# Convergence of Mann and Ishikawa Iterative Processes for Some Contractions in Convex Generalized Metric Space

Authors: Youness El Kouch; Jamal Mouline
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this paper, we consider a JS metric space endowed with convexity structure, which will allow us to examine and study convergence of Mann iteration and Ishikawa iteration for Banach type and Chatterjea type contractions defined on JS metric space.


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## Body

## 1. Introduction

Convexity structure in normed spaces and vector topological spaces has stimulated mathematicians to enlarge and generalize this important notion to metric space and generalized metric spaces. This work has been achieved by different ways and various directions. In 1970 W. Takahashi [1] opened up a new line of investigation by introducing an abstract convexity structure in metric space and obtained fixed point theorems for nonexpansive mappings; his results generalize fixed point theorems previously proved by Browder [2] and Kirk [3].In recent years, many papers studied Mann [4] and Ishikawa [5] processes in various ways for single and multivalued contractive mappings. For more details in this direction, we refer the reader to [6–13].In 2015, Jleli and Samet [14] introduced a new generalization of metric space that also recovers dislocated metric spaces [15], b-metric spaces [16], and modular metric spaces with Fatou property [17, 18]. Since then, fixed point theory has been widely studied in this new framework, and the reader may refer to [19–22].In [23], the authors have introduced a Mann type iteration method; they also obtained a result about strongly convergence of this iteration method to a fixed point of nonexpansive mappings on Banach spaces. They provide a result about strong convergence of the iteration method to a common fixed point of two mappings on uniform convex Banach spaces via a new iteration method based on Ishikawa iteration idea.Motivated by above, in this paper, we consider a JS metric space endowed with convexity structure introduced by Takahashi; this will allow us to examine and study convergence of Mann iteration [4] and Ishikawa iteration [5] for Banach type and Chatterjea type contractions defined on JS metric space.In the next, we recall the basic definitions which are used throughout in this paper. For everyx∈X, let us define the set
(1)CD,X,x=xn⊂X:limn⟶+∞Dxn,x=0.Definition 1 (see [14]).
We say thatD:X×X⟶0,+∞ is a JS metric on X if it satisfies the following conditions:
(D1) For every x,y∈X×X, we have Dx,y=0⇒x=y
(D2) For every x,y∈X×X, we have Dx,y=Dy,x
(D3) There exists C>0 such that if x,y∈X×X,xn∈CD,X,x, then
(2)x,y≤Climsupn⟶+∞Dxn,yAlso, we say the pairX,D is a JS metric space.We mention that convergent sequences and Cauchy sequences can be introduced in a similar manner as in metric space.Definition 2 (see [14]).
LetX,D be a JS metric space. Let xn be a sequence in X and x∈X.
(i)
We say thatxn converges to x, if limn⟶∞Dxn,x=0(ii)
We say thatxn is Cauchy sequence if limm,n⟶∞Dxn,xm=0(iii)
X,D is said to be complete, if every Cauchy sequence in X is convergent to some element in XIn the following, we introduce convexity structure defined on JS metric space. Notice that properties and consequences may differ from those arising from convexity structure defined on ordinary metric spaces; this dissimilarity occurs because of absence of triangular inequality in JS metric space.Definition 3.
LetX,D be a JS metric space, and I=0,1. A mapping w:X×X×I⟶X is said to be convex structure on X, if for any x,y,λ∈X×X×I and u∈X, the following inequality holds:
(3)Du,wx,y,λ≤λDu,x+1−λDu,y.IfX,D is a JS metric space with a convex structure w, then X,D,w is called a convex JS metric space.In the sequel, we are interested to study the convergence of Mann and Ishikawa iteration procedures for k-contraction mappings and Chatterjea mappings in the frame of JS metric space. We recall that(i)
Mann iteration procedure [4] is defined by(4)x0∈X,xn+1=wxn,fxn,αn,n=0,1,2⋯(ii)
Ishikawa iteration procedure [5] is defined as follows(5)x0∈X,yn=wxn,fxn,βn,n=0,1,2,⋯,xn+1=wxn,fyn,αn,where αnn≥0 and βnn≥0 satisfy αn,βn∈0,1, for all n∈ℕ.


## 2. Main Results

### 2.1. Convergence of Mann and Ishikawa Iterations for k-Contraction

We first need to state a basic lemma that can be found in [24] page 45.Lemma 4.
Letann and bnn be two bounded real sequences such that bnn converges to b≥0. Then, limsupn⟶∞anbn=blimsupn⟶∞an.Lemma 5.
Letunn and αnn be two positive sequences such that
(6)αn∈0,1andun+1≤αnun,
for alln∈ℕ. Then, we have the following:
(1)
The sequenceunn converges to some r≥0(2)
Iflimsupn⟶∞αn<1, then limn⟶∞un=0Proof.
(1)
The sequenceunn is positive; then, from (6), we get 0≤un+1≤un, for all n∈ℕ. Hence, unn is decreasing and bounded from below. As a result, unn converges to some r≥0(2)
By (1), the sequenceunn has a limit r≥0. Now, suppose that r>0, and by using Lemma 4, we get r≤rlimsupn⟶∞αn; it follows that 1≤limsupn⟶∞αn, which is a contradiction, and we conclude that limn⟶∞un=0.Lemma 6.
Letαnn be a sequence in 0,1.
Ift∈0,1 and ∑i=0∞αi diverges, then ∏i=0∞1−tαi=0.Proof.
For allt∈0,1 and x∈0,1, we have 0≤1−tx≤exp−tx, which yields 0≤∏i=0n1−tαi≤exp−t∑i=0nαi.
Since∑i=0∞αi=∞, we get limn⟶∞exp−t∑i=0nαi=0.
Hence, it follows that∏i=0∞1−tαi=0.Definition 7 (Definition 3.1 in [14]).
LetX,D be a JS metric space and f:X⟶X be a mapping. Let k∈0,1. We say that f is a k-contraction if
(7)Dfx,fy≤kDx,y,foreveryx,y∈X×X.Remark 8.
Letf be a k-contraction on X.
The ordering in inequality (7) is defined in extended real number system. Note that if we have Dfx,fy=∞ for some elements x,y∈X, then Dx,y=∞.
For everyx∈X, let
(8)δD,f,x=supDfix,fjx∣i,j∈ℕ.Theorem 9 (Theorem 3.3 in [14]).
Suppose that the following conditions hold:(i)
X,D is complete(ii)
f is a k-contraction for some k∈0,1(iii)
There existsx0∈X such that δD,f,x0<∞
Then,fnx0 converges to ω∈X, a fixed point of f. Moreover, if ω′∈X is another fixed point of f such that Dω,ω′<∞, then ω=ω′.Our first essential main result is given as follows:Theorem 10.
LetX,D,w be a complete convex JS metric space and f a k-contraction for some k∈0,1. Suppose that there exists x0∈X such that δD,f,x0<∞. If limsupn⟶∞αn<1, then Mann iterative process defined in (4) converges to a fixed point of f.Proof.
All conditions of Theorem9 are satisfied, so Fixf≠∅.
Letω∈Fixf and unn the sequence defined by un+1=wun,fun,αn and u0=x0.
Setdn≔Dun,ω for all n∈ℕ. The proof goes in two steps.
Step 1: we shall prove that dnn is bounded.
For everyp∈ℕ, we have
(9)Du0,fpx0≤δD,f,x0.
Now suppose that for everyp∈ℕ, Dun,fpx0≤δD,f,x0. We have
(10)Dun+1,fpx0=Dwun,fun,αn,fpx0≤αnDun,fpx0+1−αnDfun,fpx0≤αnDun,fpx0+k1−αnDun,fp−1x0≤αn+k1−αnδD,f,x0≤δD,f,x0.
We conclude that(11)Dun,fpx0≤δD,f,x0,foreveryn,p∈ℕ.
By Theorem9, we have fpx0∈CD,X,ω and using D3, we get
(12)dn=Dun,ω≤Climsupp⟶+∞Dun,fpx0.
Hence, by (11), we obtain
(13)dn≤CδD,f,x0,foralln∈ℕ.
Therefore,dnn is bounded.
Step 2: dnn converges to 0 (i.e., unn converges to ω).
From (3) and (7), we have
(14)dn+1=Dω,un+1=Dω,wun,fun,αn≤αnDω,un+1−αnDω,fun≤αnDω,un+1−αnDfω,fun≤αnDω,un+1−αnkDω,un≤αn+1−αnkDω,un≤k+1−kαndn.
Denoteβn=k+1−kαn, and we obtain
(15)dn+1≤βndn.
Sincelimsupn⟶∞αn<1, we get limsupn⟶∞βn<1.
Hence, by Lemma5, it follows that limn⟶+∞dn=limn⟶+∞Dω,un=0.
Therefore, Mann iterative process converges to the fixed pointω.Theorem 11.
LetX,D,w be a complete convex JS metric space and f a k-contraction for some k∈0,1. Suppose that there exists x0∈X such that δD,f,x0<∞. If ∑i=0∞1−αi diverges, then Mann iterative process defined in (4) converges to a fixed point of f.Proof.
We proceed as in the proof of Theorem10. By the first step, the sequence dnn is bounded.
Also by (15), we have
(16)dn+1≤βndnforalln∈ℕ,
withβn=k+1−kαn, which can be rewritten as follows:
(17)βn=1−1−k1−αn.
Hence, inductively, we get(18)dn+1≤d0∏0n1−1−k1−αn.
Then, sinceαnn⊂0,1, 0<1−k<1 and ∑i=0∞1−αi=∞, it follows by Lemma 6 that dn⟶0 as n⟶∞. So, Mann iterative process converges to the fixed point ω, as desired.Remark 12.
Ifx0∈X is an element such that Dx0,x0=0 and αn=0for all n∈ℕ, then Mann iteration is simply reduced to Picard iteration. Indeed, for all n∈ℕ, we have
(19)Dun+1,fn+1x0=Dwun,fun,αn,fn+1x0≤αnDun,fn+1x0+1−αnDfun,fn+1x0.
By (11), take p=n+1, and we obtain Dun,fn+1x0<∞; hence,
(20)Dun+1,fn+1x0≤kDun,fnx0⋮≤kn+1Dx0,x0=0.Therefore, by (D2), we obtain un+1=fn+1x0, for all n∈ℕ. Also, we have u0=x0; thus, un=fnx0, for all n∈ℕ.In [14], Jleli and Samet showed that every b-metric space, with coefficient s≥1, is a JS metric space for C=s, and every modular space with the Fatou property is a JS metric space for C=1. Hence, the following Corollaries are immediate consequences of Theorems 10 and 11.Corollary 13.
LetX,D,s,w be a complete convex b-metric space with constant s≥1 and f a k-contraction for some k∈0,1. If either limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges, then Mann iterative process defined in (4) converges to a fixed point of f.Remark 14.
The condition that there existsx0∈X such that δD,f,x0<∞ is omitted. Since for each x0∈X and for each i,j∈ℕ with i≤j, we have
(21)dfix0,fjx0≤sdfix0,fi+1x0+⋯+sj−idfj−1x0,fjx0≤skidx0,fx0+⋯+sj−ikj−1dx0,fx0≤ski1−skj−i1−skdx0,fx0≤s1−skdx0,fx0.Therefore, for eachx0∈X, we have δD,f,x0<∞.Corollary 15.
LetXρ,ρ be a complete convex modular space and f:X⟶X be a mapping. Suppose that the following conditions hold:
(i)
ρ satisfies the Fatou property(ii)
For somek∈0,1, we have(22)ρfx−fy≤kρx−y,foreveryx,y∈Xρ×Xρ(iii)
There existsx0∈Xρ such that(23)supρfix0−fjx0:i,j∈ℕ<∞(iv)
limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges
Then, Mann iterative process defined in (4) converges to a fixed point of f.In our next theorem, we give a convergence result of Ishikawa iteration for k-contraction considered in a complete convex JS metric space.Theorem 16.
LetX,D,w be a complete convex JS metric space and f a k-contraction for some k∈0,1. Suppose that there exists x0∈X such that δD,f,x0<∞. If either limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges, then Ishikawa iterative process defined in (5) converges to a fixed point of f.Proof.
By Theorem9, there exists some ω∈Fixf. Let xnn and ynn be the sequences defined in (5) and set dn≔Dxn,ω for all n∈ℕ. Now, we complete the proof in the following steps.
Step 1: we shall prove that dnn is bounded.
For everyp∈ℕ, we have Dx0,fpx0≤δD,f,x0.
Now suppose thatDxn,fpx0≤δD,f,x0 for every p∈ℕ.
From (3) and (7), we have
(24)Dxn+1,fpx0=Dwxn,fyn,αn,fpx0≤αnDxn,fpx0+1−αnDfyn,fpx0≤αnDxn,fpx0+k1−αnDyn,fp−1x0.
And similarly,(25)Dyn,fp−1x0=Dwxn,fxn,βn,fp−1x0≤βnDxn,fp−1x0+1−βnDfxn,fp−1x0≤βnDxn,fp−1x0+k1−βnDxn,fp−2x0.
SinceDx0,fp−1x0≤δD,f,x0 and Dx0,fp−2x0≤δD,f,x0, we get
(26)Dyn,fp−1x0≤αn+k1−αnδD,f,x0≤δD,f,x0.
It follows thatDxn+1,fpx0<δD,f,x0. Thus,
(27)Dxn,fpx0≤δD,f,x0,foreveryn,p∈ℕ.
By Theorem9, we have fpx0∈CD,X,ω, and using D3, we get
(28)dn=Dxn,ω≤Climsupp⟶+∞Dxn,fpx0.
Hence, by (27), we obtain
(29)dn≤CδD,f,x0,foralln∈ℕ.
Then,dnn is bounded as desired.
Step 2: dnn converges to 0 (i.e., xnn converges to ω).
From (3) and (7), we have
(30)dn+1=Dω,xn+1=Dω,wxn,fyn,αn≤αnDω,xn+1−αnDω,fyn≤αnDω,xn+1−αnDfω,fyn≤αnDω,xn+1−αnkDω,yn.
We have, for alln∈ℕ,
(31)Dω,yn=Dω,wxn,fxn,βn≤βnDω,xn+1−βnDω,fxn≤βnDω,xn+k1−βnDω,xn≤βn+1−βnkDω,xn≤Dω,xn.
Set(32)γn≔k+1−kαn=αn+1−αnk=1−1−k1−αn.
Now, by (30) and (31), we obtain
(33)dn+1≤γndn.
Therefore,(34)dn+1≤d0∏0n1−1−k1−αn.
Now, iflimsupn⟶∞αn<1, then we get limsupn⟶∞γn<1; hence, by (33) and Lemma 5, it follows that limn⟶+∞dn=limn⟶+∞Dω,xn=0.
If∑i=0∞1−αi=∞, then the same result follows by (34) and Lemma 6.
Consequently, Ishikawa iterative process converges to the fixed pointω.☐We have the following results, which can be deduced from Theorem16.Corollary 17.
LetX,D,s,w be a complete convex b-metric space with constant s≥1 and f a k-contraction for some k∈0,1. If either limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges, then Ishikawa iterative process defined in (4) converges to a fixed point of f.Corollary 18.
LetXρ,ρ be a complete convex modular space and f:X⟶X be a mapping. Suppose that the following conditions hold:
(i)
ρ satisfies the Fatou property(ii)
For somek∈0,1, we have(35)ρfx−fy≤kρx−y,foreveryx,y∈Xρ×Xρ(iii)
There existsx0∈Xρ such that(36)supρfix0−fjx0:i,j∈ℕ<∞(iv)
limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges
Then, Ishikawa iterative process defined in (4) converges to a fixed point of f.Remark 19.
Ifx0∈X is an element such that Dx0,x0=0 and αn=βn=0 for all n∈ℕ, then xnn in Ishikawa iteration is simply reduced to Picard iteration of f2. Indeed, for all n∈ℕ, we have
(37)Dxn+1,f2n+1x0=Dwxn,fyn,αn,f2n+2x0≤αnDxn,f2n+2x0+1−αnDfyn,f2n+2x0.
By (27), we obtain Dxn,f2n+2x0<∞ and Dxn,f2n+1x0<∞; hence,
(38)Dxn+1,f2n+1x0=Dxn+1,f2n+2x0≤kDyn,f2n+1x0≤kDwxn,fxn,βn,f2n+1x0≤kβnDxn,f2n+1x0+k1−βnDfxn,f2n+1x0≤kDfxn,f2n+1x0≤k2Dxn,f2nx0⋮≤k2n+1Dx0,x0=0.
Therefore, by (D2), we obtain xn=f2nx0 and yn=f2n+1x0, for all n∈ℕ.Example 1.
LetX=0,1 and let D be the JS metric defined by the following:
(39)Dy,x=Dx,y=∞,ifx=1ory=1,Dx,y=x+y,ifx≠1andy≠1.Definew:X×X×0,1⟶X by wx,y,λ=λx+1−λy; then, X,D is a complete convex JS metric space.Now, consider the mappingT:0,1⟶0,1 given by
(40)T1=1,Tx=6x7,ifx≠1.The mappingT is a k-contraction with k=6/7, and for every x∈0,1, we have Tx≤x. In addition, Dx,y<1 for all x,y∈0,1; consequently, δD,f,x<∞.Hence, by Theorems9, 11, and 16, T has a fixed point ω∈X that can be approximated by Picard iteration, Mann iteration with α=1/n+1, and Ishikawa iteration with αn=βn=1/n+1, and we take for initial value x0=0.5.The convergence behaviors of the three iterative schemes are displayed in Table1 and graphically in Figure 1.Table 1
Convergence behavior of the three iterative processes to the fixed point.
No. of iterationsPicard iterationMann iterationIshikawa iteration00.50000000000.50000000000.500000000010.42857142850.46428571420.448979591820.36734693870.42006802720.381785367230.31486880460.37506073850.314583351040.26988754680.33219665410.253977905450.23133218300.29264943340.202145679860.19828472830.25681480890.159207613770.16995833850.22471295780.124380948280.14567857580.19617797900.096552708890.12486735070.17095509600.0745623368⋮⋮⋮⋮450.00048569590.00085121810.0000002419460.00041631080.00073220280.0000001797470.00035683780.00062978150.0000001335480.00030586090.00054164880.0000009914490.00026216650.00046581800.0000007360500.00022471420.00040057740.0000005463Figure 1
Convergence behavior of the three iterative processes to the fixed point.


### 2.2. Convergence of Mann and Ishikawa Iterations for Chatterjea Contraction

Definition 20.
LetX,D be a JS metric space, f:X⟶X be a mapping, and k∈0,1/2. We say that f is Chatterjea contraction if
(41)Dfx,fy≤kDy,fx+Dx,fy,foreveryx,y∈X×X.In the next, we need the following result proved by El Kouch and Marhrani, in [19].Theorem 21 (Theorem 3.9 in [19]).
LetX,D be a complete generalized metric space, k∈0,1/2, and let f:X⟶X be a Chatterjea mapping.
If there exists a pointx0∈X such that δD,f,x0<∞, then the sequence fnx0n converges to some ω∈X. Moreover, if Dx0,fω<∞, then ω is a fixed point of f, and for any fixed point ω′ of f such that Dω,ω′<∞, we have ω=ω′.Now, we give a convergence result of Mann iterative process for Chatterjea mapping in complete convex JS metric space.Theorem 22.
LetX,D,w be a complete convex JS metric space and f:X⟶X be a Chatterjea mapping. Let xnn be the sequence defined by xn+1=wxn,fxn,αn, x0∈X. Suppose that the following conditions hold:
(i)
δD,f,x0<∞(ii)
For alln,m∈ℕ, Dfxn,fmx0<∞(iii)
fnx0nconverges to some ω∈X satisfying Dx0,fω<∞(iv)
limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges
Then, Mann iterative process defined in (4) converges to the fixed point ω.Proof.
All conditions of Theorem21 are satisfied and ω∈Fixf. Set dn≔Dxn,ω for all n∈ℕ. The proof goes in three steps.
Step 1: set δ=δD,f,x0, and we will prove that
(42)Foreveryn∈ℕ,thereexistspn∈ℕsuchthatm≥pnimpliesDxn,fmx0≤δ.
We will prove the statement (42) by induction on n.
Forn=0, we have Dx0,fmx0≤δ for all m∈ℕ, so we take p0=0.
Suppose that (42) is true for some n∈ℕ, that is,
(43)Thereexistspn∈ℕsuchthatm≥pnimpliesDxn,fmx0≤δ.
Now, we will prove that it is true again forn+1.
From (3), we obtain, for all m≥pn,
(44)Dxn+1,fmx0=Dwxn,fxn,αn,fmx0≤αnDxn,fmx0+1−αnDfxn,fmx0.
By successive application of the inequality (41), we have
(45)Dfxn,fmx0≤kDxn,fmx0+Dfxn,fm−1x0≤kDxn,fmx0+kDfxn,fm−1x0≤kDxn,fmx0+k2Dxn,fm−1x0+k2Dfxn,fm−2x0≤kDxn,fmx0+k2Dxn,fm−1x0+⋯+km−pnDxn,fpn+1x0+km−pn+1Dxn,fpnx0+km−pn+1Dfxn,fpn−1x0.
Since for allm≥pn, we have Dxn,fmx0≤δ. Thus, from (45), we get
(46)Dfxn,fmx0≤kδ1+k+⋯+km−pn+km−pn+1Dfxn,fpn−1x0≤kδ1−km−pn+11−k+km−pn+1Dfxn,fpn−1x0≤kδ1−k+km−pn+1Dfxn,fpn−1x0−kδ1−k.
We havek∈0,1/2 and Dfxn,fpn−1x0<∞; hence,
(47)limm⟶+∞km−pn+1Dfxn,fpn−1x0−kδ1−k=0.
Using (47) and for δ1−2k/1−k>0, there exists N∈ℕ such that for all m≥N, we have
(48)km−pn+1Dfxn,fpn−1x0−kδ1−k<δ1−2k1−k.
We takepn+1=maxpn,N; thus, from (48) and (46), we obtain, for all m≥pn+1,
(49)Dfxn,fmx0≤kδ1−k+δ1−2k1−k=δ.
From (49) and (43), we have, for all m≥pn+1, Dxn,fmx0≤δ and Dfxn,fmx0≤δ.
Let us now return to (44), and we obtain, for all m≥pn+1, Dxn+1,fmx0≤δ.
Hence, we conclude that the statement (42) is true.
Step 2: we shall prove that dnn is bounded.
By step 1, we have, for alln∈ℕ and every m≥pn,
(50)Dxn,fmx0≤δ.
Using Theorem21, we have fmx0 converges to ω and by D3, and we obtain
(51)dn=Dxn,ω≤Climsupm⟶+∞Dxn,fmx0.
Hence, by (50), we obtain
(52)dn≤Cδ,foralln∈ℕ.
Therefore,dnn is bounded.
Step 3: dnn converges to 0 (i.e., xnn converges to ω).
From (3), we have
(53)dn+1=Dω,xn+1=Dω,wxn,fxn,αn≤αnDω,xn+1−αnDω,fxn.
It follows from (41) that
(54)Dω,fxn=Dfω,fxn≤kDω,fxn+Dfω,xn≤kDω,fxn+kDω,xn.
Hence, we get(55)Dω,fxn≤k1−kDω,xn.
Setλ=k/1−k∈0,1, and we obtain from (53) and (55)
(56)dn+1≤γndn,
withγn=λ+1−λαn=1−1−λ1−αn.
Therefore,(57)dn+1≤d0∏0n1−1−λ1−αn.
Now, iflimsupn⟶∞αn<1, then we get limsupn⟶∞γn<1.
Hence, by (56) and Lemma 5, it follows that
(58)limn⟶+∞dn=limn⟶+∞Dω,un=0.
If∑i=0∞1−αi=∞, then by (57) and Lemma 6, we have the same result.
Consequently, Mann iterative process converges to the fixed pointω.☐Theorem 23.
LetX,D,w be a complete convex JS metric space and f:X⟶X be a Chatterjea mapping. Let xnn and ynn be the sequences defined in (5), x0∈X. Suppose that the following conditions hold:
(i)
δD,f,x0<∞(ii)
For alln,m∈ℕ, Dfxn,fmx0<∞ and Dfyn,fmx0<∞(iii)
fnx0nconverges to some ω∈X satisfying Dx0,fω<∞(iv)
limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges
Then, Ishikawa iterative process defined in (4) converges to the fixed point ω.Proof.
By Theorem21, we have ω∈Fixf. Let xnn and ynn be the sequences defined in (5) and set dn≔Dxn,ω for all n∈ℕ. Now, we complete the proof in the following steps.
Step 1: we prove that
(59)Foreveryn∈ℕ,thereexistspn∈ℕsuchthatm≥pnimpliesDxn,fmx0≤δ.
We will prove the statement (59) by induction on n.
Forn=0, we have Dx0,fmx0≤δ for all m∈ℕ, so we take p0=0.
Suppose that (59) is true for some n∈ℕ, that is,
(60)Thereexistspn∈ℕsuchthatm≥pnimpliesDxn,fmx0≤δ.
From (3), we obtain, for all m≥pn,
(61)Dxn+1,fmx0=Dwxn,fyn,αn,fmx0≤αnDxn,fmx0+1−αnDfyn,fmx0.
By (41), we get, for all m≥pn,
(62)Dfyn,fmx0≤kDyn,fmx0+Dfyn,fm−1x0.
On the other hand,(63)Dyn,fmx0=Dwxn,fxn,αn,fmx0≤αnDxn,fmx0+1−αnDfxn,fmx0.
According to (49), there exists sn=maxpn,N such that for all m≥sn,
(64)Dfxn,fmx0≤δ.
Then, from (60), (63), and (64), we have, for all m≥sn,
(65)Dyn,fmx0≤δ.
Hence, from (65) and (62), we get, for all m≥sn,
(66)Dfyn,fmx0≤kDyn,fmx0+k2Dyn,fm−1x0+⋯+km−snDyn,fsn+1x0+km−sn+1Dyn,fsnx0+km−sn+1Dfyn,fsn−1x0≤kδ1+k+⋯+km−sn+km−sn+1Dfyn,fsn−1x0≤kδ1−k+km−sn+1Dfyn,fsn−1x0−kδ1−k.
We havek∈0,1/2 and Dfyn,fsn−1x0<∞; hence,
(67)limm⟶+∞km−sn+1Dfxn,fsn−1x0−kδ1−k=0.
Using (67), and for δ1−2k/1−k>0, there exists N′∈ℕ such that for all m≥N′, we have
(68)km−sn+1Dfxn,fsn−1x0−kδ1−k<δ1−2k1−k.
If we takepn+1=maxsn,N′, then from (68) and (66), we obtain, for all m≥pn+1,
(69)Dfyn,fmx0≤δ.
Now, by (61), (60), and (69), we obtain, for all m≥pn+1,
(70)Dxn+1,fmx0≤δ.
It follows that the statement (59) is true.
Step 2: we shall show that dnn is bounded.
We have for everyn∈ℕ, there exists pn∈ℕ such that for all m≥pn,
(71)Dxn,fmx0≤δ.
Using Theorem21, fmx0 converges to ω, and by D3, we obtain
(72)dn=Dxn,ω≤Climsupm⟶+∞Dxn,fmx0≤Cδ.
Then,dnn is bounded.
Step 3: dnn converges to 0 (i.e., xnn converges to ω).
By (3) and (7), we have
(73)dn+1=Dω,xn+1=Dω,wxn,fyn,αn≤αnDω,xn+1−αnDω,fyn.
It follows from (41) that
(74)Dω,fyn=Dfω,fyn≤kDω,fyn+Dfω,yn≤kDω,fyn+kDω,yn.
Hence, we get(75)Dω,fyn≤k1−kDω,yn.≤λDω,yn,
withλ=k/1−k∈0,1.
Similarly, we have(76)Dω,fxn≤λDω,xn.
We have, for alln∈ℕ,
(77)Dω,yn=Dω,wxn,fxn,βn≤βnDω,xn+1−βnDω,fxn≤βnDω,xn+λ1−βnDω,xn≤βn+1−βnλDω,xn≤Dω,xn.
Set(78)γn≔λ+1−λαn=αn+1−αnλ=1−1−λ1−αn.
Now, by (73), (75), and (77), we obtain
(79)dn+1≤γndn.
Therefore,(80)dn+1≤d0∏0n1−1−λ1−αn.
Now, iflimsupn⟶∞αn<1, then we get limsupn⟶∞γn<1; hence, by (79) and Lemma 6, it follows that limn⟶+∞dn=limn⟶+∞Dω,xn=0.
If∑i=0∞1−αi=∞, then the same result follows by (80) and Lemma 6.
Consequently, Ishikawa iterative process converges to the fixed pointω.☐


## 2.1. Convergence of Mann and Ishikawa Iterations for k-Contraction

We first need to state a basic lemma that can be found in [24] page 45.Lemma 4.
Letann and bnn be two bounded real sequences such that bnn converges to b≥0. Then, limsupn⟶∞anbn=blimsupn⟶∞an.Lemma 5.
Letunn and αnn be two positive sequences such that
(6)αn∈0,1andun+1≤αnun,
for alln∈ℕ. Then, we have the following:
(1)
The sequenceunn converges to some r≥0(2)
Iflimsupn⟶∞αn<1, then limn⟶∞un=0Proof.
(1)
The sequenceunn is positive; then, from (6), we get 0≤un+1≤un, for all n∈ℕ. Hence, unn is decreasing and bounded from below. As a result, unn converges to some r≥0(2)
By (1), the sequenceunn has a limit r≥0. Now, suppose that r>0, and by using Lemma 4, we get r≤rlimsupn⟶∞αn; it follows that 1≤limsupn⟶∞αn, which is a contradiction, and we conclude that limn⟶∞un=0.Lemma 6.
Letαnn be a sequence in 0,1.
Ift∈0,1 and ∑i=0∞αi diverges, then ∏i=0∞1−tαi=0.Proof.
For allt∈0,1 and x∈0,1, we have 0≤1−tx≤exp−tx, which yields 0≤∏i=0n1−tαi≤exp−t∑i=0nαi.
Since∑i=0∞αi=∞, we get limn⟶∞exp−t∑i=0nαi=0.
Hence, it follows that∏i=0∞1−tαi=0.Definition 7 (Definition 3.1 in [14]).
LetX,D be a JS metric space and f:X⟶X be a mapping. Let k∈0,1. We say that f is a k-contraction if
(7)Dfx,fy≤kDx,y,foreveryx,y∈X×X.Remark 8.
Letf be a k-contraction on X.
The ordering in inequality (7) is defined in extended real number system. Note that if we have Dfx,fy=∞ for some elements x,y∈X, then Dx,y=∞.
For everyx∈X, let
(8)δD,f,x=supDfix,fjx∣i,j∈ℕ.Theorem 9 (Theorem 3.3 in [14]).
Suppose that the following conditions hold:(i)
X,D is complete(ii)
f is a k-contraction for some k∈0,1(iii)
There existsx0∈X such that δD,f,x0<∞
Then,fnx0 converges to ω∈X, a fixed point of f. Moreover, if ω′∈X is another fixed point of f such that Dω,ω′<∞, then ω=ω′.Our first essential main result is given as follows:Theorem 10.
LetX,D,w be a complete convex JS metric space and f a k-contraction for some k∈0,1. Suppose that there exists x0∈X such that δD,f,x0<∞. If limsupn⟶∞αn<1, then Mann iterative process defined in (4) converges to a fixed point of f.Proof.
All conditions of Theorem9 are satisfied, so Fixf≠∅.
Letω∈Fixf and unn the sequence defined by un+1=wun,fun,αn and u0=x0.
Setdn≔Dun,ω for all n∈ℕ. The proof goes in two steps.
Step 1: we shall prove that dnn is bounded.
For everyp∈ℕ, we have
(9)Du0,fpx0≤δD,f,x0.
Now suppose that for everyp∈ℕ, Dun,fpx0≤δD,f,x0. We have
(10)Dun+1,fpx0=Dwun,fun,αn,fpx0≤αnDun,fpx0+1−αnDfun,fpx0≤αnDun,fpx0+k1−αnDun,fp−1x0≤αn+k1−αnδD,f,x0≤δD,f,x0.
We conclude that(11)Dun,fpx0≤δD,f,x0,foreveryn,p∈ℕ.
By Theorem9, we have fpx0∈CD,X,ω and using D3, we get
(12)dn=Dun,ω≤Climsupp⟶+∞Dun,fpx0.
Hence, by (11), we obtain
(13)dn≤CδD,f,x0,foralln∈ℕ.
Therefore,dnn is bounded.
Step 2: dnn converges to 0 (i.e., unn converges to ω).
From (3) and (7), we have
(14)dn+1=Dω,un+1=Dω,wun,fun,αn≤αnDω,un+1−αnDω,fun≤αnDω,un+1−αnDfω,fun≤αnDω,un+1−αnkDω,un≤αn+1−αnkDω,un≤k+1−kαndn.
Denoteβn=k+1−kαn, and we obtain
(15)dn+1≤βndn.
Sincelimsupn⟶∞αn<1, we get limsupn⟶∞βn<1.
Hence, by Lemma5, it follows that limn⟶+∞dn=limn⟶+∞Dω,un=0.
Therefore, Mann iterative process converges to the fixed pointω.Theorem 11.
LetX,D,w be a complete convex JS metric space and f a k-contraction for some k∈0,1. Suppose that there exists x0∈X such that δD,f,x0<∞. If ∑i=0∞1−αi diverges, then Mann iterative process defined in (4) converges to a fixed point of f.Proof.
We proceed as in the proof of Theorem10. By the first step, the sequence dnn is bounded.
Also by (15), we have
(16)dn+1≤βndnforalln∈ℕ,
withβn=k+1−kαn, which can be rewritten as follows:
(17)βn=1−1−k1−αn.
Hence, inductively, we get(18)dn+1≤d0∏0n1−1−k1−αn.
Then, sinceαnn⊂0,1, 0<1−k<1 and ∑i=0∞1−αi=∞, it follows by Lemma 6 that dn⟶0 as n⟶∞. So, Mann iterative process converges to the fixed point ω, as desired.Remark 12.
Ifx0∈X is an element such that Dx0,x0=0 and αn=0for all n∈ℕ, then Mann iteration is simply reduced to Picard iteration. Indeed, for all n∈ℕ, we have
(19)Dun+1,fn+1x0=Dwun,fun,αn,fn+1x0≤αnDun,fn+1x0+1−αnDfun,fn+1x0.
By (11), take p=n+1, and we obtain Dun,fn+1x0<∞; hence,
(20)Dun+1,fn+1x0≤kDun,fnx0⋮≤kn+1Dx0,x0=0.Therefore, by (D2), we obtain un+1=fn+1x0, for all n∈ℕ. Also, we have u0=x0; thus, un=fnx0, for all n∈ℕ.In [14], Jleli and Samet showed that every b-metric space, with coefficient s≥1, is a JS metric space for C=s, and every modular space with the Fatou property is a JS metric space for C=1. Hence, the following Corollaries are immediate consequences of Theorems 10 and 11.Corollary 13.
LetX,D,s,w be a complete convex b-metric space with constant s≥1 and f a k-contraction for some k∈0,1. If either limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges, then Mann iterative process defined in (4) converges to a fixed point of f.Remark 14.
The condition that there existsx0∈X such that δD,f,x0<∞ is omitted. Since for each x0∈X and for each i,j∈ℕ with i≤j, we have
(21)dfix0,fjx0≤sdfix0,fi+1x0+⋯+sj−idfj−1x0,fjx0≤skidx0,fx0+⋯+sj−ikj−1dx0,fx0≤ski1−skj−i1−skdx0,fx0≤s1−skdx0,fx0.Therefore, for eachx0∈X, we have δD,f,x0<∞.Corollary 15.
LetXρ,ρ be a complete convex modular space and f:X⟶X be a mapping. Suppose that the following conditions hold:
(i)
ρ satisfies the Fatou property(ii)
For somek∈0,1, we have(22)ρfx−fy≤kρx−y,foreveryx,y∈Xρ×Xρ(iii)
There existsx0∈Xρ such that(23)supρfix0−fjx0:i,j∈ℕ<∞(iv)
limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges
Then, Mann iterative process defined in (4) converges to a fixed point of f.In our next theorem, we give a convergence result of Ishikawa iteration for k-contraction considered in a complete convex JS metric space.Theorem 16.
LetX,D,w be a complete convex JS metric space and f a k-contraction for some k∈0,1. Suppose that there exists x0∈X such that δD,f,x0<∞. If either limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges, then Ishikawa iterative process defined in (5) converges to a fixed point of f.Proof.
By Theorem9, there exists some ω∈Fixf. Let xnn and ynn be the sequences defined in (5) and set dn≔Dxn,ω for all n∈ℕ. Now, we complete the proof in the following steps.
Step 1: we shall prove that dnn is bounded.
For everyp∈ℕ, we have Dx0,fpx0≤δD,f,x0.
Now suppose thatDxn,fpx0≤δD,f,x0 for every p∈ℕ.
From (3) and (7), we have
(24)Dxn+1,fpx0=Dwxn,fyn,αn,fpx0≤αnDxn,fpx0+1−αnDfyn,fpx0≤αnDxn,fpx0+k1−αnDyn,fp−1x0.
And similarly,(25)Dyn,fp−1x0=Dwxn,fxn,βn,fp−1x0≤βnDxn,fp−1x0+1−βnDfxn,fp−1x0≤βnDxn,fp−1x0+k1−βnDxn,fp−2x0.
SinceDx0,fp−1x0≤δD,f,x0 and Dx0,fp−2x0≤δD,f,x0, we get
(26)Dyn,fp−1x0≤αn+k1−αnδD,f,x0≤δD,f,x0.
It follows thatDxn+1,fpx0<δD,f,x0. Thus,
(27)Dxn,fpx0≤δD,f,x0,foreveryn,p∈ℕ.
By Theorem9, we have fpx0∈CD,X,ω, and using D3, we get
(28)dn=Dxn,ω≤Climsupp⟶+∞Dxn,fpx0.
Hence, by (27), we obtain
(29)dn≤CδD,f,x0,foralln∈ℕ.
Then,dnn is bounded as desired.
Step 2: dnn converges to 0 (i.e., xnn converges to ω).
From (3) and (7), we have
(30)dn+1=Dω,xn+1=Dω,wxn,fyn,αn≤αnDω,xn+1−αnDω,fyn≤αnDω,xn+1−αnDfω,fyn≤αnDω,xn+1−αnkDω,yn.
We have, for alln∈ℕ,
(31)Dω,yn=Dω,wxn,fxn,βn≤βnDω,xn+1−βnDω,fxn≤βnDω,xn+k1−βnDω,xn≤βn+1−βnkDω,xn≤Dω,xn.
Set(32)γn≔k+1−kαn=αn+1−αnk=1−1−k1−αn.
Now, by (30) and (31), we obtain
(33)dn+1≤γndn.
Therefore,(34)dn+1≤d0∏0n1−1−k1−αn.
Now, iflimsupn⟶∞αn<1, then we get limsupn⟶∞γn<1; hence, by (33) and Lemma 5, it follows that limn⟶+∞dn=limn⟶+∞Dω,xn=0.
If∑i=0∞1−αi=∞, then the same result follows by (34) and Lemma 6.
Consequently, Ishikawa iterative process converges to the fixed pointω.☐We have the following results, which can be deduced from Theorem16.Corollary 17.
LetX,D,s,w be a complete convex b-metric space with constant s≥1 and f a k-contraction for some k∈0,1. If either limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges, then Ishikawa iterative process defined in (4) converges to a fixed point of f.Corollary 18.
LetXρ,ρ be a complete convex modular space and f:X⟶X be a mapping. Suppose that the following conditions hold:
(i)
ρ satisfies the Fatou property(ii)
For somek∈0,1, we have(35)ρfx−fy≤kρx−y,foreveryx,y∈Xρ×Xρ(iii)
There existsx0∈Xρ such that(36)supρfix0−fjx0:i,j∈ℕ<∞(iv)
limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges
Then, Ishikawa iterative process defined in (4) converges to a fixed point of f.Remark 19.
Ifx0∈X is an element such that Dx0,x0=0 and αn=βn=0 for all n∈ℕ, then xnn in Ishikawa iteration is simply reduced to Picard iteration of f2. Indeed, for all n∈ℕ, we have
(37)Dxn+1,f2n+1x0=Dwxn,fyn,αn,f2n+2x0≤αnDxn,f2n+2x0+1−αnDfyn,f2n+2x0.
By (27), we obtain Dxn,f2n+2x0<∞ and Dxn,f2n+1x0<∞; hence,
(38)Dxn+1,f2n+1x0=Dxn+1,f2n+2x0≤kDyn,f2n+1x0≤kDwxn,fxn,βn,f2n+1x0≤kβnDxn,f2n+1x0+k1−βnDfxn,f2n+1x0≤kDfxn,f2n+1x0≤k2Dxn,f2nx0⋮≤k2n+1Dx0,x0=0.
Therefore, by (D2), we obtain xn=f2nx0 and yn=f2n+1x0, for all n∈ℕ.Example 1.
LetX=0,1 and let D be the JS metric defined by the following:
(39)Dy,x=Dx,y=∞,ifx=1ory=1,Dx,y=x+y,ifx≠1andy≠1.Definew:X×X×0,1⟶X by wx,y,λ=λx+1−λy; then, X,D is a complete convex JS metric space.Now, consider the mappingT:0,1⟶0,1 given by
(40)T1=1,Tx=6x7,ifx≠1.The mappingT is a k-contraction with k=6/7, and for every x∈0,1, we have Tx≤x. In addition, Dx,y<1 for all x,y∈0,1; consequently, δD,f,x<∞.Hence, by Theorems9, 11, and 16, T has a fixed point ω∈X that can be approximated by Picard iteration, Mann iteration with α=1/n+1, and Ishikawa iteration with αn=βn=1/n+1, and we take for initial value x0=0.5.The convergence behaviors of the three iterative schemes are displayed in Table1 and graphically in Figure 1.Table 1
Convergence behavior of the three iterative processes to the fixed point.
No. of iterationsPicard iterationMann iterationIshikawa iteration00.50000000000.50000000000.500000000010.42857142850.46428571420.448979591820.36734693870.42006802720.381785367230.31486880460.37506073850.314583351040.26988754680.33219665410.253977905450.23133218300.29264943340.202145679860.19828472830.25681480890.159207613770.16995833850.22471295780.124380948280.14567857580.19617797900.096552708890.12486735070.17095509600.0745623368⋮⋮⋮⋮450.00048569590.00085121810.0000002419460.00041631080.00073220280.0000001797470.00035683780.00062978150.0000001335480.00030586090.00054164880.0000009914490.00026216650.00046581800.0000007360500.00022471420.00040057740.0000005463Figure 1
Convergence behavior of the three iterative processes to the fixed point.


## 2.2. Convergence of Mann and Ishikawa Iterations for Chatterjea Contraction

Definition 20.
LetX,D be a JS metric space, f:X⟶X be a mapping, and k∈0,1/2. We say that f is Chatterjea contraction if
(41)Dfx,fy≤kDy,fx+Dx,fy,foreveryx,y∈X×X.In the next, we need the following result proved by El Kouch and Marhrani, in [19].Theorem 21 (Theorem 3.9 in [19]).
LetX,D be a complete generalized metric space, k∈0,1/2, and let f:X⟶X be a Chatterjea mapping.
If there exists a pointx0∈X such that δD,f,x0<∞, then the sequence fnx0n converges to some ω∈X. Moreover, if Dx0,fω<∞, then ω is a fixed point of f, and for any fixed point ω′ of f such that Dω,ω′<∞, we have ω=ω′.Now, we give a convergence result of Mann iterative process for Chatterjea mapping in complete convex JS metric space.Theorem 22.
LetX,D,w be a complete convex JS metric space and f:X⟶X be a Chatterjea mapping. Let xnn be the sequence defined by xn+1=wxn,fxn,αn, x0∈X. Suppose that the following conditions hold:
(i)
δD,f,x0<∞(ii)
For alln,m∈ℕ, Dfxn,fmx0<∞(iii)
fnx0nconverges to some ω∈X satisfying Dx0,fω<∞(iv)
limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges
Then, Mann iterative process defined in (4) converges to the fixed point ω.Proof.
All conditions of Theorem21 are satisfied and ω∈Fixf. Set dn≔Dxn,ω for all n∈ℕ. The proof goes in three steps.
Step 1: set δ=δD,f,x0, and we will prove that
(42)Foreveryn∈ℕ,thereexistspn∈ℕsuchthatm≥pnimpliesDxn,fmx0≤δ.
We will prove the statement (42) by induction on n.
Forn=0, we have Dx0,fmx0≤δ for all m∈ℕ, so we take p0=0.
Suppose that (42) is true for some n∈ℕ, that is,
(43)Thereexistspn∈ℕsuchthatm≥pnimpliesDxn,fmx0≤δ.
Now, we will prove that it is true again forn+1.
From (3), we obtain, for all m≥pn,
(44)Dxn+1,fmx0=Dwxn,fxn,αn,fmx0≤αnDxn,fmx0+1−αnDfxn,fmx0.
By successive application of the inequality (41), we have
(45)Dfxn,fmx0≤kDxn,fmx0+Dfxn,fm−1x0≤kDxn,fmx0+kDfxn,fm−1x0≤kDxn,fmx0+k2Dxn,fm−1x0+k2Dfxn,fm−2x0≤kDxn,fmx0+k2Dxn,fm−1x0+⋯+km−pnDxn,fpn+1x0+km−pn+1Dxn,fpnx0+km−pn+1Dfxn,fpn−1x0.
Since for allm≥pn, we have Dxn,fmx0≤δ. Thus, from (45), we get
(46)Dfxn,fmx0≤kδ1+k+⋯+km−pn+km−pn+1Dfxn,fpn−1x0≤kδ1−km−pn+11−k+km−pn+1Dfxn,fpn−1x0≤kδ1−k+km−pn+1Dfxn,fpn−1x0−kδ1−k.
We havek∈0,1/2 and Dfxn,fpn−1x0<∞; hence,
(47)limm⟶+∞km−pn+1Dfxn,fpn−1x0−kδ1−k=0.
Using (47) and for δ1−2k/1−k>0, there exists N∈ℕ such that for all m≥N, we have
(48)km−pn+1Dfxn,fpn−1x0−kδ1−k<δ1−2k1−k.
We takepn+1=maxpn,N; thus, from (48) and (46), we obtain, for all m≥pn+1,
(49)Dfxn,fmx0≤kδ1−k+δ1−2k1−k=δ.
From (49) and (43), we have, for all m≥pn+1, Dxn,fmx0≤δ and Dfxn,fmx0≤δ.
Let us now return to (44), and we obtain, for all m≥pn+1, Dxn+1,fmx0≤δ.
Hence, we conclude that the statement (42) is true.
Step 2: we shall prove that dnn is bounded.
By step 1, we have, for alln∈ℕ and every m≥pn,
(50)Dxn,fmx0≤δ.
Using Theorem21, we have fmx0 converges to ω and by D3, and we obtain
(51)dn=Dxn,ω≤Climsupm⟶+∞Dxn,fmx0.
Hence, by (50), we obtain
(52)dn≤Cδ,foralln∈ℕ.
Therefore,dnn is bounded.
Step 3: dnn converges to 0 (i.e., xnn converges to ω).
From (3), we have
(53)dn+1=Dω,xn+1=Dω,wxn,fxn,αn≤αnDω,xn+1−αnDω,fxn.
It follows from (41) that
(54)Dω,fxn=Dfω,fxn≤kDω,fxn+Dfω,xn≤kDω,fxn+kDω,xn.
Hence, we get(55)Dω,fxn≤k1−kDω,xn.
Setλ=k/1−k∈0,1, and we obtain from (53) and (55)
(56)dn+1≤γndn,
withγn=λ+1−λαn=1−1−λ1−αn.
Therefore,(57)dn+1≤d0∏0n1−1−λ1−αn.
Now, iflimsupn⟶∞αn<1, then we get limsupn⟶∞γn<1.
Hence, by (56) and Lemma 5, it follows that
(58)limn⟶+∞dn=limn⟶+∞Dω,un=0.
If∑i=0∞1−αi=∞, then by (57) and Lemma 6, we have the same result.
Consequently, Mann iterative process converges to the fixed pointω.☐Theorem 23.
LetX,D,w be a complete convex JS metric space and f:X⟶X be a Chatterjea mapping. Let xnn and ynn be the sequences defined in (5), x0∈X. Suppose that the following conditions hold:
(i)
δD,f,x0<∞(ii)
For alln,m∈ℕ, Dfxn,fmx0<∞ and Dfyn,fmx0<∞(iii)
fnx0nconverges to some ω∈X satisfying Dx0,fω<∞(iv)
limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges
Then, Ishikawa iterative process defined in (4) converges to the fixed point ω.Proof.
By Theorem21, we have ω∈Fixf. Let xnn and ynn be the sequences defined in (5) and set dn≔Dxn,ω for all n∈ℕ. Now, we complete the proof in the following steps.
Step 1: we prove that
(59)Foreveryn∈ℕ,thereexistspn∈ℕsuchthatm≥pnimpliesDxn,fmx0≤δ.
We will prove the statement (59) by induction on n.
Forn=0, we have Dx0,fmx0≤δ for all m∈ℕ, so we take p0=0.
Suppose that (59) is true for some n∈ℕ, that is,
(60)Thereexistspn∈ℕsuchthatm≥pnimpliesDxn,fmx0≤δ.
From (3), we obtain, for all m≥pn,
(61)Dxn+1,fmx0=Dwxn,fyn,αn,fmx0≤αnDxn,fmx0+1−αnDfyn,fmx0.
By (41), we get, for all m≥pn,
(62)Dfyn,fmx0≤kDyn,fmx0+Dfyn,fm−1x0.
On the other hand,(63)Dyn,fmx0=Dwxn,fxn,αn,fmx0≤αnDxn,fmx0+1−αnDfxn,fmx0.
According to (49), there exists sn=maxpn,N such that for all m≥sn,
(64)Dfxn,fmx0≤δ.
Then, from (60), (63), and (64), we have, for all m≥sn,
(65)Dyn,fmx0≤δ.
Hence, from (65) and (62), we get, for all m≥sn,
(66)Dfyn,fmx0≤kDyn,fmx0+k2Dyn,fm−1x0+⋯+km−snDyn,fsn+1x0+km−sn+1Dyn,fsnx0+km−sn+1Dfyn,fsn−1x0≤kδ1+k+⋯+km−sn+km−sn+1Dfyn,fsn−1x0≤kδ1−k+km−sn+1Dfyn,fsn−1x0−kδ1−k.
We havek∈0,1/2 and Dfyn,fsn−1x0<∞; hence,
(67)limm⟶+∞km−sn+1Dfxn,fsn−1x0−kδ1−k=0.
Using (67), and for δ1−2k/1−k>0, there exists N′∈ℕ such that for all m≥N′, we have
(68)km−sn+1Dfxn,fsn−1x0−kδ1−k<δ1−2k1−k.
If we takepn+1=maxsn,N′, then from (68) and (66), we obtain, for all m≥pn+1,
(69)Dfyn,fmx0≤δ.
Now, by (61), (60), and (69), we obtain, for all m≥pn+1,
(70)Dxn+1,fmx0≤δ.
It follows that the statement (59) is true.
Step 2: we shall show that dnn is bounded.
We have for everyn∈ℕ, there exists pn∈ℕ such that for all m≥pn,
(71)Dxn,fmx0≤δ.
Using Theorem21, fmx0 converges to ω, and by D3, we obtain
(72)dn=Dxn,ω≤Climsupm⟶+∞Dxn,fmx0≤Cδ.
Then,dnn is bounded.
Step 3: dnn converges to 0 (i.e., xnn converges to ω).
By (3) and (7), we have
(73)dn+1=Dω,xn+1=Dω,wxn,fyn,αn≤αnDω,xn+1−αnDω,fyn.
It follows from (41) that
(74)Dω,fyn=Dfω,fyn≤kDω,fyn+Dfω,yn≤kDω,fyn+kDω,yn.
Hence, we get(75)Dω,fyn≤k1−kDω,yn.≤λDω,yn,
withλ=k/1−k∈0,1.
Similarly, we have(76)Dω,fxn≤λDω,xn.
We have, for alln∈ℕ,
(77)Dω,yn=Dω,wxn,fxn,βn≤βnDω,xn+1−βnDω,fxn≤βnDω,xn+λ1−βnDω,xn≤βn+1−βnλDω,xn≤Dω,xn.
Set(78)γn≔λ+1−λαn=αn+1−αnλ=1−1−λ1−αn.
Now, by (73), (75), and (77), we obtain
(79)dn+1≤γndn.
Therefore,(80)dn+1≤d0∏0n1−1−λ1−αn.
Now, iflimsupn⟶∞αn<1, then we get limsupn⟶∞γn<1; hence, by (79) and Lemma 6, it follows that limn⟶+∞dn=limn⟶+∞Dω,xn=0.
If∑i=0∞1−αi=∞, then the same result follows by (80) and Lemma 6.
Consequently, Ishikawa iterative process converges to the fixed pointω.☐


## 3. Applications

In this section, we apply Corollaries13 and 17 in order to show the existence of solution of the Fredholm integral equation:
(81)ft=∫0TKt,τ,fτdτ+gt,whereT>0, K:0,T×0,T×ℝ⟶ℝ and g:0,T⟶ℝ is continuous function.Remark 24.
The convexity of the functionfx=xpx∈ℝ,1<p<∞ implies the followings:
(i)
For alla,b∈ℝ and α∈0,1, we have αa+1−αbp≤αap+1−αbp(ii)
Particularly forα=1/2, we have a+bp≤2p−1ap+bp
LetX=C0,T and let D:X×X⟶0,+∞ be the mapping defined by
(82)Dx,y=max0≤t≤Txt−ytp.Using Remark24 above and by considering w:X×X×0,1⟶X defined by wx,y,λ=λx+1−λy, we can easily show that X,D,w is a complete convex b-metric space with C=2p−1.Theorem 25.
Suppose that the following hypotheses hold:(i)
There exists a continuous functionζ:0,T×0,T⟶ℝ+ such that for all t,s∈0,T and for all x,y∈C0,T, we have
(83)Kt,s,xs−Kt,s,ys≤ζt,sxs−ys.(ii)
maxt∈0,T∫0Tζt,sds<1.
Then, the integral Equation (81) has a unique solution on C0,T. Moreover, Mann and Ishikawa iteration processes converge to this solution for any sequences αnn,βnn∈0,1 such that limsupn⟶∞αn<1 or ∑i=0∞1−αi diverges.Proof.
Define a self-mapS on X by
(84)Sut=∫0TKt,τ,uτdτ+gt,forallu∈C0,T.Sut−Svtp=∫0TKt,s,us−Kt,s,vsdsp≤∫0TKt,s,us−Kt,s,vsdsp≤∫0Tζt,sus−vsdsp≤maxt∈0,T∫0Tζt,sdspmax0≤τ≤Tuτ−vτp≤maxt∈0,T∫0Tζt,sdspDu,v.
Therefore,(85)DSu,Sv≤maxt∈0,T∫0Tζt,sdspDu,v.
Hence,S is a k-contraction mapping on X with k=maxt∈0,T∫0Tζt,sdsp<1. All conditions of Corollaries 13 and 17 are satisfied. Hence, S has a unique fixed point. That is to say, there exists a unique solution to the integral Equation (81). Moreover, this solution could be approximated by Mann and Ishikawa iterative processes.☐


## 4. Conclusion

In this paper, we consider a JS metric space endowed with Takahashi convexity structure, and we establish convergence theorems for Mann iteration and Ishikawa iteration for Banach contraction and Chatterjea contraction. Also, we provide an example to validate the established theorems and give an application of our result to Fredholm integral equation. Finally, it remains an open problem to establish the Convergence of Mann iteration and Ishikawa iteration to fixed point for Kannan contraction.



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