# Approximating Fixed Points of Operators Satisfying (RCSC) Condition in Banach Spaces

Authors: Thabet Abdeljawad; Kifayat Ullah; Junaid Ahmad; Manuel de la Sen; Junaid Khan
Journal: Journal of Function Spaces
 (2020)

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## Abstract

LetK be a nonempty subset of a Banach space E. A mapping T:K→K is said to satisfy (RCSC) condition if each a,b∈K, 1/2a−Fa≤a−b⇒Fa−Fb≤1/3a−b+a−Fb+b−Fa. In this paper, we study, under some appropriate conditions, weak and strong convergence for this class of maps through M iterates in uniformly convex Banach space. We also present a new example of mappings with condition (RCSC). We connect M iteration and other well-known processes with this example to show the numerical efficiency of our results. The presented results improve and extend the corresponding results of the literature.


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## Body

## 1. Introduction

ℕ will denote the set of all natural numbers throughout. In 2008, Suzuki [1] introduced a new class of mappings as follow. A self-map F on a subset K of a Banach space E is said to satisfy C condition if for all a,b∈K, we have
(1)12a−Fa≤a−b⇒Fa−Fb≤a−b.Obviously, whenF is nonexpansive mapping, that is, Fa−Fb≤a−b holds for all a,b∈K, then F satisfies the C condition. However, an example in [1] shows that there exists mappings, which satisfy the C condition but not nonexpansive. A mapping with C condition is often called Suzuki-type nonexpansive mapping. The class of Suzuki-type nonexpansive mappings is extensively studied by many authors (cf. [2–12] and others).In 2012, motivated by SuzukiC condition, Karapinar [13] suggested a new condition on mappings, the so-called (RCSC) condition (or Reich-Chatterjea-Suzuki C condition). A self-map F on a subset K of a Banach space is said to satisfy the (RCSC) condition if for all a,b∈K, we have
(2)12a−Fa≤a−b⇒Fa−Fb≤13a−b+b−Fa+a−Fb.The purpose of this work is to prove some weak and strong convergence results for this class of mappings through theM iteration process [12] in the context of Banach spaces. We also give a numerical example to show the usefulness of our results. In this way, we extend and improve many well-known corresponding results of the current literature.Approximating fixed points of nonlinear mappings played an important role and solved many problems [14–20]. It is now well known that if F is nonexpansive, then the sequence of Picard iterates wn+1=Fwn may not converge to a fixed point of F. To overcome such problems and to get better a rate of convergence, many iterative processes are available in the literature. The well-known iterative processes are the Mann [21], Ishikawa [22], Noor [23], Agarwal et al.S [24], Abbas and Nazir [25], Thakur et al. [7], Ullah and Arshad M [12], and so on. Let αn,βn,γn∈0,1, n∈ℕ, and F be a self-map on a nonempty convex subset K of a Banach space.The Mann iteration process [21] is a sequence wn defined as follows:
(3)w1=w∈K,wn+1=1−αnwn+αnFwn.The Ishikawa iteration process [22] is a sequence wn defined as follows:
(4)w1=w∈K,sn=1−βnwn+βnFwn,wn+1=1−αnwn+αnFsn.The Noor iteration process [23] is a sequence wn defined as follows:
(5)w1=w∈K,vn=1−γnwn+γnFwn,sn=1−βnwn+βnFvn,wn+1=1−αnwn+αnFsn.TheS iteration process [24] is a sequence wn defined as follows:
(6)w1=w∈K,sn=1−βnwn+βnFwn,wn+1=1−αnFwn+αnFsn.The Abbas and Nazir iteration process [25] is a sequence wn defined as follows:
(7)w1=w∈K,vn=1−γnwn+γnFwn,sn=1−βnFwn+βnFvn,wn+1=1−αnFsn+αnFvn.The Thakur et al. iteration process [7] is a sequence wn defined as follows:
(8)w1=w∈K,vn=1−βnwn+βnFwn,sn=F1−αnwn+αnvn,wn+1=Fsn.TheM iteration process [12] is a sequence wn defined as follows:
(9)w1=w∈K,vn=1−αnwn+αnFwn,sn=Fvn,wn+1=Fsn.In this paper, we will present some weak and strong convergence results using theM iteration process (9) for mappings with (RCSC) condition. Similar results for the processes (3)–(8) can be proved on the same line of proofs.


## 2. Preliminaries

p∈K is called a fixed point of a self-map F on K if p=Fp. We will denote by fixF throughout the set of all fixed points of F. A Banach space E is said to satisfy Opial condition [26] if and only if for each weakly convergent sequence wn⊆E with a weak limit w∈E, we have the following property:
(10)liminfn→∞wn−w<liminfn→∞wn−zforallz∈E−w.A self-mapF on a subset K of a Banach space is said to satisfy the condition I [27] if there is nondecreasing function g:0,∞⟶0,∞ with the properties g0=0, gz>0 for every z>0, and a−Fa≥gdista,fixF for all a∈K.LetK be a nonempty subset of a Banach space E and wn a bounded sequence in E. For each x∈E, define
(i)
asymptotic radius ofwn at x by rx,wn≔limsupn→∞x−wn(ii)
asymptotic radius ofwn relative to K by rK,wn=infrx,wn:x∈K(iii)
asymptotic center ofwn relative to K by AK,wn=x∈K:rx,wn=rK,wnWhen the spaceE is uniformly convex [28], then the set AK,wn is always singleton. Notice also that the set AK,wn is convex as well as nonempty provided that K is weakly compact convex (see, e.g., [29, 30]).Lemma 1.
[13].
LetF be a self-map on a subset K of a Banach space. If F satisfies the (RCSC) condition, then for all a,b∈K, the following holds:
(11)a−Fb≤9a−Fa+a−b.The following facts are also needed.Lemma 2.
[13].
LetE be a Banach space having Opial’s property, ∅≠K⊆E and F:K⟶K. If F satisfies the condition (RCSC), then the following condition holds:
(12)wn⊆K,wn⇀w,wn−Fwn⟶0⇒Fw=w.The following lemma gives the structure of the fixed point set associated with a mapping satisfying (RCSC) condition.Lemma 3.
[13].
LetF be a self-map on a subset ∅≠K of a Banach space. If F satisfies the (RCSC) condition, then fixF is closed. Moreover, if E is strictly convex and K is convex, then fixF is also convex.Lemma 4.
[13].
LetF be a self-map on a subset ∅≠K of a Banach space. If F satisfies (RCSC) condition, then for all a∈K and p∈fixF, Fa−Fp≤a−p holds.Lemma 5.
[31].
Let0<x≤ηn≤y<1 for each n∈ℕ and vn and wn be any two sequences in a uniformly convex Banach space E such that limsupn→∞vn≤ζ, limsupn→∞wn≤ζ, and limn→∞ηnvn+1−ηnwn=ζ for some ζ≥0; then, limn→∞vn−wn=0.


## 3. Main Results

We begin this section by proving a crucial lemma.Lemma 6.
LetF be a self-map on a subset ∅≠K of a Banach space. Assume that F satisfies the (RCSC) condition and let wn be a sequence generated by (9). If fixF≠∅, then limn→∞wn−p exists for each p∈fixF.Proof.
Letp∈fixF and n∈ℕ. By Lemma 4, we have
(13)vn−p=1−αnwn+αnFwn−p≤1−αnwn−p+αnFwn−p≤1−αnwn−p+αnwn−p=wn−p,which implies that
(14)wn+1−p=sn−p≤sn−p=Fvn−p≤vn−p≤wn−p.
Hence,wn+1−p≤wn−p for all n∈ℕ and p∈fixF. Thus, wn−p is bounded and nonincreasing, which implies that limn→∞wn−p exists for each p∈fixF.Now we give the necessary and sufficient condition for the existence of a fixed point for mapping with (RCSC) condition defined on a nonempty closed convex subset of a complete uniformly convex Banach space.Theorem 7.
LetF be a self-map on a closed convex subset ∅≠K of a uniformly convex Banach space. Assume that F satisfies the (RCSC) condition and let wn be a sequence generated by (9). Then, fixF≠∅ if and only if wn is bounded and limn→∞Fwn−wn=0.Proof.
Letwn be bounded and limn→∞Fwn−wn=0. Let p∈AK,wn. By Lemma 1, we have
(15)rFp,wn=limsupn→∞wn−Fp≤9limsupn→∞Fwn−wn+limsupn→∞wn−p=limsupn→∞wn−p=rp,wn.
Hence, we conclude thatFp∈A(K,wn. Since E is uniformly convex, AK,wn consists of a unique element. Thus, we have Fp=p.
Conversely, suppose thatfixF≠∅ and p∈fixF. By Lemma 6, limn→∞wn−p exists and wn is bounded. Put
(16)limn→∞wn−p=ζ.
From (13), we have
(17)vn−p≤wn−p⇒limsupn→∞vn−p≤limsupn→∞wn−p=ζ.
By Lemma4, we have
(18)Fwn−p≤wn−p⇒limsupn→∞Fwn−p≤limsupn→∞wn−p=ζ.
From (14), we have
(19)wn+1−p≤vn−p⇒ζ=liminfn→∞wn+1−p≤liminfn→∞vn−p.
From (17) and (19), we have
(20)ζ=limn→∞vn−p.
From (20), we have
(21)ζ=limn→∞vn−p=limn→∞1−αnwn−p+αnFwn−p.
Hence,(22)ζ=limn→∞1−αnwn−p+αnFwn−p.
By Lemma5, we have
(23)limn→∞Fwn−wn=0.Now we can prove the following weak convergence theorem.Theorem 8.
LetF be a self-map on a closed convex subset ∅≠K of a uniformly convex Banach space E having Opial’s property. Assume that F satisfies the (RCSC) condition with fixF≠∅ and let wn be a sequence generated by (9). Then, wn converges weakly to a fixed point of F.Proof.
SinceE is uniformly convex, E is reflexive. By Theorem 7, wn is bounded and limn→∞Fwn−wn=0 for all n∈ℕ. By the reflexivity, one can find a weakly convergent subsequence wni of wn with a weak limit say w∈K. By Lemma 2, we have Fw=w. It is suffice to show that w is the weak limit of wn. w is not the weak limit of wn. Then, one can find another weakly convergent subsequence wnj of wn with a weak limit w′ such that w′≠w. Again by Lemma 2, Fw′=w′. By Lemma 6 together with Opial’s property, we have
(24)limn→∞wn−w=limi→∞wni−w<limi→∞wni−w′=limn→∞wn−w′=limj→∞wnj−w′<limj→∞wnj−w=limn→∞wn−w.
This is a contradiction. So, we havew=w′. Hence, w is the weak limit of wnNow we prove a strong convergence theorem as follows.Theorem 9.
LetF be a self-map on a compact convex subset ∅≠K of a uniformly convex Banach space. Assume that F satisfies the (RCSC) condition with fixF≠∅ and let wn be a sequence generated by (9). Then, wn converges strongly to a fixed point of F.Proof.
By Theorem7, limn→∞Fwn−wn=0 for all n∈ℕ. Since K is compact and convex, we can find a strongly convergent subsequence wnj of wn with a strong limit say q. By Lemma 1, we have
(25)wnj−Fq≤9wnj−Fwnj+wnj−q⟶0.
By the uniqueness of limits in Banach spaces,Fq=q. By Lemma 6, limn→∞wn−q exists and hence q is the strong limit of wn.Now we state the following theorem. Since the proof is elementary, we will not include the details.Theorem 10.
LetF be a self-map on a closed convex subset ∅≠K of a uniformly convex Banach space. Assume that F satisfies the (RCSC) condition with fixF≠∅ and let wn be a sequence generated by (9). Then, wn converges strongly to a fixed point of F provided that liminfn→∞distwn,fixF=0.The following convergence theorem is based on conditionI.Theorem 11.
LetF be a self-map on a closed convex subset ∅≠K of a uniformly convex Banach space. Assume that F satisfies the (RCSC) condition with fixF≠∅ and let wn be a sequence generated by (9). Then, wn converges strongly to a fixed point of F provided that F satisifes the condition I.Proof.
By Theorem7, it follows that liminfn→∞wn−Fwn=0. By the condition I, we have liminfn→∞distwn,fixF=0. The conclusion follows from Theorem 10.


## 4. Example

In this section, we compare the rate of convergence of theM iteration process with other iterations in the setting of mappings with (RCSC) condition.Example 1.
LetK=2,5 be endowed with the usual norm. Set Fa=2 if a=5 and Fa=2+a/2 if a≠5. We shall prove that F satisifes the (RCSC) condition. The case when a,b∈5 is trivial. We consider only the following three nontrivial cases.
Whena,b∈2,5, then Fa=2+a/2 and Fb=2+b/2. Using triangle inequality, we have
(26)∣Fa−Fb∣=12∣a−b∣≤13∣a−b∣+12∣a−b∣=13∣a−b∣+133a2−3b2=13∣a−b∣+13a−2+b2−b−2+a2≤13∣a−b∣+13a−2+b2+13b−2+a2=13a−b+a−Fb+b−Fa.
Whena∈2,5 and b∈5, then Fa=2+a/2 and Fb=2. Now
(27)∣Fa−Fb∣=a−22−2=a−22=133a−62=13a−22+a−2≤13a−22+13∣a−2∣=13a−b+b−2+a2+13a−2≤13∣a−b∣+13b−2+a2+13∣a−2∣=13∣a−b∣+∣a−Fb∣+∣b−Fa∣.
Finally, whena∈5 and b∈2,5, then Fa=2 and Fb=2+b/2. Now
(28)∣Fa−Fb∣=2−b−22=b−22=133b−62=13b−22+b−2≤13b−22+13∣b−2∣=13b−a+a−2+b2+13b−2≤13∣b−a∣+13a−2+b2+13∣b−2∣=13∣a−b∣+∣a−Fb∣+∣b−Fa∣.
Next, fora=4.2 and b=5, 1/2∣a−Fa∣<∣a−b∣ but ∣Fa−Fb∣>∣a−b∣. Hence, F does not satisfy the C condition. Let αn=0.70, βn=0.65, and γn=0.90. The strong convergence of the M (9), Thakur et al. (8), Abbas and Nazir (7), S (6), Noor (5), Ishikawa (4), and Mann (3) iterates to a fixed point p=2 is given in Table 1.Remark 12.
From Table1 and Figure 1, we see that the M iteration process converges faster to p=2 than the others.Table 1
Computation table obtained from theM, Thakur et al., Abbas and Nazir, S, Noor, Ishikawa, and Mann iterates for mapping F defined in Example 1.
nMThakur et al.Abbas and NazirSNoorIshikawaMann1333333322.16252.19312.24562.38632.48512.53632.650032.02642.03732.06032.14922.23532.28762.422542.00432.00722.01482.05762.11412.15422.274652.00072.00142.00362.02232.05542.08272.178562.00012.00032.00092.00862.02692.04432.1160722.00012.00022.00332.01302.02382.07548222.00012.00132.00632.01232.049092222.00052.00312.00682.0318102222.00022.00152.00372.0207112222.00012.00072.00202.01341222222.00032.00112.00881322222.00022.00062.00571422222.00012.00032.003715222222.00022.002416222222.00012.0016172222222.0010182222222.0007192222222.0004202222222.0003212222222.0002222222222.0001232222222Figure 1
Convergence behavior of theM (red line), Thakur et al. (green line), Abbas and Nazir (yellow line), Agarwal et al. (blue line), Noor (cyan line), Ishikawa (magenta line), and Mann (black line) iterates for mapping F defined in Example 1 where w1=3.Now using the above example, we make different choices of parametersαn and βn and intial points and also we get wn−p<10−10 as our stopping criterion where p=2 is a fixed point of F. The number of iterations forM (9) to reachp=2 is compared with the leading three steps of Thakur et al. (8) and leading two steps of S (6) iterations. The numbers in italic in Tables 2–4 show that M iteration is better than the others. The "-" represents that the number of iterations exceeds 50.Table 2
αn=2n/7n+11 and βn=1/3n+9. Number of iterations required to obtain the fixed point.
Initial pointsS (6)Thakur et al. (8)M (9)2.2231382.7241483.3251483.8251584.3261584.826158Table 3
αn=n/n+510/9 and βn=1/n+52/3. Number of iterations required to obtain the fixed point.
Initial pointsS (6)Thakur et al. (8)M (9)2.22814132.72915143.33016143.83116144.33116144.8311614Table 4
αn=n+3/5n+21/15 and βn=2n/5n+1001/4. Number of iterations required to obtain the fixed point.
Initial pointsS (6)Thakur et al. (8)M (9)2.2–11102.7–12113.3–13113.8–14114.3–15124.8–1512



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