx ),∀x,y∈A,∀x,y∈A,∀j∈J;

38 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
(ii)There exists a comparison function ΥandL1≥0such that
/rho1j(Tx,Ty )≤Υ(/rho1r(j)(x,y)) +L1·ρr(j)(x,Tx ),∀x,y∈A,∀x,y∈A,∀j∈J.
.
Then
(1)Thas a unique ﬁxed point, i.e. F(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 (2.5.1);
(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 2.5.22 .We obtain Theorem 2.5.17 as a particular of Theorem 2.5.21 if
we setdj≡/rho1j.
In order to expand the class of almost local ϕ- contractions, we try at ﬁrst with the
quasiϕ- contractions, as shown below.
Definition 2.5.23 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX. We letτbe the weak topology on Xdeﬁned by the family D. Letrbe a
function from JtoJ, and let the subset A⊂Xbe aτ-bounded sequencially τ-complete
andT- invariant subset of X. A mapping T:A→Ais called quasi ϕ- contraction if
there exist a comparison function ϕsuch that we have:
(2.5.26) dj(Tx,Ty )≤ϕ(dr(j)(x,y)),∀x,y∈A,∀j∈J.
Theorem 2.5.24 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned on X. In order to underline the local character of these type of contractions,
we letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned by the family
D. Letrbe a function from JtoJandT:A→Abe a quasiϕ- contraction, i.e. a
mapping which satisﬁes (2.5.26)withϕa comparison function. Then Thas a unique
ﬁxed point if there exists x0∈Asuch that the Picard iteration {xn}∞
n=0deﬁned by
xn+1=Txn, n∈N
is bounded.

6. (B)-ALMOST LOCAL CONTRACTIONS 39
Proof:If the sequence of the successive approximations {xn}∞
n=0is bounded, for a
certainx0∈A, it means there exists a constant k>0and an element u∈Asuch that
(2.5.27) 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 (2.5.26) 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 (2.5.2), we conclude that the
sequence (xn)n∈Nisdj- Cauchy for each j∈J.
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 applying again the triangle inequality, we obtain
0≤dj(Tx∗,x∗)≤dj(Tx∗,Txn) +dj(Txn,x∗)≤
≤ϕ/parenleftBig
dr(j)(x∗,xn)) +dr(j)(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 2.5.25 .The converse of theorem 2.5.24 is also true:
If the mapping Thas a unique ﬁxed point, noted by x∗, then the sequence of the suc-
cessive approximations is bounded. Indeed, by choosing x0=x∗, the sequence (xn)is
constant, so it is bounded.
6. (B)-almost local contractions
In this section our goal is to combat the inconvenience of having non symmetric
relations for the deﬁnition of almost local contractions, that is
ALC 1 dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈A
and, respectively,
ALC 2 dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,Ty ),∀x,y∈A

40 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
under the assumptions of Deﬁnition 2.1.7 given for almost local contractions.
Remark 2.6.1.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 2.6.2.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:
Lemma 2.6.3.LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
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 2.6.1, it
also satisﬁes (ALC 2).
The inequality (ALC 1)can be rewritten in equivalent form:
(2.6.1)1
L[dj(Tx,Ty )−θ·dr(j)(x,y)]≤dr(j)(y,Tx ),∀x,y∈A
The other condition, ALC 2also can be written as:
(2.6.2)1
L[dj(Tx,Ty )−θ·dr(j)(x,y)]≤dr(j)(x,Ty ),∀x,y∈A
The inequalities (2.6.1) and (2.6.2) 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
(m−ALC ). Then, obviously it also fulﬁlls (ALC 1).
Hence, the conditions (ALC 1)and(m−ALC )are equivalent.
This completes the proof.

7. APPROXIMATE FIXED POINTS 41
7. 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 suﬃcient to approximate
the solution. So the existence of ﬁxed points is not strictly required, but that of
proximity ﬁxed points. This approximation is also used when the conditions imposed
for the existence of the ﬁxed points are too strong for the real problem that waiting
for solution.
It is a natural consequence to introduce the concepts of ε-ﬁxed point (orapproximate
ﬁxed point ), which, in fact, represent a proximity ﬁxed point, and that of function with
theapproximate ﬁxed point property and to establish qualitive and quantitive theorems
for various types of almost local contractions.
In this section, the starting point is represented by the article of Tijs, Torre and
Branzei (see [ 95]) and also the paper of M. Berinde (see [ 7]).
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 needed in order to deal with approximate ﬁxed points, it
was published in [ 48], probably attended by the work of Granas,Dugundji ([ 43]):
Definition 2.7.1.Let(E,τ)be a topological space, αan open covering of Eand
f:E→Ean operator. Then x∈Eis anα-ﬁxed point of fif there exists U∈αsuch
thatxandf(x)are inU.
The following deﬁnition was given by V.R.Klee [ 56] and it was mentioned also by
van der Walt [ 96]:
Definition 2.7.2.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))represent the diameter of the set f(U).
Definition 2.7.3.(see[95]) 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.
Remark 2.7.4.The set of all ε-ﬁxed points of f, for a given ε, shall be denoted by
Fε(f) ={x∈X|x is anε-ﬁxed point of f}

42 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Remark 2.7.5.Clearly, any ﬁxed point of fis also anε-ﬁxed point of f:
x∈Ff⇒x∈Fε(f),
but the converse is not always true, as shown in the next example (Example 3.1.1in
[67]):
Example 2.7.6.LetX= [0,1]with the usual metric and f: [0,1]→[0,1]be
deﬁned by
f(x) =

1ifx∈[0,1)
0ifx= 1
Then:
(1)Ff=?;
(2)Fε(f) = [1−ε,1), for anyε>0.
Proof:1) Conclusion 1) is obvious.
2) Letε>0and choose xanε-ﬁxed point of f, which means
(2.7.1) |x−f(x)|≤ε.
According to the deﬁnition of f, the only possibility is that x∈[0,1),in this case we
havef(x) = 1.Then, using (2.7.1), we get
|x−1|≤ε⇔1−x≤ε⇔x>1−ε.
The last inequality means that: x∈[1−ε,1), it results Fε(f) = [1−ε,1), for any
ε>0.
Definition 2.7.7.([95]) Letf:X→X. Thenfhas the approximate ﬁxed point
property if
∀ε>0,Fε(f)/negationslash=?.
Remark 2.7.8.The concept of asymptotically regular operator was ﬁrst introduced
in[30]in metric spaces,see also [55]and[82]. It seems to be very useful concept in
this section.
Lemma 2.7.9.([7]) 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 2.7.9 could be extended to a larger, more general space, such as the pseu-
dometric space, as in in the foolowing result:

7. APPROXIMATE FIXED POINTS 43
Lemma 2.7.10 .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 for 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
We shall using the notation:
y0=Tn(x0),
it follows that
∀ε>0,∃y0∈Asuch that dj(y0),T(y0))<ε,
it results : for each ε >0, there exists an ε- ﬁxed point of Tin the subset A⊂X,
which isy0.
So we proove that Thas the approximate ﬁxed point property.
Remark 2.7.11 .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 2.7.12 .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 shall denote by δ(A)the diameter of the nonempty set A, namely:
δ(A) = sup{d(x,y)|x,y∈A}.
Lemma 2.7.13 .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)Fε(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∈Fε(T).
Then:
δ(Fε(T))≤ϕ(2ε).

44 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Proof:Fixε>0andx,y∈Fε(T). By means of deﬁnition 2.7.3, we have:
dj(x,T(x))<ε,dj(y,T(y))<ε.
By applying 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(Fε(T))≤ϕ(2ε).
This completes the proof.
Remark 2.7.14 .According to Lemma 2.7.10, condition (i) from Lemma 2.7.13 can
be replaced by the asymptotic regularity condition.
Ergo, Lemma 2.7.13 can be reformulated in the form:
Lemma 2.7.15 .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 such that for
ε>0the following hold:
(i)
d(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∈Fε(f).
Then:
δ(Fε(T))≤ϕ(2ε).
A. Qualitative results for mappings in pseudometric spaces
Our ﬁrst goal is to state and prove, using Lemma 2.7.10, 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 2.7.16 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned on X. In order to underline the local character of these type of contractions,
we letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned by the family
D. Letrbe a function from JtoJ, withJa family of indices. Let T:A→A

7. APPROXIMATE FIXED POINTS 45
a quasi-almost local contraction, i.e., a mapping satisfying conditions from Deﬁnition
2.4.17.
Then:
∀ε>0,Fε(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→∞.
So, we have for all j∈J:
dj(Tn(x),Tn+1(x))→0,asn→∞,∀x∈A.
According to Lemma 2.7.10, it follows that Fε(T)/negationslash=?,∀ε>0
Theorem 2.7.17 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned
by the familyD. Letrbe a function from JtoJ, withJa family of indices. Let
T:A→Aana-local contraction, i.e., a mapping for which there exist a∈[0,1)such
that:dj(Tx,Ty )≤a·dr(j)(x,y),for all x,y∈A.
Then:
∀ε>0,Fε(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 2.7.10, it follows that Fε(T)/negationslash=?,∀ε>0.
This completes the proof.
Theorem 2.7.18 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned
by the familyD. Letrbe a function from JtoJ, withJa family of indices. Let
T:A→Aa Kannan-type almost local contraction, i.e., a mapping for which there
exista∈[0,1
2)such that
(2.7.2) dj(Tx,Ty )≤a·[dr(j)(x,T(x)) +dr(j)(y,T(y))],for all x,y∈A,

46 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
The pseudometric satisﬁes the monotonicity property:
dr(j)(x,y)≤dj(x,y),∀j∈J,∀x,y∈A,∀j∈J,
Then:
∀ε>0,Fε(T)/negationslash=?.
Proof:Letε > 0andx∈A. After following similar computations as in the
previous theorem, and by using the monotonicity of the pseudometric ((2.4.10))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))
Having in view that a∈[0,1
2), this meansa
1−a∈[0,1)so,/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 2.7.10, it follows that Fε(T)/negationslash=?,∀ε>0.
This completes the proof.
Theorem 2.7.19 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned on X. In order to underline the local character of these type of contractions,
we letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned by the family
D. Letrbe a function from JtoJ, withJa family of indices.
LetT:A→Aa Chatterjea-type almost local contraction, i.e., a mapping satisfying
the condition: there exist a constant 0≤c<1
2such that
dj(Tf,Tg )≤c[dr(j)(f,Tg ) +dr(j)(g,Tf )],∀f,g∈A.
If the monotonicity is valid for the pseudometric, namely (2.4.10),
then:
∀ε>0,Fε(T)/negationslash=?.

7. APPROXIMATE FIXED POINTS 47
Proof:Letε>0andx∈A.
Aftersimplecomputations,andbyusingthemonotonicityofthepseudometric((2.4.10)),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))] =
=a·dr(j)(Tn−1(x),Tn+1(x)).
Using the triangle inequality, 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−a)dj(Tn(x),Tn+1(x))≤a·dr(j)(Tn−1(x),Tn(x)),
which is equivalent with:
dj(Tn(x),Tn+1(x))≤a
1−adr(j)(Tn−1(x),Tn(x))≤···≤
≤/parenleftBiga
1−a/parenrightBigndr(j)(x,T(x))
Remind that a∈[0,1
2), soa
1−a∈[0,1), which means:/parenleftBig
a
1−a/parenrightBign→0asn→∞.
We obtain, for all j∈J:
dj(Tn(x),Tn+1(x))→0,asn→∞,∀x∈A.
According to Lemma 2.7.10, it follows that Fε(T)/negationslash=?,∀ε>0.
The proof is complete.
Theorem 2.7.20 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned
by the familyD. Letrbe a function from JtoJ, withJa family of indices.
LetT:A→Aa Zamﬁrescu-type almost local contraction, i.e., 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(Tx,Ty )≤a·dr(j)(x,y);
(ii)dj(Tx,Ty )≤k·[dr(j)(x,T(x)) +dr(j)(y,T(y))];
(iii)dj(Tx,Ty )≤c·[dr(j)(x,T(y)) +dr(j)(y,T(x))].
Assume the monotonicity property ( (2.4.10)) assured for the pseudometric.
Then:
∀ε>0,Fε(T)/negationslash=?.

48 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Proof:Our ﬁrst goal is to merge the independent conditions (i),(ii) and (iii) into a
singleoneforaneasierproofoftheapproximateﬁxedpointpropertyfortheZamﬁrescu-
type almost local contractions.
Letx,y∈A.
a) Assuming (ii) is valid, we obtain by applying the monotonicity 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)) +dr(j)(x,y) +k·dr(j)(T(x),T(y))≤
≤2k·dr(j)(x,T(x)) +dr(j)(x,y) +k·dj(T(x),T(y))
This implies
(2.7.3) 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(Tx,Ty )≤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(x)) +c·dr(j)(x,y)≤
≤c·dj(T(x),T(y)) + 2c·dr(j)(y,T(x)) +c·dr(j)(x,y)
By dividing with 1−c, we have that:
(2.7.4) dj(T(x),T(y))≤2c
1−cdr(j)(y,T(y)) +c
1−cdr(j)(x,y).
Continuing in this manner, 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
(2.7.5) dj(T(x),T(y))≤2c
1−cdr(j)(x,T(x)) +c
1−cdr(j)(x,y).
Now, having in view (i),(2.7.3), (2.7.4), and (2.7.5); for a more concentrated approach,
we denote:
δ= max/braceleftBig
a,k
1−k,c
1−c/bracerightBig
.
Obviously, we have δ∈[0,1].

7. APPROXIMATE FIXED POINTS 49
The mapping Tsatisfying at least one of the inequalities (i),(ii), (iii); we conclude:
(2.7.6) dj(T(x),T(y))≤2δ·dr(j)(x,T(x)) +δ·dr(j)(x,y)
and
(2.7.7) dj(T(x),T(y))≤2δ·dr(j)(y,T(y)) +δ·dr(j)(x,y)
Using conditions (2.7.6) and (2.7.7), we have for a given x∈X:
dj(Tn(x),Tn+1(x)) =dj(T(Tn−1(x),T(Tn(x)))≤
≤2δ·dr(j)(Tn−1(x),T(Tn−1(x))) +δ·dr(j)(Tn−1(x),Tn(x)) =
= 3δ·dr(j)(Tn−1(x),Tn(x))≤···≤ (3δ)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 2.7.10, it follows that Fε(T)/negationslash=?,∀ε>0.
Now, the proof is complete.
Remark 2.7.21 .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 2.7.17, 2.7.18 and 2.7.19 are actually contained in Theorem 2.7.20
Theorem 2.7.22 .Under the assumptions of Theorem 2.7.20, let T:A→Aan
almost local contraction (or weak contraction), i.e., a mapping satisfying condition
(2.1.3)from Deﬁnition 2.1.7 .
Then:
∀ε>0,Fε(T)/negationslash=?.
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 a∈[0,1), which implies that we have for all j∈J:
dj(Tn(x),Tn+1(x))→0,asn→∞,∀x∈A.
According to Lemma 2.7.10, it follows that Fε(T)/negationslash=?,∀ε>0.
This completes the proof.

50 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
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 2.7.13.
Theorem 2.7.23 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned
by the familyD. Letrbe a function from JtoJ, withJa family of indices. Let
T:A→Aana-local contraction and assume the monotonicity property ( (2.4.10))
assured for the pseudometric.
Then:
δ(Fε(T))≤2ε
1−a,∀ε>0.
Proof:Letε>0. Condition (i) from Lemma 2.7.13 was proved in Theorem 2.7.17.
It remain to show that (ii) holds for a-contractions.
Fixγ >0andx,y∈Fε(T). We make the assumption
dj(x,y)−dj(T(x),T(y))≤γ.
In order to obtain the conclusion of the Lemma, we shall proove that there exists
anϕ(γ)>0such thatdj(x,y)≤ϕ(γ). It results, after applying the monotonicity
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.
This way, 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 2.7.13 it results that
δ(Fε(T))≤ϕ(2ε),∀ε>0.
The last inequality actually means that
δ(Fε(T))≤2ε
1−a,∀ε>0.
This completes the proof.
Theorem 2.7.24 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned
by the familyD. Letrbe a function from JtoJ, withJa family of indices. Let
T:A→Aa Kannan-type almost local contraction and assume the monotonicity
property ( (2.4.10)) assured for the pseudometric.

7. APPROXIMATE FIXED POINTS 51
Then:
δ(Fε(T))≤2ε(1 +a),∀ε>0.
Proof:Letε>0. Again, condition (i) from Lemma 2.7.13 is satisﬁed, as one can
see in the proof of Theorem 2.7.17.
It remain to show that (ii) holds for Kannan-type ALC-s.
Fixγ >0andx,y∈Fε(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,dj(y,T(y))<ε,
which implies
dj(x,y)≤γ+dj(T(x),T(y))≤γ+adr(j)(x,T(x)) +adr(j)(y,T(y))≤
≤γ+adj(x,T(x)) +adj(y,T(y))
⇒dj(x,y)≤2aε+γ.
So∀γ >0,∃ϕ(γ) =γ+ 2aε> 0such that
dj(x,y)−dj(T(x),T(y))≤γ⇒dj(x,y)≤γ.
Now, by Lemma 2.7.13 it results that
δ(Fε(T))≤ϕ(2ε),∀ε>0.
The last inequality leads us to the conclusion:
δ(Fε(T))≤2ε(1 +a),∀ε>0
The proof is complete.
Theorem 2.7.25 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned
by the familyD. Letrbe a function from JtoJ, withJa family of indices. Let
T:A→Aa Chatterjea-type almost local contraction and assume the monotonicity
property ( (2.4.10)) assured for the pseudometric.
Then:
δ(Fε(T))≤2ε(1 +a)
1−2a,∀ε>0.
Proof:Letε>0. Again, condition (i) from Lemma 2.7.13 is satisﬁed, as one can
see in the proof of Theorem 2.7.17.
It remain to show that (ii) holds for Chatterjea-type ALC-s.
Letγ >0andx,y∈Fε(T). Assume that
dj(x,y)−dj(T(x),T(y))≤γ.

52 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
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+adj(x,T(y)) +adj(y,T(x))≤γ+a[dj(x,y) +dj(y,T(x))] +a[dj(y,x) +dj(x,T(x))].
Butx,yare approximate ﬁxed point 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
So∀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 2.7.13 it results that
δ(Fε(T))≤ϕ(2ε),∀ε>0.
The last inequality leads us to the conclusion:
δ(Fε(T))≤2ε(1 +a)
1−2a,∀ε>0.
This completes the proof.
Theorem 2.7.26 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned
by the familyD. Letrbe a function from JtoJ, withJa family of indices. Let
T:A→Aa Zamﬁrescu-type almost local contraction and assume the monotonicity
property ( (2.4.10)) assured for the pseudometric.
Then:
dj(T(x),T(y))≤2ρ·dj(x,T(x)) +ρ·dj(x,y)
and
dj(T(x),T(y))≤2ρ·dj(y,T(y)) +ρ·dj(x,y)
hold.
Proof:Letε>0. Again, condition (i) from Lemma 2.7.13 is satisﬁed, as one can
see in the proof of Theorem 2.7.17.
It remain to show that (ii) holds for Zamﬁrescu-type ALC-s.
Letγ >0andx,y∈Fε(T). Assume that
dj(x,y)−dj(T(x),T(y))≤γ.

7. APPROXIMATE FIXED POINTS 53
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.
So for allγ >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 2.7.13 it follows that
δ(Fε(T))≤ϕ(2ε),∀ε>0.
The last inequality leads us to the conclusion:
δ(Fε(T))≤2ε(1 +ρ)
1−ρ,∀ε>0.
This completes the proof.
Remark 2.7.27 .The case of almost contractions is very similar to the previous
contraction, but we need an additional condition, which is θ+L<1in order to get the
result, as we can see in the next theorem.
Theorem 2.7.28 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, letA⊂Xa subset of X. We letτbe the weak topology on Xdeﬁned
by the familyD. Letrbe a function from JtoJ, withJa family of indices. Let
T:A→Aan almost local contraction with θ+L < 1.Assume the monotonicity
property ( (2.4.10)) assured for the pseudometric.
Then:
δ(Fε(T))≤2 +L
1−θ−L·ε,∀ε>0.
Proof:Letε>0. Again, condition (i) from Lemma 2.7.13 holds.
It remain to show that (ii) holds for ALC-s.
Letγ >0andx,y∈Fε(T). Assume that
dj(x,y)−dj(T(x),T(y))≤γ.
By using the triangle inequality and the monotonicity 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

54 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
After that step, we jump to the conclusion :
⇒(1−θ−L)dj(x,y)≤Lε+γ⇒dj(x,y)≤Lε+γ
1−θ−L.
So, for allγ >0,∃ϕ(γ) =Lε+γ
1−θ−Lsuch that
dj(x,y)−dj(T(x),T(y))≤γ⇒dj(x,y)≤γ.
Again, by Lemma 2.7.13 it follows that
δ(Fε(T))≤ϕ(2ε),∀ε>0.
The last inequality leads us to the conclusion:
δ(Fε(T))≤2 +L
1−θ−L·ε,∀ε>0.
Now, the proof is complete.
8. Almost local contractions in b-pseudometric spaces
The generalisation of metric and respectively metric space was given by many au-
tors, the most importants: [ 84], [31], [4], [36], and recent works, amongst which we
mention [28] , [29]. The concept of b-metric space was introduced by Czerwik in
[36]. Since then several publications were studied the ﬁxed point of single valued and
multivalued operators in b-metric spaces (see [ 21], [91], [36]).
Definition 2.8.1.(see[84]) 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.
Example 2.8.2.(see[4]) 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 2.8.3.In the sequel, our goal is to expand the b-metrics to the more general
case ofb-pseudometrics , as it follows:

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 55
Definition 2.8.4.LetXbe a nonempty set.
A mapping d:X×X→R+is called b-pseudometric if the following hold:
(1)d(x,y) =d(y,x),∀x,y∈X;
(2)d(x,z)≤b·[d(x,y) +d(y,z)],∀x,y,z∈X,
whereb≥1is a given real number
(3)x=yimpliesd(x,y) = 0
A nonempty set Xendowed with a b-pseudometric db:X×X→R+is called b-
pseudometric space.
In the beginning of this section, we shall present a set of concepts which help us
to establish a theory of some ﬁxed point theorems, related to a various type of almost
local contractions in b-pseudometric spaces.
The following concepts are given in metric spaces, they appear originally in [ 85], but
we shall present them in the framework of a b-pseudometric space.
Definition 2.8.5.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 2.8.6.For the particular case b= 1, from Deﬁnition2.8.5 we get the
well-known deﬁnition of good (weakly) Picard operator in a metric space.
Definition 2.8.7.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.
Remark 2.8.8.For the particular case b= 1, from Deﬁnition2.8.7 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 [38] was studied also in [ 70], [86] and [60]. In ab- metric space was introduced in
[67] by M.Păcurar, we introduce it in a b-pseudometric space as follows:
Definition 2.8.9.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→∞,

56 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
then we say that the ﬁxed point problem for the operator fis well posed.
Remark 2.8.10 .For the particular case b= 1, from Deﬁnition2.8.9 we obtain the
well-known deﬁnition of a well posed ﬁxed point problem in a metric space.
Definition 2.8.11 .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 2.8.12 .For the particular case b= 1, from Deﬁnition2.8.11 we obtain the
usual deﬁnition of limit shadowing property for an operator on a metric, respectively
b-metric space.
Example 2.8.13 .(see[85]) 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.
8.1.ϕ-contractions in b-pseudometric spaces
The type of ϕ-contractions have been studied in pseudometric spaces (see sec-
tion 5 in chapter 4). In the sequel, we analyses them in the more general case of
b-pseudometric spaces, as it follows:
Theorem 2.8.14 .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,
sequentially τ- complete, and T- invariant subset A⊂X.
Letf:X→Xa quasiϕ-contraction, as in (2.5.26).
Thenfhas a unique ﬁxed point if and only if there exists x0∈Xsuch that the
Picard iteration{xn}n≥0given by (1.2.2)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 exists a
constantc>0and an element y∈Xsuch that
dj(xn,y)≤c,∀n∈N,∀j∈J.
Form,n∈N, by using the (2) property of the b-pseudometric, we get
dj(xn,xm)≤b[dj(xn,y) +dj(y,xm)]≤2b·c

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 57
Applying the condition for the quasi ϕ-contraction, we obtain
dj(xn,xn+p) =dj(f(xn−1),f(xn+p−1))≤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.
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 for the
topologyτtox∗, which implies
x∗= limn→∞xn⇒limn→∞dj(xn,x∗) = 0
After simple computations involving 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 unicity of the ﬁxed
point is based on reductio ad absurdum method.
The proof is complete.
At this point (see [ 21]), ﬁxd≥1a real number and let ϕ:R+→R+be a
comparison function for which there exists a series of positive terms∞/summationtext
n=0vn, which is
convergent and a number 0≤α<1such that
(2.8.1) bk+1ϕk+1(t)≤α·bk·ϕk(t) +vk,∀t∈R,
for eachk≥N, withNﬁxed.
By using (2.8.1), the series
(2.8.2)∞/summationdisplay
k=0bkϕk(t),
converges for each t∈R+and its sum, denoted by sb(t)is monotone increasing and
continuous at zero.
Theorem 2.8.15 .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,
sequentially τ- complete, and T- invariant subset A⊂X.
Letf:X→Xa quasiϕ-contraction, with ϕsatisfying condition (2.8.1).
Ifx0∈Xis chosen such that the sequence of successive approximations is bounded and
F(f) ={x∗}, then we have
(2.8.3) dj(xn,x∗)≤b·sb(dj(xn,xn+1)),n≥0,
wheresb(t)is the sum of the series (2.8.2).

58 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Proof:By means of contraction condition (2.5.26), 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
(2.8.4) dj(xn−1,xn)≤ϕ(dr(j)(xn−2,xn−1))
Having in view the monotonicity 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, using (2.8.4):
(2.8.5) dj(xn,xn+p)≤bp−1/summationdisplay
k=0bkϕk(dr(j)(xn,xn+1))
Then, by taking p→∞in (2.8.5), we get
(2.8.6) dj(xn,x∗)≤b·sb(dj(xn,xn+1)),n≥0,
The proof is complete.
Theorem 2.8.16 .Let(X,d)be ab-pseudometric space with b≥1,ϕ:R+→R+
ab-comparison function, consider a function r:J→J, also consider a nonempty, τ-
bounded, sequentially τ- complete, and T- invariant subset A⊂X.
Letf:X→Xa quasiϕ-contraction, with ϕsatisfying condition (2.8.1).
Then:
(1)fis a Picard operator;
(2) the following a priori and a posteriori estimates are available:
dj(xn,x∗)≤b·sb(ϕn(dj(x0,x1))),n≥0, (2.8.7)
dj(xn,x∗)≤b·sb(dj(xn,xn+1)),n≥0, (2.8.8)
wheresb(t)is the sum of the series (2.8.2).
(3) for any x∈X:
(2.8.9) dj(x,x∗)≤b·sb(dj(x,f(x))).
Proof:1) Fixx0∈X, then considering 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 obtain by induction with related to nthe following inequality:
(2.8.10) dj(xn,xn+1)≤ϕn(dr(j)(x0,x1)).

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 59
Having in view that dis ab-pseudometric, for n≥0,p≥1, we get:
(2.8.11)dj(xn,xn+p)≤bdj(xn,xn+1) +b2dj(xn+1,xn+2) +···+bpdj(xn+p−1,xn+p)
Accordingly (2.8.10), we obtain:
(2.8.12)dj(xn,xn+p)≤bϕn(dj(x0,x1))+b2ϕn+1(dj(x0,x1)+···+bpϕn+p−1(dj(x0,x1))
which yields
(2.8.13) dj(xn,xn+p)≤1
bn−1[bnϕn(dj(x0,x1)) +···+bn+p−1ϕn+p−1(dj(x0,x1))].
By using the notation
(2.8.14) Sn=n/summationdisplay
k=0bkϕk(dj(x0,x1),n≥1,
the inequality (2.8.13) becomes:
(2.8.15) dj(xn,xn+p)≤1
bn−1[Sn+p−1−Sn−1], n≥1,p≥1.
But the series
∞/summationdisplay
k=0bkϕk(dj(x0,x1))
is convergent, hence there is
(2.8.16) S= limn→∞Sn∈R+.
Having in view that b≥1, by (2.8.15) 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:
(2.8.17) dj(xn+1,f(x∗)) =dj(f(xn),f(x∗))≤ϕ(dj(xn,x∗)),n≥0,n≥0
Remind that ϕis continuous at zero, and by taking n→∞we get that:
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 2.5.4 and the monotonicity
((2.4.10))of the pseudometric, we have:
dj(x∗,y∗) =dj(f(x∗),f(y∗))≤ϕ(dr(j)(x∗,y∗))<dr(j)≤dj(x∗,y∗)

60 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
The above inequality is obviously a contradiction. So we proove the unicity of the ﬁxed
point, which means that fis a Picard operator.
2) By using (2.8.12), it results that
dj(xn,xn+p)≤b[ϕ0(ϕn(dj(x0,x1))) +bϕ(ϕn(dj(x0,x1))) +
+···+bp−1ϕp−1(ϕn(dj(x0,x1)))], (2.8.18)
wheren≥0,p≥1.At this point, we let p→∞in (2.8.18), and we get the ﬁrst
required estimate:
dj(xn,x∗)≤b·sb(ϕn(dj(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:
(2.8.19) dj(xn+k,xn+k+1)≤ϕk(dr(j)(xn,xn+1)),n≥1,k≥0.
Now, by replacing (2.8.19) in (2.8.11), we get
dj(xn,xn+p)≤b[dj(xn,xn+1) +bϕ(dj(xn,xn+1)) +
+···+bp−1ϕp−1(dj(xn,xn+1))], (2.8.20)
forn≥0,p≥1.Lettingp→∞in (2.8.20) we obtain the second required estimate:
dj(xn,x∗)≤b·sb(dj(xn,xn+1)),n≥0.
3) Lettingxn=x, the a posteriori estimate (2.8.16) can be written 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 section.
Theorem 2.8.17 .Letf:A→Abe as in Theorem 2.8.16.
Thenfis a good Picard operator.
Proof:Choosex0∈X.By using (2.8.12) from the proof of Theorem 2.8.16, we
get:
(2.8.21) 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))

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 61
Finally, according to Lemma 2.5.5 and the inequality/summationtext
n∈Nd(fn(x),fn+1(x))<∞, we
conclude that fis a good Picard operator.
This completes the proof.
Theorem 2.8.18 .Letf:A→Abe as in Theorem 2.8.16. Then the ﬁxed point
problem for fis well posed.
Proof:Let the sequence{zn}n∈N⊂Xsuch that
(2.8.22) dj(zn,f(zn))→0asn→∞.
By (2.8.9), if we choose x=zn,n∈N, we get:
(2.8.23) dj(zn,x∗)≤b·sb(dj(zn,f(zn))),n∈N.
Remind that sbis continuous at zero, according to Lemma 2.5.5. Letting n→∞in
(2.8.23), and combining it with (2.8.22), we obtain:
dj(zn,x∗)→0,n→∞,
which means that the ﬁxed point problem for fis well posed.
Theorem 2.8.19 .Letf:A→Abe as in Theorem 2.8.16. If ϕsatisﬁes:
(2.8.24) ϕ(a1t1+a2t2)≤a1ϕ(t1) +a2ϕ(t2)
for anya1,a2,t1,t2∈R+, thenfpossess the limit shadowing property.
Proof:Let the sequence{zn}n∈N⊂Xsuch that
(2.8.25) dj(zn+1,f(zn))→0asn→∞.
Forn≥0we obtain:
(2.8.26) 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, inequality (2.8.26) take the equivalent form:
(2.8.27) 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 (2.8.27), by using (2.8.24) 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∗)).

62 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
This can be written in the equivalent form:
dj(zn+1,x∗)≤bn/summationdisplay
k=0bkϕk(dj(zn−k+1,f(zn−k))) +
+bn+2ϕn+1(dj(z0,x∗)). (2.8.28)
Considering Lemma 2.5.6 with the substitution: an=dj(zn+1,f(zn)), it results that
(2.8.29)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 2.5.5. At this point, we let
n→∞in (2.8.28) and it follows that
(2.8.30) dj(zn+1,x∗)→0,n→∞.
By using Theorem 2.8.16, we conclude that for any x∈Athe Picard iteration
{fn(x)}n≥0converges to x∗.This means that the following inequalities holds:
(2.8.31) dj(zn+1,fn(x))≤dj(zn+1,x∗) +dj(x∗,fn(x)),n≥0
for somex∈X. Taking the limit n→∞in (2.8.31), we have that
dj(zn+1,fn(x))→0,asn→∞,
which actually means that fhas the limit shadowing property.
The next Theorem state and prove the data dependence of the ﬁxed point for quasi
ϕ-contractions on b-pseudometric spaces with ϕab-comparison function:
Theorem 2.8.20 .Letf:A→Abe as in Theorem 2.8.16 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
(2.8.32) dj(f(x),g(x))≤η,for anyx∈X
Then
dj(x∗,y∗)≤b·sb(η),
whereFf={x∗}andsbwas introduced in Lemma 2.5.5.
Proof:From 2.8.9 in Theorem 2.8.16, 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 2.5.5, it follows from that: sbis increasing, which by (2.8.32)
yields
dj(x∗,y∗)≤b·sb(η).

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 63
This completes the proof.
8.2. Fixed points of almost local contractions in b-pseudometric spaces
In this chapter, almost local contractions were introduced in a pseudometric space
setting. The framework in this subsection shall be also the b-pseudometric spaces,
providing some interesting new results.
Theorem 2.8.21 .LetXbe a set and letD= (dj)j∈Jbe a family of b- pseudometrics
deﬁned onX, with constant b≥1. We choose a subset A⊂Xand we letτbe the weak
topology on Xdeﬁned by the family D. Letf:A→Aan almost local contraction with
constantsθ∈[0,1
b)andL≥0.
Then:
(i)fis a weakly Picard operator;
(ii) for any x∈Xthe following error estimates hold:
dj(fn(x),f∞(x))≤bθn
1−bθdj(x,f(x)),n≥1; (2.8.33)
dj(fn(x),f∞(x))≤bθ
1−bθdj(fn−1(x),fn(x)),n≥1; (2.8.34)
Proof:In the beginning, we shall 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∈Xand
{xn}n≥0be the Picard iteration which starts from x0. Applying the deﬁnition of the
Picard iteration and also the deﬁnition of an ALC, 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 regard to n:
(2.8.35) 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 (2.8.35) 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, (2.8.36)

64 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
forn≥0,p≥1.
Remind that θ∈[0,1
b), withb≥1, it is obvious that 0≤bθ < 1, which yields from
(2.8.36) the conclusion that {xn}n≥0is adj−Cauchy sequence in the b-pseudometric
space. This means that it is convergent with his limit denoted by
(2.8.37) x∗= limn→∞xn
Applying the deﬁnition of the b-pseudometric, we get:
dj(x∗,f(x∗))≤b[dj(x∗,f(xn+1)) +dj(f(xn),f(x∗))].
After using the deﬁnition of an almost local contraction and the monotonicity property
(2.4.10) we obtain from the last inequality:
dj(x∗,f(x∗))≤bdj(x∗,f(xn)) +bθdr(j)(xn,x∗) +bLdr(j)(x∗,f(xn))≤
≤bdr(j)(x∗,f(xn)) +bθdr(j)(xn,x∗) +bLdr(j)(x∗,f(xn)) =
=b(1 +L)dr(j)(x∗,xn+1) +bθd(xn,x∗).
Having in view (2.8.37) and letting n→∞it results that
dj(x∗,f(x∗)) = 0,
which means that x∗is a ﬁxed point of f. Sofis a weakly Picard operator.
In the sequel, remind that 0≤bθ< 1and letting p→∞in (2.8.36) we get the a priori
error estimate (2.8.33). By using the induction in (2.8.35), we obtain:
(2.8.38) 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).
But0≤bθ< 1, lettingp→∞inthelastinequality, itresultstheaposterioriestimate,
namely the second relation in (2.8.35).
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 2.8.22 .LetXbe a set and letD= (dj)j∈Jbe a family of b- pseudo-
metrics deﬁned on X, with constant b≥1. We choose again a subset A⊂Xand
we letτbe the weak topology on Xdeﬁned by the family D. Letf:A→Aa strict
almost local contraction with constants θ∈[0,1
b)andL≥0, andθu∈[0,1
b)andLu≥0.
Then:

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 65
(i)fis a Picard operator;
(ii) the following error estimates hold:
(2.8.39) dj(xn,x∗)≤bθn
1−bθdj(x0,x1),n≥1,
(2.8.40) dj(xn,x∗)≤bθ
1−bθdj(xn−1,xn),n≥1;
(iii) the rate of convergence of the Picard iteration is given by
(2.8.41) 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 2.8.21. In order to prove the uniqueness of the
ﬁxedpoint,supposethat fhastwodiﬀerentﬁxedpoints x∗,y∗∈X. Then,applyingthe
monotonicity condition (2.1.13) for the b- pseudometric and the uniqueness condition
(2.1.14), 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∗},sofis a Picard operator.
(ii)Theaprioriandaposterioriestimates((2.8.39), (2.8.40))followbyTheorem2.8.21.
(iii) From (2.1.14) 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.

In the sequel, we shall make a comparison to other type of contractive conditions
inb-pseudometric spaces.
Lemma 2.8.23 .In ab-pseudometric space any Kannan-type ALC with constant
k∈[0,1
2b)is an almost local contraction with θ=kb
1−kbandL=2kb
1−kb.
Proof:Supposef:A→Ais a Kannan-type ALC with constant k∈[0,1
2b). This
means:
(2.8.42) dj(f(x),f(y))≤k[dr(j)(x,f(x)) +dr(j)(y,f(y))],∀x,y∈A.

66 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Applying the deﬁnition of the b-pseudometric and the monotonicity property, we can
write:
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 almost local contraction, i.e.
it satisﬁes (2.1.3) with θ=kb
1−kb∈[0,1)andL=2kb
1−kb≥0.
This completes the proof of the Lemma.
Lemma 2.8.24 .In ab-pseudometric space any Kannan-type ALC with constant
k∈[0,1
b(b+1))is a strict almost local contraction with θ=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,
which means the conclusion of Lemma 2.8.23 is valid. Furthermore, from (2.8.42), it
results:
dj(f(x),f(y))≤kdr(j)(x,f(x)) +kbdr(j)(f(y),f(x)) +kbdr(j)(f(x),y)≤
≤kdr(j)(x,f(x)) +kbdr(j)(f(y),f(x)) +kb2dr(j)(f(x),x) +kb2dr(j)(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 2.8.25 .In ab-pseudometric space any Ćirić-Reich-Rus -type ALC with
constantsα,β∈R+such thatα+ 2bβ < 1is an almost local contraction with θ=α+bβ
1−bβ
andL=2bβ
1−bβ.
Proof:Denotef:A→Aa Ćirić-Reich-Rus -type ALC, where Ais a subset of
theb-pseudometric space X. Letα,β∈R+,α+ 2bβ < 1, such that
(2.8.43) dj(f(x),f(y))≤αdj(x,y) +β[dj(x,f(x)) +dj(y,f(y))],
for anyx,y∈A.
At this point, 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)),

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 67
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 2.8.26 .Forb= 1in Lemma 2.8.25, the values from Theorem 2.4.21 are
obtained.
Lemma 2.8.27 .In ab-pseudometric space any Ćirić-Reich-Rus -type ALC with
constantsα,β∈R+such thatα+b(b+ 1)β < 1is a strict almost local contraction
withθ=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 2.8.25 holds. Furthermore, according to (2.8.43), we obtain:
dj(f(x),f(y))≤αdj(x,y) +βdj(x,f(x)) +βdj(f(y),y)≤
≤αdj(x,y) +βdj(x,f(x)) +bβdj(f(y),f(x)) +b2βdj(f(x),x) +b2βdj(x,y),
so
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 (2.1.14) with θu=α+b2β
1−bβ∈[0,1)andLu=β(b2+1)
1−bβ,
which means that fis a strict almost local contraction.
Lemma 2.8.28 .In ab-pseudometric space, any Chatterjea-type ALC with constant
c∈[0,1
b(b+1))is an almost local contraction with θ=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
(2.8.44) dj(f(x),f(y))≤c[dj(x,f(y)) +dj(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)),
so
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 2.8.29 .Forb= 1in Lemma 2.8.28, we obtain the conclusion of Theorem
2.4.23.

68 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Lemma 2.8.30 .In ab-pseudometric space, any Chatterjea-type ALC with constant
c∈[0,1
b(b+1))is a strict almost local contraction with θ=cb2
1−cbandL=c(b2+1)
1−cband,
respectively, θu=cb
1−cbandLu=2cb
1−cb.
Proof:The conclusions of Lemma 2.8.28 holds and from (2.8.44) we have that:
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 (2.1.14) with θu=cb
1−cbandLu=2cb
1−cb≥0.
Having in view that c∈[0,1
b(b+1))andb≥1, clearly, it results that c <1
2b, which
meansθu∈[0,1).
In the sequel, it is natural to study the Zamﬁrescu-type ALC-s in this new space
setting.
Lemma 2.8.31 .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 almost local contraction with
θ= max/braceleftBig
α,bk
1−bk,b2c
1−bc/bracerightBig
andL= max/braceleftBig
2bk
1−bk,(b2+1)c
1−bc/bracerightBig
.
Lemma2.8.32 .In ab-pseudometric space, any Zamﬁrescu-type ALC with constants
α∈[0,1),k∈[0,1
b(b+1))andc∈[0,1
2b)satisﬁes the inequality (2.1.14)withθu=
max/braceleftBig
α,b2k
1−bk,bc
1−bc/bracerightBig
andLu= max/braceleftBig(b2+1)k
1−bk,2bc
1−bc/bracerightBig
.
Lemma 2.8.33 .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 almost local contrac-
tion withθ= 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.
Lemma 2.8.34 .In ab-pseudometric space, any quasi- ALC with constant
h∈[0,1
b(b+1))is an almost local contraction 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
(2.8.45)
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. We shall use the notation
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))}.

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 69
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)),
it results
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),
so
dj(f(x),f(y))≤b2h
1−hbdr(j)(x,y) +b2h
1−hbdr(j)(y,f(x)).
From that, we may state that for the previously chosen xandy, the ALC (2.1.3)
condition is veriﬁed with θ= max{b2h
1−bh,0}=b2h
1−bh∈[0,1)andL= 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)

70 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
and
L= max/braceleftBig
0,bh
1−bh,b2h
1−bh/bracerightBig
=b2h
1−bh≥0.
The proof is complete.
In the sequel, we shall return to the class of strict almost local contractions.
Theorem 2.8.35 .LetXbe a set and letD= (dj)j∈Jbe a family of b- pseudo-
metrics deﬁned on X, with constant b≥1. We choose again a subset A⊂Xand
we letτbe the weak topology on Xdeﬁned by the family D. Letf:A→Aa strict
almost local contraction with constants θ∈[0,1
b)andL≥0, andθu∈[0,1
b)andLu≥0.
Thenfis a good Picard operator.
Proof:In the proof of Theorem , we have from (2.4.23) the following relation:
(2.8.46) 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 2.8.5.
Theorem 2.8.36 .Letf:A→Aan operator as appear in Theorem 2.8.22. Then
fis a special Picard operator.
Proof:We prove in Theorem 2.8.22 the following error estimate:
dj(xn,x∗)≤bθn
1−bθdj(x0,x1),n≥1.
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 2.8.7. This
completes the proof.

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 71
Theorem 2.8.37 .Letf:A→Aan operator as appear in Theorem 2.8.22.
Then the ﬁxed point problem is well posed.
Proof:Choose the point yn∈A,n∈Nsuch that
(2.8.47) dj(yn,f(yn))→0asn→∞
According to Theorem 2.8.22, the operator fpossess 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 (2.8.47), we have:
yn→x∗,asn→∞,
which means, according to Deﬁnition 2.8.9, the ﬁxed point problem is well posed.
Theorem 2.8.38 .Letf:A→Aan operator as appear in Theorem 2.8.22.
Thenfhas the limit shadowing property.
Proof:Choose the point yn∈A,n∈Nsuch that
(2.8.48) dj(yn+1,f(yn))→0asn→∞
According to Theorem 2.8.22, 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,
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 almost local contractions, 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, it results that
dj(yn,x∗)≤bdj(yn,f(yn−1)) +bθudr(j)(yn−1,x∗).

72 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Having in view that bθu∈[0,1), we get from (2.8.48) and from Lemma 2.2.2 by taking
an=dj(yn,x∗),q=bθu,bn=bdj(yn+1,f(yn)),we get the following conclusion:
(2.8.49) dj(yn,x∗)→0,asn→∞.
Note thatx∗∈Ais the limit of the Picard iteration:
(2.8.50) x∗= limn→∞fn(x0),
for somex0∈A, thus we obtain that
dj(yn,fn(x0))≤bdj(yn,x∗) +bdj(x∗,fn(x0)).
By using (2.8.49) and (2.8.50), the last inequality leads us to:
dj(yn,fn(x0))→0,asn→∞,
which means, according to Deﬁnition 2.8.11, 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 2.8.39 .LetXbe a set and letD= (dj)j∈Jbe a family of b- pseudometrics
deﬁned onX, with constant b≥1. We choose again a subset A⊂Xand we let τbe
the weak topology on Xdeﬁned by the family D. Letf:A→Aan operator as appear
in Theorem 2.8.22 and g:A→Aa mapping satisfying:
(i)ghas at least one ﬁxed point, denoted by x∗
g∈Fg;
(ii) there exists η>0such that
(2.8.51) 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.
Proof:Theorem 2.8.22 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)).
By using the uniqueness condition for almost local contractions, the (2.8.51) inequality,
and the monotonicity property (2.1.13), 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.

8. ALMOST LOCAL CONTRACTIONS IN b-PSEUDOMETRIC SPACES 73
In Theorem 2.8.22 we have the condition 1−bθu>0, it follows that:
dj(x∗
f,x∗
g)≤bη
1−bθu.
Now the proof is complete.
Theorem 2.8.40 .LetXbe a set and letD= (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 appear in
Theorem 2.8.22 and fu:A→Aa mapping satisfying:
(i) the operator fuhas at least a ﬁxed point x∗
n∈Ffn,n∈N;
(ii)fnconverges uniformly to fasn→∞.
Thenxn→x∗asn→∞, whereFf={x∗}.
Proof:By using Theorem 2.8.22, the operator fhas a unique ﬁxed point, denoted
byx∗∈A.Having in view that fnconverges uniformly to fasn→∞,there exist
ηn∈R+,n∈Nsuch thatηn→0asn→∞and
dj(fn(x),f(x))≤ηn,
for anyx∈A.According to Theorem 2.8.39, it results that
dj(x∗
n,x∗)≤ηn·b
1−bθu,n∈N,
for each mapings fandfn.
From that, by taking ηn→0,n→∞, so the proof is complete.
Theorem 2.8.41 .LetXbe a set and letD= (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 appear in
Theorem 2.8.22 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→∞.
Thenx∗
n→x∗asn→∞, whereFfn={x∗
n},n∈NandFf={x∗}.
Proof:According to Theorem 2.8.22, the operators fn,n∈Nandfhave a unique
ﬁxed point, thus we shall denote the sets of their ﬁxed points by Ff={x∗}andFfn=
{x∗
n}, respectively. Applying the deﬁnition of the b-pseudometric and the monotonicity

74 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
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 section we present a Maia type result related to strict almost
local contractions on b-pseudometric spaces.
Theorem 2.8.42 .LetXbe a set, we let τbe the weak topology on Xdeﬁned by
the familyD. Consider dandρ b- pseudometrics deﬁned on X, with constant b≥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)Ff={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 2.8.21. 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
Almost local contractions in dynamic programming
applications
1.k-almost local contractions
Thetheoryofdynamicprogramming(DP)withuncountablestatespacebeginswith
the essential work of Blackwell (see [ 27]). His paper was extended in a large number
of directions, with important applications in economics, 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 the preliminary concepts are taken from
Rincon-Zapatero [ 76] and Matkowski-Nowak [ 61].
Xwill denote a topological space such that
(3.1.1) X=∞/uniondisplay
j=1Kj,
with{Kj}an increasing sequence of compact subsets of X. Suppose that
X=∞/uniondisplay
j=1Int(Kj).
DenoteC(X)thesetofallreal-valued, continuousfunctionsover X. Weshallintroduce
a countable family of pseudometrics, as follows:
(3.1.2) dj(f,g) := max
x∈Kj|f(x)−g(x)|, j∈N.
Let the operator d, deﬁned by
(3.1.3) 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 3.1.1.[76]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.
75

76 3. ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING APPLICATIONS
In the sequel, 0denotes the function ϕsuch thatϕ(x) = 0for allx∈X.
[76] The setA⊆C(X)is said to be bounded if there exists 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 [ 61]:
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, we shall use the notation Fm(X)for the set of all operators ϕ∈F(X)
such that
(3.1.4)∞/summationdisplay
j=1/bardblϕ/bardblj
mjcj<∞.
Note that the operator /bardbl·/bardbl:Fm(X)→Rdeﬁned by
(3.1.5) /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 3.1.2.[61]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 3.1.3.It is obvious from the deﬁnition that a 0−LCis also a 1−LC.
Next, we develop the concept of k−LC-s to the more general case of k-almost local
contractions (k-ALC).
Definition 3.1.4.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 exists
the constants θj,Lj∈[0,1)such that
(3.1.6)/bardblTϕ−Tψ/bardblj≤θj/bardblϕ−ψ/bardblj+k+Lj/bardblψ−Tϕ/bardblj+k∀,ϕ,ψ∈A,j∈N.

1.k-ALMOST LOCAL CONTRACTIONS 77
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 3.1.5 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 3.1.5.a) Let the subset A⊆Fm(X)andT:A→Aan operator.
IfTis a0−ALC, then for each ϕ,ψ∈Athere exists the constants θ,L∈[0,1), such
that
(3.1.7)/bardblTϕ−Tψ/bardbl≤θ/bardblϕ−ψ/bardbl+L/bardblψ−Tϕ/bardbl ∀,ϕ,ψ∈A,j∈N.
IfT0∈Fm(X), thenTmapsFm(X)into itself. Furthermore, Tpossess a ﬁxed point
ϕ∗∈Fm(X).
b) If we add the conditions
/bardblT0/bardblj≤(1−θ−L)·m,∀j∈N
θ+L < 1, (3.1.8)
thenTmapsFmb(X)into itself, i.e. T:Fmb(X)→Fmb(X).
Proof:a) Chooseϕ,ψ∈A.For the proof of (3.1.7), apply the deﬁnition (3.1.5)
and (3.1.4), 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,

78 3. ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING APPLICATIONS
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 yealds:
/bardblTϕ/bardbl<∞.
b) For the desired conclusion, suppose that for every j∈Nand∀ϕ∈Fmb(X), condi-
tions (3.1.8) 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 (3.1.8), 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 3.1.6.Let the subset A⊆Fm(X)andT:A→Aa1−ALC. Denote
(3.1.9) 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.
This way, we proove that Tis an almost local contraction with the ﬁxed point ϕ∗∈
A.

2. USEFUL DEVELOPMENT IN DYNAMIC PROGRAMMING 79
2. Useful development in dynamic programming
Before proceeding further, we present some preliminary concepts, borrowed from
[61] and [58].
Denote with (X,Σ)a measurable space, Ya separable metric space. The mapping
B:X→Sis considered (weakly) measurable if B−1(D) :={x∈X:B(x)∩D/negationslash=
Φ}∈Σ, for an open set D⊂Y.In this context, Srepresent the family of nonempty
subsets ofY.
In the sequel, let Xbe a metric space and let Ea set-valued mapping. Then Eis
called continuous if E−1(D)is closed for every closed set D⊂Yand open for each set
D⊂Y.Note that every measurable mapping Bwith nonempty compact values B(x)
for allx∈Xpossess a measurable selector, see [ 58],(1965).
LetBa measurable compact set-valued mapping and denote
(3.2.1) C:={(x,a) :x∈X,a∈E(x)}.
According to Himmelberg’s work ([ 47]),Cis a measurable subset of X×Yequipped
with the product σ-algebra.
Proposition 3.2.1.(see[61]) 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, fol-
lowing[61]: Thediscrete-time Markov decision process includetheobjects: X,Y,{A(x)}x∈X,u,q,β
such that:
M1:Xrepresent 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)⊂Ysigniﬁes the set
of all actions available in state x∈X.The mapping Cis deﬁned according to
(3.2.1).
M3:u:C→Rrepresent the (product) measurable instantaneous return function.
M4:qsigniﬁes a transition probability from CtoX, termed the law of motion among
states.
M5: The coeﬃcient β∈(0,1)represent the discount factor.

80 3. ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING APPLICATIONS
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 shall 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:
(3.2.2) 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
xsigniﬁes
theexpectation operator regarding to the unique conditional probability measure Pπ
x
deﬁnedbyπandthetransitionprobability qstatedbytheIonescuTulceaTheorem(see
[64]). Inthesequel, weassumethattheexpectedreturns(3.2.2)arewell-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
(3.2.3) 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).
Inthecaseof Xnotbeeingatopologicalspace, weuseanotherregularitycondition:
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
(3.2.4) uj(x) := max
a∈A(x)|u(x,a)|, x∈Kj;rj:= sup
x∈Kjuj(x).
We shall use the sequences {mj}and{Kj}mentioned before, in the previous section.
Suppose that (3.1.1) 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
(3.2.5) ζ:=θc·sup/braceleftBigmj+1
mj,j∈N/bracerightBig
, ξ :=Lc·sup/braceleftBigmj+1
mj,j∈N/bracerightBig
, ζ,ξ < 1.

2. USEFUL DEVELOPMENT IN DYNAMIC PROGRAMMING 81
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 3.2.2.Actually, (3.2.5)leads us 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. His form appear in many papers, we shall use the
Bellman equation stated by Matkowski and Nowak ([ 61]), as follows:
(3.2.6) Lv(x,a) :=u(x,a) +β/integraldisplay
Xv(y)q(dy|x,a),(x,a)∈C.
for the integrable function v:X→R.
The Bellman equation can be put in the form
(3.2.7) v∗(x) = max
a∈A(x)Lv∗(x,a), x∈X.
In this section, 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.
Theorem 3.2.3.a) We accept condition P1. If condition R1 (or R2 with rj<∞
for everyj∈N) is valid, then there exist 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 exist 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) [ 25] assure 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) + (3.2.8)
+L/integraldisplay
X(v−Tv)(y)·q(dy|x,a)/parenrightBig
withv∈Cmb(X),x∈X.Note that (3.2.8) take the form of an integral equation.
DenoteH(y) := (v−Tv)(y), then (3.2.8) becomes:
Tv(x) := max
a∈A(x)/parenleftBig
(1−θ)u(x,a) +θ/integraldisplay
Xv(y)·q(dy|x,a) + (3.2.9)
+L/integraldisplay
XH(y)·q(dy|x,a)/parenrightBig

82 3. ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING APPLICATIONS
Again, by the maximum theorem of Berge ([ 25]),Tvis continuous on each set Kj.
According to (3.2.3), 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
shallbestudiedinchapter5. Inthelightoftheseconsiderations, wecaneasilyconclude
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 2.4.21) and by using Proposition 3.1.5, 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 3.1.5. Obviously, the new ﬁxed point is v∗∈Mm(X).
Theorem 3.2.4.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 (3.2.8) shall
be deﬁned for any v∈Cm(X),hence/bardblv/bardbl<∞.Next, we introduce the notation
ρ:=/bardblg/bardbl, u∗(x) := max
a∈A(x)|u(x,a)|
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:
(3.2.10) 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 Tit seems 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 3.1.6, and by using the previous result that any Ćirić-Reich-
Rus type ALC is an ALC (theorem 2.4.21), leads us to the conclusion that the mapping

2. USEFUL DEVELOPMENT IN DYNAMIC PROGRAMMING 83
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.
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 [ 59]) and also Rincon-Zapatero, Rodriguez-Palmero (2003) (see [ 76]), it is
our aim to develop their results and to obtain new ones.
First, we need some concepts and notations used in DP.
Definition 3.2.5.(see[76]) 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,…) (3.2.11)
x0∈Xﬁxed,
whereXis a subset of R,U:Graph (Γ)→Rrepresent 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 3.2.6.(The Bellman operator, see [76]) LetVbe the set of functions
fromXto[−∞,∞). The Bellman operator BonVis deﬁned by
(3.2.12)Bf(x) = max
y∈Γ(x){U(x,y) +βf(y)}, x∈X,f∈V.
The Bellman equation is: Bf=f, which is very similar to the ﬁxed point condition.
The Bellman operator possess various properties in the literature, such as: monotonic-
ity and discounting (see [ 50], [92]).
In the sequel, we set as the real metric dRthe Euclidian distance: dR(x,y) =|x−y|.

84 3. ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING APPLICATIONS
By using this metric (norm) we obtain a family of semidistances (seminorms) {dj}on
C(X), deﬁned as
(3.2.13) dj(f,g) = max
x∈Kj|f(x)−g(x)|=/bardblf−g/bardblKj (dj(f,0) =/bardblf/bardblKj)
We shall use the same metric d, as in the previous subsection:
d(f,g) =∞/summationdisplay
j=12−j/bardblf−g/bardblKj
1 +/bardblf−g/bardblKj.
ThefollowingTheoremwasprovedbyRincon-ZapateroandRodriguez-Palmeroin[ 76]:
Theorem 3.2.7.([76]) 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→∞.
Starting from theorem 3.2.7, the main result of this subsection is presented in the
next theorem:
Theorem 3.2.8.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, we shall use:
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).
(b) The conclusion from (b) is immediate from the above inequalities: /bardblBf−g/bardbl≤
/bardblBg−g/bardbl+β·/bardblf−g/bardbl, for allf,g∈C(X).

2. USEFUL DEVELOPMENT IN DYNAMIC PROGRAMMING 85
(c) We know from the proof of theorem 3.2.7 ([ 76]) 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 3.2.9.By introducing a new operator T:=B−I(Ibeeing the identity
function), we can reformulate (c) from the previous theorem:
(3.2.14) /bardblTf/bardbl≤/bardblTg/bardbl+ (β+ 1)/bardblf−g/bardbl,∀f,g∈C(X).
Next, we extend the classical contractions to the more general case of almost local
contractions. To this end, we start from the papers of Bertsekas (2012)(see [ 23], [24]).
In the beginning, we need some concepts borrowed from Bertsekas ([ 23]):
Let us consider 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 functionµ:X→U, whereµ(x)∈U(x), for allx∈X, which will be a ”policy”.
Letusconsiderthesetofallpolicies M, andletR(X)bethesetofreal-valuedfunctions
J:X→R. LetH:X×U×R(X)→Rbe a given mapping. We also introduce the
mappingTdeﬁned by
(3.2.15) (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 mapping
Tµ:R(X)→R(X)as follows:
(3.2.16) (TµJ)(x) =H(x,µ(x),J),∀x∈X.
The purpose of this construction is to ﬁnd a function J∗∈R(X)such that
(3.2.17) 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 ([ 23]) introduced the weighted sup-norm, in the fol-
lowing manner:
Take a function v:X→R, with positive values:
v(x)>0,∀x∈X,

86 3. ALMOST LOCAL CONTRACTIONS IN DYNAMIC PROGRAMMING APPLICATIONS
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):
(3.2.18) /bardblJ/bardbl= sup
x∈X|J(x)|
v(x).
In the sequel, we give a few examples, according to ([ 23]).
Example 3.2.10 .([23]) (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 represent 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 his 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
,
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 shall
apply policy iteration algorythms for a more favorably computation of the value x.
Example 3.2.11 .([23])(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 Grepresent cost per stage, and mxy(u)are nonnegative numbers
withn/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 3.2.12 .([23])(Minimax Problems)
Consider a minimax version of Example 3.2.10, where an antagonistic player chooses
vfrom a setV(x,u). We shall use the mapping
H(x,u,J ) = sup
v∈V(x,u)/bracketleftBig
g(x,u,v ) +αJ(f(x,u,v ))/bracketrightBig
.

2. USEFUL DEVELOPMENT IN DYNAMIC PROGRAMMING 87
Again, the equation J=TJis Bellmann’s equation for an inﬁnite horizon minimax
DP problem. A generalisation 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.
In his research paper, Bertsekas work with contractions:
Definition 3.2.13 .([23]) For allJ∈B(X)andµ∈M, the functions TµJand
TJbelong toB(X). Moreover, for some α∈(0,1), we have
(3.2.19) /bardblTµJ−TµJ/prime/bardbl≤α/bardblJ−J/prime/bardbl,∀J,J/prime∈B(X),µ∈M.
Condition (3.2.19) could be written in the equivalent form:
|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 (3.2.19) leads us to
(3.2.20) /bardblTJ−TJ/prime/bardbl≤α/bardblJ−J/prime/bardbl,∀J,J/prime∈B(X).
Note that the contraction assumption (3.2.19) is valid for the mapping Hin examples
3.2.10, 3.2.11, 3.2.12. with v(x)≡1, which means that vrepresent the unit function.
In this section, we extend the model of Bertsekas, the strong correlation with dy-
namic programming justify the introduction of a new type of contraction, namely the
weighted sup-norm almost local contractions (ALC).
Definition 3.2.14 .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 (3.2.18). 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 exists the constants α∈(0,1)andL≥0such that:
(3.2.21)/bardblTµJ−TµJ/prime/bardblv≤α/bardblJ−J/prime/bardblu+L/bardblJ/prime−TµJ/bardblu,∀J,J/prime∈B(X),µ∈M.
(3.2.22)/bardblTJ−TJ/prime/bardblv≤α/bardblJ−J/prime/bardblu+L/bardblJ/prime−TJ/bardblu,∀J,J/prime∈B(X),
where
(3.2.23) /bardblJ/bardblv= sup
x∈X|J(x)|
v(x),/bardblJ/bardblu= sup
x∈X|J(x)|
u(x).
Remark 3.2.15 .It can be easily seen that the functions TandTµrepresent in fact
two almost local contractions, as they are deﬁned in 2.1.7.

CHAPTER 4
MULTIVALUED SELF ALMOST LOCAL
CONTRACTIONS
1. Preliminaries
The notion of multivalued contraction was ﬁrst introduced by Nadler in [ 63] fol-
lowing are borrowed from Nadler [ 63]
Definition 4.1.1.Let(X,d)be a metric space, we shall 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}}, the Pompeiu-Hausdorﬀ
metric onCB(X)induced by d.
We know thatCB(X)form a metric space with the Pompeiu-Hausdorﬀ distance
functionH. It is also known,that if (X,d)is a complete metric space then ( CB(X),H)
is a complete metric space, too. (Rus [ 82])
LetP(X)be the family of all nonempty subsets of Xand letT:X→P (X)be a
multi-valued mapping. An element x∈Xwithx∈T(x)is called a ﬁxed point of T.
We shall denote Fix(T)the set of all ﬁxed points of T, i.e.,
Fix(T) ={x∈X:x∈T(x)}
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 F(f), (resp.F(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 [ 82], it is useful for the next theorem.
88

1. PRELIMINARIES 89
Lemma 4.1.2.(see[63]) Let (X,d)be a metric space, let A,B⊂Xandq>1.
Then, for every a∈A, there exists b∈Bsuch that
(4.1.1) d(a,b)≤qH(A,B)
Definition 4.1.3.(see[14]) Let (X,d)be a metric space and T:X→P (X)be a
multi-valued operator. Tis said to be a multi-valued weak contraction or a multi-valued
(θ,L)-weak contraction if there exist two constants θ∈(0,1),L≥0such that
(4.1.2) H(Tx,Ty )≤θ·d(x,y) +L·D(y,Tx ),∀x,y∈X
Remark 4.1.4.Because of the simmetry of the distance dandH, the almost con-
traction condition (4.1.2)includes the following dual one:
(4.1.3) 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 (4.1.2)
and(4.1.3).
Theorem 4.1.5.(Berinde V., Berinde M. [14]) Let (X,d)be a metric space and
T:X→P (X)be a (θ,L)-weak 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:
(4.1.4) d(xn,u)≤hn
1−hd(x0,x1), n = 0,1,2…
(4.1.5) d(xn,u)≤h
1−hd(xn−1,xn), n = 1,2…
for a certain constant h<1.
We shall use the assumptions from the deﬁnition of almost local contractions (see
2.1.7 and 2.1.6) and we make the following notations:
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 4.1.6.From the deﬁnition of Dj, we have the following result:
ifDj(a,B) = 0, this implies that a∈B
The main result of this section is the extension to the case of multivalued ALC-s,
starting from the multivalued almost contractions. To this end, we shall improve a lot
of corresponding results in literature [ 37], [40], [51], [52], [57], [62], [45], [66], [71]
and many others.

90 4. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS
Definition 4.1.7.Letrbe a function from JtoJ, letS⊂Xbe aτ-bounded
sequencially τ-complete and T- invariant subset of X. An operator T:S→P (S)is
called a multivalued almost local contraction (ALC) related to ( D,r) if, for every j∈J,
there exists the constants θ∈(0,1)andL≥0such that
(4.1.6) Hj(Tx,Ty )≤θ·dr(j)(x,y) +L·Dr(j)(y,Tx ),∀x,y∈S,∀j∈J.
The following deﬁnition represent an extension of generalized almost contractions,
concept introduced by Berinde -Berinde in [ 14] to the more general case of generalized
ALC-s.
Definition 4.1.8.Letrbe a function from JtoJ, letS⊂Xbe aτ-bounded
sequencially 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
everyj∈J, there exists a function α: [0,∞)→[0,1)satisfying lim supr→t+α(r)<1
for 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. (4.1.7)
Theorem 4.1.9.Letrbe a function from JtoJ, letS⊂Xbe aτ-bounded
sequencially τ-complete and T- 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 4.1.5 by replacing the term
θdr(j)(x,y)d(x,y)by 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 (4.1.7) holds.
Therefore we do not present the proof of this Theorem.
In order to prove a ﬁxed point theorem related to these type of multivalued ALC-s,
we state and prove the following Lemma.
Lemma4.1.10 .Letrbe a function from JtoJ,S⊂Xbe aτ-bounded sequencially
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.

1. PRELIMINARIES 91
At this point, by considering ε=/parenleftBig
1
b−1/parenrightBig
dj(x,Tx )>0, we obtain the conclusion of
this Lemma.
Theorem 4.1.11 .Letrbe a function from JtoJ, letS⊂Xbe aτ-bounded
sequencially τ-complete and T- 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;
(ii) there exist L≥0,b∈(0,1)andϕ: (0,∞)→[0,b)such that for all t∈(0,∞),
(4.1.8) 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. (4.1.9)
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 4.1.10 that there exists y∈Txsuch
thaty∈Ix
b,i.e.,
(4.1.10) 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 (4.1.10) and
assumption (ii), there exists x2∈Tx1,x2/negationslash=x1,satisfying:
(4.1.11) bdr(j)(x1,x2)≤dj(x1,Tx 1).
The last inequality can be write, using (4.1.8) in the equivalent form:
(4.1.12) 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, (4.1.11) and (4.1.12) 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
(4.1.13) bdr(j)(x2,x3)≤dj(x2,Tx 2)

92 4. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS
such that
(4.1.14) dj(x3,Tx 3)≤ϕ/parenleftBig
dr(j)(x2,x3))dr(j)(x2,x3),ϕ(dr(j)(x2,x3)/parenrightBig
<b.
Again, by (4.1.13) and (4.1.14) 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.
By (4.1.13) and (4.1.14) 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 exists xn+1∈Txn,xn/negationslash=xn+ 1,
such that
(4.1.15) bdr(j)(xn,xn+1)≤dj(xn,Txn)
and, similar to (4.1.14), satisfying
(4.1.16)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 (4.1.15) and (4.1.16), 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 (4.1.17)
and also we get:
(4.1.18) 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 yields they are both convergent.
By applying (4.1.7), it results that there exists s∈[0,b)such that
(4.1.19) ϕ(dr(j)(xn,xn+1))<b0,∀n>n 0.
Ifweintroducenowthenotation a=b−b0,wegetfrom(4.1.17)thefollowinginequality:
(4.1.20) dj(xn,Txn)−dj(xn+1,Txn+1)≥a/parenleftBig
dr(j)(xn,xn+1)/parenrightBig
,∀n>n 0.

1. PRELIMINARIES 93
At this point, we use (4.1.15),(4.1.16) and (4.1.19) 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,
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 (4.1.20), 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
, (4.1.21)
whichmeansthatthesequence {xn}n∈Nisdj-Cauchyforeach j∈J. Thesubset S⊂X
is assumed to be sequentially τ-complete, there exists x∗inSsuch that{Tnx}n∈Nis
τ- convergent to x∗. Besides, the sequence {Tnx}n∈Nconverges for the topology τto
x∗, 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 4.1.12 .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
(4.1.22) dj(a,b)≤qHj(A,B)

94 4. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS
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 exist b∈Bsuch thatdj(a,b) = 0.
The inequality (4.1.22) is valid, i.e., 0≤0.
IfHj(A,B)>0, then let us denote
(4.1.23) 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
(4.1.24) dj(a,b)≤qDj(a,B) +ε≤Hj(A,B) +ε
Combining (4.1.23) and (4.1.24), we get the conclusion of the lemma.
Theorem 4.1.13 .With the assumptions of Deﬁnition 4.1.7, let T:S→P (S)be
a multivalued ALC. Then we have:
(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:
(4.1.25) dj(xn,u)≤hn
1−hdj(x0,x1), n = 0,1,2…
(4.1.26) 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:
(4.1.27) 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 4.1.12, there exists x2∈Tx1such that
(4.1.28) dj(x1,x2)≤qHj(Tx0,Tx 1)
By (4.1.6) 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

1. PRELIMINARIES 95
and we obtain dj(x1,x2)<h·dj(x0,x1).
IfHj(Tx1,Tx 2) = 0thenDj(x2,Tx 2) = 0, that means x2∈Tx2using Remark 4.1.6.
LetHj(Tx1,Tx 2)/negationslash= 0. Again, using Lemma 4.1.12, there exists x3∈Tx2such that
(4.1.29) dj(x2,x3)≤qh·dj(x1,x2),∀j∈J.
This way, we obtain an orbit {xn}∞
n=0ofTat the point x0satisfying
(4.1.30) dj(xn,xn+1)≤h·dj(xn−1,xn),∀j∈J, n = 1,2,…
By (4.1.30), we inductively obtain
(4.1.31) dj(xn,xn+1)≤hndj(x0,x1),∀j∈J,
and, respectively,
(4.1.32) dj(xn+k,xn+k+1)≤hk+1dj(xn−1,xn), k∈N
Using the inequality (4.1.31), we obtain
(4.1.33) dj(xn,xn+p)≤hn(1−hp)
1−hdj(x0,x1), n,p∈N
Recall 0< h < 1, conditions (4.1.32),(4.1.33) 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:
(4.1.34) 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 (4.1.6) yields
(4.1.35) 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 (4.1.34),
Dr(j)(u,Txn)→0, asn→∞. We get
Dr(j)(u,Tu ) = 0
SinceTuis closed, this implies u∈Tu.
We letp→∞in (4.1.33) to obtain (4.1.25). Using (4.1.32), we get
(4.1.36) d(xn,xn+p)≤h(1−hp)
1−hd(xn−1,xn), p∈N,n≥1
and letting p→∞in (4.1.36), we obtain (4.1.26).
The proof is complete.

96 4. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS
Definition 4.1.14 .(see[8]) 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 [ 75]) instead of Hhas been used the
functionalD.
The next Theorem shows that any multivalued ALC is continuous at the ﬁxed point.
Theorem 4.1.15 .With the assumptions of Deﬁnition 4.1.7, let T:S→P (S)be
a multivalued ALC, i.e., a mapping for which there exists the constants
θ∈(0,1)andL≥0such that, for every j∈J, the next inequality is valid:
(4.1.37) 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 ﬁrst part of the conclusion follows by Theorem 4.1.13.
Let{yn}∞
n=0be any sequence in the subset Sconverging to the ﬁxed point p. Then by
takingy:=ynandx:=pin the multivalued ALC condition (4.1.37), we get
(4.1.38) 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)
By replacing Dr(j)(yn,Tp)from (4.1.38) with dj(yn,p), we get:
(4.1.39) dj(Tyn,Tp)≤(δ+L)·dj(yn,p),n= 0,1,2,…
Now, by letting n→∞in (4.1.39) we get Tyn→Tpasn→∞, that means: Tis
continuous at p.
Example 4.1.16 .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
we shall use for the application r:J→J,J=Nthe identity map,i.e., r(j) =j,∀j∈J.
We can write: Hj(Tx,Ty )≤
frac13·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 in
every ﬁxed point of T, thus Theorem 4.1.15 is valid.
The following Theorem proves that any generalized strict multivalued ALC is con-
tinuous at the ﬁxed point.

1. PRELIMINARIES 97
Theorem 4.1.17 .With the assumptions of Deﬁnition 4.1.7, let T:S→CB (S)be
a multivalued (θ,L)-strict ALC, i.e., a mapping for which there exists θ∈[0,1)and
someL≥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 4.1.13, with minor diﬀerences, but shall be presented below:
We consider q >1, letx0∈Xandx1∈Tx0. IfHj(Tx0,Tx 1) = 0, that means from
the deﬁnition of DjandHj:
(4.1.40) 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 4.1.12, there exists x2∈Tx1such that
(4.1.41) dj(x1,x2)≤qHj(Tx0,Tx 1)
By (4.1.17) we have
dj(x1,x2)≤q[θ·dr(j)(x0,x1) +L·Dr(j)(x1,Tx 0)] =qθ·dr(j)(x0,x1),
since 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) = 0thenTx1=Tx2, that means x2∈Tx2using Remark 4.1.6.
LetHj(Tx1,Tx 2)/negationslash= 0. Again, using Lemma 4.1.12, there exists x3∈Tx2such that
(4.1.42) dj(x2,x3)≤qh·dj(x1,x2),∀j∈J.
This way, we obtain an orbit {xn}∞
n=0ofTat the point x0satisfying
(4.1.43) dj(xn,xn+1)≤h·dj(xn−1,xn),∀j∈J, n = 1,2,…
By (4.1.43), we inductively obtain
(4.1.44) dj(xn,xn+1)≤hndj(x0,x1),∀j∈J,
and, respectively,
(4.1.45) dj(xn+k,xn+k+1)≤hk+1dj(xn−1,xn), k∈N

98 4. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS
Using the inequality (4.1.44), we obtain
(4.1.46) dj(xn,xn+p)≤hn(1−hp)
1−hdj(x0,x1), n,p∈N
Recall 0< h < 1, conditions (4.1.45),(4.1.46) 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:
(4.1.47) 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 (4.1.17) yields
(4.1.48) 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 (4.1.47),
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
(4.1.17), 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:
(4.1.49) dj(Tyn,Tp)≤θdr(j)(yn,p),n= 0,1,2,···,
If we letn→∞in (4.1.49), we obtain Tyn→Tp, which means that Tis continuous
atp.
Now, the proof is complete.
Theorem 4.1.17 can be extended to the more general case of generalized almost
local contractions, as it follows:
Theorem 4.1.18 .With the assumptions of Deﬁnition 4.1.7, let T:S→CB (S)
be a generalized multivalued (θ,L)- almost local contraction, i.e., a mapping for which
there exists θ∈[0,1)and someL≥0such that∀x,y∈Sand∀j∈J:
(4.1.50) Hj(Tx,Ty )≤θdr(j)(x,y) +Ldr(j)(y,Tx ).
ThenFix(T)/negationslash=φand for any p∈Fix(T),Tis continuous at p.

2. APPROXIMATE FIXED POINTS OF MULTIVALUED ALMOST LOCAL CONTRACTIONS 99
Proof:The existence of the ﬁxed point follows from Theorem 4.1.13.
Let{yn}∞
n=0be any sequence in the subset Sconverging to p. By taking y:=ynand
x:=pin the generalized ALC ((4.1.50)) condition, we get:
dj(Tp,Tyn)≤θdr(j)(yn,p) +Ldr(j)(yn,Tp),n= 0,1,2,···.
Note that: Tp=p,which implies:
(4.1.51) dj(Tyn,Tp)≤(θ+L)·dr(j)(yn,p),n= 0,1,2,···,
If we letn→∞in (4.1.51), we obtain Tyn→Tpasn→∞, which means that Tis
continuous at p.
The following concept was published by Rus ([ 87]), respectively Rus et al. ([ 88]).
Definition 4.1.19 .Let(X,d)be a metric space and T:X→P (X)be a multi-
valued operator. Tis said to be a multivalued weakly Picard (breaﬂy 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 4.1.20 .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 point x0.
In the sequel, a few example of MWP operators shall be presented borrowed from [88].
Example 4.1.21 .(see[63]) 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 4.1.22 .(see Reich [71],[72],[73]) 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. Approximate ﬁxed points of multivalued almost local contractions
In this section our main goal is to extend the approximate ﬁxed point property to
the class of multivalued ALC-s.

100 4. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS
Definition 4.2.1.(see[6]) A multivalued mapping T:X→P (X)is said to have
the approximate ﬁxed point property provided
(4.2.1) inf
x∈Xd(x,Tx ) = 0
or, equivalently, for any ε>0, there exists z∈Xsuch that
(4.2.2) d(z,Tz )≤ε
or, equivalently, for any ε>0, there exists xε∈Xsuch that
(4.2.3) T(xε)∩B(xε,ε)/negationslash=φ,
whereB(x,r)denotes a closed ball of radius rcentered at x.
In the beginning, we shall prove that every generalized multivalued ALC has the
approximate ﬁxed point property.
Theorem 4.2.2.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
(4.2.4) 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 (4.2.4) and the deﬁnition 4.1.8, 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 ).
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.

2. APPROXIMATE FIXED POINTS OF MULTIVALUED ALMOST LOCAL CONTRACTIONS 101
Remark 4.2.3.Theorem 4.2.2 is valid also for multivalued (θ,L)-almost local con-
tractions, as shown in the next Theorem. The proof is very similar to that of Theorem
4.2.2, 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.
Theorem 4.2.4.Every multivalued (θ,L)-almost local contraction has the approx-
imate ﬁxed point property,where θ∈[0,1)andL≥0.

CHAPTER 5
NON-SELF SINGLE VALUED ALMOST LOCAL
CONTRACTIONS
LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics deﬁned on X. 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
(5.0.1) d(x,Tx ) =d(x,y) +d(y,Tx ),y∈∂K
In general, the set Yof pointsysatisfying condition (5.0.1) above may contain more
than one element.
We shall need the following concept, borrowed from [ 15]
Definition 5.0.1.([15]) LetXbe a set and let Ka nonempty closed subset of X
,T:K→Xa non-self mapping. Let x∈KwithTx /∈Kand lety∈∂Kbe the
corresponding elements given by (5.0.1). If, for any such elements x, we have
(5.0.2) d(y,Ty )≤d(x,Tx ),
for at least one of the corresponding y∈Y, then we say that Thas property (M).
Next, we give an example of non-self mapping Tthat have property (M). Next,
we exit from the usual metric space to our pseudometric space, by giving the new
deﬁnition of property (M).
Definition 5.0.2.LetXbe a set and let Ka nonempty closed subset of X,
T:K→Xa non-self mapping. Denote 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
(5.0.3) dj(x,Tx ) =dj(x,y) +dj(y,Tx ),yj∈∂K
If, for the aformentioned elements xj, we have
(5.0.4) dj(y,Ty )≤dj(x,Tx ),
102

5. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS 103
for at least one of the corresponding yj∈Y, then we say that Thas property (M).
Example 5.0.3.ConsiderX= [1,2]∪{4}with the usual norm, let
K={1,2,4}⊂Xand take the nonself 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 his corresponding set
isY={2}, having in view that (5.0.1)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, (5.0.2)holds, therefore Thave property (M).
The next Theorem state and prove the existence of the ﬁxed point for nonself single
valued ALC.
Theorem 5.0.4.LetXbe a set 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)and
L≥0such that
(5.0.5) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈K
Assume that conditions from Theorem 2.1.10 are veriﬁed, also assume again that the
monotonicity property for the pseudometric is valid.
IfThas property (M) and satisﬁes Rothe’s boundary condition
(5.0.6) 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 2.1.10 for X=K. It is natural to consider the case
T(K)/negationslash⊂K. Letx0∈∂K. Using (5.0.6) we know that Tx0∈K. We construct a
sequence (xn)n∈Nin the following way:
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}constructed 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

104 5. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
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 (5.0.6), 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.
CaseI.xn,xn+1∈P. Havinginviewthedeﬁnitionoftheset P, wehavexn=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
(5.0.7) 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 (5.0.7).
Case III.xn∈Q,xn+1∈P.
In this case, we have xn−1∈P. The mapping Thas property (M), this means: there
exists an indice j∈Jsuch that
(5.0.8) 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 (5.0.8), we obtain
(5.0.9) dj(xn,xn+1)≤θ·dr(j)(xn−2,xn−1)

5. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS 105
At this point, after analysing all three cases, and using (5.0.7), (5.0.9), and the mono-
tonicity 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)} (5.0.10)
for alln≥2. Now, by induction for n≥2from (5.0.10) 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
(5.0.11) 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 rule 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
(5.0.12) dj(x∗,Tx∗)≤dj(xn+1,x∗) +θ·dr(j)(xn,x∗) +L·dr(j)(x∗,Txn)
for alln≥0. Lettingn→∞in (5.0.12), 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 5.0.5.A mapping Tsatisfying (2.1.3), i.e.Tis a non-self ALC, the
mappingTmay be discontinuous (see Example 5.0.6), 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∗)

106 5. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
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.
The next example show a non-self mapping with property (M), but discontinuous
and which is not an almost local contraction.
Example 5.0.6.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 (5.0.2), becauseThas property (M), Tis discontinuous in9
10, the
unique ﬁxed point of Tis0, andTis continuous in 0.Thas property (M). Indeed,
ifx=9
10∈K,Tx=−1
10/negationslash∈K, then using the condition (5.0.2)we have/vextendsingle/vextendsingle/vextendsingley
y+1/vextendsingle/vextendsingle/vextendsingle≤1.
This is valid for both y∈{0,1}, soy∈∂K. However, Tdoes not 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
The following Theorem assure the uniqueness of the ﬁxed point for non-self ALC.
Theorem 5.0.7.LetXbe a set 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)and
L1≥0such that
(5.0.13) dj(Tx,Ty )≤θ·dr(j)(x,y) +L1·dr(j)(y,Tx ),∀x,y∈K
Assume that an additional condition is veriﬁed:
(U) for every ﬁxed j∈Jthere exists:
(5.0.14) limn→∞(θ+L)ndiamrn(j)(z,A) = 0,∀x,y∈X
IfThas property (M) and satisﬁes Rothe’s boundary condition
(5.0.15) T(∂K)⊂K,
thenThas a unique ﬁxed point in K
Remark 5.0.8.The proof is quite similar to the case of single valued self almost
local contractions (see Theorem 2.1.16)

5. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS 107
Example 5.0.9.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}
andT2 =1
2. The only element x∈KwithTx/negationslash∈Kisx= 2and the corresponding set
isY={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.
Very recently, Rus,I.A. and Serban,M.A published their work (see [ 89]), where they
studied the nonself operators.
Definition 5.0.10 .Let(X,d)be a metric space, Y∈Pcl(X)andf:Y→Xbe a
continuous nonself operator. The maximal displacement functional corresponding to f
represent the functional Ef:P(Y)→R+∪{+∞}deﬁned by
(5.0.16) 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 5.0.11 .(see[89]) An operator f:Y→Xisα-graphic contraction if
0≤α<1andx∈Y,f(x)∈Yimply
(5.0.17) d(f2(x),f(x))≤αd(x,f(x)).
Example 5.0.12 .Iff:Y→Xisα-contraction then fisα-graphic contraction.
Example 5.0.13 .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.
Starting from the work of Rus-Șerban, our main aim is to extend these notions to
the more general case of nonself single valued almost local contractions.
Definition 5.0.14 .LetXbe a set and letD= (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 X. An operator T:K→Xisα-graphic local
contraction if 0≤α<1andx∈K,f(x)∈Kimply
(5.0.18) dj(T2x,Tx )≤αdr(j)(x,Tx ),∀j∈J.
In the sequel, we shall study several types of non-self ALC-s, in order to establish
if they areα-graphic local contractions.

108 5. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
Theorem 5.0.15 .Use the assumptions of Deﬁnition 5.0.14, and assume the mono-
tonicity property (2.1.13)valid for the pseudometric. Let K∈Pcl(X)andT:K→X
a nonself Ćirić-Reich-Rus operator, then Tis a nonself 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
(5.0.19) 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 monotonicity property (2.1.13) 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 5.0.16 .Use the assumptions of Deﬁnition 5.0.14, and assume the mono-
tonicity property (2.1.13)valid for the pseudometric. Let K∈Pcl(X)and
T:K→Xa nonself almost local contraction, then Tis a nonself 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 nonself α-graphic local contraction with α=θ.
This completes the proof.
Remark 5.0.17 .A nonself generalized ALC is also a nonself α-graphic local con-
traction with α=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 )},

5. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS 109
if we repeat the substitution y:=Tx, we obtain exactly the same inequality as in the
case of nonself ALC:
dj(Tx,T2x)≤θ·dr(j)(x,Tx ),∀x,y∈K,∀j∈J,
which shows that Tis a nonself α-graphic local contraction with α=θ.
Theorem 5.0.18 .A nonself generalized Berinde-type ALC is also a nonself α-
graphic local contraction with α=θ. Under the assumptions of Deﬁnition 5.0.14, 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 nonself α-graphic local contraction with α=θ.
This completes the proof.
Theorem 5.0.19 .Use the assumptions of Deﬁnition 5.0.14, and assume the mono-
tonicity property (2.1.13)valid for the pseudometric. Let K∈Pcl(X)and
T:K→Xa nonself Chatterjea-type ALC, then Tis a nonself α-graphic local con-
traction with α=c
1−c.
Proof:Remind that a nonself Chatterjea-type ALC with regard to ( D,r) or(δ,L)-
Chatterjea contraction is a mapping T:K→Xfor which there exist a constant
0≤c<1
2such that
(5.0.20) dj(Tx,Ty )≤c·[dr(j)(x,Ty ) +dr(j)(y,Tx )],∀x,y∈K.
Letx∈Ksuch thatTx∈K. We use (5.0.20), and after applying the substitution
y:=Tx, by using the triangle inequality, also the monotonicity 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 nonself α-graphic local contraction with α=c
1−c.

CHAPTER 6
NON-SELF MULTIVALUED ALMOST LOCAL
CONTRACTIONS
1. NON-SELF MULTIVALUED ALMOST CONTRACTIONS
In [14], M. Berinde and V. Berinde introduce the non-self multivalued almost con-
tractions.
Definition 6.1.1.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
(6.1.1) H(Tx,Ty )≤δ·d(x,y) +L·D(y,Tx ),∀x,y∈K
Definition 6.1.2.A metric space (X,d)is convex if for each x,y∈Xwithx/negationslash=y
there exists x∈X,x/negationslash=z/negationslash=ysuch that
(6.1.2) d(x,y) =d(x,z) +d(z,y)
Remark 6.1.3.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 [2])
Proposition 6.1.4.(Assad and Kirk [2]) LetKbe a closed subset of a closed and
convex metric space X. Ifx∈Kandy /∈K, then there exists a point z∈∂Ksuch
that
(6.1.3) d(x,y) =d(x,z) +d(z,y)
Lemma 6.1.5.(see[2]) 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
(6.1.4) d(x,y)≤H(A,B) +α.
2. FIXED POINT THEOREMS FOR NON-SELF MULTIVALUED ALC
The aim of this section is to prove a ﬁxed point theorem for multivalued non-self
almost local contractions.
110

2. FIXED POINT THEOREMS FOR NON-SELF MULTIVALUED ALC 111
Definition 6.2.1.LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX. We letτbe the weak topology on X, which is convex space, deﬁned by
the familyD, letKa nonempty closed subset of X. Letrbe a function from Jto
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 exists the constants θ∈(0,1)andL≥0
such that
(6.2.1) Hj(Tx,Ty )≤θ·dr(j)(x,y) +L·Dr(j)(y,Tx ),∀x,y∈K
Theorem 6.2.2.Under the assumptions of deﬁnition 6.2.1, suppose that
T:K→CB(X)is a multivalued almost local contraction, that is,
(6.2.2) 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 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.
Proof:We shall use two sequences {xn}and{yn}by constructing them as it fol-
lows:
Letx0∈Kandy1∈Tx0. Ify1∈K, letx1=y1. Ify1/∈K, then there exists x1∈∂K
such that
(6.2.3) dj(x0,x1) +dj(x1,y1) =dj(x0,y1)
We havex1∈Kand using Lemma 6.1.5 with α=θ, we choose y2∈Tx1such that
(6.2.4) dj(y1,y2)≤Hj(Tx0,Tx 1) +θ
Again, ify2∈K, putx2=y2. Ify2/∈K, then there exists x2∈∂Ksuch that
(6.2.5) dj(x1,x2) +dj(x2,y2) =dj(x1,y2)
Therefore,x2∈K, and by Lemma 6.1.5 and α=θ2, we can choose y3∈Tx2such that
(6.2.6) dj(y2,y3)≤Hj(Tx1,Tx 2) +θ2
Continuing in this manner, 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
(6.2.7) dj(xn−1,xn) +dj(xn,yn) =dj(xn−1,yn)

112 6. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS
Our next goal is to prove that {xn}is a Cauchy sequence. For the simplicity, we use
two notations:
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.
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

2. FIXED POINT THEOREMS FOR NON-SELF MULTIVALUED ALC 113
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
(6.2.8) dj(xn,xn+1)<h·dj(xn−2,xn−1) +h·θn−2+θn
This way, combining all three cases, we get
(6.2.9) dj(xn,xn+1)≤αdj(xn−1,xn) +αn
or the other possibility:
(6.2.10) αdj(xn−2,xn−1) +αn−1+αn
where
(6.2.11) α= max{θ,h}=h
We obtain by induction with related to n:
(6.2.12) dj(xn,xn+1)≤hn−1
2·ω+hn
2,
where
(6.2.13) 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.

114 6. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS
These relations show us that the sequence (xn)n∈Nisdj- Cauchy for each j∈J. The
subsetAis assumed to be sequentially τ-complete, there exists zinKsuch that
(6.2.14) z= limn→∞xn
As we constructed the sequence (xn)n∈N, there is a subsequence {xp}such that
(6.2.15) 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
(6.2.16) 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, which means z∈Tz.
The proof is complete.
Note that, by Theorem 6.2.2 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.
Corollary 6.2.3.Under the assumptions of deﬁnition 6.2.1, suppose that
T:K→CB(X)is a multivalued contraction, that is,
(6.2.17) 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 6.2.4.Corollary 6.2.3 represent a particular case of Theorem 6.2.2 by
takingL= 0, therefore we skip over the proof.
Another interesting result related to multivalued nonself mappings will be stated
and proved in the following Theorem.

2. FIXED POINT THEOREMS FOR NON-SELF MULTIVALUED ALC 115
Theorem 6.2.5.LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX, where (X,d)is convex pseudometric space. We let τbe the weak topology
onX, deﬁned by the family D, with the monotonicity property (2.4.10)assured. Let
Ka nonempty closed subset of X. Takerbe a function from JtoJ,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 (6.2.18)
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:Again, as in the proof of the previous Theorem, we shall use 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
(6.2.19) dj(x0,x1) +dj(x1,y1) =dj(x0,y1)
Picky2∈Tx1and, according to Lemma 6.1.5, we have:
(6.2.20) 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
(6.2.21) dj(x1,x2) +dj(x2,y2) =dj(x1,y2)
Continuing in this manner, we construct two sequences {xn}and{yn}such that
(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
(6.2.22) 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. For the simplicity, we use
two notations:
P={xi∈{xn}:xi=yi}
Q={xi∈{xn}:xi/negationslash=yi}

116 6. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS
We distinguish three cases as it follows:
Case 1. Ifxn,xn+1∈P, having in view (6.2.18), 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,
we denotek:=(α+β+γ)
1−γ.
Case 2. Ifxn∈P,xn+1∈Q, from (6.2.18), 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.
We observe that: if in the coeﬃcient of β, the maximum is dr(j)(xn−1,xn), and if the
monotonicity condition is veriﬁed, 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
monotonicity condition is veriﬁed, then we get:
dj(xn,yn+1)≤(α+γ)dj(xn−1,xn) + (1−β−γ)εn
1−β−γ.
After analysing all cases, we conclude:
(6.2.23) dj(xn,xn+1)≤k·dr(j)(xn−1,xn) +εn,∀j∈J.

2. FIXED POINT THEOREMS FOR NON-SELF MULTIVALUED ALC 117
Case 3. Ifxn∈Q,xn+1∈P, thenyn/negationslash=xn,yn+1=xn+1,yn−1=xn−1andyn∈Txn−1.
After using (6.2.18), 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 rule, 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).
From that, we can write:
dj(xn−1,xn+1) +dj(xn,yn)≤
Therfefore
dj(xn,xn+1)≤max{(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}≤
≤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
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),

118 6. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS
which means that (xn)n∈Nisdj- Cauchy for each j∈J.
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,which
means that Thas the ﬁxed point z.
The proof is complete.

CHAPTER 7
COMPLEX VALUED b-METRIC SPACES AND
RATIONAL LOCAL CONTRACTIONS
1. Complex valued b-metric
The notion of b-metric space was ﬁrst introduce by Bakhtin [ 4] which represent a
more general form of metric space. Furthermore, the complex valued metric space has
been introduced by Azam et al. [ 3].
Definition 7.1.1.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.[ 69]
Definition 7.1.2.LetXbe a nonempty set 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.
Example 7.1.3.(see[69]) IfX= [0,1], the mapping ddd:X×X→Cis deﬁned by
ddd(x,y) =|x−y|2+i|x−y|2for allx,y∈X. Then (X,ddd)is complex valued b-metric
space withs= 2.
119

120 7. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS
The following deﬁnition remind the convergent sequence and the Cauchy sequence.
Definition 7.1.4.(see[69]) 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→∞=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 7.1.5.(see[69]) 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 7.1.6.(see[69]) 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)|→0
asn→∞, wherem∈N.
2. Fixed point theorems for rational local contractions
Definition 7.2.1.LetXbe a set and letD= (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 coeﬃcient s≥1. The mapping T:X→Xis called complex valued rational local
contraction if the following inequality hold true:
(7.2.1) 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)
for allx,y∈Xsuch thatx/negationslash=y,dddr(j)(x,Ty ) +dddr(j)(y,Tx ) +dddr(j)(x,y)/negationslash= 0, whereα,β
represent nonnegative real numbers.
Next, we extend the study of ﬁxed point theorems in complex valued b-metric
spaces to the most general case of the ﬁxed points for rational local contractions.
Theorem 7.2.2.Let(X,ddd)be a complete complex valued b-metric space with the
coeﬃcients≥1and let two mappings S,T:X→Xsatisfying
(7.2.2) 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 nonnegative 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.

2. FIXED POINT THEOREMS FOR RATIONAL LOCAL CONTRACTIONS 121
Proof:For any arbitrary x0∈X, deﬁne a sequence {xn}inXsuch that
x2n+1=Sx2n (7.2.3)
x2n+2=Tx2n+1 (7.2.4)
forn= 0,1,2,3,…
Our ﬁrst goal is to prove that the sequence {xn}is Cauchy sequence.
Takex=x2nandy=x2n+1in (7.2.2); we have
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) + (7.2.5)
+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)
This way, we obtain
|dddj(x2n+1,x2n+2)|≤
≤α|dddr(j)(x2n,x2n+1)|+ (7.2.6)
+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
(7.2.7)|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)|
So, we get
(7.2.8) |dddj(x2n+1,x2n+2)|≤(α+β)·|dddr(j)(x2n,x2n+1)|
In a similar way, we obtain
(7.2.9) |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. By using the notation
δ=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)|≤···≤ (7.2.10)
≤δ2n+1·|dddr(j)(x0,x1)|

2. FIXED POINT THEOREMS FOR RATIONAL LOCAL CONTRACTIONS 123
From that, we conclude
(7.2.15) |dddj(xn,xm)|≤(sδ)n
1−sδ|dddr(j)(x0,x1)|→0
asm,n→∞.
This way we prove that {xn}is a Cauchy sequence in X. SinceXis complete, there
exists some u∈Xsuch thatxn→uasn→∞.
Next, we show that the mapping Shas the ﬁxed point u. To this end, assume the
contrary, which is
(7.2.16) |dddj(u,Su )|=|z|>0
According to the triangular inequality and (7.2.2), 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)
which implies that
|z|=|dddj(u,Su )|≤
≤s|dddr(j)(u,x 2n+2)|+sα·|dddr(j)(u,x 2n+1)|+ (7.2.17)
+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 (7.2.17) as n→∞, we conclude that
|z|=|dddj(u,Su )|≤0
and this is a contradiction with (7.2.16). So |z|= 0, henceSu=u.Proceeding in a
similar way, we obtain Tu=u.
Now, 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∗) + (7.2.18)
+β[dddr(j)(u,Su )dddr(j)(u∗,Tu∗)]
dddr(j)(u,Tu∗) +dddr(j)(u∗,Su) +dddr(j)(u,u∗)

124 7. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS
From that, we conclude
|dddj(u,u∗)| ≤
≤α|dddr(j)(u,u∗)|+ (7.2.19)
+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∗, so the uniqueness of the ﬁxed point is proved.
In the second case:
(7.2.20) dddr(j)(x,Ty ) +dddr(j)(y,Sx ) +dddr(j)(x,y) = 0,
putx=x2nandy=x2n+1in the last condition and we get:
(7.2.21) 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 K1
andl1such thatK1=Sl1=l1, whereK1=x2n+1andl1=x2n. Continuing in
this manner, 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 (7.2.20)),
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.
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 (7.2.20), 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 7.2.3.Let(X,ddd)be a complete complex valued b-metric space with the
coeﬃcients≥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 )] (7.2.22)
for allx,y∈Xsuch thatx/negationslash=y, whereα,β,γare nonnegative reals satisfting the
conditionα+ 2sβ+ 2γ <1ordddj(Sx,Ty ) = 0ifdddr(j)(x,Ty ) +dddr(j)(y,Sx ) = 0.
ThenSandThave a unique common ﬁxed point.

2. FIXED POINT THEOREMS FOR RATIONAL LOCAL CONTRACTIONS 125
Proof:For any arbitrary x0∈X, deﬁne a sequence {xn}inXsuch that
x2n+1=Sx2n (7.2.23)
x2n+2=Tx2n+1 (7.2.24)
forn= 0,1,2,3,…
Our ﬁrst goal is to prove that the sequence {xn}is Cauchy sequence.
Takex=x2nandy=x2n+1in (7.2.22); 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)|]
Due to triangular inequality, we obtain
(7.2.25)|dddj(x2n+1,x2n+2)|≤|dddj(x2n+1,x2n)|+|dddj(x2n,x2n+2)|
and hence
|dddj(x2n+1,x2n+2)| ≤α·|dddr(j)(x2n,x2n+1)|+
+β·|dddr(j)(x2n,x2n+2)|+
+γ·[|dddr(j)(x2n,x2n+1)|+|dddr(j)(x2n+1,x2n+2)|].
We also know that
(7.2.26)|dddj(x2n,x2n+2)|≤s/bracketleftBig
|dddj(x2n,x2n+1)|+|dddj(x2n+1,x2n+2)|/bracketrightBig

2. FIXED POINT THEOREMS FOR RATIONAL LOCAL CONTRACTIONS 127
At this point, by using (7.2.30), 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)|+
+sm−n−1δm−2|dddr(j)(x0,x1)|+
+sm−nδm−1|dddr(j)(x0,x1)|=
=m−n/summationdisplay
i=1siδi+n−1|dddj(x0,x1)|.
By continuing in this manner, we get
|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)|≤ (7.2.32)
≤∞/summationdisplay
t=n(sδ)t|dddr(j)(x0,x1)|=
=(sδ)n
1−sδ|dddr(j)(x0,x1)|.
which means
(7.2.33) |dddj(xn,xm)|≤(sδ)n
1−sδ|dddr(j)(x0,x1)|→0
asm,n→∞.
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
(7.2.34) dddj(u,Su ) =|z|>0.

128 7. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS
According to the triangular inequality and (7.2.2), 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)] = (7.2.35)
=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)|+ (7.2.36)
+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 (7.2.36) as n→∞, we conclude that
|z|=|dddj(u,Su )|≤0
and this is a contradiction with (7.2.34). 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∗) + (7.2.37)
+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∗).
which is a contradiction. So u=u∗, and this way we prove the uniqueness of the
common ﬁxed point in X.
For the second case dddj(Sx,Ty ) = 0ifdddr(j)(x,Ty )+dddr(j)(y,Sx ) = 0, the proof of unique
common ﬁxed point is quitq similar to that appearing in Theorem 7.2.2 This completes
the proof of the theorem.

2. FIXED POINT THEOREMS FOR RATIONAL LOCAL CONTRACTIONS 129
As a natural consequence of theorem 7.2.3, two corollary-s could be stated as it
follows:
Corollary 7.2.4.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 )] (7.2.38)
for allx,y∈Xsuch thatx/negationslash=y, whereα,β,γare nonnegative reals satisfting 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 7.2.3, by taking S=T.
Corollary 7.2.5.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)+ (7.2.39)
+γ[dddr(j)(x,Tnx) +dddr(j)(y,Tny)]
for allx,y∈Xsuch thatx/negationslash=y, whereα,β,γare nonnegative reals satisfting 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 7.2.3, 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)+ (7.2.40)
+γ[dddr(j)(Tu,TnTu) +dddr(j)(u,Tnu)] =
= (α+ 2β)dddj(Tu,u )
By taking modulus of (7.2.40) and using α+ 2β <1, we get
|dddj(Tu,u )|≤(α+ 2β)|dddj(Tu,u )|<|dddj(Tu,u )|
which is obviously a contradiction. So Tu=u.
HenceTu=Tnu=u. So, we prove the uniqueness of the ﬁxed point. This completes
the proof.

130 7. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS
Proof:Our ﬁrst goal is to prove the existence (E) of the ﬁxed point.
(E) Letgan element in A. From the deﬁnition of local contraction T, for every pair
of integersq>n> 0, we have
dj(Tqg,Tng)≤βjdr(j)(Tq−1g,Tn−1g)≤···
≤βjβr(j)···βrn−1(j)drn(j)(Tq−ng,g)
SinceAisT- invariant, Tq−ngbelongs toA, which yields
dj(Tqg,Tng)≤βjβr(j)···βrn−1(j)diamrn(j)(A).
Condition (1.4.1) show us that the sequence (Tng)n∈Nisdj-Cauchy for each j. The
subsetAis assumed to be sequentially τ-complete, there exists f∗inAsuch that
(Tng)n∈Nisτ- convergent to f∗. Besides, the sequence (Tng)n∈Nconverges for the
topologyτtof∗, which implies
∀j∈J, dj(Tf∗,f∗) = limn→∞(dj(Tf∗,Tn+1g).
Recall that the operator Tis a local contraction with regarding to ( D,r). From that,
we have
∀j∈J, dj(Tf∗,f∗)≤βjlimn→∞(dr(j)(f∗,Tng).
The convergence for the τ- topology implies convergence for the semidistance dr(j), we
obtaindj(Tf∗,f∗) = 0for everyj∈J. This way, we prove that Tf∗=f∗sinceτis
Hausdorﬀ. So, we prove the (E) condition.
All we have to do is to prove the stability (S) criterion.
(S) Fix an arbitrary h∈F. For eachj∈Jand everyn∈N∗, we have
dj(Tn+1h,Tn+1f∗)≤βjdr(j)(Tnh,Tnf∗)
≤βjβr(j)···βrn(j)drn+1(j)(h,f∗)
≤βjβr(j)···βrn(j)[drn+1(j)(h,A) +diamrn+1(j)(A)].
SinceTf∗=f∗, the sequence (Tnh)n∈Nisdj- convergent to f∗, from (1.4.1) and (1.4.2).
That way, we have proved that (Tng)n∈Nisτ- convergent to f∗.
This completes the proof.

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