# Fixed Point Theorems for a New Class of Mappings in Modular Spaces Endowed with a Graph

Authors: Nour-eddine El Harmouchi; Karim Chaira; El Miloudi Marhrani
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

In this paper, we discuss a class of mappings more general thanρ-nonexpansive mapping defined on a modular space endowed with a graph. In our investigation, we prove the existence of fixed point results of these mappings. Then, we also introduce an iterative scheme for which proves the convergence to a fixed point of such mapping in a modular space with a graph.


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## Body

## 1. Introduction

The abstract definition of modular spaces was introduced by Nakano [1] in 1950. Further, Musielak and Orlicz [2] redefined the modular spaces in such a way that a modular function space is seen as a vector space endowed with a modular function.Fixed point theorems for monotone mappings in metric spaces endowed with partial ordering are considered firstly by Ran and Reurings [3] in 2004. They gave a generalization of Banach’s contraction theorem in partially ordered metric spaces. Their results were extended by Neito and Rodríguez-López [4]. Following this line of research, fixed point theory rapidly developed in the framework of spaces equipped with partial ordering. Some related works are [5–7].Graphs are an ordered structure that contains and generalizes partial ordering as a particular case. The use of graphs, in the study of fixed points, was introduced by Jachymski [8]. The main idea of his work is that the contraction inequality satisfied only for some connected vertices of a graph. Therefore, he generalized Banach’s theorem in metric spaces with a graph. This work opened a new direction of research for the development of fixed point theory, in which many papers appeared [9–13].In the last 10 years, many authors have considered the approach of Jachymski in the setting of modular spaces. Öztürk et al. [14] presented some fixed point results for mappings satisfying the integral contraction condition in a vector modular space equipped with a graph. Also, Alfuraidan [15] proved a generalization of Banach’s theorem in modular metric spaces endowed with a graph. Afterwards, Alfuraidan examined in [16] the existence of fixed points for multivalued monotone G-contraction and G-nonexpansive mappings in modular function spaces.Over the past few decades, several iterative methods have been proposed by many mathematicians to find and approximate fixed points of nonlinear mappings. Some references to iterative processes are [17–23].In 2016, Ullah and Arshad [24] introduced a new iterative algorithm named AK iteration, defined by the following: for a given x0∈C,
(1)xn+1=Tyn,yn=T1−αnzn+αnTzn,zn=T1−βnxn+βnTxn,n∈ℕ,where αnn and βnn are in 0,1. According to Ullah and Arshad, this scheme converges to fixed points of contractions and also has a better converge rate than all previous processes.Subsequently, many authors extended and studied the convergence of iterative methods for nonlinear maps in various spaces endowed with a graph.In 2012, Aleomraninejad et al. [25] proved the convergence of certain iterative processes for G-contraction and G-nonexpansive mapping acting on a graph. Alfuraidan [16] justified the existence and convergence for multivalued mapping in modular function spaces endowed with a graph. He also gave some results on the approximation of fixed points of Ćirić quasicontraction mappings in metric modular spaces equipped with a graph (see [15]).In this paper, we start by introducing a new generalization ofρ-nonexpansive mappings. Using Mann’s [20] iteration, we prove the existence of fixed points of this large class in modular spaces endowed with an oriented graph. Afterwards, we establish some convergence results for a new iterative process (see (55)) which can be considered an accelerated version of the AK iteration scheme.


## 2. Preliminaries

Throughout this work,X stands for a linear vector space on the field R.Definition 1 (see [26]).
A functionρ:X⟶0,+∞ is called a modular if the following holds:
(1)
ρx=0 if and only if x=0(2)
ρ−x=ρx(3)
ραx+1−αy≤ρx+ρy for any α∈0,1for any x and y in X.
If (3) is replaced by (3′) ραx+1−αy≤αρx+1−αρy for any α∈0,1 and x and y in X, then ρ is called a convex modular.
A modularρ defines a corresponding modular space, i.e., the vector space Xρ given by
(2)Xρ=x∈X:limλ→0ρλx=0.
Letρ be a convex modular, and the modular space Xρ is equipped with a norm called the Luxemburg norm, defined by
(3)xρ=infλ>0:ρxλ≤1.In the next definition, we give the basic definitions needed throughout this work.Definition 2 (see [26]).
Letρ be a modular defined on a vector space X.
(1)
We say that a sequencexnn∈ℕ⊂Xρ is ρ-convergent to x∈Xρ if and only if ρxn−x converges to 0 when n goes to infinity. Note that the limit is unique(2)
A sequencexnn⊂Xρ is called ρ-Cauchy if ρxn−xm⟶0 as n,m⟶+∞(3)
We say thatXρ is ρ-complete if and only if any ρ-Cauchy sequence is ρ-convergent(4)
A subsetC of Xρ is ρ-closed if the ρ-limit of a ρ-convergent sequence of C always belong to C(5)
A subsetC of Xρ is ρ-bounded if we have δρC=supρx−y:x,y∈C<∞(6)
A subsetK of Xρ is ρ-sequentially compact if any sequence xnn of C has a subsequence ρ-convergent to a point x∈C(7)
We say thatρ satisfies the Fatou property if ρx−y≤limn→+∞ρx−yn whenever ynnρ-converges to y for any x, y, and yn in Xρ
Note thatρ-convergence does not imply ρ-Cauchy condition. Also, we have xn⟶ρx which does not imply in general λxn⟶ρλx for every λ>1An important property associated with a modular which plays a powerful role in modular spaces is the propertiesΔ2-condition and Δ2-type condition.Definition 3 (see [26, 27]).
Letρ be a modular defined on a vector space X. We say that ρ satisfies
(i)
theΔ2-condition, if ρ2xn⟶0 whenever ρxn⟶0 as n goes to +∞(ii)
theΔ2-type condition, if there exists K>0 such that ρ2x≤Kρx for all x∈Xρ
Ifρ is assumed to be convex and to satisfy Δ2-type condition, we can define the growth function (see [28]):
(4)ω:0,∞⟶0,∞,ωt=supρtxρx:0<ρx<∞.
Then,1<ω2. The arguments for this statement can be extracted from Lemma 2.6 in [10]. The minimal possible value of K is usually denoted by ω2.In the sequel, we assume thatρ is a convex modular.Definition 4 (see [26]).
Letρ be a modular and r>0 and ε>0. Define, for i∈1,2, that
(5)Dir,ε=x,y∈Xρ×Xρ:ρx≤r,ρy≤r,ρx−yi≥rε.
IfDir,ε≠∅, let
(6)δir,ε=inf1−1rρx+y2:x,y∈Di.
IfDir,ε=∅, we set δir,ε=1:
(i)
We say thatρ is uniformly convex (UCi) if for every r>0 and ε>0, we have δir,ε>0(ii)
We say thatρ is uniquely uniformly convex (UUCi) if for all s≥0 and ε>0, there exists ηs,ε>0 such that δir,ε>ηs,ε, for r>s(iii)
We say thatρ is strictly convex (SC), if for every x,y∈Xρ such that ρx=ρy and ρx+y/2=ρx+ρy/2, we have x=yThe following proposition characterizes the relationship between the above notions.Proposition 5 (see [26]).
(a)
(UUCi) implies (UCi) for i=1,2(b)
δ1r,ε≤δ2r,ε for r>0 and ε>0(c)
(UC1) implies (UC2) implies (SC)(d)
(UUC1) implies (UUC2)The following property plays a similar role as the reflexivity in Banach spaces for modular spaces.Definition 6 (see [27]).
Letρ be a modular. We say that the modular space Xρ satisfies the property R if and only if every decreasing sequence Cnn∈ℕ of nonempty ρ-closed convex and ρ-bounded subsets of Xρ has a nonempty intersection.Lemma 7 (see [26].
Letρ be a convex modular that satisfies the Fatou property. Assume that Xρ is ρ-complete and ρ is (UUC2). Then, Xρ satisfies the property R.Definition 8 [26].
Letxnn be a sequence in Xρ and K be a nonempty subset of Xρ. The function τ:K⟶0,∞, defined by τx=limsupn→∞ρxn−x, is called a ρ-type function.
Note that theρ-type function τ is convex since ρ is convex. A sequence cnn⊂K is called a minimizing sequence of τ if limn→+∞τcn=infx∈Kτx.The following lemmas play an important role in the proof of the next fixed point theorem.Lemma 9 (see [26]).
Letρ be a convex modular (UUC1) satisfying the Fatou property and Xρ a ρ-complete modular space. Let C be a nonempty ρ-closed convex subset of Xρ. Consider the ρ-type function τ:C⟶0,+∞ generated by a sequence xnn in Xρ. Assume that τ0=infτx:x∈C<∞. Then, all the minimizing sequences of τ are ρ-convergent to the same limit.Lemma 10 (see [29]).
Suppose thatρ is convex and satisfies the property (UUC1), and let tnn be a sequence in a,b, where 0<a≤tn≤b<1. If there exists a positive real number r such that limsupn→∞ρxn≤r, limsupn→∞ρyn≤r, and limn→∞ρ1−tnxn+tnyn=r, then limn→∞ρxn−yn=0.Lemma 11 (see [30]).
Letrnn and tnn be two real bounded sequences. Then,
(i)
limsupn→∞maxrn,tn=maxlimsupn→∞rn,limsupn→∞tn(ii)
letzn=1−αnrn+αntn, with αnn⊂0,1 convergent to a real α∈0,1. Then,
(7)limsupn→∞zn≤1−αlimsupn→∞rn+αlimsupn→∞tn.Let us finish this section with a few of graph theory terminology and basic definitions which will be needed throughout. LetC be a nonempty subset of a modular space Xρ and Δ=x,x:x∈C be the loop set. Consider a directed graph G such that the set of vertices VG coincides with C and the set of its edges EG contains all loops, i.e., Δ⊂EG. Assume that G has no parallel edges, so it can be identified to the pair VG,EG.Letx and y be vertices of a graph G. A path from x to y of length N∈ℕ is a finite sequence xn of N+1 elements for which x0=x, xN=y, and xi,xi+1∈EG for i=0,⋯,N−1.A graphG is said to be connected if there is a path between any two vertices of the graph G. A directed graph G=VG,EG is said to be transitive if for any x, y, and z in VG such that x,yEG and y,z are in EG, then x,z∈EG. Moreover, the conversion of a graph G, denoted G−1, is the graph obtained by reversing the direction of the edges of the graph G. Thus, we have EG−1=y,x∈X×X:x,y∈EG. For more details on graph theory, we refer readers to the book [31].Definition 12.
LetXρ be a modular space. A graph G is said to be convex if and only if for any x, y, z, and w in Xρ and λ∈0,1, we have x,y∈EG and z,w∈EG which lead to 1−λx+λz,1−λy+λw∈EG.Definition 13.
LetC be a nonempty subset of a modular space Xρ, and let G=VG,EG be a directed graph such that VG=C. We say that C have the property P: if each sequence xnn in Cρ-converges to x and xn,xn+1∈EG, then xn,x∈EG for all n≥0.


## 3. Results and Discussion

Throughout, we assume thatXρ is a modular space, where ρ is a convex modular. Let C be a nonempty ρ-closed convex subset of Xρ. Let G=VG,EG be a directed transitive graph such that VG=C,EG is convex and Δ⊆EG. Moreover, we assume that G-intervals are ρ-closed. Recall that a G-interval is any of the subsets a,⟶=x∈X:a,x∈EG, ⟵,b=x∈X:x,b∈EG, and a,b=a,⟶∩⟵,b.Definition 14.
LetC be a nonempty subset of a modular space Xρ. Let T:C⟶C be a mapping and γ∈0,1 and μ∈0,1/2 such that 2μ≤γ. The mapping T is said to
(1)
preserve edges ifTx,Ty∈EG whenever x,y∈EG for any x,y∈C(2)
satisfy the conditionG−Mγ,μ if T preserves edges γρx−Tx≤ρx−y+μρy−Ty which leads to ρTx−Ty≤1−γρx−y+ρμx−Ty+ρμy−Tx, for all x,y∈C such that x,y∈EG
Note that ifγ=μ=0, then T is ρ-nonexpansive.Lemma 15.
Letρ be a convex modular and Xρ be a modular space endowed with a directed transitive graph such that VG=C,EG is convex and Δ⊆EG. Let T:C⟶C satisfy the condition G−Mγ,μ on C. If z is a fixed point of T, then ρz−Tx≤ρz−x for all x∈C such that x,z∈EG.Proof.
Letz be a fixed point of T and x∈C such that x,z∈EG. We have
(8)0=ρz−Tz≤ρx−z≤ρx−z+μρx−Tx.
By conditionG−Mγ,μ,
(9)ρTx−Tz=ρz−Tx≤1−γρx−z+ρμx−Tz+ρμz−Tx≤1−γρx−z+μρx−z+μρz−Tx.
Hence,1−μρz−Tx≤1−γ+μρz−x, which implies
(10)ρz−Tx≤1−γ+μ1−μρz−x.
Since2μ≤γ, then ρz−Tx≤ρz−x.It is easy to obtain the following consequence of condition (3′), which we need in the sequel, by using the convexity of the modular ρ, and we prove by induction on n≥1 the inequality of the property.Property 16.
Letρ be a convex modular. For a1,⋯,an in ℝ+ such that ∑i=1nai=1 and x1,⋯,xn in Xρ, we have
(11)ρ∑i=1naixi≤∑i=1naiρxi.Proposition 17.
Letρ be a convex modular and Xρ be a modular space endowed with a directed transitive graph such that VG=C,EG is convex and Δ⊆EG. Let T be a self-mapping on C monotone and satisfy the condition G−Mγ,μ on C, where γ≤1/ω2. Then,
(i)
for allx∈C such that x,Tx∈EG, we have ρTx−T2x≤ρx−Tx(ii)
ifρ satisfies the Δ2-type condition, then for all x and y in C such that x,Tx and Tx,y are in EG and for c∈0,1, we have (a) or (b):
(a)
c/ω2ρx−Tx≤ρx−y(b)
c/ω2ρTx−T2x≤ρTx−yProof.
(i)
Letx∈C such that x,Tx∈EG. Then, we have γρx−Tx≤ρx−Tx≤ρx−Tx+μρTx−T2x. By the condition G−Mγ,μ for y=Tx, we have
(12)ρTx−T2x≤1−γρx−Tx+ρμx−T2x+ρμTx−Tx≤1−γρx−Tx+ρμx−Tx+Tx−T2x≤1−γρx−Tx+μρx−Tx+μρTx−T2x,which implies 1−μρTx−T2x≤1−γ+μρx−Tx. Since 2μ≤γ, then ρTx−T2x≤ρx−Tx.
(ii)
Assume thatρ satisfies the Δ2-type condition, that is, ρ2x≤ω2ρx for all x∈Xρ. Let x,y∈C such that x,Tx and Tx,y are in EG. Assume on the contrary that c/ω2ρx−Tx>ρx−y and c/ω2ρTx−T2x>ρTx−y. We have
(13)ρx−Tx≤ω22ρx−y+ω22ρTx−y<cω2ω22ρTx−T2x+cω2ω22ρx−Tx≤cρx−Tx≤ρx−Tx.
Then,ρx−Tx<ρx−Tx which is impossible. Therefore, (a) or (b) holds.
LetT:C⟶C be a mapping preserving edges. Let αnn be a sequence 0,1. Let x0∈C such that x0,Tx0∈EG (resp., Tx0,x0∈EG). By induction, we construct a sequence xnn in C, called the Mann iteration, defined by
(14)xn+1=1−αnxn+αnTxn.
Recall thatxn,xn+1, xn,Txn, and Txn,Txn+1 (resp., xn+1,xn, Txn,xn, and Txn+1,Txn) are in EG, for any n≥0.In the following result, we prove the existence of fixed points ofG−Mγ,μ mappings using the Mann iteration defined by (14) such that the sequence αnn converges to α∈0,1.Theorem 18.
Letρ be a (UUC1) convex modular that satisfies the Fatou property and the Δ2-type condition, and Xρ is a ρ-complete modular space endowed with a directed transitive graph, where VG=C is a nonempty ρ-closed convex and ρ-bounded, EG is convex, and Δ⊆EG. LetT:C⟶C be a mapping satisfying the condition G−Mγ,μ with γ∈0,1/ω2 and ρ-continuous on C. Assume that there exists x0∈C such that x0,Tx0∈EG (resp., Tx0,x0∈EG). Then, T has a fixed point z in C such that x0,z∈EG (resp., z,x0∈EG) if and only if limn→∞ρxn−Txn=0.Proof.
Assume thatT has a fixed point p∈C such that x0,p∈EG. Using the convexity of the modular ρ, we have
(15)ρxn+1−p≤1−αnρxn−p+αnρTxn−p≤1−αnρxn−p+αnρxn−p≤ρxn−p.
It follows thatρxn−pn is a decreasing nonnegative sequence. Since C is convex, then xn∈C for all n∈ℕ. Moreover, Cρ-bounded implies that ρxn−p<∞ for all n∈ℕ. These lead to the conclusion that ρxn−pn is convergent. Let limnρxn−p=r. We have ρTxn−p≤ρxn−p. Then,
(16)limsupn→∞ρTxn−p≤limsupn→∞ρxn−p=r.
Moreover,(17)limn→∞ρ1−αnxn−p+αnTxn−p=limn→∞ρxn+1−p=r.
Thus, by Lemma10, we get limn→∞ρxn−Txn=0. In the following, let us prove the converse statement. Let x0∈C such that x0,Tx0∈EG. Set K=y∈C:xn,y∈EG,∀n≥0. The subset K is a nonempty ρ-closed convex. In fact, set Cn=y∈C:xn,y∈EG for all n≥0. We have Cnn which is a decreasing sequence of nonempty ρ-closed convex ρ-bounded subsets of Xρ. Indeed, it is easy to verify that Cn is nonempty and ρ-bounded for all n≥0. Let λ∈0,1 and y1,y2∈Cn. Then, xn,λy1+1−λy2∈EG. Hence, λy1+1−λy2∈Cn; that is, Cn is convex for all n≥0. Moreover, Cn is ρ-closed for all n≥0. In fact, let ypp⊂Cnρ-converge to y∈Xρ. By the definition of Cn, we have Cn=C∩xn,⟶. Since C and xn,⟶ are supposed to be ρ-closed, then y∈Cn for all n∈ℕ. Since xn,xn+1∈EG, then Cn+1⊂Cn; that is, Cnn is a decreasing sequence. By Lemma 7, the modular space Xρ satisfies the property R. Then, K=⋂nCn is a nonempty ρ-closed convex.
Letτ,τ¯:K⟶0,+∞ be two ρ-type functions generated by the sequences xnn and Txnn, respectively; that is, τy=limsupn→∞ρxn−y and τ¯y=limsupn→∞ρTxn−y. Let us prove that for all y∈K such that xn,y∈EG and Txn,y∈EG, one has
(18)τTy≤τ¯Ty≤τy≤τ¯y.Case 1.
Ifxnn satisfies (ii, a) of Proposition 17 for some n≥0, that is, c/ω2ρxn−Txn≤ρxn−y, then by condition G−Mγ,μ for c=γω2∈0,1, we have
(19)ρTxn−Ty≤1−γρxn−y+ρμTxn−y+ρμTy−xn≤1−γρxn−y+μρxn−y+μρTxn−xn+μρxn−Txn+μρTxn−Ty,for n≥0. Then,
(20)1−μρTxn−Ty≤1−γ+μρxn−y+2μρxn−Txn,for all n≥0. Hence,
(21)ρTxn−Ty≤1−γ+μ1−μρxn−y+2μ1−μρxn−Txn≤ρxn−y+2μ1−μρxn−Txn,for all n≥0, since 2μ≤γ.Case 2.
Ifxnn satisfies (ii, b) of Proposition 17 for some n≥0, that is,
(22)cω2ρTyn−yn=cω2ρT2xn−Txn≤ρTxn−y=ρyn−y,cω2ρTxn−xn>ρxn−y,where yn=Txn. By the condition G−Mγ,μ for c=γω2∈0,1, we have
(23)ρTyn−Ty=ρT2xn−Ty≤1−γρyn−y+ρμTyn−y+ρμyn−Ty≤1−γρyn−y+μρyn−Tyn+μρTyn−Ty+μρy−yn+μρyn−Tyn,for all n≥0. It follows that 1−μρTyn−Ty≤1−γ+μρyn−y+2μρyn−Tyn for all n≥0. Since 2μ≤γ, then
(24)ρTyn−Ty≤ρyn−y+2μ1−μρyn−Tyn,for all n≥0. Moreover, for all n≥0,
(25)ρTxn−Ty=ρyn−Ty≤ω22ρyn−Tyn+ρTyn−Ty≤ω22ρyn−Tyn+ρyn−y+2μ1−μρyn−Tyn≤ω22ρyn−y+1+μ1−μρyn−Tyn≤ω221+μ1−μρyn−Tyn+ω22ρyn−xn+ω22ρxn−y<ω221+μ1−μρyn−Tyn+ω22ρyn−xn+ω22cω2ρyn−xn≤ω221+μ1−μρyn−Tyn+c+ω22ρxn−yn.
By Proposition17, we have ρTyn−yn=ρT2xn−Txn≤ρxn−Txn. Then,
(26)ρyn−Ty=ρTxn−Ty<ω221+μ1−μρxn−Txn+c+ω22ρxn−Txn<ω223+ω2+c2ρxn−Txn,for all n≥0. Thus,
(27)ρyn−Ty=ρTxn−Ty<ω26+c+ω24ρxn−Txn,for all n≥0. Therefore, by (21) and (27), we get
(28)ρyn−Ty=ρTxn−Ty<maxρxn−y+2μ1−μρxn−Txn,ω26+c+ω24ρxn−Txn.
By thelimsup as n goes to infinity, we have
(29)τ¯Ty≤maxτy,0=τy,since limn→∞ρxn−Txn=0. Otherwise, ρxn+1−y≤αnρTxn−y+1−αnρxn−p for all n≥0. By limsup as n goes to infinity, we get
(30)τy≤τ¯y.
Thus, by (29) and (30), one has
(31)τTy≤τ¯Ty≤τy≤τ¯y,for all y∈K. Let zpp be a minimizing sequence of the type function τ. By Lemma 9, zppρ-converges to a point z∈K. By (31), the sequence Tzpp is also a minimizing sequence of τ. Hence, Tzpp converges also to z. Since T is ρ-continuous, then Tzppρ-converges to Tz. Thus, the uniqueness of the limit implies that z=Tz.Example 1.
For the functionp:ℕ⟶1,∞, consider the modular space
(32)ℓp·=xnn∈ℝℕ:∑n=0+∞1pnλxnpn<∞forsomeλ>0,where the modular ρ is given by ρx=ρxnn=∑n=0+∞1/pnxnpn such that 1<p−≤pn≤p+<∞ for all n≥0, where p−=infn∈ℕpn and p+=supn∈ℕpn.
We haveρ which is a convex modular satisfying the Fatou property and (UUC2) (for more details, see [32]). Moreover, ρ satisfies the Δ2-type condition. In fact, let x∈ℓp.. Since pn≤p+, we have 2pn≤2p+. Thus, 2pn/pnxnpn≤2p+/pnxnpn. Therefore, ∑n=0q2pn/pnxnpn≤∑n=0q2p+/pnxnpn for all q∈ℕ. Taking limn→∞, one has
(33)ρ2x=∑n=0+∞2pnpnxnpn≤∑n=0+∞2p+pnxnpn≤2p+∑n=0+∞1pnxnpn=2p+ρx.
Recall that ifρ satisfies the Δ2-type condition, then ∥·∥ρ convergence is equivalent to ρ-convergence (see [33]). Since ℓp.,∥·∥ρ is a Banach space (see [32]), then ℓp. is a ρ-complete modular space.
Recall the following two inequalities:(i)
For all1<p≤2, we have(34)a+b2p+pp−12a−b∣a∣+∣b∣2−pa−b2p≤12ap+bp,for all a,b∈ℝ such that ∣a∣+∣b∣≠0.
(ii)
For allp≥2, we have(35)a+b2p+a−b2p≤12ap+bp,for all a,b∈ℝ.
The modularρ is (UUC1). In fact, let r>0, ε>0, and x,y∈ℓp· such that ρx≤r, ρy≤r, and ρx−y≥rε. Let us partition the set ℕ into two subsets E and F given as follows: E=n∈ℕ:pn≥2 and F=n∈ℕ:pn<2. Using the definition of the modular ρ, we will denote ρx=ρEx+ρFx, because the two sums on E and F converge, with
(36)ρEx=∑n∈E1pnxnpnρFx=∑n∈F1pnxnpn,for all x∈ℓp·. Thus, ρx−y≥rε leads to ρEx−y≥rε/2 or ρFx−y≥rε/2.Case 1.
Assume thatρEx−y≥rε/2. Since pn≤p+<∞, then
(37)ρEx−y2≥12p+ρEx−y≥12p+rε2=rε2.2p+.
Sincepn≥2, then by the inequality (ii), we get
(38)ρEx+y2+ρEx−y2≤12ρEx+ρEy.
By the convexity ofρF, we have
(39)ρx+y2+ρEx−y2≤12ρx+ρy≤r.
Thus,(40)ρx+y2≤r−ρEx−y2≤r−rε2.2p+=r1−ε2.2p+.Case 2.
Assume thatρFx−y≥rε/2. Let A be a subset of F defined by A=n∈F:∣xn−yn∣>c∣xn∣+∣yn∣, where c=min1/2;ε/8. Let B=F\A. Then, ∣xn−yn∣≤c∣xn∣+∣yn∣ for all n∈B. Thus,
(41)∑n∈B1pnxn−ynpn≤∑n∈B1pnc∣xn∣+∣yn∣pn≤∑n∈B1pn2cpn∣xn∣+∣yn∣2pn≤∑n∈B1pn2cpnxnpn+ynpn2≤12ρ2cx+ρ2cy≤2c2ρx+ρysincec≤12≤4cr2=2cr.
Thus,∑n∈B1/pnxn−ynpn≤2rc≤rε/4 since c≤ε/8. Hence,
(42)∑n∈A1pnxn−ynpn=ρFx−y−∑n∈B1pnxn−ynpn≥rε2−rε4=rε4,since B=F\A. Otherwise,
(43)pnpn−1≥p−p−−1≥p−−1,xn−yn∣xn∣+∣yn∣2−pn≥c2−pn≥c,for all n∈A. Thus, by the inequality (i),
(44)xn+yn2pn+cp−−12xn−yn2pn≤12xnpn+ynpn,for all n∈A. Thus,
(45)∑n∈A1pnxn+yn2pn+cp−−12∑n∈A1pnxn−yn2pn≤12∑n∈A1pnxnpn+∑n∈A1pnynpn.
Hence,(46)ρAx+y2+cp−−12ρAx−y2≤12ρAx+ρAy.
Thus,(47)ρAx+y2≤12ρAx+ρAy−cp−−12ρAx−y2.
Sincepn<2, then
(48)ρAx−y2=∑n∈A1pnxn−yn2pn≥14∑n∈A1pnxn−ynpn≥14ρAx−y.
Therefore,ρAx+y/2≤1/2ρAx+ρAy−cp−−1/2·ρAx−y/4. It follows
(49)ρAx+y2≤12ρAx+ρAy−εrcp−−132,since ρAx−y≥rε/4. Let A¯=ℕ\A. Since ρA¯ is convex, we have
(50)ρx+y2=ρAx+y2+ρA¯x+y2≤12ρAx+ρAy+ρA¯x+ρA¯y−εrcp−−132≤12ρx+ρy−εrcp−−132≤r1−εrcp−−132.
Therefore, by (40) and (50), we put η1s,ε=minε/2.2p+;εcp−−1/32. Thus, ρ is (UUC1).
Consider the nonempty setC=x=xnn∈ℓp.:0≤xn≤2−n,∀n≥0. It is clear that C is convex and ρ-bounded, and it is ρ-closed. Indeed, let xqq be a sequence in C which ρ-converges to a point x=x0,x1,⋯∈ℓp·, where xq=x0q,x1q,⋯, for all q≥0. We have
(51)1p+xnq−xnm≤∑n=0+∞1pnxnq−xnpn,for all q≥0, where m=p+if∣xnq−xn∣≥1m=p−if∣xnq−xn∣≤1. Since xqq≥0ρ-converges to x, then by (51), limq→∞∣xnq−xn∣=0, i.e., limq→∞xnq=xn. Thus, 0≤xn≤1/2n because 0≤xnq≤1/2n for all n≥0.
Let the convex directed graphG=VG,EG such that VG=C and EG=x,y∈C×C:xn≤yn,∀n≥2. The graph G is transitive such that EG is convex, Δ⊂EG, and G-intervals are ρ-closed convex. Let T:C⟶C be a mapping defined by Txnn=xn/4.2p+−1n. The mapping T satisfies the condition G−Mγ,μ for γ=1/2p+∈0,1/ω2 and μ=1/2.2p+. In fact, let x and y in C such that x,y∈EG. It is quite easy to check that T preserves edges and is ρ-continuous. Moreover,
(52)1−γρx−y+ρμx−Ty+ρμTx−y≥∑n=0+∞1pn12.2p+pnxn−yn4.2p+−1pn+yn−xn4.2p+−1pn≥∑n=0+∞1pn12.2p+pnxn−yn+xn−yn4.2p+−1pn≥∑n=0+∞1pn12.2p+pn4.2p+pn4.2p+−1pnxn−ynpn≥∑n=0+∞2p−pnxn4.2p+−1−yn4.2p+−1pn≥∑n=0+∞1pnxn4.2p+−1−yn4.2p+−1pn=ρTx−Ty.
For the sequel, we considerpn=2+2n/n2+2. We have p−=2 and p+=p1. Let x0=1/2nn≥0∈C, and we have Tx0,x0∈EG. Set αq=λ∈0,a, for all q≥0, where a=4.2p+−1/4.2p+. By induction on q, we construct the sequence xqq≥0 given as follows: xq=1−λ4.2p+/4.2p+−1qx0 for all q≥0. Moreover, we have
(53)ρxq−Txq=∑n≥01pnxnq−xnq4.2p+−1pn≤∑n≥01pn1−λ4.2p+4.2p+−1qpnxn0pn≤∑n≥01pn1−λ4.2p+4.2p+−1qp+xn0pn≤1−λ4.2p+4.2p+−1p+qρx0→p→∞0.
Thus,limq→∞ρxq−Txq=0. Therefore, T has a fixed point z such that z,x0∈EG.In the sequel, we will use the fixed point set with the partial ordersFx0GT given by
(54)Fx0GT=p∈C:Tp=pandx0,p∈EG.Now, we discuss the convergence of a modified version of the AK iteration scheme in order to get an accelerated iteration. The modified version is given by the following: for a givenx0∈C,
(55)xn+1=T1−αnyn+αnTyn,yn=T1−βnzn+βnTzn,zn=T1−δnxn+δnTxn,n∈ℕ,where αnn, βnn, and δnn are in 0,1 such that there exists a,b∈0,1 with a≤δn≤b, C a convex subset of a modular space Xρ, and T satisfying the condition G−Mγ,μ on C.In the proof of the following lemma, we use the notationszn¯=1−δnxn+δnTxn, yn¯=1−βnzn+βnTzn, and xn+1¯=1−αnyn+αnTyn for all n≥0.Lemma 19.
Letρ be a convex modular and Xρ be a modular space endowed with a directed transitive graph, where VG=C a nonempty ρ-closed convex and ρ-bounded, EG is convex, and Δ⊆EG. Also, let T:C⟶C be a mapping preserving edges. Assume that the sequence xnn is defined by iteration (55) and x0,Tx0∈EG. Then, xn,Txn, Txn,xn+1, and xn,xn+1 are in EG for all n≥0.Proof.
Letx0∈C such that x0,Tx0∈EG. We have the G-interval x0,Tx0=x0,⟶∩⟵,Tx0 which is convex. Then, z0¯∈x0,Tx0; that is, x0,z0¯ and z0¯,Tx0 are in EG. Since T preserves edges, then Tx0,Tz0¯ and Tz0¯,T2x0 are in EG. Thus, Tx0,z0 and z0,T2x0 are in EG. Since Tx0,z0 and x0,Tx0 are in EG, we have x0,z0∈EG and T2x0,Tz0∈EG since T preserves edges. Using the transitivity of G for z0,T2x0 and T2x0,Tz0, one finds z0,Tz0∈EG. Since z0,Tz0 is convex, then z0,y0¯ and y0¯,Tz0 are in EG. Again, T preserves edges, then Tz0,Ty0¯ and Ty0¯,T2z0 are in EG; that is, Tz0,y0 and y0,T2z0 are in EG. By transitivity of G for z0,Tz0 and Tz0,y0, one has z0,y0∈EG. From Tz0,y0 and since T preserves edges, we get T2z0,Ty0∈EG. Combining y0,T2z0 and T2z0,Ty0, we obtain y0,Ty0∈EG. Again, the G-interval y0,Ty0 is convex. Then, y0,x1¯ and x1¯,Ty0 are in EG. Since T preserves edges, one has Ty0,Tx1¯ and Tx1¯,T2y0 which are in EG. That is, Ty0,x1 and x1,T2y0 are in EG. Using the transitivity of the graph G for y0,Ty0 and Ty0,x1, one finds y0,x1∈EG. Again, T preserves edges; then, Ty0,x1 implies that T2y0,Tx1∈EG. Thus, combining with x1,T2y0 and the transitivity of G, we get x1,Tx1∈EG. Again, for the transitivity of G for the edges x0,z0, z0,y0, and y0,x1, one has x0,x1∈EG. Moreover, from Tx0,z0, z0,y0, and y0,x1, we get Tx0,x1∈EG. Therefore, we have Tx0,x1, x0,x1, and x0,Tx1 which are in EG.
Next, we assume thatxk,Txk∈EG. Using the same argument as before, we deduce that xk,zk and xk,Tzk are in EG since zk¯∈xk,Txk. Again, T preserves edges which implies that Txk,zk and zk,T2xk are in EG. Applying transitivity of G for xk,Txk and Txk,zk, we obtain xk,zk∈EG. Since T preserves edges, then from Txk,zk, we have T2xk,Tzk∈EG. Again, transitivity of G implies that for zk,T2xk and T2xk,Tzk, we have zk,Tzk∈EG.
From the convexity ofzk,Tzk, we have zk,yk¯ and yk¯,Tzk which are in EG. Since T preserves edges, then Tzk,Tyk¯ and Tyk¯,T2zk are in EG. Thus, Tzk,yk and yk,T2zk are in EG. Applying transitivity of G for zk,Tzk and Tzk,yk, we get zk,yk∈EG. Since T preserves edges, then T2zk,Tyk∈EG. Again, by transitivity for yk,T2zk and T2zk,Tyk, one has yk,Tyk∈EG. Since yk,Tyk is convex, then xk+1¯∈yk,Tyk; that is, yk,xk+1¯ and xk+1¯,Tyk are in EG. Since the mapping T preserves edges, then Tyk,Txk+1¯ and Txk+1¯,T2yk are in EG; that is, Tyk,xk+1 and xk+1,T2yk are in EG. Again, the transitivity of G for yk,Tyk and Tyk,xk+1 implies that yk,xk+1∈EG.
SinceT preserves edges, then Tyk,xk+1 implies that T2yk,Txk+1∈EG. Transitivity of G for xk+1,T2yk and T2yk,Txk+1 leads to xk+1,Txk+1∈EG. Moreover, from xk,zk, zk,yk, and yk,xk+1, we deduce that xk,xk+1∈EG. And from Txk,zk, zk,yk, and yk,xk+1, one finds Txk,xk+1∈EG. By xk,Txk and xk,xk+1 and since T preserves edges, we obtain xk,Txk+1∈EG.
Therefore,xk,xk+1, xk+1,Txk+1, Txk,xk+1, and xk,Txk+1 are in EG. Finally, we conclude, by induction, that xk,Txk, xk,xk+1, and Txk,xk+1 are in EG for all k≥0.Lemma 20.
Letρ be a convex modular and Xρ be a modular space endowed with a directed transitive graph G=VG,EG, where VG=C is a nonempty ρ-closed convex subset, EG is convex, and Δ⊂EG. Let T:C⟶C be a mapping satisfying G−Mγ,μ on C. Assume that there exists x0∈C such that x0,Tx0∈EG and Fx0GT is nonempty. Then, limnρxn−p exists for all p∈Fx0GT, where xnn is defined by (55).Proof.
Assume that there existsx0∈C such that x0,Tx0∈EG and Fx0GT is nonempty. Fix p∈Fx0GT. Using the convexity of EG, one can prove by induction that xn,p, yn,p, and zn,p are in EG for all n≥0.
Moreover, we have(56)ρxn+1−p=ρT1−αnyn+αnTyn−p≤1−αnρyn−p+αnρTyn−p≤ρyn−p≤ρT1−βnzn+βnTzn−p≤ρzn−p≤ρT1−δnxn+δnTxn−p≤ρxn−p,for all n≥1. Hence, ρxn−pn is a decreasing sequence and bounded in ℝ+. Thus, limnρxn−p exists.Theorem 21.
Letρ be a convex modular (UUC1) satisfying Δ2-type condition and Xρ be a modular space endowed with a directed transitive graph G=VG,EG, where VG=C is a nonempty ρ-closed convex subset, EG is convex, and Δ⊂EG. Let T:C⟶C be a mapping satisfying G−Mγ,μ on C. Assume that there exists x0∈C such that x0,Tx0∈EG and Fx0GT is nonempty. Then, limnρxn−Txn=0.Proof.
Letx0∈C such that x0,Tx0∈EG and Fx0GT is nonempty. Fix p∈Fx0GT. By Lemma 20, limnρxn−p=r≥0 exists. We assume that r>0, because if r=0, there is nothing to prove. We have
(57)limsupn→∞ρTxn−p≤limsupn→∞ρxn−p=r.
Otherwise, by the inequalities (56), we have
(58)ρxn+1−p≤ρ1−δnxn−p+δnTxn−p≤ρxn−p,for all n≥1. Hence,
(59)limnρδnTxn−p+1−δnxn−p=r.
By Lemma10 and Equations (57) and (59), we obtain limnρxn−Txn=0.Theorem 22.
Letρ be a convex modular (UUC1) satisfying Δ2-type condition and Xρ be a modular space endowed with a directed transitive graph G=VG,EG, where VG=C is a nonempty ρ-sequentially compact convex and ρ-bounded subset having property P, EG is convex, and Δ⊂EG. Also, let T:C⟶C be a mapping satisfying G−Mγ,μ on C with γ∈0,1/ω2. Assume that there exists x0∈C such that x0,Tx0∈EG and Fx0GT is nonempty. Then, the sequence xnn given by (55) ρ-converges to a fixed point of T.Proof.
Assume that there existsx0∈C such that x0,Tx0∈EG. Since C is ρ-sequentially compact, then xnn has a subsequence xφnn which ρ-converges to z∈C, where φ is an increasing function on ℕ. Moreover, xφn,xφn+1∈EG for all n≥0. Thus, the property P implies that xφn,z∈EG for all n≥0. Next, let us prove that z is a fixed point of T.Case 1.
If there exists a subsequencexφ∘σ1nn such that γρxφ∘σ1n−Txφ∘σ1n≤ρxφ∘σ1n−z for all n≥0, that is, T satisfies (ii, a) of Proposition 17 for the subsequence xφ∘σ1nn and c=γω2. By the condition G−Mγ,μ, we obtain
(60)ρTxφ∘σ1n−Tz≤1−γρxφ∘σ1n−z+ρμxφ∘σ1n−Tz+ρμz−Txφ∘σ1n.
For alln≥0,
(61)ρTxφ∘σ1n−Tz≤1−γρxφ∘σ1n−z+μρxφ∘σ1n−Txφ∘σ1n+ρTxφ∘σ1n−Tz+μρxφ∘σ1n−Txφ∘σ1n+ρxφ∘σ1n−z≤1−γ+μρxφ∘σ1n−z+2μρxφ∘σ1n−Txφ∘σ1n+μρTxφ∘σ1n−Tz.
Hence,1−μρTxφ∘σ1n−Tz≤1−γ+μρxφ∘σ1n−z+2μρxφ∘σ1n−Txφ∘σ1n for all n≥0. It follows
(62)ρTxφ∘σ1n−Tz≤ρxφ∘σ1n−z+2μ1−μρxφ∘σ1n−Txφ∘σ1n,since 2μ≤γ. Taking limn→∞, we obtain limnρTxφ∘σ1n−Tz=0. Otherwise,
(63)ρxφ∘σ1n−Tz≤ω22ρxφ∘σ1n−Txφ∘σ1n+ω22ρTxφ∘σ1n−Tz,for all n≥1. Therefore, again by limn→∞ and by Theorem 21, limnρxφ∘σ1n−Tz=0.Case 2.
If there exists a subsequencexφ∘σ2nn of xφnn such that γρT2xφ∘σ2n−Txφ∘σ2n≤ρTxφ∘σ2n−z for all n≥0, that is, T satisfies (ii, b) of Proposition 17 for the subsequence xφ∘σ2nn and c=γω2. By the condition Mγ,μ, we obtain
(64)ρT2xφ∘σ2n−Tz≤1−γρTxφ∘σ2n−z+ρμTxφ∘σ2n−Tz+ρμT2xφ∘σ2n−z.
Hence,(65)ρT2xφ∘σ2n−Tz≤1−γρTxφ∘σ2n−z+μρTxφ∘σ2n−T2xφ∘σ2n+ρT2xφ∘σ2n−Tz+μρT2xφ∘σ2n−Txφ∘σ2n+ρTxφ∘σ2n−z,for all n≥0. This implies 1−μρT2xφ∘σ2n−Tz≤1−γ+μρTxφ∘σ2n−z+2μρTxφ∘σ2n−T2xφ∘σ2n for all n≥0. Hence,
(66)ρT2xφ∘σ2n−Tz≤ρTxφ∘σ2n−z+2μ1−μρTxφ∘σ2n−T2xφ∘σ2n≤ω22ρTxφ∘σ2n−xφ∘σ2n+ω22ρxφ∘σ2n−z+2μ1−μρxφ∘σ2n−Txφ∘σ2n,since 2μ≤γ for all n≥1. Taking the limit as n goes to infinity, we get limnρT2xφ∘σ2n−Tz=0. Moreover, for all n≥0, we have
(67)ρxφ∘σ2n−Tz≤ω22ρxφ∘σ2n−Txφ∘σ2n+ω22ρTxφ∘σ2n−Tz≤ω22ρxφ∘σ2n−Txφ∘σ2n+ω224ρTxφ∘σ2n−T2xφ∘σ2n+ω224ρT2xφ∘σ2n−Tz≤ω221+ω22ρxφ∘σ2n−Txφ∘σ2n+ω224ρT2xφ∘σ2n−Tz.
Bylimn→∞, we get limnρxφ∘σ2n−Tz=0. Thus, from both cases, we conclude that Tz=z, since xφnnρ-converges to z. Hence, z is a fixed point of the mapping T.
In order to complete the proof, let us show thatxnnρ-converges to z. By Lemma 20, limnρxn−p exists, for all p∈Fx1±T. Since xnn is a G-decreasing sequence, then x0,xφn∈EG for all n≥0. Moreover, we have xφn,z∈EG for all n≥0. Thus, z∈Fx0GT. Hence, limnρxn−z exists. Since xφnnρ-converges to z, then limnρxφn−z=0. Thus, limnρxn−z=0. Therefore, xnnρ-converges to z a fixed point of T.


## 4. Numerical Example

Example 2.
LetXρ=ℝ, where the modular ρ is defined by ρx=∣x∣, for all x∈Xρ, with ω2=2, and let the graph G=VG,EG such that VG=−1,1 and EG=x,y∈−1,1×−1,1:x≤yand∣x−y∣≤1. Consider the mapping T:−1,1⟶−1,1 defined by
(68)Tx=x+23if−1≤x≤−12,x+13if−12≤x≤1.
The mappingT preserves edges and satisfies the condition Mγ,μ for γ=1/2 and μ=1/4. Indeed, if x and y are in −1,−1/2 or in −1/2,1 such that x,y∈EG, we have T which is ρ-nonexpansive and preserves edges. Then, it is G−Mγ,μ. Otherwise, if x∈−1,−1/2 and y∈−1/2,1, then x,y∈EG, and we have ρTx−Ty=1/3y−x−1/3. Otherwise,
(69)1−γρx−y+ρμx−Ty+ρμy−Tx≥1−12y−x+141−2y3=13y−12x+14≥13y−x−13.
Ifx1=1, we have Tx1,x1∈EG and all the hypotheses of Theorem 18 are fulfilled. Then, the mapping T has a fixed point x∗=1/2.To illustrate the convergence of the proposed algorithm, we provide some numerical results and a comparison with other recent iterations: AK iteration [24], M iteration [34], K iteration [35], and the iteration of Piri [22]. Firstly, we show the convergence behavior of the iterative process (55) with different initial points. To do this, we take αn=3n+8/9n+128, βn=n+4/4n+5, δn=2n/3n+5, and ρxn−x∗<10−15 as a stop criterion. From the given initial points x11=−1, x12=−0.5, x13=0, and x14=1, the convergence behaviors of algorithm (55) are displayed in Table 1 and graphicaly in Figures 1 and 2. The numerical results of Table 1 show that the decrease of the initial point x1 affects the convergence speed. That is, the sequence xnn generated by algorithm (55) will converge in less number of iterations to a fixed point when x1 is decreased.Table 1
Influence of initial points.
Iter. (n)x11x12x13x141−1−0.50120.4574089028729470.4819310497036750.4909655248518370.50903447514816330.4992268300941900.4996719885248080.4998359942624040.50016400573759640.4999861619806960.4999941293251440.4999970646625720.50000293533742850.4999997565439440.4999998967156120.4999999483578060.50000005164219460.4999999957867960.4999999982125800.4999999991062900.50000000089371070.4999999999281670.4999999999695250.4999999999847630.50000000001523780.4999999999987910.4999999999994870.4999999999997440.50000000000025690.4999999999999800.4999999999999920.4999999999999960.500000000000004100.50.50.50.5Figure 1
Influence of initial pointsx11=−1.Figure 2
Influence of initial pointsx13=0.We show, in Table2, the stability of the iterative process (55) based on different iteration parameters αn, βn, and δn. Moreover, for more precision of the convergence to the fixed point x∗, we take also ρxn−x∗<10−15 as a stop criterion. Table 2 shows that different parameters αn, βn, and δn which have a small effect on the number of iterations needed to achieve the fixed point x∗. Finally, we compare the iteration numbers of the proposed scheme (55) with the other processes. The numerical results of Table 2 prove that our scheme is advantageous because it required less number of iterations for the convergence than the other schemes.Table 2
Influence of parameters and initial points.
Param.-init. pointsAK iter. [24]M iter. [34]K iter. [35]Piri [22]Iter. (55)Param. 1:αn=0.7, βn=0.2, δn=0.9−191311138−0.591311128091311127191311127Param. 2:αn=3n/8n+42, βn=n/4n+2, δn=2n/3n+5−1111611159−0.510161115901016111591101611159Param. 3:αn=4n2+n3/2n3+10, βn=n2/2n+62, δn=1−6n+2/n+1−191311128−0.591211128091211127191211128


## 5. Conclusions

Throughout the paper, we have established an existence result and some convergence theorems of a new iterative scheme for the class of mappings satisfying the conditionG−Mγ,μ in the framework of modular spaces endowed with a directed graph G. Further, we gave an example of this class of mapping to space ℓp· endowed with a directed graph G. Our results are generalization of several results as in relevant items from the reference section of this paper, as well as in the literature in general.



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