# Some Convergence Results for a Class of Generalized Nonexpansive Mappings in Banach Spaces

Authors: Thabet Abdeljawad; Kifayat Ullah; Junaid Ahmad; Manuel de la Sen; Muhammad Naveed Khan
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

This paper investigates fixed points of Reich-Suzuki-type nonexpansive mappings in the context of uniformly convex Banach spaces through anM∗ iterative method. Under some appropriate situations, some strong and weak convergence theorems are established. To support our results, a new example of Reich-Suzuki-type nonexpansive mappings is presented which exceeds the class of Suzuki-type nonexpansive mappings. The presented results extend some recently announced results of current literature.


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## Body

## 1. Introduction

Fixed point theory and applications played an important role in many areas of applied sciences and solved many problems rising in engineering, mathematical economics and optimization. When the problem cannot be solved by ordinary analytical methods, we transform it to the form of fixed point problems and apply an appropriate iterative method to obtain the required fixed point. In 1922, Banach [1] proved that if T is contraction mapping on a closed subset K of a Banach space, that is, Ts−Ts′≤αs−s′ for a fixed α∈0,1 and s,s′∈K, then T possesses a unique fixed point, which can be obtained by using the Picard [2] iteration process. In 1930, Caccioppoli [3] extended the Banach result to the frame work of complete metric spaces. The Banach-Caccioppoli results is an important tool for solving many problems in fractional calculus, mathematical biology, mathematical economics, and engineering. Nevertheless, the conclusions of Banach-Caccioppoli result no more holds if one replace the contraction condition of T by a nonexpansive condition of T, that is, if one choose the value of α=1 in the contraction inequality, even the underlying space is a Banach space. The first basic result concerning the existence of a fixed point for the class of nonexpansive mappings was independently proved by Browder [4] and Göhde [5]. They proved that any nonexpansive mapping on a closed bounded convex subset of a uniformly convex Banach space (in short UCBS) always possesses a fixed point. The class of nonexpansive mappings is an important extension of the class of contraction mappings. On the other hand, Suzuki [6] introduced the notion of generalized nonexpansive mappings by restricting the range of elements that satisfy the nonexpansive inequality: a selfmap T on a subset K of a Banach space is said to be Suzuki-type nonexpansive if for all s,s′∈K satisfying the condition 1/2s−Ts≤s−s′, one has the inequality Ts−Ts′≤s−s′. Obviously, any nonexpansive mapping T belongs to the class of Suzuki-type nonexpansive mappings. Recently in 2019, Pant and Pandey [7] introduced an interesting generalization of Suzuki-type nonexpansie mappings: a selfmap T on a subset K of a Banach space is said to be Reich-Suzuki-type nonexpansive if for all s,s′∈K, satisfying the condition 1/2s−Ts≤s−s′, one can find a real constant c∈0,1, such that Ts−Ts′≤cs−Ts+s′−Ts′+1−2cs−s′. The class of Reich-Suzuki-type nonexpansive mappings is important, because it properly contains the class of Suzuki-type nonexpansive mappings as shown by an example in this paper. Unlike contraction mappings, Picard iterative process does not always converge to a fixed point of a nonexpansive mapping T even FT=p∈K:Tp=p≠∅. For finding fixed points of nonexpansive and generalized nonexpansive mappings, many iterative processes are available in the literature (see, e.g., Mann [8], Ishikawa [9], Agarwal et al. [10], Noor [11], Abbas and Nazir [12], and Thakur et al. [13]). For more details on this direction, we shall refer the reader to [14–18].Recently, Ullah and Arshad [19] introduced a new iterative process called M∗ iterative process, as follows:
(1)s1∈K,zn=1−ρnsn+ρnTsn,yn=T1−δnsn+δnTzn,sn+1=Tyn.They proved thatM∗ iterative process can be used for finding fixed points of Suzuki-type nonexpansive mappings. With the help of a numerical example, they proved that M∗ iterative process converges faster that all of the Mann, Ishikawa, Agarwal, Noor, Abbas, and Thakur iterative processes. In this paper, we extend their results to the general setting of Reich-Suzuki-type nonexpansive mappings.


## 2. Preliminaries

This section is consists of some basic definitions and earlier results, which are needed in sequel.Definition 1.
Letsn be a bounded sequence in a Banach space X and ∅≠K⊆X. The asymptotic radius of sn wrt to K is rK,sn=inflimsupn⟶∞sn−w:w∈K. Moreover, the asymptotic center of sn relative to K is the set AK,sn=w∈K:limsupn⟶∞sn−w=rK,sn.Remark 2.
In Banach spaces, the setAK,sn is singleton provided that X is uniformly convex [20]. Moreover, the set AK,sn is convex provided that K is weakly compact and convex (see, e.g., [21, 22]).Definition 3 (see [23]).
A Banach spaceX is said to be endowed with an Opial’s property provided that for any sequence sn in X which weakly converges to s∈X and for each element w of X-s, it follows that
(2)limsupn⟶∞sn−s<limsupn⟶∞sn−w.The following result gives many numbers of Reich-Suzuki-type nonexpansive mappings.Lemma 4 (see [7]).
LetK be a nonempty subset of a Banach space and let T:K⟶K be a Reich-Suzuki-type nonexpansive mapping. Then, T is also Suzuki-type nonexpansive mapping with a real constant c=0.Lemma 5 (see [7]).
LetK be a nonempty subset of a Banach space and let T:K⟶K be a Reich-Suzuki-type nonexpansive mapping. Then, for all p∈FT and s∈K, we have Ts−p≤s−p.Lemma 6 (see [7].
LetK be a nonempty subset of a Banach space and let T:K⟶K be a Reich-Suzuki-type nonexpansive mapping. Then, for all s,s′∈K, we have s−Ts′≤3+c/1−cs−Ts+s−s′.Lemma 7 (see [15]).
LetT be a Reich-Suzuki-type nonexpansive mapping on a subset K of a Banach space X with the Opial property. If sn converges weakly to q and limn⟶∞sn−Tsn=0, then q∈FT.We recall an important result of Schu [24].Lemma 8.
LetX be a UCBS and 0<a≤dn≤b<1 for all n∈ℕ. If gn and hn are two sequences in X such that limsupn⟶∞gn≤ξ, limsupn⟶∞hn≤ξ, and limn⟶∞dngn+1−dnhn=ξ for some ξ≥0, then limn⟶∞gn−hn=0.


## 3. Convergence Theorems in Uniformly Convex Banach Spaces

This section deals with some weak and strong convergence results. First, we give the following key lemma, which will play an important role in the sequel.Lemma 9.
LetK be a nonempty closed convex subset of a Banach space X and let T:K⟶K be a Reich-Suzuki-type nonexpansive mapping with FT≠∅. For arbitrarily chosen s1∈K, let the sequence sn be defined by (1); then, limn⟶∞sn−q exists for all q∈FT.Proof.
Supposeq∈FT. By Lemma 5, we have
(3)sn+1−q=Tyn−q≤yn−q=T1−δnsn+δnTzn−q≤1−δnsn+δnTzn−q≤1−δnsn−q+δnzn−q=1−δnsn−q+δn1−ρnsn+ρnTsn−q≤1−δnsn−q+δn1−ρnsn−q+ρnTsn−q≤1−δnsn−q+δn1−ρnsn−q+ρnsn−q=1−δnsn−q+δnsn−q=sn−q.
Thus,sn−q is bounded and nonincreasing, which implies that limn⟶∞sn−q exists for each q∈FT.Now, we establish the following theorem, which will be used in the upcoming strong and weak convergence results.Theorem 10.
LetK be a nonempty closed convex subset of a UCBS X and let T:K⟶K be a Reich-Suzuki-type nonexpansive mapping. Let sn be the sequence defined by (1). Then, FT≠∅ if and only if sn is bounded and limn⟶∞sn−Tsn=0.Proof.
Assume that the sequencesn is bounded and limn⟶∞sn−Tsn=0. Fix q∈AK,sn. We need to prove that q is the element of FT. Using Lemma 6, we have
(4)rTq,sn=limsupn⟶∞sn−Tqlimsupn⟶∞3+c1−csn−Tsn+limsupn⟶∞sn−q=limsupn⟶∞sn−q=rq,sn.
Hence,Tq∈AK,sn. Since the set AK,sn is singleton, we must have Tq=q. Hence, p is the element of FT and so FT≠∅.
Conversely, we take the setFT≠∅ and prove that the sequence sn is bounded and limn⟶∞sn−Tsn=0. Fix q∈FT. By Lemma 9, limn⟶∞sn−q exists and sn is bounded. Let us put
(5)limn⟶∞sn−q=ξ.
In the proof of Lemma9, we see that
(6)zn−q≤sn−q⇒limsupn⟶∞zn−q≤limsupn⟶∞sn−q=ξ.
Using Lemma5 and (5), we obtain the following:
(7)limsupn⟶∞Tsn−q≤limsupn⟶∞sn−q=ξ.
Again in the from the proof of Lemma9, we see that
(8)sn+1−q≤1−δnsn−q+δnzn−q.
It follows that(9)sn+1−q−sn−q≤sn+1−q−sn−qδn≤zn−q−sn−q.
So, we can getsn+1−q≤zn−q.
(10)⇒ξ≤liminfn⟶∞zn−q.
From (6) and (10), we get
(11)ξ=limn⟶∞zn−q.
Using (11), we have
(12)ξ=limn⟶∞zn−q=limn⟶∞1−ρnsn+ρnTsn−q=limn⟶∞1−ρnsn−q+ρnTsn−q.
Hence(13)ξ=limn⟶∞1−ρnsn−q+ρnTsn−q.
Now bearing (5), (7), and (13) and Lemma 8, we obtain
(14)limn⟶∞Tsn−sn=0.
Now, we state and prove the following strong convergence results, under the strong assumption of compactness. Notice that it is the extension of the ([19], Theorem 3.3) from the setting of Suzuki-type nonexpansive mappings to the general setting of Reich-Suzuki-type nonexpansive mappings.Theorem 11.
LetK be a nonempty convex compact subset of a UCBS X and let T and sn be as in Theorem 10 and FT≠∅. Then, sn converges strongly to the fixed point of T.Proof.
By Theorem10, limn⟶∞Tsn−sn=0. The compactness of K follows that the sequence sn has a strongly convergent subsequence, namely, snl with a strong limit, say, v∈K. We need to prove that v is the strong limit of sn and v∈FT. Using Lemma 6, we have
(15)snl−Tv≤3+c1−csnl−Tsnl+snl−v.
If we applyl⟶∞, then we obtain Tv=v, that is, v∈FT. By Lemma 9, limn⟶∞sn−v exists. Hence, v is the strong limit of sn.
Now, we shall only state the following result. We will not include the proof here because the proof is elementary.Theorem 12.
LetK be a nonempty closed convex subset of a UCBS X and let T and sn be as in Theorem 10. If FT≠∅ and liminfn⟶∞infp∈FTsn−p=0, then sn converges strongly to fixed point of T.The following definition is essentially due to Sentor and Dotson [25].Definition 13.
A mappingT on a subset K of a Banach space X is said to satisfy condition I if there exists some nondecreasing function η:0,∞⟶0,∞ satisfying η0=0, ηa>0 for every a>0 and s−Ts≥ηinfq∈FTs−q for all s∈K.
The strong convergence result using conditionI is given as follows. Notice that it is the extension of the [19], Theorem 3.4, from the setting of Suzuki-type nonexpansive mappings to the general setting of Reich-Suzuki-type nonexpansive mappings.Theorem 14.
LetK be a nonempty closed convex subset of UCBS X and let T and sn be as in Theorem 10 and FT≠∅. If T satisfies condition I, then sn converges strongly to a fixed point of T.Proof.
From Theorem10, it follows that
(16)liminfn⟶∞Tsn−sn=0.
SinceT is mapping satisfying condition (I), so we have
(17)sn−Tsn≥ηinfq∈FTsn−q.
Using (16) and (17), we have
(18)liminfn⟶∞ηinfq∈FTsn−q=0.
But the functionη is nondecreasing satisfying η0=0, so we have
(19)liminfn⟶∞infq∈FTsn−q=0.
The conclusions follows from Theorem12.We finish this section with following weak convergence result. Notice that it is the extension of the [19], Theorem 3.2, from the setting of Suzuki-type nonexpansive mappings to the general setting of Reich-Suzuki-type nonexpansive mappings.Theorem 15.
LetX be a UCBS with Opial’s property, K a nonempty closed convex subset of X, and T and sn as in Theorem 10 and FT≠∅. Then sn converges weakly to a fixed point of T.Proof.
By Theorem10, sn is bounded and limn⟶∞Tsn−sn=0. The uniform convexity of X follows the reflexivity of X. Hence, the sequence sn must have a weakly convergenet subsequence, namely, snk with a weak limit say, u1∈K. By Lemma 7, u1 is the element of the set FT. We need only to show that u1 is the weak limit of the sequence sn itself. If u1 is not the weak limit of sn, then one can choose a subsequence, namely, snl of sn and u2∈K such that snl converges weakly to u2 and u2≠u1. Again by Lemma 7, u2 is the element of FT. By Lemma 9 and the Opial property, we have
(20)limn⟶∞sn−u1=limk⟶∞snk−u1<limk⟶∞snk−u2=limn⟶∞sn−u2=liml⟶∞snl−u2<liml⟶∞snl−u1=limn⟶∞sn−u1.
This is a contradiction. Therefore, the proof is complete.


## 4. Numerical Example

In this section, first we present a new example of Reich-Suzuki-type nonexpansive mappings, which exceeds the class of Suzuki-type nonexpansive mappings as follows.Example 16.
DefineT:−2,2⟶−2,2 by
(21)Ts=−s2,fors∈−2,0\−140,fors=−14−s5fors∈0,2.
If we chooses=−1/4 and s′=−2/5, then 1/2s−Ts<s−s′ and Ts−Ts′>s−s′. On the other hand, T is Reich-Suzuki-type nonexpansive with the constant c=1/2. We shall include only nontrivial cases here.Case 1.
Fors,s′∈−2,0\−1/4, we have
(22)Ts−Ts′=12s−s′≤12s+12s′≤34s+34s′=12s+s2+12s′+s′2≤cs−Ts+cs′−Ts′+1−2cs−s′.Case 2.
Fors,s′∈0,2, we have
(23)Ts−Ts′=15s−s′≤15s+15s′≤610s+610s′=12s+s′5+12s′+s′5=cs−Ts+cs′−Ts′+1−2cs−s′.Case 3.
Fors∈−2,0\−1/4 and s′∈0,2, we have
(24)Ts−Ts′=s5−s′2≤12s+15s′≤34s+610s′=12s+s2+12s′+s′5=cs−Ts+cs′−Ts′+1−2cs−s′.Case 4.
Fors∈−2,0\−1/4 and s′=−1/4, we have
(25)Ts−Ts′=12s≤34s≤34s+18=12s−Ts+12s′−Ts′=cs−Ts+cs′−Ts′+1−2cs−s′.Case 5.
Fors∈0,2 and s′=−1/4, we have
(26)Ts−Ts′=15s≤610s≤610s+18=12s−Ts+12s′−Ts′=cs−Ts+cs′−Ts′+1−2cs−s′.
Thus,T is a Reich-Suzuki-type nonexpansive mapping with FT≠∅.
Now, using the above example, andδn=1/n+5 and ρn=n/n+77/4, we get sn−p<10−15, our stopping criterion where p=0 is a fixed point of T. The graphs in Figure 1 show that the sequence sn generated by M∗ iteration process converges faster than the sequence sn generated by the leading Thakur [13] iteration process.Figure 1
Rate of convergence ofM∗ (red line) and Thakur (blue line) under different initial points.Finally, we compare the numbers of iteration required to obtain a fixed points ofM∗ iteration with leading Thakur and S iterations. Set sn−p<10−10 as a stopping criterion where p=0 is a fixed point of the mapping T.Remark 17.
Figure1 and Table 1 suggest that M∗ is better than the Thakur and S iterative processes.Table 1
Influence of parameters: comparison of various iteration processes.
IterationsInitial points−1.9−1.6−1.11.21.61.9Forδn=n/n+110/9,ρn=1/n+32/3S161616151616Thakur998999M∗776666Forδn=1−1/5n+3,ρn=1/n3S181818151515Thakur888999M∗555555Forδn=1/n,ρn=1/n+24S18181 8171818Thakur999999M∗888888Forδn=n/4n+3,ρn=1/4n+93/4S181818181818Thakur109991010M∗877477



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