JMATHJournal of Mathematics2314-47852314-4629Hindawi10.1155/2020/21696522169652Research ArticleApproximating Fixed Points of Reich–Suzuki Type Nonexpansive Mappings in Hyperbolic SpacesUllahKifayat1https://orcid.org/0000-0002-8119-9546AhmadJunaid1https://orcid.org/0000-0001-9320-9433De La SenManuel2KhanMuhammad Naveed1JaballahAli1Department of MathematicsUniversity of Science and TechnologyBannu 28100Khyber PakhtunkhwaPakistanustb.edu.pk2Institute of Research and Development of ProcessesUniversity of the Basque CountryCampus of Leioa (Bizkaia)P. O. Box 644-BilbaoBarrio SarrienaLeioa 48940Spainehu.eus20202272020202025052020170620202406202022720202020Copyright © 2020 Kifayat Ullah et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this work, we prove some strong and Δ convergence results for Reich-Suzuki type nonexpansive mappings through M iterative process. A uniformly convex hyperbolic metric space is used as underlying setting for our approach. We also provide an illustrate numerical example. Our results improve and extend some recently announced results of the metric fixed-point theory.

Spanish GovernmentRTI2018-094336-B-I00Basque GovernmentIT1207-19
1. Introduction

A self-map S on a subset B of a metric space X=X,p is called contraction if there exists some constant θ0,1 such that for all a,bB it follows that pSa,Sbθpa,b. If pSa,Sbpa,b for all a,bB, then S is called nonexpansive. A point wB is called a fixed point of S whenever w=Sw. Banach  theorem (1922) states that any contraction map S on a complete metric space has a unique fixed point which is the limit of the sequence xk generated by the Picard iterates, that is, xk+1=Sxk. In 1965, Kirk , Browder , and Göhde  independently proved that any self nonexpansive mapping S defined on a bounded closed convex subset B of a uniformly convex Banach space always has a fixed point. Now, a natural question which comes in mind is that whether the sequence xk of Picard iterates converges to a fixed point of a self nonexpansive mapping. The answer of this question in general is negative. Therefore, there is a need to construct some new procedures to overcome such situations and to obtain a better rate of convergence, for example, Mann , Ishikawa , Noor , S , Abbas and Nazir , and Thakur et al.  iterative processes are often used to approximate fixed points of nonexpansive mappings. In 2008, Suzuki  introduced a weaker notion of nonexpansive mappings: a self-map S on a subset B of a metric space is said to be Suzuki type nonexpansive if for every two elements a,b in B, pSa,Sbpa,b holds whenever 1/2pa,Sapa,b. It is easy to observe that the class of Suzuki type nonexpansive mappings properly includes the class of nonexpansive mappings.

The class of Suzuki type nonexpansive mappings was studied extensively by many authors (cf. [10, 1221]). Very recently, Pant and Pandey  introduced Reich–Suzuki type nonexpansive mappings which in turn include the class of Suzuki type nonexpansive mappings.

Definition 1.

(see ). Let B be a nonempty subset of a metric space. A map S:BB is said to be Reich–Suzuki type nonexpansive if for all a,bB, there is some constant t0,1 such that(1)12pa,Sapa,bpSa,Sbtpa,Sa+tpb,Sb+12tpa,b.

Approximation of fixed points of nonexpansive and generalized nonexpansive mappings is an active area of research on its own . Recently, Pant and Pandey  used Thakur et al.  iterative process to approximate fixed points of Reich–Suzuki type nonexpansive mappings. The purpose of this paper is to prove strong and Δ convergence results for Reich–Suzuki type nonexpansive mappings under M iterative process , which is known to converge faster than the Thakur et al.  iterative process. In this way, we improve and extend to many results of the current literature.

Kohlenbach  suggested the concept of generalized metric spaces and so-called hyperbolic metric spaces. This type of metric spaces includes normed spaces, the Hilbert ball with the hyperbolic metric, Cartesian products of Hilbert balls, metric trees, Hadamard manifolds, and CAT (0) spaces in the sense of Gromov. The definition is given as follows:

A triplet X,p, is called a hyperbolic metric space whenever X,p is a metric space and :X×X×0,1X is a function such that for all a,b,w,sX and μ,ξ0,1, the following conditions hold.

ps,a,b,μ1μps,a+μps,b

pa,b,μ,a,b,ξ=μξpa,b

a,b,μ=b,a,1μ

pa,s,μ,b,w,μμps,w+1μpa,b

The set sega,ba,b,μ:μ0,1 is known as metric segment with endpoints a and b. Throughout, we will write a,b,μ=1μaμb. A subset B of X is called convex provided that 1μaμbB, for every a,bB and μ0,1. When there is no ambiguity, we will write X,p for X,p,.

Definition 2.

Let X,p be a hyperbolic metric space. For any sX,c>0 and λ>0. Set(2)σc,λ=inf11cp12a12b,s:pa,sc,pb,sc,pa,bcλ.

We say that X is uniformly convex whenever σc,λ>0, for every c>0 and λ>0.

Definition 3.

(see [29, 30]). A hyperbolic metric space X,p is said to be strictly convex provided that for every a,b,sX and μ0,1 such that pμa1μb,s=pa,s=pb,s follows the condition a=b. By , every uniformly convex hyperbolic metric space is strictly convex.

Recently, Ullah and Arshad  introduced M-iteration process in Banach spaces. The hyperbolic space version of M iteration process reads as follows:(3)x1=xB,zk=1δkxkδkSxk,yk=Szk,xk+1=Syk,k1,where δkk=1 is a sequence in 0,1. Ullah and Arshad  proved some weak and strong convergence results of M iterative process for Suzuki type nonexpansive mappings in the context of Banach spaces. Ullah et al.  extended the results of Ullah and Arshad  to the setting of CAT (0) spaces. In this paper, we study M iteration process for Reich–Suzuki type nonexpansive mappings in the setting of hyperbolic spaces.

2. Preliminaries

Let B be a nonempty subset of hyperbolic metric space X,p and xk be a bounded sequence in X. For each aX, define

Asymptotic radius of xk at a as(4)rxk,alimsupkpxk,a.

Asymptotic radius of xk relative to B as(5)rxk,Binfrxk,a;aB.

Asymptotic center of xk relative to K by(6)Axk,BaB:rxk,a=rxk,B.

We know that in a complete hyperbolic space with monotone modulus of uniform convexity, every bounded sequence xk has a unique asymptotic center with respect to every nonempty closed convex subset B of X.

Definition 4.

(see ). Assume that xk is a bounded sequence in a hyperbolic space X,p. Then, xk is said to be Δ-convergent to a point sX, if s is a unique asymptotic center of each subsequence xki of xk.

The following lemma gives many numbers of Reich‐Suzuki type nonexpansive mappings.

Lemma 1 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Let B be a nonempty subset of a hyperbolic space and S:BB. If S is Suzuki nonexpansive, then S is Reich–Suzuki type nonexpansive with constant t=0.

Fixed-point set structure of Reich–Suzuki type maps is as follows.

Lemma 2 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Let B be a nonempty subset of a hyperbolic space and S:BB. If S is Reich–Suzuki type nonexpansive, then FS is closed. Furthermore, if the space X is strictly convex and the set B is convex, then FS is also convex.

Lemma 3 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Let B be a nonempty subset of a hyperbolic metric space and S:BB is a Reich–Suzuki type nonexpansive. Then, for every aB and wFS, it implies that pSw,Sapw,a.

Lemma 4 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Let B be a nonempty subset of a hyperbolic metric space and S:BB is a Reich–Suzuki type nonexpansive. Then, for all a,bB, it follows that(7)pa,Sb3+t1tpa,Sa+pa,b.

Lemma 5 (see [<xref ref-type="bibr" rid="B33">33</xref>]).

Let X,p be a complete hyperbolic space with a monotone modulus of uniform convexity and xX. If 0<aδkb<1 and xk, yk are sequences in X such that limsupkdxk,xξ, limsupkpyk,xξ, and limkpδkxk1δkyk,x=ξ for some ξ0, then limkpxk,yk=0.

3. Convergence Results in Hyperbolic Spaces

Throughout this section, the letter X will stand for a hyperbolic space with a monotone modulus of uniform convexity.

Lemma 6.

Let B be a nonempty closed convex subset of X. Let S:BB be a Reich–Suzuki type nonexpansive map with FS. Let xk be the sequence generated by (3). Then, limkpxk,w exists for wFS.

Proof.

Let wFS. By Lemma 3, we have(8)pxk+1,w=pSyk,wpyk,w=pSzk,wpzk,w=p1δkxkδkSxk,w1δkpxk,w+δkpSxk,w1δkpxk,w+δkpxk,w=pxk,w.

Hence, the sequence pxk,w is bounded below and decreasing. Thus, limkpxk,w exists for wFS.

The following theorem will be used in the upcoming results.

Theorem 1.

Let B be a nonempty closed convex subset of X and let S:BB be a Reich–Suzuki type nonexpansive map. Let xk be the sequence defined by (3). Then, FS if and only if xk is bounded and limkpSxk,xk=0.

Proof.

We assume that xk is bounded and limkpSxk,xk=0. Let wAB,xk. We shall prove that Sw=w. By Lemma 4, we have(9)rSw,xk=limsupkpxk,Swlimsupk3+t1tpSxk,xk+limsupkpxk,w=limsupkpxk,w=rw,xk.

Hence, rSw,xk=rw,xk and AB,xk is singleton set. We must have Sw=w. Hence, FS.

Conversely, we assume that FS and wFS. We shall prove that xk is bounded and limkpxk,Sxk=0. By Lemma 6, limkpxk,w exists and xk is bounded. Put(10)limkpxk,w=ξ.

From the proof of Lemma 6, it follows that(11)pzk,wpxk,w,(12)limsupkpzk,wlimsupkpxk,w=ξ.

By Lemma 3, we have(13)limsupkpSxk,wlimsupkpxk,w=ξ.

Again from the proof of Lemma 6, it follows that(14)pxk+1,wpzk,w,(15)ξliminfkpzk,w.

From (12) and (15), we get(16)ξ=limkpzk,w.

From (16), we have(17)ξ=limkpzk,w,ξ=limkp1δkxkδkSxk,w.

Now from (10), (13), and (17) together with Lemma 5, we obtain(18)limkpSxk,xk=0.

The Δ convergence result is as follows.

Theorem 2.

Let B be a nonempty closed convex subset of X. If S:BB is a Reich–Suzuki type nonexpansive mapping with FS, then xk is defined by (3). Δ converges to an element of FS.

Proof.

By Theorem 1, the sequence xk is bounded. Hence, one can find a Δ-convergent subsequence of xk. Next, it is our aim to prove that each Δ-convergent subsequence of xk has a unique Δ limit in the set FS. For this purpose, we assume that xk has two Δ-convergent subsequences, namely, xkj and xkl, with Δ limits d1 and d2, respectively. In view of Theorem 1, the sequence xkj is bounded and limjpSxkj,xkj=0. We claim that d1FS. Now,(19)rxk,Sd1=limsupkpxk,Sd1.

Using Lemma 4, we have(20)rxkj,Sd1=limsupjpxkj,Sd13+t1tlimsupjpxkj,Sxkj+limsupjpxkj,d1rxkj,d1.

Since the asymptotic center of xkj has a unique element, Sd1=d1. Similarly, Sd2=d2. By the uniqueness of asymptotic center of a sequence, we have(21)limsupkpxk,d1=limsupjpxkj,d1<limsupjpxkj,d2=limsupkpxk,d2=limsuplpxkl,d2<limsuplpxkl,d1=limsupkdxk,d1,which is a contradiction. Hence, the conclusions are reached.

Now, we establish a strong convergence theorem for Reich–Suzuki type nonexpansive maps using iteration process (3).

Theorem 3.

Let B be a nonempty compact convex subset of X and S:BB be a Reich–Suzuki type nonexpansive map with FS. If xk is generated by (3), then xk converges strongly to the fixed point of S.

Proof.

By Theorem 1, limkpSxk,xk=0. By the compactness assumption, there exists a subsequence xki of xk such that xki converges strongly to some z in B. By Lemma 4, we have(22)pxki,Sz3+t1tpxki,Sxki+pxki,z0.

By the uniqueness of limits in metric spaces, we must have z=Sz. By Lemma 6, limkpxk,z exists, and hence z is the strong limit of xk.

Example 1.

Let B=2,2 be a subset of endowed with the usual metric, that is, pa,b=ab. Define S:BB by(23)Sa=a2,if a2,0\14,0,if a=14,a4,if a0,2.

Here, first we shall show that the mapping S is not Suzuki type nonexpansive. To do this, we choose a=1/4 and b=2/5. The straightforward calculations give(24)12pa,Sa<pa,b,pSa,Sb>pa,b.

Next, we shall show that the mapping S is Reich–Suzuki type nonexpansive. To do this, we choose t=1/2. We consider different situations as given below.

Case 1.

For a,b2,0\=1/4, we have(25)pSa,Sb=12ab12a+12b34a+34b=12a+a2+12b+b2tpa,Sa+tpb,Sb+12tpa,b.

Case 2.

For a,b0,2, we have(26)pSa,Sb=14ab14a+14b58a+58b=12a+a4+12b+b4=tpa,Sa+tpb,Sb+12tpa,b.

Case 3.

For a2,0\1/4 and b0,2, we have(27)pSa,Sb=a2b412a+14b34a+58b=12a+a2+12b+b4=tpa,Sa+tpb,Sb+12tpa,b.

Case 4.

For a2,0\1/4 and b=1/4, we have(28)pSa,Sb=12a34a34a+18=12pa,Sa+12pb,Sb=tpa,Sa+tpb,Sb+12tpa,b.

Case 5.

For a2,0 and b=1/4, we have(29)pSa,Sb=14a58a58a+18=12pa,Sa+12pb,Sb=tpa,Sa+tpb,Sb+12tpa,b.

Thus, S is Reich–Suzuki type nonexpansive mapping with FS. Set δk=1/k+1k=1. Hence, the requirements of Theorem 2 are fulfilled. Now, the conclusions of Theorems 2 and 3 are reached. However, we cannot directly apply any result in [9, 10, 14, 16, 17, 1922, 34] and references cited therein because in this situation, S is not Suzuki type nonexpansive mapping.

For the next strong convergence result, compactness assumption is not necessary; however, the following condition will be added.

Definition 5.

Let B be a nonempty subset of X. A sequence xk in X is called Fejer monotone with respect to B, if(30)pxk+1,wpxk,w,for every wB and k1.

A mapping S:BB is said to satisfy condition I  if one can construct a nondecreasing function Ω:0,0, with properties Ω0=0 and Ωc>0 for every c0, such that pa,SaΩpa,FS for each aB.

The following facts can be found in .

Proposition 1.

Let B be a nonempty closed subset of X. Let xk be a Fejer monotone sequence with respect to B. Then, xk converges strongly to some point of B if and only if limkpxk,B=0.

Theorem 4.

Let B be a nonempty closed convex subset of X and S:BB be a generalized Reich–Suzuki type nonexpansive map with FS. If S satisfies condition (I), then xk generated by (3) converges strongly to the fixed point of S.

Proof.

From Theorem 1, we have(31)limkpSxk,xk=0.

Since the mapping S satisfies the condition I, it follows that(32)limkpxk,FS.

By Lemma 2, the set FS is closed. It follows from Lemma 6 that xk is Fejer monotone sequence with respect to the set FS. The conclusions are based on Proposition 1.

4. Conclusions

Our results extend the corresponding results of Ullah and Arshad  in two ways: (i) from the class of Suzuki type nonexpansive maps to the class of Reich–Suzuki type nonexpansive maps and (ii) from Banach spaces to the general setting of hyperbolic spaces.

Our results extend the corresponding results of Ullah et al.  in two ways: (i) from the class of Suzuki type nonexpansive maps to the class of Reich–Suzuki type nonexpansive maps and (ii) from CAT (0) spaces to the general setting of hyperbolic spaces.

Our results also extend and improve the corresponding results proved in [9, 10, 14, 16, 17, 1922, 34] and references cited therein.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The authors are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19.

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