# Iterative Approximations for a Class of Generalized Nonexpansive Operators in Banach Spaces

Authors: Thabet Abdeljawad; Kifayat Ullah; Junaid Ahmad; Nabil Mlaiki
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

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.


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## Body

## 1. Introduction

LetX be a Banach space, ∅≠C⊆X and J:C⟶C. An element p∈C 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 ℕ the set of all natural numbers. When J is nonexpansive, that is, for all z,z′∈C, one has Jz−Jz′≤z−z′, then FixJ is nonempty provided that X is uniformly convex and C is convex closed bounded (see [1–3] and others). In 2008, Suzuki [4] introduced a new class of nonlinear mappings, which is the generalization of the class of nonexpansive mappings. A mapping J:C⟶C is said to satisfy condition (C) (or Suzuki mapping) if for all z,z′∈C,(1)12z−Jz≤z−z′one hasJz−Jz′≤z−z′.In 2011, Garcia-Falset et al. [5] extended condition (C) to the general formulations as follows. A mapping J:C⟶C is said to satisfy condition Eμ if there exists some μ≥1 such that(2)z−Jz′≤μz−Jz+z−z′for allz,z′∈C.A mappingJ is said to satisfy condition (E) (or Garcia-Falset mapping) whenever J satisfies condition Eμ for some μ≥1. Garcia-Falset et al. [5] 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 [6] 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., [7–11] 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:C⟶C. Assume that an,bn,cn∈0,1 for all n∈ℕ. Then, the well-known Picard, Mann [12], Ishikawa [13], Noor [14], Agarwal [15], Abbas and Nazir [16], Thakur [17], M∗ [18], and AK [19] iterative schemes are, respectively, read as follows:(3)z1=z∈C,zn+1=Jzn,(4)z1=z∈C,zn+1=1−anzn+anJzn,(5)z1∈C,wn=1−bnzn+bnJzn,zn+1=1−anzn+anJwn,(6)z1=z∈C,vn=1−cnzn+cnJzn,wn=1−bnzn+bnJvn,zn+1=1−anzn+anJwn,(7)z1=z∈C,wn=1−bnzn+bnJzn,zn+1=1−anJzn+anJwn,(8)z1=z∈C,vn=1−cnzn+cnJzn,wn=1−bnJzn+bnJvn,zn+1=1−anJwn+anJvn,(9)z1=z∈C,vn=1−bnzn+bnJzn,wn=J1−anzn+anvn,zn+1=Jwn,(10)z1=z∈C,vn=1−bnzn+bnJzn,wn=J1−anzn+anJvn,zn+1=Jwn,(11)z1=z∈C,vn=J1−bnzn+bnJzn,wn=J1−anvn+anJvn,zn+1=Jwn.In [15], Agarwal et al. proved that the iterative scheme (7) converges faster than the iterative schemes (3)–(5) for contraction mappings. In [17], 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 [18], 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 [19], 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

LetC be any nonempty subset of a Banach space X and let zn be a bounded sequence in X. For z∈X, we set(12)rz,zn=limsupn⟶∞z−zn.The asymptotic radius ofzn relative to C is given by(13)rC,zn=infrz,zn:z∈C.The asymptotic center ofzn relative to C is the set(14)AC,zn=z∈C:rz,zn=rC,zn.When the spaceX is uniformly convex [20], 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 spaceX has Opial’s property [23] if and only if for all zn in C which weakly converges to z∈X and for every w∈X-z, one has(15)lim supn⟶∞zn−z<lim supn⟶∞zn−w.The following lemma gives many examples of Garcia-Falset mappings.Lemma 1.
(see [5]). 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 [5]). Let J be a mapping on a subset C of a Banach space. If J satisfies condition (E), then for all p∈FixJ and z∈C, we have Jz−p≤z−p.Lemma 3.
(see [5]). 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 limn⟶∞zn−Jzn=0, then q∈FixJ.In 1991, Schu [24] proved the following useful fact.Lemma 4.
LetX be a uniformly convex Banach space and 0<u≤θn≤v<1 for all n∈ℕ. If zn and wn are two sequences in X such that limsupn⟶∞zn≤λ, limsupn⟶∞wn≤λ, and limn⟶∞θnzn+1−θnwn=λ for some λ≥0, then limn⟶∞zn−wn=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.
LetC be a nonempty closed convex subset of a Banach space X and let J:C⟶C be a mapping satisfying condition (E) with FixJ≠∅. Let the sequence zn be defined by (11), then limn⟶∞zn−p exists for all p∈FixJ.Proof.
Supposep∈FixJ. By Lemma 2, we have(16)vn−p=J1−bnzn+bnJzn−p≤1−bnzn+bnJzn−p≤1−bnzn−p+bnJzn−p≤1−bnzn−p+bnzn−p≤zn−p,which implies that(17)zn+1−p=Jwn−p≤wn−p=J1−anvn+anJvn−p≤1−anvn+anJvn−p≤1−anvn−p+anJvn−p≤1−anvn−p+anvn−p=vn−p≤zn−p.
Thus,zn−p is bounded and nonincreasing, which implies that limn⟶∞zn−p exists for each p∈FixJ.
The following theorem will be used in the upcoming results.Theorem 1.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X and let J:C⟶C be a mapping satisfying condition (E). Let zn be the sequence defined by (11). Then, FixJ≠∅ if and only if zn is bounded and limn⟶∞Jzn−zn=0.Proof.
Letzn be bounded and limn⟶∞Jzn−zn=0. Let p∈AC,zn. We shall prove that Jp=p. Since J satisfies condition (E), we have(18)rJp,zn=lim supn⟶∞zn−Jp≤μlim supn⟶∞Jzn−zn+lim supn⟶∞zn−p=lim supn⟶∞zn−p=rp,zn.
It follows thatJp∈AC,zn. Since AC,zn is a singleton set, we have Jp=p. Hence, FixJ≠∅.
Conversely, we assume thatFixJ≠∅ and p∈FixJ. We shall prove that zn is bounded and limn⟶∞zn−Jzn=0. By Lemma 5, limn⟶∞zn−p exists and zn is bounded. Put(19)limn⟶∞zn−p=λ.
By (16), we have(20)vn−p≤zn−p⟹lim supn⟶∞vn−p≤lim supn⟶∞zn−p=λ.
By Lemma2, we have(21)Jzn−p≤zn−p,(22)⟹lim supn⟶∞Jzn−p≤lim supn⟶∞zn−p=λ.
By (17), we have(23)zn+1−p≤vn−p.
So, we can get(24)⟹λ≤lim infn⟶∞vn−p.
From (20) and (24), we get(25)λ=limn⟶∞vn−p.
Using (19) and (25), we have(26)λ=limn⟶∞vn−p=limn⟶∞J1−bnzn+bnJzn−p≤limn⟶∞1−bnzn+bnJzn−p=limn⟶∞1−bnzn−p+bnJzn−p≤1−bnlimn⟶∞zn−p+bnlimn⟶∞Jzn−p≤1−bnlimn⟶∞zn−p+bnlimn⟶∞zn−p=λ.
Hence,(27)λ=limn⟶∞1−bnzn−p+bnJzn−p.
Applying Lemma4, we obtain(28)limn⟶∞Jzn−zn=0.
First, we discuss the strong convergence ofzn defined by (11) for mappings with condition (E).Theorem 2.
LetC 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 Theorem1, limn⟶∞Jzn−zn=0. By compactness of C, we can find a subsequence znl of zn such that znl converges strongly to q∈C for some q. Since J satisfies condition (E), there exists some μ≥1, such that(29)znl−Jq≤μznl−Jznl+znl−q.
Lettingl⟶∞, we get Jq=q. By Lemma 5, limn⟶∞zn−q exists. Hence, q is the strong limit of zn.
Proof of the following result is elementary and hence omitted.Theorem 3.
LetC 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 liminfn⟶∞distzn,FixJ=0, then zn converges strongly to a fixed point of J.Now, we establish a strong convergence result for Garcia-Falset mappings using theAK iteration process with the help of condition (I).Definition 1.
(see [25]). Let C be a nonempty subset of a Banach space X. A mapping J:C⟶C is said to satisfy condition (I) if there is a function α:0,∞⟶0,∞ satisfying α0=0 and αt>0 for all t∈0,∞ such that z−Jz≥αdistz,FixJ for all z∈C.Theorem 4.
LetC 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 Theorem1, it follows that(30)lim infn⟶∞Jzn−zn=0.
SinceJ satisfies condition (I), we have(31)zn−Jzn≥αdistzn,FixJ.
From (30), we get(32)lim infn⟶∞αdistzn,FixJ=0.
α:0,∞⟶0,∞ is a nondecreasing function with α0=0 and αt>0 for each t∈0,∞. Hence,(33)lim infn⟶∞distzn,FixJ=0.
The conclusion follows from Theorem3.
Finally, we establish a weak convergence ofzn for mappings with condition (E).Theorem 5.
LetX 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 Theorem1, zn is bounded and limn⟶∞Jzn−zn=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 q1∈C. By Lemma 3, we have q1∈FixJ. 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 q2∈C such that znl converges weakly to q2 and q2≠q1. Again by Lemma 3, q2∈FixJ. By Lemma 5 together with the Opial property, we have(34)limn⟶∞zn−q1=limk⟶∞znk−q1<limk⟶∞znk−q2=limn⟶∞zn−q2=liml⟶∞znl−q2<liml⟶∞znl−q1=limn⟶∞zn−q1.
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 theAK iterative scheme with the other iterative schemes. This example also shows that the converse of Lemma 1 may not hold in general.Example 1.
LetC=0,1. Set J on C as follows:(35)Jz=0,if0≤z<1150,2z3,if1150≤z≤1.
Letμ=3. We shall prove that z−Jz′≤3z−Jz+z−z′ for all z,z′∈C.Case (i): ifz,z′∈0,1/150, then Jz=Jz′=0. Now,(36)z−Jz′=z≤3z=3z−Jz≤3z−Jz+z−z′.Case (ii): ifz,z′∈1/150,1, then Jz=2z/3 and Jz′=2z′/3. Now,(37)z−Jz′≤z−Jz+Jz−Jz′=z−Jz+2z3−2z′3=z−Jz+23z−z′≤z−Jz+z−z′≤3z−Jz+z−z′.Case (iii): ifz∈1/150,1 and z′∈0,1/150, then Jz=2z/3 and Jz′=0. Now,(38)z−Jz′=z=3z3=3z−Jz≤3z−Jz+z−z′.
From the above cases, we conclude thatJ satisfies condition (E). Now choose z=1/250 and z′=1/150. Then, 1/2z−Jz<z−z′ but Jz−Jz′>z−z′. Hence, J is not a Suzuki mapping. For all n∈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.Table 1
Computation table obtained from the AK,M∗, Thakur, Abbas, Agarwal, Noor, Ishikawa, and Mann iteration process using Example 1.
nAKM∗ThakurAbbasAgarwalNoorIshikawaMann10.90.90.90.90.90.90.90.920.16010.26620.33930.39060.50900.54440.59990.690030.02840.07870.12790.16950.28780.32930.39860.529040.00500.02320.04820.07350.16280.19910.26530.4055500.00680.01810.03190.09200.12040.17660.31096000.00680.01380.05200.07280.11750.238370000.00600.02940.04400.07820.1827800000.01660.02660.05200.1401900000.00940.01610.03460.10741000000.00530.00970.02300.082311000000.00290.01530.063112000000.00080.01020.048413000000.00020.00670.0371140000000.00200.0284150000000.00010.02181600000000.01671700000000.01281800000000.0098Figure 1
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 [18], Abbas and Nazir [16], Thakur et al. [17], Phuengrattana [26], and many others. Moreover, our results extend the idea of Ullah and Arshad [19] from the setting of contraction mappings to the general setting of Garcia-Falset mappings.



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