DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi10.1155/2020/46272604627260Research ArticleIterative Approximations for a Class of Generalized Nonexpansive Operators in Banach Spaceshttps://orcid.org/0000-0002-8889-3768AbdeljawadThabet123UllahKifayat4https://orcid.org/0000-0002-8119-9546AhmadJunaid4https://orcid.org/0000-0002-7986-886XMlaikiNabil1GalewskiMarek1Department of Mathematics and General SciencesPrince Sultan UniversityP.O. Box 66833Riyadh 11586Saudi Arabiapsu.edu.sa2Department of Medical ResearchChina Medical UniversityTaichung 40402Taiwancmu.edu.cn3Department of Computer Sciences and Information EngineeringAsia UniversityTaichungTaiwanasia.edu.tw4Department of MathematicsUniversity of Science and TechnologyBannu 28100Khyber PakhtunkhwaPakistanustb.edu.pk202067202020202403202001062020080620206720202020Copyright © 2020 Thabet Abdeljawad 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 article, we prove some weak and strong convergence theorems for mappings satisfying condition (E) using the AK iterative scheme in the setting of Banach spaces. We offer a new example of mapping with condition (E) in support of our main result. Our results extend and improve many well-known corresponding results of the current literature.

Prince Sultan UniversityRG-DES-2017-01-17
1. Introduction

Let X be a Banach space, CX and J:CC. An element pC is called a fixed point for J if Jp=p. We denote by FixJ the set of all fixed points of the map J. Throughout the work, we will denote by the set of all natural numbers. When J is nonexpansive, that is, for all z,zC, one has JzJzzz, then FixJ is nonempty provided that X is uniformly convex and C is convex closed bounded (see  and others). In 2008, Suzuki  introduced a new class of nonlinear mappings, which is the generalization of the class of nonexpansive mappings. A mapping J:CC is said to satisfy condition (C) (or Suzuki mapping) if for all z,zC,(1)12zJzzzone hasJzJzzz.

In 2011, Garcia-Falset et al.  extended condition (C) to the general formulations as follows. A mapping J:CC is said to satisfy condition Eμ if there exists some μ1 such that(2)zJzμzJz+zzfor allz,zC.

A mapping J is said to satisfy condition (E) (or Garcia-Falset mapping) whenever J satisfies condition Eμ for some μ1. Garcia-Falset et al.  proved that every Suzuki mapping satisfies condition (E) with μ=3. Notice also that the class of Garcia-Falset mappings also includes many other classes of generalized nonexpansive (see  for details). In this paper, we study in deep this general class of mapping.

The iterative approximation of fixed points for nonlinear operators is an active research area nowadays (see, e.g.,  and others). The Banach contraction principle suggests a Picard iterative scheme for finding the unique fixed point of a given contraction mapping. However, the Picard iterative scheme does not always converge to a fixed point of a nonexpansive mapping. Thus, to overcome such difficulties and to obtain a better speed of convergence, many iterative schemes are available in the literature. Let C be a nonempty convex subset of a Banach space X and J:CC. Assume that an,bn,cn0,1 for all n. Then, the well-known Picard, Mann , Ishikawa , Noor , Agarwal , Abbas and Nazir , Thakur , M , and AK  iterative schemes are, respectively, read as follows:(3)z1=zC,zn+1=Jzn,(4)z1=zC,zn+1=1anzn+anJzn,(5)z1C,wn=1bnzn+bnJzn,zn+1=1anzn+anJwn,(6)z1=zC,vn=1cnzn+cnJzn,wn=1bnzn+bnJvn,zn+1=1anzn+anJwn,(7)z1=zC,wn=1bnzn+bnJzn,zn+1=1anJzn+anJwn,(8)z1=zC,vn=1cnzn+cnJzn,wn=1bnJzn+bnJvn,zn+1=1anJwn+anJvn,(9)z1=zC,vn=1bnzn+bnJzn,wn=J1anzn+anvn,zn+1=Jwn,(10)z1=zC,vn=1bnzn+bnJzn,wn=J1anzn+anJvn,zn+1=Jwn,(11)z1=zC,vn=J1bnzn+bnJzn,wn=J1anvn+anJvn,zn+1=Jwn.

In , Agarwal et al. proved that the iterative scheme (7) converges faster than the iterative schemes (3)–(5) for contraction mappings. In , Thakur et al. with the help of a numerical example proved that the iterative scheme (9) converges faster than all of the iterative schemes (3)–(8) for the general setting of Suzuki mappings. In , Ullah and Arshad with the help of a numerical example proved that the iterative scheme (10) is better than the leading iterative scheme (9) for Suzuki mappings in Banach spaces. In , the authors proved that the AK iterative scheme (11) is stable and converges faster than many well-known iterative schemes for contraction mappings. The purpose of this research is to study the iterative scheme (11) for the generalized class of Garcia-Falset mappings. We also give a new example of the Garcia-Falset mapping and show that its AK iteration process is more efficient than all of the above schemes.

2. Preliminaries

Let C be any nonempty subset of a Banach space X and let zn be a bounded sequence in X. For zX, we set(12)rz,zn=limsupnzzn.

The asymptotic radius of zn relative to C is given by(13)rC,zn=infrz,zn:zC.

The asymptotic center of zn relative to C is the set(14)AC,zn=zC:rz,zn=rC,zn.

When the space X is uniformly convex , then the set AC,zn is singleton. Notice also that the set AC,zn is convex as well as nonempty provided that C is weakly compact convex (see, e.g., [21, 22]).

We say that a Banach space X has Opial’s property  if and only if for all zn in C which weakly converges to zX and for every wX-z, one has(15)lim supnznz<lim supnznw.

The following lemma gives many examples of Garcia-Falset mappings.

Lemma 1.

(see ). Let J be a mapping on a subset C of a Banach space. If J satisfies condition (C), then J also satisfies condition (E) with μ=3.

Lemma 2.

(see ). Let J be a mapping on a subset C of a Banach space. If J satisfies condition (E), then for all pFixJ and zC, we have Jzpzp.

Lemma 3.

(see ). Let J be a mapping on a subset C of a Banach space X having the Opial property. Assume that J satisfies the condition (E). If zn converges weakly to q and limnznJzn=0, then qFixJ.

In 1991, Schu  proved the following useful fact.

Lemma 4.

Let X be a uniformly convex Banach space and 0<uθnv<1 for all n. If zn and wn are two sequences in X such that limsupnznλ, limsupnwnλ, and limnθnzn+1θnwn=λ for some λ0, then limnznwn=0.

3. Convergence Theorems in Uniformly Convex Banach Spaces

In this section, we shall state and prove our main results. First, we give the following key lemma.

Lemma 5.

Let C be a nonempty closed convex subset of a Banach space X and let J:CC be a mapping satisfying condition (E) with FixJ. Let the sequence zn be defined by (11), then limnznp exists for all pFixJ.

Proof.

Suppose pFixJ. By Lemma 2, we have(16)vnp=J1bnzn+bnJznp1bnzn+bnJznp1bnznp+bnJznp1bnznp+bnznpznp,which implies that(17)zn+1p=Jwnpwnp=J1anvn+anJvnp1anvn+anJvnp1anvnp+anJvnp1anvnp+anvnp=vnpznp.

Thus, znp is bounded and nonincreasing, which implies that limnznp exists for each pFixJ.

The following theorem will be used in the upcoming results.

Theorem 1.

Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let J:CC be a mapping satisfying condition (E). Let zn be the sequence defined by (11). Then, FixJ if and only if zn is bounded and limnJznzn=0.

Proof.

Let zn be bounded and limnJznzn=0. Let pAC,zn. We shall prove that Jp=p. Since J satisfies condition (E), we have(18)rJp,zn=lim supnznJpμlim supnJznzn+lim supnznp=lim supnznp=rp,zn.

It follows that JpAC,zn. Since AC,zn is a singleton set, we have Jp=p. Hence, FixJ.

Conversely, we assume that FixJ and pFixJ. We shall prove that zn is bounded and limnznJzn=0. By Lemma 5, limnznp exists and zn is bounded. Put(19)lim nznp=λ.

By (16), we have(20)vnpznplim supnvnplim supnznp=λ.

By Lemma 2, we have(21)Jznpznp,(22)lim supnJznplim supnznp=λ.

By (17), we have(23)zn+1pvnp.

So, we can get(24)λlim infnvnp.

From (20) and (24), we get(25)λ=limnvnp.

Using (19) and (25), we have(26)λ=limnvnp=limnJ1bnzn+bnJznplimn1bnzn+bnJznp=limn1bnznp+bnJznp1bnlimnznp+bnlimnJznp1bnlimnznp+bnlimnznp=λ.

Hence,(27)λ=limn1bnznp+bnJznp.

Applying Lemma 4, we obtain(28)limnJznzn=0.

First, we discuss the strong convergence of zn defined by (11) for mappings with condition (E).

Theorem 2.

Let C be a nonempty convex compact subset of a uniformly convex Banach space X and let J and zn be as in Theorem 1 and FixJ. Then, zn converges strongly to a fixed point of J.

Proof.

By Theorem 1, limnJznzn=0. By compactness of C, we can find a subsequence znl of zn such that znl converges strongly to qC for some q. Since J satisfies condition (E), there exists some μ1, such that(29)znlJqμznlJznl+znlq.

Letting l, we get Jq=q. By Lemma 5, limnznq exists. Hence, q is the strong limit of zn.

Proof of the following result is elementary and hence omitted.

Theorem 3.

Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let J and zn be as in Theorem 1. If FixJ and liminfndistzn,FixJ=0, then zn converges strongly to a fixed point of J.

Now, we establish a strong convergence result for Garcia-Falset mappings using the AK iteration process with the help of condition (I).

Definition 1.

(see ). Let C be a nonempty subset of a Banach space X. A mapping J:CC is said to satisfy condition (I) if there is a function α:0,0, satisfying α0=0 and αt>0 for all t0, such that zJzαdistz,FixJ for all zC.

Theorem 4.

Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let J and zn be as in Theorem 1 and FixJ. If J satisfies condition (I), then zn converges strongly to a fixed point of J.

Proof.

From Theorem 1, it follows that(30)lim infnJznzn=0.

Since J satisfies condition (I), we have(31)znJznαdistzn,FixJ.

From (30), we get(32)lim infnαdistzn,FixJ=0.

α:0,0, is a nondecreasing function with α0=0 and αt>0 for each t0,. Hence,(33)lim infndistzn,FixJ=0.

The conclusion follows from Theorem 3.

Finally, we establish a weak convergence of zn for mappings with condition (E).

Theorem 5.

Let X be a uniformly Banach space with the Opial property, C a nonempty closed convex subset of X, and let J and zn be as in Theorem 1 and FixJ. Then, zn converges weakly to a fixed point of J.

Proof.

By Theorem 1, zn is bounded and limnJznzn=0. By the Milman–Pettis theorem, the space X is reflexive. Thus, by Eberlin’s theorem, there exists a subsequence znk of zn such that znk converges weakly to some q1C. By Lemma 3, we have q1FixJ. It is sufficient to show that zn converges weakly to q1. In fact, if zn does not converge weakly to q1. Then, there exists a subsequence znl of zn and q2C such that znl converges weakly to q2 and q2q1. Again by Lemma 3, q2FixJ. By Lemma 5 together with the Opial property, we have(34)limnznq1=limkznkq1<limkznkq2=limnznq2=limlznlq2<limlznlq1=limnznq1.

This is a contradiction. Hence, the conclusions are reached.

4. Numerical Example

In this section, we present a new example of the Garcia-Falset mapping which is not a Suzuki mapping. Using this example, we compare the rate of convergence of the AK iterative scheme with the other iterative schemes. This example also shows that the converse of Lemma 1 may not hold in general.

Example 1.

Let C=0,1. Set J on C as follows:(35)Jz=0,if0z<1150,2z3,if1150z1.

Let μ=3. We shall prove that zJz3zJz+zz for all z,zC.

Case (i): if z,z0,1/150, then Jz=Jz=0. Now,(36)zJz=z3z=3zJz3zJz+zz.

Case (ii): if z,z1/150,1, then Jz=2z/3 and Jz=2z/3. Now,

(37)zJzzJz+JzJz=zJz+2z32z3=zJz+23zzzJz+zz3zJz+zz.

Case (iii): if z1/150,1 and z0,1/150, then Jz=2z/3 and Jz=0. Now,

(38)zJz=z=3z3=3zJz3zJz+zz.

From the above cases, we conclude that J satisfies condition (E). Now choose z=1/250 and z=1/150. Then, 1/2zJz<zz but JzJz>zz. Hence, J is not a Suzuki mapping. For all n, let an=0.70, bn=0.65, and cn=0.90. Table 1 and Figure 1 show that the AK iteration scheme converges faster to the fixed point p=0 of the mapping J as compared with the other known iterative schemes.

Computation table obtained from the AK, M, Thakur, Abbas, Agarwal, Noor, Ishikawa, and Mann iteration process using Example 1.

nAKMThakurAbbasAgarwalNoorIshikawaMann
10.90.90.90.90.90.90.90.9
20.16010.26620.33930.39060.50900.54440.59990.6900
30.02840.07870.12790.16950.28780.32930.39860.5290
40.00500.02320.04820.07350.16280.19910.26530.4055
500.00680.01810.03190.09200.12040.17660.3109
6000.00680.01380.05200.07280.11750.2383
70000.00600.02940.04400.07820.1827
800000.01660.02660.05200.1401
900000.00940.01610.03460.1074
1000000.00530.00970.02300.0823
11000000.00290.01530.0631
12000000.00080.01020.0484
13000000.00020.00670.0371
140000000.00200.0284
150000000.00010.0218
1600000000.0167
1700000000.0128
1800000000.0098

Convergence behavior of AK (red line), M (orange line), Thakur (green line), Abbas (yellow line), Agarwal (blue line), Noor (cyan line), Ishikawa (magenta line), and Mann (black line) iterates for mapping J defined in Example 1 where z1=0.9.

5. Conclusions

We have proved several strong and weak convergence results for mappings with condition (E) (Garcia-Falset mappings) in the context of Banach spaces. In view of the above discussion, the results for an operator satisfying condition (C) or else nonexpansive are special cases of our new results. Hence, our results are more general than the results of Ullah and Arshad , Abbas and Nazir , Thakur et al. , Phuengrattana , and many others. Moreover, our results extend the idea of Ullah and Arshad  from the setting of contraction mappings to the general setting of Garcia-Falset mappings.

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 first author and the last author would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (group number RG-DES-2017-01-17).

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