# Common Fixed Points of TwoG-Nonexpansive Mappings via a Faster Iteration Procedure

Authors: Sabiya Khatoon; Izhar Uddin; Javid Ali; Reny George
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this work, we study the convergence of a new faster iteration in which twoG-nonexpansive mappings are involved in the setting of uniformly convex Banach spaces with a directed graph. Moreover, by constructing a numerical example, we show the fastness of our iteration procedure over other existing iteration procedures in the literature.


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## Body

## 1. Introduction

In 1922, Banach originated a great tool for solving the existence problems of nonlinear mappings, which is familiar as Banach’s contraction principle [1]. After it, this principle has been generalized in so many different directions.Over the last 50 years, many authors introduced and studied various iteration schemes for different classes of contractive and nonexpansive mappings. In 2008, Jachymski [2] introduced the concept of joining fixed point theory and graph theory and established the Banach contraction principle in a complete metric space endowed with a directed graph. In 2012, Aleomraninejad et al. [3] presented some iterative scheme for G-contraction and G-nonexpansive mappings in a Banach space with a graph. Tiammee et al. [4] familiarized Browder’s convergence theorem for G-nonexpansive mappings in a Hilbert space with a directed graph. In 2016, Tripak [5] showed the convergence of a sequence developed by the Ishikawa iteration to some common fixed points of two G-nonexpansive mappings in a Banach space combined with a graph. In 2017, Suparatulatorn et al. [6] introduced and studied the modified S-iteration for two G-nonexpansive mappings in a uniformly convex Banach space associated with a graph. Recently, Thianwan and Yambangwai [7] introduced a new two-step iteration process involving two G-nonexpansive mappings and studied its convergence analysis in a uniformly convex Banach space endowed with a graph. There is vast literature in this direction for more details (see [8–19] and references therein).Recently, Ullah et al. [20] introduced a new three-step iteration process known as the K-iteration process and proved its convergence for Suzuki’s generalized nonexpansive mapping.Inspired by the above work, we proposed a modified iteration process containing twoG-nonexpansive mappings, by generating the sequence rn as follows:LetC be a nonempty convex subset of a Banach space X, for any random r0∈C,
(1)tn=1−ςnrn+ςnP2rn,(2)sn=P21−σnP1rn+σnP2tn,(3)rn+1=P1sn,where σn and ςn are appropriate real sequences in 0,1 and P1,P2:C→C are G-nonexpansive mappings. Under some certain conditions, we demonstrate the convergence analysis of (1) for approximating common fixed points of two G-nonexpansive mappings in a uniformly convex Banach space X with graph. We also construct a numerical example, and by using MATLAB R2018a, we clarify that the proposed iteration procedure converges faster than modified Ishikawa iteration, modified S-iteration, and Thianwan’s new iteration (see [5–7]).


## 2. Preliminaries

In this part, we gather some familiar concepts and applicable conclusions which will be used often.LetC be a nonempty subset of a Banach space X. Let Δ denote the diagonal of the cartesian product C×C, i.e., Δ=r,r:r∈C.VG denotes the set of vertices that coincides with C in a directed graph G, and the set EG of its edges contains all loops, i.e., EG⊇Δ. By assuming G has no parallel edge to identify the graph G with the pair (VG,EG. G−1 denotes the conversion of a graph G. So we have
(4)EG−1=r,s∈C×C:s,r∈EG.A setB is said to be dominated by r0 if for each r∈B,r0,r∈EG and dominates r0 if for each r∈B,r,r0∈EG.LetP:C→C be a self map. An edge preserving mapping, i.e., r,s∈EG⇒Pr,Ps∈EG, is said to be G-nonexpansive if
(5)∥Pr−Ps∥≤∥r−s∥∀r,s∈EG.A mappingP:C→C is said to be G-demiclosed at 0, if for any sequence rn in C such that rn⇀r for all rn,rn+1∈EG and Prn→0 then Pr=0.Let us recall that a Banach spaceX is said to satisfy Opial’s property if rn⇀r and then
(6)limsupn→∞∥rn−r∥<limsupn→∞∥rn−s∥,∀s≠r.Lemma 1 [21].
LetC be a subset of a metric space X,d. A mapping P:C→C is semicompact if for a sequence rn in C with limn→∞drn,Prn=0 there exists a subsequence rnj of rn such that rnj→p∈C.LetC be a subset of a normed space X and let G=VG,EG be a directed graph such that VG=C. Then, C is said to have property WG (SG) if for each sequence rn in C converging weakly (strongly) to r∈C and rn,rn+1∈EG there is a subsequence rnj of rn such that rnj,r∈EG for all n∈ℕ.Lemma 2 [6].
Suppose thatX is the Banach space having Opial’s condition, C has property WG, and let P:C→C be a G-nonexpansive mapping. Then, I−P is G-demiclosed at 0, i.e., if rn⇀r and rn−Prn→0, then r∈FP, where FP is the set of fixed points of P.Lemma 3 [22].
LetX be uniformly convex Banach space and σn a sequence in δ,1−δ for some δ∈0,1. Suppose that the sequences rn and sn in X are such that limsupn→∞∥rn∥≤c, limsupn→∞∥sn∥≤c, and limsupn→∞∥σnrn+1−σnsn∥=c for some c≥0. Then, limn→∞∥rn−sn∥=0.Lemma 4 [23].
LetX be the Banach space that satisfies Opial’s condition and let rn be a sequence in X. Let u,v∈X be such that limn→∞∥rn−u∥ and limn→∞∥rn−v∥ exist. If rnj and rnk are subsequences of rn that converges weakly to u and v, respectively, then u=v.Lemma 5 [24].
Letrn be a bounded sequence in a reflexive Banach space X. If for any weakly convergent subsequences rnj of rn, both rnj and rnj+1 converge weakly to the same point in X, then the sequence rn is weakly convergent.Lemma 6 [7].
LetC be a nonempty closed convex subset of a uniformly convex Banach space X and suppose that C has property WG. Let P be a G-nonexpansive mapping on C. Then, I−P is G-demiclosed at 0.


## 3. Main Results

We initiate this section by proving the following proposition.Proposition 7.
LetP1 and P2 be two self G-nonexpansive mappings on C with F=FP1∩FP2 nonempty, where C is a nonempty closed convex subset of a uniformly convex Banach space X endowed with a directed graph. Let VG=C, EG is convex and the graph G is transitive. For random r0∈C, define the sequence rn by (1). Let p0∈F be such that r0,p0, p0,r0 are in EG. Then, rn,p0, sn,p0, tn,p0, p0,rn, p0,sn, p0,tn, rn,sn, rn,tn, and rn,rn+1 are in EG.Proof.
We go ahead by induction. By using the edge preserving property ofP2 and assumption r0,p0∈EG, we get P2r0,p0∈EG. By convexity of EG, we obtain t0,p0∈EG. Again, by edge preservingness of P2 and t0,p0∈EG, we have P2t0,p0∈EG. By convexity of EG and P1r0,p0,P2t0,p0∈EG and applying the edge preservingness of P2 again, we have s0,p0∈EG. Again, by using the property of edge preserving of P1 and s0,p0∈EG, we get r1,p0∈EG. Again, by applying edge preserving of P2, P2r1,p0∈EG, we get t1,p0∈EG as EG is convex. Thus, by edge preserving of P2, P2t1,p0∈EG. Again, by using convexity of EGP1r1,p0,P2t1,p0∈EG and edge preservingness of P2, we get s1,p0∈EG. By edge preserving of P1, we get P1s1,p0∈EG, and we get r2,p0∈EG. Next, we assume that rk,p0∈EG. By edge preserving of P2 and convexity of EG, we get P2rk,p0∈EG and tk,p0∈EG. By applying edge preserving of P2 on tk,p0∈EG, we get P2tk,p0∈EG. By using convexity of EG and P1rk,p0,P2tk,p0∈EG and edge preserving property of P2, we have sk,p0∈EG. As P1 is edge preserving, we get rk+1,p0∈EG. Owing to edge preserving of P2, we obtain P2rk+1,p0∈EG and so tk+1,p0∈EG, since EG is convex. By convexity of EG and P1rk+1,p0,P2tk+1,p0∈EG and applying the edge preservingness of P2 again, we have sk+1,p0∈EG. Therefore, rn,p0,sn,p0,tn,p0∈EG for all n≥1. Using a similar argument, we can show that p0,rn,p0,sn,p0,tn∈EG under the assumption that p0,r0∈EG. By using transitivity of G, we get rn,sn,rn,tn,sn,tn,rn,rn+1∈EG. This completes the proof.Lemma 8.
LetX, C, F, P1, P2, and rn be the same as in Proposition 7. Suppose that σn and ςn are real sequences in 0,1 and r0,p0,p0,r0∈EG for arbitrary r0∈C and p0∈F. Then,
(i)
limn→∞∥rn−p0∥ exists(ii)
limn→∞∥P1rn−rn∥=0=limn→∞∥P2rn−rn∥Proof.
(i)
Letp0∈F. By Proposition 7, we have rn,p0,sn,p0,tn,p0∈EG. As P1 and P2 are G-nonexpansive mappings and using the iterative sequence rn, we have(7)∥tn−p0∥=∥1−ςnrn+ςnP2rn−p0∥≤1−ςn∥rn−p0∥+ςn∥P2rn−p0∥≤1−ςn∥rn−p0∥+ςn∥rn−p0∥=∥rn−p0∥,(8)∥sn−p0∥=∥P21−σnP1rn+σnP2tn−p0∥≤∥1−σnP1rn+σnP2tn−p0∥≤1−σn∥P1rn−p0∥+σn∥P2tn−p0∥≤1−σn∥rn−p0∥+σn∥tn−p0∥≤1−σn∥rn−p0∥+σn∥rn−p0∥≤∥rn−p0∥,(9)∥rn+1−p0∥=∥P1sn−p0∥≤∥sn−p0∥≤∥rn−p0∥.
This implies that sequence∥rn−p0∥ is decreasing and bounded below for all p0∈FP. Hence, limn→∞∥rn−p0∥ exists.
(ii)
Assume thatlimn→∞∥rn−p0∥=c. If c=0; then, by G-nonexpansiveness of P1 and P2, we get(10)∥rn−Pirn∥≤∥rn−p0∥+∥p0−Pirn∥≤∥rn−p0∥+∥p0−rn∥.
Therefore, the result follows.
Suppose thatc>0.
Taking the lim sup on both sides in the inequality (7) and (8), we obtain
(11)limsupn→∞∥tn−p0∥≤limsupn→∞∥rn−p0∥=c,(12)limsupn→∞∥sn−p0∥≤limsupn→∞∥rn−p0∥=c.
On the other hand, using (1), we have
(13)∥rn+1−p0∥=∥P1sn−p0∥≤∥sn−p0∥≤∥P21−σnP1rn+σnP2tn−p0∥≤∥1−σnP1rn+σnP2tn−p0∥=∥1−σnP1rn−p0+σnP2tn−p0∥≤1−σn∥rn−p0∥+σn∥tn−p0∥≤∥rn−p0∥−σn∥rn−p0∥+σn∥tn−p0∥.
This implies that(14)∥rn+1−p0∥−∥rn−p0∥σn≤∥tn−p0∥−∥rn−p0∥.
So(15)∥rn+1−p0∥−∥rn−p0∥≤∥rn+1−p0∥−∥rn−p0∥σn≤∥tn−p0∥−∥rn−p0∥.
This implies(16)∥rn+1−p0∥≤∥tn−p0∥.
By taking lim inf both sides, we have(17)c≤liminfn→∞∥tn−p0∥.
From (11) and (17), we get
(18)c=limn→∞∥tn−p0∥,(19)c=limn→∞∥1−ςnrn+ςnP2rn−p0.
By using (11) and (19) and Lemma 3, we get
(20)limn→∞∥P2rn−rn∥=0,(21)∥rn+1−p0∥≤∥P1sn−p0∥≤∥sn−p0∥.
By taking lim inf both sides, we have(22)c≤liminfn→∞∥sn−p0∥.
By using (12) and (22), we have
(23)limn→∞∥sn−p0∥=c,limn→∞∥P21−σnP1rn+σnP2tn−p0∥=c.We also have
(24)∥P21−σnP1rn+σnP2tn−p0∥≤∥1−σnP1rn+σnP2tn−p0∥.
By taking limit infimum on both sides, we get(25)c≤liminfn→∞∥1−σnP1rn+σnP2tn−p0∥.
By using edge preserving property ofP1 and P2, we have
(26)∥P1rn−p0∥≤∥rn−p0∥,(27)∥P2tn−p0∥≤∥tn−p0∥≤∥rn−p0∥.
By taking limit sup on both sides in (26) and (27), we get
(28)limsupn→∞∥P1rn−p0∥≤c,(29)limsupn→∞∥P2tn−p0∥≤c.
By using (28) and (29), we obtain
(30)∥1−σnP1rn+σnP2tn−p0∥≤1−σn∥P1rn−p0∥+σn∥P2tn−p0∥≤1−σn∥rn−p0∥+σn∥tn−p0∥≤∥rn−p0∥.
By taking limit sup on both sides in (30), we get
(31)limsupn→∞∥1−σnP1rn+σnP2tn−p0∥≤c.
By using (25) and (31), we have
(32)limn→∞∥1−σnP1rn+σnP2tn−p0∥=c.
By (28), (29), and (32) and Lemma 3, we have
(33)limn→∞∥P1rn−P2tn∥=0.
In addition,(34)∥tn−rn∥=∥1−ςnrn+ςnP2rn−rn∥=∥ςnP2rn−rn∥→0asn→∞.
By using (20), (33), and (34), we have
(35)∥P1rn−rn∥=∥P1rn−P2tn+P2tn−rn∥≤∥P1rn−P2tn∥+∥P2tn−rn∥≤∥P1rn−P2tn∥+∥P2tn−P2rn∥+∥P2rn−rn∥≤∥P1rn−P2tn∥+∥tn−rn∥+∥P2rn−rn∥→0asn→∞.
Therefore, we concludelimn→∞∥P1rn−rn∥=0=limn→∞∥P2rn−rn∥.The next proof is for the weak convergence of the sequence generated by (1) in a uniformly convex Banach space with directed graph satisfying Opial’s condition.Theorem 9.
LetX, C, F, P1, P2, and rn be the same as in Proposition 7 with X satisfying Opial’s condition and C has property WG. Suppose that σn and ςn are real sequences in 0,1 and r0,p0,p0,r0∈EG for arbitrary r0∈C and p0∈F and then rn weakly converges to a common fixed point of P1 and P2.Proof.
Letp0∈F be such that r0,p0,p0,r0∈EG. From Lemm 8(i), limn→∞∥rn−p0∥ exists, so rn is bounded. It follows from Lemma 8(ii) that limn→∞∥P1rn−rn∥=0=limn→∞∥P2rn−rn∥. Since X is uniformly convex and rn is bounded, we may assume that rn⇀u as n→∞, without loss of generality. By Lemma 2, we have u∈F. Suppose that subsequences rnk and rnj of rn converges weakly to u and v, respectively. By Lemma 8(ii), we obtain that ∥Pirnk−rnk∥→0 and ∥Pirnj−rnj∥→0 as k,j→∞. Using Lemma 2, we have u,v∈F. By Lemma 8(i), limn→∞∥rn−u∥ and limn→∞∥rn−v∥ exist. It follows from Lemma 4 that u=v. Therefore, rn converges weakly to a common fixed point of P1 and P2.
Next, we prove weak convergence of the sequencern generated by (1) without assuming Opial’s condition in a uniformly convex Banach space with a directed graph.Theorem 10.
LetX, C, F, P1, P2, and rn be the same as in Proposition 7 with C having property WG, σn and ςn are real sequences in 0,1, F is dominated by r0, and F dominates r0. Then, rn weakly converges to a common fixed point of P1 and P2.Proof.
Letp0∈F be such that r0,p0,p0,r0∈EG. From Lemma 8(i), limn→∞∥rn−p0∥ exists, so rn is bounded in C. Since C is nonempty closed convex subset of a uniformly convex Banach space X, it is weakly compact and hence there exists a subsequence rnj of the sequence rn which converges weakly to some point p∈C. By Lemma 8(ii), we obtain that
(36)∥rnj−P1rnj∥=0=∥rnj−P2rnj∥.
By using (20), (34), and (35), we have
(37)∥sn−rn∥=∥P21−σnP1rn+σnP2tn−rn∥≤∥P21−σnP1rn+σnP2tn−P2rn∥+∥P2rn−rn∥≤∥1−σnP1rn+σnP2tn−rn∥+∥P2rn−rn∥≤∥1−σnP1rn−rn+σnP2tn−rn∥+∥P2rn−rn∥≤1−σn∥P1rn−rn∥+σn∥P2tn−rn∥+∥P2rn−rn∥≤1−σn∥P1rn−rn∥+σn∥P2tn−P2rn∥+∥P2rn−rn∥≤1−σn∥P1rn−rn∥+σn∥tn−rn∥+∥P2rn−rn∥→0asn→∞.
In addition,(38)∥P2tn−tn∥≤∥P2tn−P1rn∥+∥P1rn−tn∥≤∥P2tn−P1rn∥+∥P1rn−rn∥+∥rn−tn∥→0asn→∞.And by using (35) and (37),
(39)∥P1sn−sn∥≤∥P1sn−P1rn∥+∥P1rn−sn∥≤∥sn−rn∥+∥P1rn−rn∥+∥rn−sn∥≤2∥sn−rn∥+∥P1rn−rn∥→0asn→∞.
Using Lemma6, I−P1 and I−P2 are G-demiclosed at 0 so that p∈F. To complete the proof, it suffices to show that rn converges weakly to p. To this end, we need to show that rn satisfies the hypothesis of Lemma 5. Let rnj be a subsequence of rn which converges weakly to some q∈C. By similar argument as above q∈F. Now, for each j≥1, using (1), we have
(40)rnj+1=P1snj.
By using (34), we get
(41)P1rnj=P1rnj−rnj+rnj⇀q.
By using (20), we have
(42)P2rnj=P2rnj−rnj+rnj⇀q.We have
(43)tnj=1−ςnjrnj+ςnjP2rnj⇀q.
It follows from (38)
(44)P2tnj=P2tnj−tnj+tnj⇀q.
By using (41) and (44), we get
(45)snj=P21−σnjP1rnj+σnjP2tnj⇀q.
It follows from (39) and (45), we get
(46)P1snj=P1snj−snj+snj⇀q,rnj+1=P1snj⇀q.
Therefore, the sequencern satisfies the hypothesis of Lemma 5 which in turn implies that rn weakly converges to q so that p=q.Next, we recall condition (B) for strong convergence.LetC be a nonempty closed convex subset of a uniformly convex Banach space X. The mappings P1 and P2 on C are said to satisfy condition (B) [21] if there exists a nondecreasing function f:0,∞→0,∞ with f0=0 and fz>0 for all z>0 such that for all r∈C,
(47)max∥r−P1r∥,∥r−P2r∥≥fdr,F,where dr,F=inf∥r−q∥:q∈F.Theorem 11.
LetX, C, F, P1, P2, and rn be the same as in Proposition 7. Suppose that σn and ςn are real sequences in 0,1, Pii=1,2 satisfy condition (B), F is dominated by r0, and F dominates r0. Then, the sequence rn converges strongly to a common fixed point of P1 and P2.Proof.
From Lemma8(i), limn→∞∥rn−q∥ exists and so limn→∞drn,F exists for any q∈F. Also, from Lemma 8(ii), limn→∞∥rn−P1rn∥=0=limn→∞∥rn−P2rn∥. Owing to condition (B),
(48)fdr,F≤max∥r−P1r∥,∥r−P2r∥.We have limn→∞fdrn,F=0. As f:0,∞→0,∞ is a nondecreasing function satisfying f0=0,fz>0 for all z∈0,∞, we obtain that limn→∞drn,F=0.
Hence, we can find a subsequencernj of rn and a sequence uj⊂F such that ∥rnj−uj∥≤1/2j. Put nj+1=nj+h for some h≥1. Then,
(49)∥rnj+1−uj∥≤∥rnj+h−1−uj∥≤∥rnj−uj∥≤12j,∥uj+1−uj∥≤∥uj+1−rnj+1∥+∥rnj+1−uj∥≤12j+1+12j<12j−1→0asj→∞.
Souj is a Cauchy sequence. We assume that uj→q0∈C as j→∞. Since F is closed, we get q0∈F. So we have rnj→q0 as j→∞. Since limn→∞∥rn−q0∥ exists, we get rn→q0.We prove another strong convergence theorem as follows.Theorem 12.
LetX, C, F, P1, P2, and rn be the same as in Proposition 7 with C having property SG, σn and ςn are real sequences in 0,1, F is dominated by r0, and F dominates r0. If one of Pii=1,2 is semicompact, then rn converges strongly to a common fixed point of P1 and P2.Proof.
It follows from Lemma8 that rn is bounded and limn→∞∥P1rn−rn∥=0=limn→∞∥P2rn−rn∥. Since one of P1 and P2 is semicompact, then there exist subsequences rnj of rn such that rnj→q∈C as j→∞. Since C has property SG and transitivity of graph G, we obtain rnj,q∈EG. Notice that, for each i∈0,1, limj→∞∥rnj−Pirnj∥=0.
Then,(50)∥q−Piq∥≤∥q−rnj∥+∥rnj−Pirnj∥+∥Pirnj−Piq∥≤∥q−rnj∥+∥rnj−Pirnj∥+∥rnj−q∥→0asj→∞.
Hence,q∈F. Thus, limn→∞drn,F exists by Theorem 11. We note that drnj,F≤drnj,q→0 as j→∞. Hence, limn→∞drn,F=0. It follows, as in the proof of Theorem 11, that rn converges strongly to a common fixed point of P1 and P2.


## 4. Numerical Examples

This section contains a numerical example which supports our main theorem. It is worth mentioning here that this example is motivated by [25].Example 1.
LetX=ℝ and C=0,2. Let G=VG,EG be a directed graph defined by VG=C and r,s∈EG if and only if 0.50≤r≠s≤1.70 or r=s∈C. Define mappings P1,P2:C→C(51)P1r=r1/3,P2r=4062arcsinr−1+1,for any r∈C.It is easy to show thatP1,P2 are G-nonexpansive mappings but P1,P2 are not nonexpansive mappings because
(52)∣P1r−P1s∣>r−s,∣P2u−P2v∣>u−v,when r=0.5, s=.03, u=1.9, and v=1.5. Choose σn=n/n+5, ςn=n/n+5, for all n∈N and initial points r0=s0=t0=0.5. In Table 1 and Figure 1, we have shown the convergence rate of the modified Ishikawa iteration, modified S-iteration, Thianwan iteration, and proposed iteration (1).Table 1
Convergence of iterative schemes.
Iteration no.Modified Ishikawa iterationModified S-iterationThianwan’s new iterationProposed iteration1.00.50000000000000.50000000000000.50000000000000.50000000000003.00.58397294650790.90453032420560.91602902759940.99535076718265.00.72287812595090.98311244918430.98753895159210.99995895838857.00.84832102538600.99705904027160.99820548549750.99999960905139.00.92993363568280.99950590804440.99975199712460.999999996150311.00.97202899507660.99992080727300.99996731709790.999999999961613.00.99016110176810.99998793438750.99999589698920.999999999999615.00.99690243673230.99999825229320.99999950854331.00000000000000Figure 1
Numerical experiment of Example1 using the modified Ishikawa iteration, modified S-iteration, Thianwan new iteration, and proposed iteration.Figure1 shows the convergence of the modified Ishikawa iteration, modified S-iteration, Thianwan new iteration, and proposed iteration (1) to the common fixed point of P1 and P2 which is 1 in this numerical experiment, and it is clear that the proposed iteration process converges faster than others.


## 5. Conclusion

The purpose of this paper was to study the convergence of a new faster iteration in which twoG-nonexpansive mappings were involved in the setting of uniformly convex Banach spaces with a directed graph. Also, we constructed a numerical example to show the fastness of our iteration procedure over other existing iteration procedures in the literature.



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