Technical University of Cluj-Napoca [632189]

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Technical University of Cluj-Napoca [632189]

Byadmin ianuarie 1, 2024

Technical University of Cluj-Napoca
Noth University Center of Baia Mare
Faculty of Sciences
Ph.D. Thesis
TITLUL
Scientiﬁc Advisor: Ph.D. Student: [anonimizat]. Univ. Dr. Vasile Berinde Zakany Monika
Baia Mare
2017

Contents
Chapter 1. Introduction……………………………………………….. 1
1. Deﬁnire scop…………………………………………………….. 1
2. Strict contractions, Picard operators………………………………… 3
3. Almost contractions………………………………………………. 4
4. Local contractions………………………………………………… 7
Chapter 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS…. 10
1. Introduction …………………………………………………….. 10
2. Continuity of almost local contractions……………………………… 13
3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS …………… 16
4. ALMOST LOCAL ϕ-CONTRACTIONS ……………………………. 23
Chapter 3. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS…… 30
1. Preliminaries…………………………………………………….. 30
Chapter4. NON-SELFSINGLEVALUEDALMOSTLOCALCONTRACTIONS 35
Chapter 5. NON-SELF MULTIVALUED ALMOST LOCAL CONTRACTIONS 40
1. NON-SELF MULTIVALUED ALMOST CONTRACTIONS…………… 40
2. FIXED POINT THEOREMS FOR NON-SELF MULTIVALUED ALC…. 40
Chapter 6. COMPLEX VALUED b-METRIC SPACES AND RATIONAL
LOCAL CONTRACTIONS …………………………………. 45
1. Complex valued b-metric………………………………………….. 45
2. Fixed point theorems for rational local contractions ………………….. 46
Bibliography ………………………………………………………….. 56
1

CHAPTER 1
Introduction
1. Deﬁnire scop
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 com-
plete 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 the
most used tool in obtaining existence and uniqueness results for nonlinear functional
questions.
However, any Banach contraction is continuous on X, due to the contraction condition
itself:
(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 analysis
have challenged researchers to obtain its conclusions unther weaker assumptions
than (1.2), which do not force the continuity of the operator T.
The ﬁrst achievement in this respect has been reported by Kannan in 1968 [ 25],
who obtained a ﬁxed point theorem for discontinuous mappings. Chatterjea [ 19],
Bianchini [ 18], Reich, Rus [ 34],[34], Ciric [20], Zamﬁrescu [ 43] and many other
researchers 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 [ 11] introduced a large class of contractive mappings, called
weak contractions [in [ 7]] and later almost contractions (in [ 6] 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 Ciric
[21]. 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 [ 27], 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 [ 27] 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 [ 27] 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 is given by the next Theorem.
Theorem 1.2.1.(see[5]) Let (X,d)be a complete metric space and T:X→Xa
map satisfying
(1.2) 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
(1.3) xn+1=Txn, n = 0,1,2···
converges to p, for anyx0∈X.
(iii) The following a priori and a posteriori error estimates hold:
(1.4) d(xn,x∗)≤an
1−ad(x0,x1),n= 0,1,2…
(1.5) d(xn,x∗)≤a
1−ad(xn−1,xn),n= 1,2…
(iv) The rate of convergence of Picard iteration is given by
(1.6) d(xn,x∗)≤a·d(xn−1,x∗),n= 1,2…
A map satisfying (i) and (ii) is said to be a Picard operator, see [ 35] ,[37].
A mapping satisfying (1.2) is usually called strict contraction or a-contraction. Hence
Theorem 1.2.1 shows that any contraction is a Picard operator.
A mapping satisfying condition (1.2) 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.
In1968, R.Kannan[ 25]ﬁndapositiveanswertothisproblembyproovingthefollowing
ﬁxed point Theorem. He extends Theorem 1.2.1 to mappings that need not to be

4 1. INTRODUCTION
continuous, by replacing condition (1.2) with the following one: there exists 0≤b<1
2
such that
(1.7) d(Tx,Ty )≤b[d(x,Tx ) +d(y,Ty )],∀x,y∈X.
3. Almost contractions
Definition 1.3.2.(see[11]) 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
(1.8) d(Tx,Ty )≤δ·d(x,y) +L·d(y,Tx ),∀x,y∈X.
Remark 1.3.3.The term of almost contraction is the same as weak contraction,
and it was ﬁrst introduced by V. Berinde in [14].
Remark 1.3.4.([11]) Because of the simmetry of the distance, the almost contrac-
tion condition (1.8)includes the following dual one:
(1.9) d(Tx,Ty )≤δ·d(x,y) +L·d(x,Ty ),∀x,y∈X,
obtained from (1.8)by replacing d(Tx,Ty )byd(Ty,Tx )andd(x,y)byd(y,x)respec-
tively, and after that step, changing xwithy, and viceversa. Obviously, to prove the
almost contractiveness of T, it is necessary to check both (1.8)and(1.9).
Remark 1.3.5.A strict contraction satisﬁes (1.8), withδ=aandL= 0, therefore
it is an almost contraction with a unique ﬁxed point.
Other examples of almost contractions are given in [ 9], [10], [7], [14]. There are
manyotherexamplesofcontractiveconditionswhichimpliesthealmostcontractiveness
condition, see for example Taskovic [ 41], Rus [36].
We present an existence theorem 1.3.6, then an existence and uniqueness theorem
1.3.7, as they are presented in [ 14]. Their main merit is that they extend Banach’s
contraction principle and Zamﬁrescu’s ﬁxed point Theorem (1972 in [ 43]). They also
show us a method for approximating the ﬁxed point, for whitch both a priori and a
posteriori error estimates are available.
Theorem 1.3.6.Let(X,d)be a complete metric space and T:X→Xalmost
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
(1.10) d(xn,x∗)≤δn
1−δd(x0,x1), n = 0,1,2…

3. ALMOST CONTRACTIONS 5
(1.11) d(xn,x∗)≤δ
1−δd(xn−1,xn), n = 1,2…
hold, where δis the constant appearing in (1.8).
Theorem 1.3.7.([14]) 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
(1.12) d(Tx,Ty )≤θ·d(x,y) +L1·d(x,Tx ),∀x,y∈X
Then
(1)Thas a unique ﬁxed point,i.e., Fix(T) ={x∗};
(2) For any x0∈X, the Picard iteration {xn}∞
n=0converges to x∗;
(3) The a priori and a posteriori error estimates
d(xn,x∗)≤δn
1−δd(x0,x1), n = 0,1,2…
d(xn,x∗)≤δ
1−δd(xn−1,xn), n = 1,2…
hold.
(4) The rate of convergence of the Picard iteration is given by
(1.13) d(xn,x∗)≤θ·d(xn−1,x∗), n = 1,2…
Remark 1.3.8.(1) The study of almost contractions emerged as an area of intense
research activity, thanks to the further development of mappings satisfying the condi-
tion(1.8). Such examples of almost contraction was given by V. Berinde in [14], for
example it was proved that:
– any Zamﬁrescu mapping from Theorem Z in [43]is an almost contraction;
– any quasi-contraction with 0<h<1
2is an almost contraction;
– any Kannan mapping (in [25]) is the same kind of almost contraction
(2) The weak contractive condition has been used in a large class of applications, see
for example Taskovic [41], Rus[36]for some of them.
(3) Almost contractions need not have a unique ﬁxed point, however,the almost con-
tractions possess other important properties, amongst which we mention
a) In the class of almost contractions a method for constructing the ﬁxed points – i.e.
the Picard iteration – is always available;
b) It seems to be easy to checked in concrete applications the almost contractive condi-
tion(1.8)and(1.9).
c) Moreover, for this method of approximating the ﬁxed points, both a priori and a
posteriori error estimates are available. In order to decide when to stop the iterative
process, the Picard iteration is very useful.
(4) The ﬁxed point x∗attained by the Picard iteration depends on the initial guess

6 1. INTRODUCTION
x0∈X. Therefore, the class of weak contractions provides a large class of weakly
Picard operators.
Recall, see Rus [36],[38]that an operator T:X→Xis said to be a weakly Picard
operator if the sequence {Tnx0}∞
n=0converges for all x0∈Xand the limits are ﬁxed
points ofT.
(v) Condition (1.8)implies the so called 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 [23], Ivanov [24], Rus[35]and Taskovic [41].
The next Theorem show that an almost contraction is continuous at any ﬁxed point
of it, according to [ 6].
Theorem 1.3.9.Let(X,d)be a complete metric space and T:X→Xbe an
almost contraction. Then Tis continuous at p, for anyp∈Fix(T).
Definition 1.3.10 .(see[40]) 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
(1.14) d(Tx,Ty )≤r·d(x,y) +b(y)·d(y,Tx ),∀x,y∈X
Definition 1.3.11 .Let(X,d)be a metric space. Any mapping T:X→Xis
called Ćirić-Reich-Rus contraction if it is satisﬁed the condition:
(1.15) d(Tx,Ty )≤α·d(x,y) +β·[d(x,Tx ) +d(y,Ty )],∀x,y∈X,
whereα,β∈R +andα+ 2β <1
Corollary 1.3.12 .[31]. Let (X,d)be a metric space. Any Ćirić-Reich-Rus con-
traction,i.e., any mapping T:X→Xsatisfying the condition (1.15), represent an
almost contraction.
Theorem 1.3.13 .A mapping satisfying the contractive condition:
there exists 0≤h<1
2such that
(1.16)d(Tx,Ty )≤h·max{d(x,y),d(x,Tx ),d(y,Ty ),d(x,Ty ),d(y,Tx )},∀x,y∈X
is a weak contraction.
An operator satisfying (1.16)with 0<h< 1is called quasi-contraction.
Theorem 1.3.14 .Any mapping satisfying the condition: there exists 0≤b<1/2
such that
(1.17) d(Tx,Ty )≤b[d(x,Tx ) +d(y,Ty )],∀x,y∈X

4. LOCAL CONTRACTIONS 7
is a weak contraction.
A mapping satisfying (1.17)is called Kannan mapping.
A kind of dual of Kannan mapping is due to Chatterjea [ 19]. The new contractive
condition is similar to (1.17) there exists 0≤c<1
2such that
(1.18) d(Tx,Ty )≤c[d(x,Ty ) +d(y,Tx )],∀x,y∈X,
Theorem 1.3.15 .Any mapping Tsatisfying the Chatterjea contractive condition,
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.
Example 1.3.16 .LetT: [0,1]→[0,1]a mapping given by Tx=2
3forx∈[0,1),
andT1 = 0. ThenThas the following properties:
1)Tsatisﬁes (1.16)withh∈[2
3,1), i.e.Tis quasi-contraction;
2)Tsatisﬁes (1.8), withδ≥2
3andL≥0, i.e.Tis also weak contraction;
3)Thas a unique ﬁxed point, x∗=2
3.
4)Tis not continuous.
4. Local contractions
The concept of local contraction was ﬁrst introduced by Martins da Rocha and
Filipe Vailakis in [ 27] (2010), where they studied the existence and uniqueness of ﬁxed
points for the local contractions.
Definition 1.4.17 .(see[44]) The function d(x,y) :X×X→Ris said to be
semimetric on Xif:
(1)d(x,y)≥0,∀x,y∈X;
(2)d(x,y) =d(y,x),∀x,y∈X;
(3)d(x,y) = 0if and only if x=y.
Note that the triangle inequality is not necessarily satisﬁed in that case.
Definition 1.4.18 .(see[33]) LetXbe a topological vector space over the topologi-
cal ﬁeldK.Xis aKvector space equipped with a topology so that vector addition and
scalar multiplication are continuous. The weak topology on Xis the initial topology
with respect to the dual space X∗, which consist of all linear functions from Xinto the
base ﬁeldKthat are continuous with respect to the given topology. Precisely, the weak
topology is the coarsest topology (the topology with the fewest open sets) such that each
element of X∗remains a continuous function.

8 1. INTRODUCTION
Inordertoavoidtheconfusionbetweentheweaktopologyandtheoriginaltopology
onX, the original topology on Xis often called the strong topology on X.
Definition 1.4.19 .([33])
The sequence{xn}fromXis weakly convergent to xif
ϕ(xn)→ϕ(x)
asn→∞for allϕ∈X∗.
The sequence{xn}fromXis a weakly Cauchy sequence if ϕ(xn)converges to a scalar
limitL(f), for every ϕ∈X∗.
The space Xis called sequencially σ- complete if every weakly Cauchy sequence is
weakly convergent in Xwith respect to the weak topology σdeﬁned onX.
Definition 1.4.20 .(see[27]) LetFbe a set and letD= (dj)j∈Ja family of
semidistances 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.
Letrbe a function from JtoJ.
IfAis a nonempty subset of F, then for each hinA, we let
dj(h,A)≡inf{dj(h,g) :g∈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
∀f,g∈A, dj(Tf,Tg )≤βjdr(j)(f,g)
Definition 1.4.21 .([27]) The subset Aisσ- 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.
The existence and uniqueness of ﬁxed points for the local contractions was estab-
lished in [27] by Martins da Rocha and Filipe Vailakis.
Theorem 1.4.22 .Consider a function r:J→Jand letT:A→Abe a local
contraction with respect to (D,r). Consider a nonempty, σ- bounded, sequentially σ-
complete, and T- invariant subset A⊂F.
E:If the condition
(1.19) ∀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:Moreover, if h∈Asatisﬁes
(1.20) ∀j∈J, limn→∞βjβr(j)···βrn+1(j)drn+1(j)(h,A) = 0
then the sequence (Tnh)n∈Nis σ- convergent to f∗.

4. LOCAL CONTRACTIONS 9
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 (1.19) show us that the sequence (Tng)n∈Nisdj-Cauchy for each j. The
subsetAis assumed to be sequentially σ-complete, there exists f∗inAsuch that
(Tng)n∈Nisσ- convergent to f∗. Besides, the sequence (Tng)n∈Nconverges for the
topologyσtof∗, which implies
∀j∈J, dj(Tf∗,f∗) = limn→∞(dj(Tf∗,Tn+1g).
Recall that the operator Tis a local contraction with respect to ( D,r). From that, we
have
∀j∈J, dj(Tf∗,f∗)≤βjlimn→∞(dr(j)(f∗,Tng).
The convergence for the σ- topology implies convergence for the semidistance dr(j), we
obtaindj(Tf∗,f∗) = 0for everyj∈J. This way, we prove that Tf∗=f∗sinceσis
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 (1.19) and (1.20).
That way, we have proved that (Tng)n∈Nisσ- convergent to f∗.
This completes the proof.

CHAPTER 2
SINGLE VALUED SELF ALMOST LOCAL
CONTRACTIONS
1. Introduction
We try to combine these two diﬀerent type of contractive mappings: the almost
and local contractions, to study their ﬁxed points. This new type of mappings was
ﬁrst introduced in [ 42]
Definition 2.1.23 .The mapping d(x,y) :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)x=yimpliesd(x,y) = 0
Remark 2.1.24 .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 2.1.25. Also, observe the diﬀerence from a pseudometric and a semimetric
(deﬁnition 1.4.17 and 2.1.23).
Example 2.1.25 .
Definition 2.1.26 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX. We letσbe the weak topology on Xdeﬁned by the family D.
A sequence (xn)n∈N∗is said to be σ−Cauchyif it isdj-Cauchy,∀j∈J.
The subset AofXis said to be sequencially σ-complete if every σ-Cauchy sequence in
Xconverges in Xfor theσ-topology.
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.
Definition 2.1.27 .Letrbe a function from JtoJ. An operator T:A→A
is called an almost local contraction with respect ( D,r) if, for every j, there exist the
constantsθ∈(0,1)andL≥0such that
(2.21) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈A
Remark 2.1.28 .1)The almost contractions represent a particular case of almost
local contractions, by taking (X,d)metric space instead of the pseudometrics djand
10

1. INTRODUCTION 11
dr(j)deﬁned onX. Also, to obtain the almost contractions, we take in (2.21)forrthe
identity function, so we have r(j) =j.
2) Because of the simmetry of the pseudodistance, the almost local contraction condition
(2.21)includes the following dual one:
(2.22) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(x,Ty ),∀x,y∈A
Definition 2.1.29 .The spaceXisσ- 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}.
Theorem 2.1.30 is an existence ﬁxed point theorem for almost local contractions,
as they appear in [ 42].
Theorem 2.1.30 .Consider a function r:J→Jand letT:A→Abe an almost
local contraction with respect to ( D,r). Consider a nonempty, σ- bounded, sequentially
σ- complete, and T- invariant subset A⊂X. If the condition
(2.23) ∀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 (2.21) to obtain
dj(Txn−1,Txn)≤θ·dr(j)(xn−1,xn)
which yields
(2.24) dj(xn,xn+1)≤θ·dr(j)(xn−1,xn),∀j∈J
Using (2.21), we obtain by induction with respect to n:
(2.25) dj(xn,xn+1)≤θn·dr(j)(x0,x1), n = 0,1,2,···
According to the triangle rule, by (2.25) 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 σ-complete, there exists x∗inAsuch that
(Tnx)n∈Nisσ- convergent to x∗. Besides, the sequence (Tnx)n∈Nconverges for the
topologyσtox∗, which implies
∀j∈J, dj(Tx∗,x∗) = limn→∞dj(Tx∗,Tn+1x).

12 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Recall that the operator Tis an almost local contraction with respect to ( D,r). From
that, we have
∀j∈J, dj(Tx∗,x∗)≤βjlimn→∞dr(j)(x∗,Tnx).
The convergence for the σ- topology implies convergence for the pseudometric dr(j), we
obtaindj(Tx∗,x∗) = 0for everyj∈J.
This way, we prove that Tf∗=f∗, sinceσis Hausdorﬀ.
So, we prove the existence of the ﬁxed point for almost local contractions.
Remark 2.1.31 .ForTveriﬁes(2.21)withL= 0, andr:J→Jthe identity
function, we ﬁnd Theorem Vailakis [27]by takingθ=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.
Remark 2.1.32 .In Theorem 2.1.30, 2.1.35 the coeﬃcient of contraction θ∈(0,1)
is constant, but local contractions have a coeﬃcient of contraction θj∈[0,1)whitch
depends on j∈J. Our ﬁrst goal is to extend the local almost contractions to the most
general case of θj∈(0,1).
One extend Deﬁnition 2.1.27 to the case of almost local contractions with variable
coeﬃcient of contraction.
Definition 2.1.33 .Letrbe a function from JtoJ. An operator T:A→Ais
called almost local contraction with respect ( D,r)or(θj,Lj)- contraction, if there exist
a constantθj∈(0,1)and someLj≥0such that
(2.26) dj(Tx,Ty )≤θj·dr(j)(x,y) +Lj·dr(j)(y,Tx ),∀x,y∈A
Theorem 2.1.34 .With the assumptions of Theorem 2.1.30, if we modify the con-
dition(2.23)with the following one:
(2.27) ∀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)

2. CONTINUITY OF ALMOST LOCAL CONTRACTIONS 13
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 σ-complete, there exists x∗inAsuch that (Tnx)n∈Nisσ- convergent
tox∗. Besides, the sequence (Tnx)n∈Nconverges for the topology σtox∗, which implies
∀j∈J, dj(Tx∗,x∗) = limn→∞dj(Tx∗,Tn+1x).
Recall that the operator Tis an almost local contraction with respect to ( D,r). From
that, we have
∀j∈J, dj(Tx∗,x∗)≤θjlimn→∞dr(j)(x∗,Tnx) +Ljdr(j)(Tnx,Tx∗).
The convergence for the σ- topology implies convergence for the pseudodistance dr(j),
we obtaindj(Tx∗,x∗) = 0for everyj∈J. This way, we prove that Tx∗=x∗sinceσ
is Hausdorﬀ. So, we prove the existence of the ﬁxed point.
The next Theorem represent an existence and uniqueness theorem for the almost
local contractions with constant coeﬃcient of contraction.
Theorem 2.1.35 .If to the conditions of Theorem 2.1.30, we add:
(U) for every ﬁxed j∈Jthere exists:
(2.28) limn→∞(θ+L)ndiamrn(j)(z,A) = 0,∀x,y∈X
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∗)≤θdr(j)(x∗,y∗) +Ldr(j)(y∗,Tx∗) =
= (θ+L)·dr(j)(x∗,y∗)≤···≤ (θ+L)ndrn(j)(x∗,y∗)≤
≤(θ+L)ndiamrn(j)(z,A)
Now, letting n→∞, we obtained a contradiction with condition (2.28), i.e. the ﬁxed
point is unique.
2. Continuity of almost local contractions
ThissectioncanberegardedasanextensionofV.BerindeandM.Pacurar( 2015,[6])
analysis about the continuity of almost contractions in their ﬁxed points. The main
results are given by Theorem 2.2.36, which give us the answer about the continuity of
local almost contractions in their ﬁxed points.

14 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
Theorem 2.2.36 .LetXbe a set andD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX; letT:X→Xbe an almost local contraction satisfying condition (2.23),
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
(2.29) dj(Tx,Ty )≤θ·dr(j)j(x,y) +L·dr(j)(y,Tx ),∀x,y∈X
For any sequence {yn}∞
n=0inXconverging to f, we takey:=yn,x:=fin (2.29), and
we get
(2.30) 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:
(2.31) dj(Tyn,Tf)≤θ·dr(j)(f,yn) +L·dr(j)(yn,f),n= 0,1,2,…
Now by letting n→∞in (2.31) 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 2.1.27, 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 2.2.37 .LetX= [1,n]×[1,n]⊂R2, 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= [1,n]×[1,n]⊂R2is given by the diagonal line of the
square whose four sides have length n−1.
We shall use the pseudometric:
(2.32) dj((x1,y1),(x2,y2)) =|x1−x2|·ej,∀j∈Q,
whereQis a subset of N. This is a pseudometric, but not a metric, take for example:
dj((1,4),(1,3)) =|1−1|·ej= 0, however (1,4)/negationslash= (1,3)
In this case, the mapping Tis a contraction, which implies that Tis an almost local
contraction, with the unique ﬁxed point (0,0).
According to Theorem 2.2.36, Tis continuous at (0,0)∈Fix(T), but is not continuous
at(1,0)∈X.

2. CONTINUITY OF ALMOST LOCAL CONTRACTIONS 15
Example 2.2.38 .With the assumptions in Example 2.2.37 and the pseudometric
deﬁned by (2.32), we get another example for almost local 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:
(2.33) dj(Tx,Ty )≤θ·dj(x,y),
is not valid∀x,y∈X, and for any θ∈(0,1). Indeed, (2.33)is equivalent with:
|x1−x2|·ej≤θ·|x1−x2|·ej
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|·ej≤θ·|x1−x2|·ej
2+L·|x2−x1|·ej
2
which is equivalent to : ej
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 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 (2.21)from
Theorem 2.1.30
Theorem 2.2.36 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 2.2.39 .LetXthe set of positive functions: X={f|f: [0,∞)→[0,∞)}
anddj(f,g) =|f(0)−g(0)|·ej,∀f,g∈X.
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 (2.21)for almost
local contractions:
|f(0)−g(0)|·ej≤θ·|f(0)−g(0)|·ej
2+L·|g(0)−f(0)|·ej
2
which is equivalent to: ej/2≤θ+L
This inequality becames true if j <0, θ =1
4∈(0,1), L = 3>0
However,Tis also not a contraction, because the contractive condition (2.33)leads us
again to the false assumption: 1≤θ. The mapping Thas inﬁnite number of ﬁxed
points:FixT ={f∈X}, by taking:
|f(x)|=f(x),∀f∈X,x∈[0,∞)

16 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS
In this section we present the extension for generalized ALC, Ćirić-Reich-Rus type
ALC, Chatterjea type ALC, Zamﬁrescu type ALC.
a)Generalized ALC
Definition 2.3.40 .Letrbe a function from JtoJ. LetA⊂Fbe aσ-bounded
sequencially σ-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 )} (2.34)
Remark 2.3.41 .It is obvious that any generalized almost local contraction is an
almost contraction, i.e., it does satisfy the inequality (1.8).
Theorem 2.3.42 .LetT:A→Abe a generalized almost local contraction, i.e.,
a mapping satisfying (2.34), and also verifying the condition (2.62)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 (2.34) is satisﬁed. We know by Theorem (2.1.35)
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 (2.34), we get
(2.35) 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 (2.35), we get Tyn→Tfwhich shows that Tis continuous
atf.
b)Ćirić type almost local contraction
Definition 2.3.43 .(see[17]) 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
(2.36) d(Tx,Ty )≤α·M(x,y) +L·d(y,Tx ),for all x,y∈X,

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 17
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.
Definition 2.3.44 .Under the assumptions of deﬁnition 2.1.27, the operator
T:A→Ais called Ćirić-type almost local contraction with respect ( D,r) if, for every
j∈J, there exist the constants θ∈[0,1)andL≥0such that
(2.37) dj(Tf,Tg )≤θ·Mr(j)(f,g) +L·dr(j)(g,Tf ),for all f,g∈A,
where
Mr(j)(f,g) = max{dr(j)(f,g),dr(j)(f,Tf ),dr(j)(g,Tg ),dr(j)(f,Tg ),dr(j)(g,Tf )}.
Theorem 2.3.45 .Consider a function r:J→J, let a nonempty, σ- bounded,
sequentially σ- complete, and T- invariant subset A⊂Xand letT:A→Abe Ćirić-
type almost local contraction with respect 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:
(2.38) dj(xn,x∗)≤δn
1−δ2dj(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 (2.37) 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 we have
dj(Tnx,Tn+1x) =dj(TTn−1x,T2Tn−1x)≤θ·δ[O(Tn−1x,2)].
By using Remark 2.42, we can easily conclude: there exist a positive integer k1∈{12}
such that
δ[O(Tn−1x,2)] =dj(Tn−1x,Tk1Tn−1x)
and therefore
dj(xn,xn+1)≤θ·dj(Tn−1x,Tk1Tn−1x).

18 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
By using once again Lemma 2.42, we obtain, for n≥2,
dj(Tn−1x,Tk1Tn−1x) =dj(TTn−2x,Tk1+1Tn−2x)≤
≤θ·δ[O(Tn−2x,k1+ 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
(2.39) dj(Tnx,Tn+1x)≤θ·δ[O(Tn−1x,2)]≤···≤θn·δ[O(x,n+ 1)].
At this point, by Lemma 2.38, we obtain
δ[O(x,n+ 1)]≤δ[O(x,∞)]≤1
1−θdj(x,Tx ),
which by (2.39) yields
(2.40) dj(Tnx,Tn+1x)≤θn
1−θdj(x,Tx ).
The inequality (2.39) and the triangle inequality can be merged to obtain the following
estimate
(2.41) dj(Tnx,Tn+px)≤θn
1−θ·1−θp
1−θdj(x,Tx ).
Let us remind the fact that 0≤θ≤1, then, by using (2.41), we can conclude that
{xn}∞
n=0is a Cauchy sequence. The subset Ais assumed to be sequentially σ-complete,
there exists x∗inAsuch that{xn}isσ- convergent to x∗. After simple computations
involving the triangular inequality and the Deﬁnition (2.37), 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)]. (2.42)
Lettingn→∞in (2.42) we obtain
dj(x∗,Tx∗) = 0,

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 19
which means that x∗is a ﬁxed point of T. The estimate (2.38) can be obtained from
(2.41) by letting p→∞.
This completes the proof.
Remark 2.3.46 .1) Theorem 2.3.45 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 1.4 below.
2) Let us remind (see [37],[38]) that an operator T:X→Xis said to be a weakly
Picard operator (WPO) if the sequence {Tnx0}∞
n=0converges for all x0∈Xand the
limits are ﬁxed point of T. The main merit of Theorem 2.3.45 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 es-
timate(2.38)obtained in Theorem 2.3.45 is weaker as (1.4),(1.5)from the Banach’s
contraction principle or the estimate given in Theorem 2.44.
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 (2.37),
according to the next theorem.
Theorem 2.3.47 .With the assumptions of Theorem 2.3.45, let T:A→Abe a
Ćirić type almost local contraction with the additional inequality:
(2.43) 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
(2.44) 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 (2.38)holds;
(4) The rate of the convergence of the Picard iteration is given by
(2.45) dj(xn,f∗)≤θdr(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 (2.44), and condition (2.43) for every ﬁxed j∈Jwith

20 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
f:=f∗,g:=g∗we get:
dj(f∗,g∗)≤θ·dr(j)(f∗,g∗)≤dj(f∗,g∗)⇔(1−θ)·dj(f∗,g∗)≤0,
which is obviously a contradiction with dj(f∗,g∗)>0. So, we prove the uniqueness of
the ﬁxed point.
The proof for 2) and 3) is quite similar to the proof from the Theorem 2.3.45.
4) At this point, letting g:=xn,f:=f∗in (2.44), it results the rate of convergence
given by (2.45). The proof is complete.
The contractive conditions (2.37) and (2.44) can be merged to maintain the unicity
of the ﬁxed point, stated by the next theorem.
Theorem 2.3.48 .Under the assumptions of deﬁnition 2.3.44, 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 )}, (2.46)
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 (2.38)holds.
c)Ćirić-Reich-Rus type almost local contraction
Definition 2.3.49 .Under the assumptions of deﬁnition 2.1.27, the operator
T:A→Ais called Ćirić-Reich-Rus type almost local contraction with respect ( D,r)
if the mapping T:A→Asatisfying the condition
(2.47) dj(Tf,Tg )≤δ·dr(j)(f,g) +L·[dr(j)(f,Tf ) +dr(j)(g,Tg )],
for allf,ginA, whereδ,L∈R +andδ+ 2L<1
Theorem 2.3.50 .If the pseudometric dsatisfy the condition:
dr(j)(f,g)<dj(f,g),∀j∈J,∀f,g∈Athen any Ciric- Reich- Rus type almost local
contraction, i.e. any mapping T:A→Asatisfying the condition (2.47)withL/negationslash= 1is
an almost local contraction.

3. NEW CLASSES OF ALMOST LOCAL CONTRACTIONS 21
Proof:Using condition (2.47) 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:
(2.48) (1−L)·dj(Tf,Tg )≤(δ+L)·dj(f,g) + 2L·dr(j)(g,Tf )
and which implies
(2.49) dj(Tf,Tg )≤δ+L
1−L·dj(f,g) +2L
1−L·dr(j)(g,Tf ),∀f,g∈A
Considering δ,L∈R +andδ+ 2L<1, the inequality (2.47) holds, with
δ+L
1−L∈(0,1)and2L
1−L≥0. Therefore, anyCiric-Reich-Rustypealmostlocalcontraction
with the condition for the pseudometric, is an almost local contraction.
d)Chatterjea type ALC
Definition 2.3.51 .Under the assumptions of deﬁnition 2.1.27, the operator
T:A→Ais called Chatterjea type almost local contraction with respect ( D,r) or
(δ,L)- Chatterjea contraction if there exist a constant 0≤c<1
2such that
(2.50) dj(Tf,Tg )≤c·[dr(j)(f,Tg ) +dr(j)(g,Tf )],∀f,g∈A.
Theorem 2.3.52 .If the pseudometric dsatisfy the condition:
(2.51) 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 (2.50)is an almost local contraction.
Proof:Using condition (2.50) 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:
(2.52) dj(g,Tf )<dr(j)(g,Tf )
From this point, we get after simple computations:
(2.53) dj(Tf,Tg )≤c[dr(j)(f,g) +dr(j)(g,Tf )] +c[dr(j)(g,Tf ) +dj(Tf,Tg )]

22 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
(1−c)dj(Tf,Tg )≤c·dr(j)(f,g) + 2c·dr(j)(g,Tf )
and which implies
(2.54) 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 (2.50) 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.
e)Zamﬁrescu type ALC
Definition 2.3.53 .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.
Now, it is natural to state the next result:
Proposition 2.3.54 .Any Zamﬁrescu type almost local contraction is an almost
local contraction.
f)Berinde type ALC
Definition 2.3.55 .Under the assumptions of deﬁnition 2.1.27, the operator
T:A→Ais called generalized Berinde type ALC with respect ( D,r) if there exist a
constantθ∈[0,1)and a function bfrom the subset Ainto[0,∞)such that
(2.55) dj(Tx,Ty )≤θ·dr(j)(x,y) +b(y)·dr(j)(y,Tx ),∀x,y∈A,∀j∈J.
Theorem 2.3.56 .We shall use assumptions from Deﬁnition 2.1.27.
LetTbe a mapping on the subset A. Consider a function bfromAinto[0,∞). Assume
that there exists θ∈[0,1)such that
(2.56)
(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
(2.57) (1 +θ)−1dj(x,Tx )≤dj(x,Tx ),∀j∈J,
and we get
(2.58)dj(Tx,T2x)≤θdr(j)(x,Tx ) +b(Tx)dr(j)(Tx,Tx ) =θdr(j)(x,Tx ),∀x∈A

4. ALMOST LOCAL ϕ-CONTRACTIONS 23
Letu∈A, then from (2.58) 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 σ-complete, there exists zinAsuch that (Tnx)n∈Nisσ- convergent to
z∈A.
Using (2.58), 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 (2.56), 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.
Corollary 2.3.57 .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 2.3.58 .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.
4. 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 ([ 36]) and Berinde ([ 8]).
Definition 2.4.59 .[36]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

24 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
(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 [ 15]),ϕsatisﬁes (iiiϕ)if and only if there exist
0<c< 1and a convergent series of positive terms,/summationtext∞
n=0unsuch that
ϕk+1(t)≤cϕk(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,
(2.59) s(t) =∞/summationdisplay
k=0ϕk(t),t∈R+
is monotone increasing and continuous at zero, and that any (c)- comparison function
is a comparison function.
Weakϕ- contractions was ﬁrst introduced by V. Berinde in [ 16]. The aim of this
section is to extend weak ϕ- contractions to the more general case of weak ALC- s.
Definition 2.4.60 .[16]Let(X,d)be a metric space. A self operator T:X→X
is said to be a weak ϕ- contraction or (ϕ,L)- weak contraction, provided that there exist
a comparison function ϕand someL≥0, such that
(2.60) d(Tx,Ty )≤ϕ(d(x,y)) +Ld(y,Tx ),for allx,y∈X.
From this point,it is natural the extension of weak ϕ- contractions to the weak
ALC- s.
Definition 2.4.61 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned onX. We letσbe the weak topology on Xdeﬁned by the family D. Letrbe a
function from JtoJ, and letA⊂Xbe aσ-bounded sequencially σ-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:
(2.61) dj(Tx,Ty )≤ϕ(dr(j)(x,y)) +L·dr(j)(y,Tx ),∀x,y∈A,∀j∈J.
Remark 2.4.62 .It is obvious that any almost local contraction became an almost
localϕ- 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 respect to the same pseudometric, according to Example 2.4.63
Example 2.4.63 .Letϕ(t) =t
t+ 1, t∈R+, withr(j) =jand letTan almost
localϕ- contraction, i.e. a mapping which satisﬁes (2.61).
Thenϕis a nonlinear comparison function, but not veriﬁes the condition for the (c)-

4. ALMOST LOCAL ϕ-CONTRACTIONS 25
comparison function. In this case, Tis an almost local ϕ- contraction without being an
ALC.
Remark 2.4.64 .Similar to the case of ALC-s, if Tsatisﬁes(2.25), for allx,y∈A,
does imply that the following dual inequality
(2.62) dj(Tx,Ty )≤ϕ(dr(j)(x,y)) +L·dr(j)(x,Ty ),∀x,y∈A
obtained from (2.25)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 (2.25)and(2.62)inequalities.
Remark 2.4.65 .The class of almost local ϕ- contractions includes a very large
type of mappings, see for example 2.4.66, which contain a mapping with not one, but
inﬁnite number of ﬁxed points.
Example 2.4.66 .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 (2.25)
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. The next two theorems represent an existence theorem
and, respectively, a uniqueness theorem for the almost local ϕ- contractions.
Theorem 2.4.67 .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 letσbe 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
(2.63) xn+1=Txn, n = 0,1,2,…
converges to a ﬁxed point x∗∈Fix(T)
(3) The a posteriori estimate
(2.64) dj(xn,x∗)≤s(dj(xn,xn+1)), n = 0,1,2…,∀x,y∈A,∀j∈J
holds, where s(t)is given by (2.59).

26 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
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
(2.65) 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 (2.65) to obtain
dj(Txn−1,Txn)≤ϕ(dr(j)(xn−1,xn))
which yields
(2.66) dj(xn,xn+1)≤ϕ(dr(j)(xn−1,xn)),∀j∈J,∀n= 1,2,…
Sinceϕis non decreasing, by (2.66) we have
(2.67) dj(xn+1,xn+2)≤ϕ(dr(j)(xn,xn+1)),∀x,y∈A,∀j∈J.
From that relation, we obtain by induction with respect to n:
(2.68) dj(xn+k,xn+k+1)≤ϕk(dr(j)(xn,xn+1)),∀x,y∈A,∀j∈J.
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 (2.69)
where we denoted r=dj(xn,xn+1). Again, by (2.67) we conclude
(2.70) dj(xn,xn+1)≤ϕn(dr(j)(x0,x1)),n= 0,1,2,…
which, by property (iiϕ)from the deﬁnition (2.4.59) of a comparison function implies
(2.71) limn→∞dj(xn,xn+1) = 0.
Having in view that ϕis positive, it is clear that
(2.72) 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 (2.72) and (2.69) we get
(2.73) dj(xn,xn+p)≤s(dr(j)(xn,xn+1)),n∈N,p∈N,∀x,y∈A,∀j∈J.

4. ALMOST LOCAL ϕ-CONTRACTIONS 27
Sincesis continuous at zero, (2.72) and (2.71) implies that the sequence (xn)n∈Nisdj-
Cauchy for each j∈J. The subset Ais assumed to be sequentially σ-complete, there
existsx∗inAsuch that the sequence (xn)isσ- convergent to x∗. We shall prove that
x∗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 (2.25) we have
dj(Txn,Tx∗)≤ϕ(dr(j)(xn,x∗)) +L·dr(j)(x∗,Txn),∀x,y∈A,∀j∈J
and hence
(2.74) 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 (2.74) 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 (2.64) follows from (2.69) by taking p→∞.
This completes the proof.
Remark 2.4.68 .1)The a posteriori error estimates (2.64)and(2.70)leads us to
the a priori estimate for the Picard iteration {xn}∞
n=0.
2)An almost local ϕ- contraction can be discontinuous, as shown by Example 2.4.69.
3) By taking ϕ(t) =θ·t,t∈R+,0<θ< 1, from Theorem 2.4.67 we obtain the same
result for almost local contractions, i.e. Theorem 2.1.30.
Example 2.4.69 .
Remark 2.4.70 .An almost local ϕ- contraction may have more than one ﬁxed
point, as shown by Example 2.4.66. In Theorem 2.4.67, 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 2.4.71.
Theorem 2.4.71 .LetXandTas in Theorem 2.4.67, A⊂X. SupposeTalso
satisﬁes the following condition: there exist a comparison function Υand someL1>0
such that
(2.75) 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

28 2. SINGLE VALUED SELF ALMOST LOCAL CONTRACTIONS
(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 (2.59);
(3) The rate of the convergence of the Picard iteration is given by
(2.76) 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 (2.75), by taking x:=x∗andy:=y∗, we obtain
dj(x∗,y∗)≤Υdj(x∗,y∗),∀x,y∈A,∀j∈J,
which by induction with respect to n, yields
(2.77) dj(x∗,y∗)≤Υn·dj(x∗,y∗),n= 1,2,…
Lettingn→∞in (2.77), 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 (2.76), all we have to do is to change x:=x∗andy:=xnin the
inequality (2.75).
The proof is complete.
Remark 2.4.72 .1) Having in view the pairs of dual conditions (2.21)and(2.22),
(2.61)and(2.62), condition (2.75)holds if and only if its dual
(2.78) dj(Tx,Ty )≤Υ(dr(j)(x,y)) +L1·dr(j)(y,Ty ),
is valid for all x,y∈A,∀x,y∈A,∀j∈J.
2) Condition (2.75)is not necessary for the unicity of the ﬁxed point, according to
Example 2.4.73.
3) IfThas a unique ﬁxed point x∗and the Picard iteration {Tnx0}∞
n=0converges to x∗
for allx0∈A, then by Theorem 2.37, for any a∈(0,1), there exist a pseudometric
/rho1onAsuch that (X,/rho1)isσ- complete and Tis ana- contraction with respect to the
pseudometric /rho1.
Therefore, condition (2.75)can be reformulated using another pseudometric, which
leads us to a more general result.
Example 2.4.73 .LetX= [1,n]×[1,n]⊂R2. We shall use the pseudometric:
(2.79) dj((x1,y1),(x2,y2)) =|x1−x2|·ej,∀j∈Q.

4. ALMOST LOCAL ϕ-CONTRACTIONS 29
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}
Theorem 2.4.74 .LetXbe a nonempty set, let A⊂Xbe aσ-bounded sequencially
σ-complete and T- invariant subset of Xandd,/rho1two pseudometrics on Asuch that
(X,d)isσ- 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;
(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 (2.59);
(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 2.4.75 .We obtain Theorem 2.4.71 as a particular of Theorem 2.4.74 if
we setdj≡/rho1j.

CHAPTER 3
MULTIVALUED SELF ALMOST LOCAL
CONTRACTIONS
1. Preliminaries
The notion of multivalued contraction was ﬁrst introduced by Nadler in [ 28] fol-
lowing are borrowed from Nadler [ 28]
Definition 3.1.76 .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 [ 37])
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 can be found in Rus [ 37], it is useful for the next theorem.
30

1. PRELIMINARIES 31
Lemma 3.1.77 .[28]Let(X,d)be a metric space, let A,B⊂Xandq>1.
Then, for every a∈A, there exists b∈Bsuch that
(3.80) d(a,b)≤qH(A,B)
Definition 3.1.78 .[12]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
(3.81) H(Tx,Ty )≤θ·d(x,y) +L·D(y,Tx ),∀x,y∈X
Remark 3.1.79 .Because of the simmetry of the distance dandH, the almost
contraction condition (3.81)includes the following dual one:
(3.82) H(Tx,Ty )≤θ·d(x,y) +L·D(x,Ty ),∀x,y∈X
Obviously, to prove the almost contractiveness of T, it is necessary to check both (3.81)
and(3.82).
Theorem 3.1.80 .(Berinde V., Berinde M. [12]) 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:
(3.83) d(xn,u)≤hn
1−hd(x0,x1), n = 0,1,2…
(3.84) 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 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 3.1.81 .From the deﬁnition of Dj, we have the following result:
ifDj(a,B) = 0, implies that a∈B
Definition 3.1.82 .Letrbe a function from JtoJ, letS⊂Xbe aσ-bounded
sequencially σ-complete and T- invariant subset of X. An operator T:S→P (S)
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≥0such that
(3.85) Hj(Tx,Ty )≤θ·dr(j)(x,y) +L·Dr(j)(y,Tx ),∀x,y∈S,∀j∈J.

32 3. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS
Lemma 3.1.83 .LetSbe a subset of Xand letD= (dj)j∈Jbe a family of pseudo-
metrics deﬁned on X. We letσbe the weak topology on Xdeﬁned by the family D.
LetA,B⊂Sandq>1.
Then, for every j∈Janda∈A, there exists b∈Bsuch that
(3.86) dj(a,b)≤qHj(A,B)
Proof:IfHj(A,B) = 0, then for every a∈A, we have:
Hj(A,B)≥Dj(a,B)⇒Dj(a,B) = 0
From that, we conclude: there exist b∈Bsuch thatdj(a,b) = 0.
The inequality (3.86) is valid, i.e., 0≤0.
IfHj(A,B)>0, then let us denote
(3.87) ε= (h−1−1)H(A,B)>0
Using the deﬁnition of Hj(A,B)andDj(a,B), we conclude that for any ε >0there
existsb∈Bsuch that
(3.88) dj(a,b)≤qDj(a,B) +ε≤Hj(A,B) +ε
Combining (3.87) and (3.88), we get the conclusion of the lemma.
Theorem 3.1.84 .With the assumptions of Deﬁnition 3.1.82, 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:
(3.89) dj(xn,u)≤hn
1−hdj(x0,x1), n = 0,1,2…
(3.90) dj(xn,u)≤h
1−hdj(xn−1,xn), n = 1,2…
for a certain constant h<1.
Proof:We consider q >1, letx0∈Xandx1∈Tx0. IfHj(Tx0,Tx 1) = 0, that
means from the deﬁnition of DjandHj:
(3.91) 0 =Hj(Tx0,Tx 1)≥Dj(x1,Tx 1)
and that is possible only if Dj(x1,Tx 1) = 0, from here, we conclude x1∈Tx1, which
leads us to the conclusion Fix(T)/negationslash=φ.
LetHj(Tx0,Tx 1)/negationslash= 0. According to Lemma 3.1.83, there exists x2∈Tx1such that
(3.92) dj(x1,x2)≤qHj(Tx0,Tx 1)

1. PRELIMINARIES 33
By (3.85) we have
dj(x1,x2)≤q[θ·dr(j)(x0,x1) +L·Dr(j)(x1,Tx 0)] =qθ·dr(j)(x0,x1).
sincex1∈Tx0,Dr(j)(x1,Tx 0) = 0.
We takeq>1such that
h=qθ< 1
and we obtain dj(x1,x2)<h·dj(x0,x1).
IfHj(Tx1,Tx 2) = 0thenDj(x2,Tx 2) = 0, that means x2∈Tx2using Remark 3.1.81.
LetHj(Tx1,Tx 2)/negationslash= 0. Again, using Lemma 3.1.83, there exists x3∈Tx2such that
(3.93) dj(x2,x3)≤qh·dj(x1,x2),∀j∈J.
This way, we obtain an orbit {xn}∞
n=0ofTat the point x0satisfying
(3.94) dj(xn,xn+1)≤h·dj(xn−1,xn),∀j∈J, n = 1,2,…
By (3.94), we inductively obtain
(3.95) dj(xn,xn+1)≤hndj(x0,x1),∀j∈J,
and, respectively,
(3.96) dj(xn+k,xn+k+1)≤hk+1dj(xn−1,xn), k∈N
Using the inequality (3.95), we obtain
(3.97) dj(xn,xn+p)≤hn(1−hp)
1−hdj(x0,x1), n,p∈N
Recall 0< h < 1, conditions (3.96),(3.97) show us that the sequence (xn)n∈Nisdj-
Cauchyforeachj,whichshowsthat {xn}∞
n=0isaCauchysequence. Thatmeans {xn}∞
n=0
is convergent with the limit u:
(3.98) u= limn→∞xn
After simple computations, we get for all j∈J:
Dr(j)(u,Tu )≤Dr(j)(u,xn+1) +Dr(j)(xn+1,Tu)≤dr(j)(u,xn+1) +Hr(j)(Txn,Tu)
which, by (3.85) yields
(3.99) Dr(j)(u,Tu )≤dr(j)(u,xn+1) +θdr(j)(xn,u) +L·Dr(j)(u,Txn),∀j∈J
Lettingn→∞and using the fact that xn+1∈Txnimplies by (3.98),
Dr(j)(u,Txn)→0, asn→∞. We get
Dr(j)(u,Tu ) = 0
SinceTuis closed, this implies u∈Tu.
We letp→∞in (3.97) to obtain (3.89). Using (3.96), we get
(3.100) d(xn,xn+p)≤h(1−hp)
1−hd(xn−1,xn), p∈N,n≥1

34 3. MULTIVALUED SELF ALMOST LOCAL CONTRACTIONS
and letting p→∞in (3.100), we obtain (3.90). The proof is complete.
The next Theorem shows that any multivalued ALC is continuous at the ﬁxed
point.
Theorem 3.1.85 .With the assumptions of Deﬁnition 3.1.82, 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:
(3.101) 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 3.1.84.
Let{yn}∞
n=0be any sequence in the subset Sconverging to the ﬁxed point p. Then by
takingy:=ynandx:=pin the multivalued ALC condition (3.101), we get
(3.102) 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 (3.102) with dj(yn,p), we get:
(3.103) dj(Tyn,Tp)≤(δ+L)·dj(yn,p),n= 0,1,2,…
Now, by letting n→∞in (3.103) we get Tyn→Tpasn→∞, that means: Tis
continuous at p.

CHAPTER 4
NON-SELF SINGLE VALUED ALMOST LOCAL
CONTRACTIONS
LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics deﬁned on X. We
letσbe 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 (2.21).
Ifx∈Kis verifying Tx /∈K, then we can always choose an y∈∂K(the boundary of
K) such that y= (1−λ)x+λ·Tx, (0<λ< 1), which actually means that
(4.104) d(x,Tx ) =d(x,y) +d(y,Tx ),y∈∂K
In general, the set Yof pointsysatisfying condition (4.104) above may contain more
than one element.
We shall need the following concept, borrowed from [ 13]
Definition 4.0.86 .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 (4.104). If, for any such elements x, we have
(4.105) d(y,Ty )≤d(x,Tx ),
for at least one of the corresponding y∈Y, then we say that Thas property (M).
The next Theorem state and prove the existence of the ﬁxed point for nonself single
valued ALC.
Theorem 4.0.87 .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)andL≥0such that
(4.106) dj(Tx,Ty )≤θ·dr(j)(x,y) +L·dr(j)(y,Tx ),∀x,y∈K
Assume that conditions from Theorem 2.1.30 are veriﬁed. If Thas property (M) and
satisﬁes Rothe’s boundary condition
(4.107) T(∂K)⊂K,
thenThas a ﬁxed point in K.
35

36 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
Proof:IfT(K)⊂K, thenTis in fact a self mapping on the closed set Kand the
conclusion follows by Theorem 2.1.30 for X=K. It is natural to consider the case
T(K)/negationslash⊂K. Letx0∈∂K. Using (4.107) 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
By introducing the following notations, we shall simplify our proof:
P={xk∈{xn}:xk=Txk−1}
Q={xk∈{xn}:xk/negationslash=Txk−1}
It is obvious that {xn}⊂Kand also, if xk∈Q, then both xk−1andxk+1belong to the
setP. Moreover, according to (4.107), 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 aσ-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 (2.21) we get
dj(xn+1,xn) =dj(Txn,Txn−1)≤θ·dr(j)(xn,xn−1) +L·dr(j)(xn,Txn−1),
which means
(4.108) 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 (2.21), we get
dj(xn,xn+1)≤θ·dr(j)(xn,xn−1) +L·dr(j)(xn,Txn−1) =θ·dr(j)(xn,xn−1)
which yields again inequality (4.108).

4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS 37
Case III.xn∈Q,xn+1∈P.
In this case, we have xn−1∈P. The mapping Thas property (M), this means
dj(xn,xn+1) =dj(xn,Txn)≤dj(xn−1,Txn−1)
Sincexn−1∈P, we havexn−1=Txn−2and by (2.21) 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, we obtain
(4.109) dj(xn,xn+1)≤θ·dr(j)(xn−2,xn−1)
At this point, after analysing all three cases, and using (4.108) and (4.109), it follows
that the sequence {xn}verify the inequality:
(4.110) dj(xn,xn+1)≤θ·max{dr(j)(xn−2,xn−1),dr(j)(xn−1,xn)}
for alln≥2. Now, by induction for n≥2from (4.110) 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}isσ-Cauchy sequence.
Note that{xn}⊂KandKis closed, which means that {xn}converges to some point
inK.
Denote
(4.111) 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 (2.21), we obtain
dj(Txn,Tx∗)≤θ·dr(j)(xn,x∗) +L·dr(j)(x∗,Txn),
and hence
(4.112) dj(x∗,Tx∗)≤dj(xn+1,x∗) +θ·dr(j)(xn,x∗) +L·dr(j)(x∗,Txn)
for alln≥0. Lettingn→∞in (4.112), we get the ﬁnal conclusion for our proof,i.e.
dj(x∗,Tx∗) = 0
which shows that x∗is a ﬁxed point of T.

38 4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS
Remark 4.0.88 .A mapping Tsatisfying (2.21), i.e.Tis a non-self ALC, the
mappingTmay be discontinuous (see Example 4.0.89), however Tis continuous at the
ﬁxed point. For the argumentation, if {yn}is a sequence in K, convergent to x∗=Tx∗,
then by(2.21)we get
dj(Tyn,x∗) =dj(Tx∗,Tyn)≤θ·dr(j)(x∗,yn) +L·dr(j)(yn,Tx∗)
From that inequality, by letting n→∞, we obtain the continuity of Tat the ﬁxed point
x∗, that is:dj(Tyn,x∗)→0asn→∞, which means Tyn→x∗. The proof is complete.
The next example show a non-self mapping with property (M), but discontinuous
and which is not an almost local contraction.
Example 4.0.89 .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 ?,Tis discontinuous in9
10, the unique ﬁxed point of Tis0, and
Tis continuous in 0.Thas property (M). Indeed, if x=9
10∈K,Tx=−1
10/negationslash∈K,
then using the condition (4.105)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
10
andy=x
x+1in(2.21)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 (2.21), 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 4.0.90 .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)andL1≥0such that
(4.113) 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:
(4.114) limn→∞(θ+L)ndiamrn(j)(z,A) = 0,∀x,y∈X
IfThas property (M) and satisﬁes Rothe’s boundary condition
(4.115) T(∂K)⊂K,
thenThas a unique ﬁxed point in K

4. NON-SELF SINGLE VALUED ALMOST LOCAL CONTRACTIONS 39
Remark 4.0.91 .The proof is quite similar to the case of single valued self ALC- s
(see Theorem 2.1.35)
Example 4.0.92 .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.

CHAPTER 5
NON-SELF MULTIVALUED ALMOST LOCAL
CONTRACTIONS
1. NON-SELF MULTIVALUED ALMOST CONTRACTIONS
In [12], M. Berinde and V. Berinde introduce the non-self multivalued almost con-
tractions.
Definition 5.1.93 .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
(5.116) H(Tx,Ty )≤δ·d(x,y) +L·D(y,Tx ),∀x,y∈K
Definition 5.1.94 .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
(5.117) d(x,y) =d(x,z) +d(z,y)
Remark 5.1.95 .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 5.1.96 .(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
(5.118) d(x,y) =d(x,z) +d(z,y)
Lemma5.1.97 .(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
(5.119) 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.
40

2. FIXED POINT THEOREMS FOR NON-SELF MULTIVALUED ALC 41
Definition 5.2.98 .LetXbe a set and letD= (dj)j∈Jbe a family of pseudometrics
deﬁned on X. We letσbe the weak topology on Xdeﬁned by the family D, letKa
nonempty closed subset of X. Letrbe a function from JtoJ.
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
(5.120) Hj(Tx,Ty )≤θ·dr(j)(x,y) +L·Dr(j)(y,Tx ),∀x,y∈K
Theorem 5.2.99 .Under the assumptions of deﬁnition 5.2.98, suppose that
T:K→CB(X)is a multivalued almost local contraction, that is,
(5.121) 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
(5.122) dj(x0,x1) +dj(x1,y1) =dj(x0,y1)
We havex1∈Kand using Lemma 5.1.97 with α=θ, we choose y2∈Tx1such that
(5.123) dj(y1,y2)≤Hj(Tx0,Tx 1) +θ
Again, ify2∈K, putx2=y2. Ify2/∈K, then there exists x2∈∂Ksuch that
(5.124) d(x1,x2) +d(x2,y2) =d(x1,y2)
Therefore,x2∈K, and by Lemma 5.1.97 and α=θ2, we can choose y3∈Tx2such
that
(5.125) 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
(5.126) dj(xn−1,xn) +dj(xn,yn) =dj(xn−1,yn)

42 5. 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 43
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
(5.127) dj(xn,xn+1)<h·dj(xn−2,xn−1) +h·θn−2+θn
This way, combining all three cases, we get
(5.128) dj(xn,xn+1)≤αdj(xn−1,xn) +αn
or the other possibility:
(5.129) αdj(xn−2,xn−1) +αn−1+αn
where
(5.130) α= max{θ,h}=h
We obtain by induction with respect to n:
(5.131) dj(xn,xn+1)≤hn−1
2·ω+hn
2,
where
(5.132) 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.

44 5. 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 σ-complete, there exists zinKsuch that
(5.133) z= limn→∞xn
As we constructed the sequence (xn)n∈N, there is a subsequence {xp}such that
(5.134) yp=xp∈Txp−1
In what follows, we propose ton prove that z∈Tz. In fact, by (i), xp∈Txp−1. Since
xp→zasp→∞, we have
(5.135) 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 5.2.99 we obtain a ﬁxed point theorem for multivalued
nonselfalmostcontractionsstatedbyMaryamA.Alghamdi, VasileBerinde, andNaseer
Shahzad [1] as a particular case by letting r(j) =j,θ=δ, and (X,d)be a complete
convex metric space.

CHAPTER 6
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 6.1.100 .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.[ 32]
Definition 6.1.101 .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 6.1.102 .(see[32]) IfX= [0,1], the mapping ddd:X×X→Cis deﬁned
byddd(x,y) =|x−y|2+i|x−y|2for allx,y∈X. Then (X,ddd)is complex valued b-metric
space withs= 2.
45

46 6. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS
The following deﬁnition remind the convergent sequence and the Cauchy sequence.
Definition 6.1.103 .(see[32]) 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 6.1.104 .(see[32]) 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)|→ 0as
n→∞.
Lemma 6.1.105 .(see[32]) 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)|→
0asn→∞, wherem∈N.
2. Fixed point theorems for rational local contractions
Definition 6.2.106 .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:
(6.136) 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 6.2.107 .Let(X,ddd)be a complete complex valued b-metric space with the
coeﬃcients≥1and let two mappings S,T:X→Xsatisfying
(6.137) 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 47
Proof:For any arbitrary x0∈X, deﬁne a sequence {xn}inXsuch that
x2n+1=Sx2n (6.138)
x2n+2=Tx2n+1 (6.139)
forn= 0,1,2,3,…
Our ﬁrst goal is to prove that the sequence {xn}is Cauchy sequence.
Takex=x2nandy=x2n+1in (6.137); 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) + (6.140)
+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)|+ (6.141)
+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
(6.142)|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
(6.143) |dddj(x2n+1,x2n+2)|≤(α+β)·|dddr(j)(x2n,x2n+1)|
In a similar way, we obtain
(6.144) |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)|≤···≤ (6.145)
≤δ2n+1·|dddr(j)(x0,x1)|

2. FIXED POINT THEOREMS FOR RATIONAL LOCAL CONTRACTIONS 49
From that, we conclude
(6.150) |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
(6.151) |dddj(u,Su )|=|z|>0
According to the triangular inequality and (6.137), 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)|+ (6.152)
+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 (6.152) as n→∞, we conclude that
|z|=|dddj(u,Su )|≤0
and this is a contradiction with (6.151). 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∗) + (6.153)
+β[dddr(j)(u,Su )dddr(j)(u∗,Tu∗)]
dddr(j)(u,Tu∗) +dddr(j)(u∗,Su) +dddr(j)(u,u∗)

50 6. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS
From that, we conclude
|dddj(u,u∗)| ≤
≤α|dddr(j)(u,u∗)|+ (6.154)
+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:
(6.155) dddr(j)(x,Ty ) +dddr(j)(y,Sx ) +dddr(j)(x,y) = 0,
putx=x2nandy=x2n+1in the last condition and we get:
(6.156) 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 (6.155)),
impliesdddr(j)(Sl1,Tl2) = 0. Thus,K1=Sl1=l1=Tl2=K2. So, we get from here
K1=Sl1=l1=SK1. In a quite similar way, one can have K2=TK2. ButK1=K2
impliesSK1=KT1=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.
Fromthecondition(6.155),wehave dddr(j)(K1,TK∗
1)+dddr(j)(K∗
1,SK 1)+dddr(j)(K1,K∗
1) = 0,
thereforedddr(j)(K1,K∗
1) =dddr(j)(SK1,TK∗
1) = 0, and this means that K1=K∗
1.
This completes the proof of the theorem.
Theorem 6.2.108 .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 )] (6.157)
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 51
Proof:For any arbitrary x0∈X, deﬁne a sequence {xn}inXsuch that
x2n+1=Sx2n (6.158)
x2n+2=Tx2n+1 (6.159)
forn= 0,1,2,3,…
Our ﬁrst goal is to prove that the sequence {xn}is Cauchy sequence.
Takex=x2nandy=x2n+1in (6.157); 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
(6.160)|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
(6.161)|dddj(x2n,x2n+2)|≤s/bracketleftBig
|dddj(x2n,x2n+1)|+|dddj(x2n+1,x2n+2)|/bracketrightBig

2. FIXED POINT THEOREMS FOR RATIONAL LOCAL CONTRACTIONS 53
At this point, by using (6.165), 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)|≤ (6.167)
≤∞/summationdisplay
t=n(sδ)t|dddr(j)(x0,x1)|=
=(sδ)n
1−sδ|dddr(j)(x0,x1)|.
which means
(6.168) |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
(6.169) dddj(u,Su ) =|z|>0.

54 6. COMPLEX VALUED B-METRIC SPACES AND RATIONAL LOCAL CONTRACTIONS
According to the triangular inequality and (6.137), 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)] = (6.170)
=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)|+ (6.171)
+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 (6.171) as n→∞, we conclude that
|z|=|dddj(u,Su )|≤0
and this is a contradiction with (6.169). 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∗) + (6.172)
+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 6.2.107 This
completes the proof of the theorem.

2. FIXED POINT THEOREMS FOR RATIONAL LOCAL CONTRACTIONS 55
As a natural consequence of theorem 6.2.108, two corollary-s can be stated as it
follows:
Corollary 6.2.109 .Let(X,ddd)be a complete complex valued b-metric space with
the coeﬃcient s≥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 )] (6.173)
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 6.2.108, by taking S=T.
Corollary 6.2.110 .Let(X,ddd)be a complete complex valued b-metric space with
the coeﬃcient s≥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)+ (6.174)
+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 6.2.108, 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)+ (6.175)
+γ[dddr(j)(Tu,TnTu) +dddr(j)(u,Tnu)] =
= (α+ 2β)dddj(Tu,u )
By taking modulus of (6.175) 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.

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