Copyright © 2009 Yisheng Song and Xiao Liu. 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.

Abstract

We discuss the following viscosity approximations with the weak contraction A for a non-expansive mapping sequence {Tn}, yn=αnAyn+(1−αn)Tnyn, xn+1=αnAxn+(1−αn)Tnxn. We prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Mouda's viscosity approximations with the weak contraction.

1. Introduction

The following famous theorem is referred to as the Banach Contraction Principle.Theorem 1.1 (Banach [1]). Let be a complete metric space and let be a contraction on , that is, there exists such that (1.1) Then has a unique fixed point. In 2001, Rhoades [2] proved the following very interesting fixed point theorem which is one of generalizations of Theorem 1.1 because the weakly contractions contains contractions as the special cases .Theorem 1.2 (Rhoades[2], Theorem  2). Let be a complete metric space, and let be a weak contraction on , that is, (1.2) for some is a continuous and nondecreasing function such that is positive on and . Then has a unique fixed point. The concept of the weak contraction is defined by Alber and Guerre-Delabriere [3] in 1997. The natural generalization of the contraction as well as the weak contraction is nonexpansive. Let be a nonempty subset of Banach space is said to be nonexpansive if (1.3) One classical way to study nonexpansive mappings is to use a contraction to approximate a nonexpansive mapping. More precisely, take and define a contraction by where is a fixed point. Banach Contraction Principle guarantees that has a unique fixed point in , that is, (1.4) Halpern [4] also firstly introduced the following explicit iteration scheme in Hilbert spaces: for (1.5) In the case of having a fixed point, Browder [5] (resp. Halpern [4]) proved that if is a Hilbert space, then (resp. ) converges strongly to the fixed point of , that is, nearest to . Reich [6] extended Halpern's and Browder's result to the setting of Banach spaces and proved that if is a uniformly smooth Banach space, then and converge strongly to a same fixed point of , respectively, and the limit of defines the (unique) sunny nonexpansive retraction from onto . In 1984, Takahashi and Ueda [7] obtained the same conclusion as Reich's in uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Recently, Xu [8] showed that the above result holds in a reflexive Banach space which has a weakly continuous duality mapping . In 1992, Wittmann [9] studied the iterative scheme (1.5) in Hilbert space, and obtained convergence of the iterations. In particular, he proved a strong convergence result [9, Theorem  2] under the control conditions (1.6) In 2002, Xu [10, 11] extended wittmann's result to a uniformly smooth Banach space, and gained the strong convergence of under the control conditions and (1.7) Actually, Xu [10, 11] and Wittmann [9] proved the following approximate fixed points theorem. Also see [12, 13].Theorem 1.3. Let be a nonempty closed convex subset of a Banach space . provided that is nonexpansive with , and is given by (1.5) and satisfies the condition and (or ). Then is bounded and In 2000, for a nonexpansive selfmapping with and a fixed contractive selfmapping , Moudafi [14] introduced the following viscosity approximation method for (1.8) and proved that converges to a fixed point of in a Hilbert space. They are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations. Xu [15] extended Moudafi's results to a uniformly smooth Banach space. Recently, Song and Chen [12, 13, 16–18] obtained a number of strong convergence results about viscosity approximations (1.8). Very recently, Petrusel and Yao [19], Wong, et al. [20] also studied the convergence of viscosity approximations, respectively.

In this paper, we naturally introduce viscosity approximations (1.9) and (1.10) with the weak contraction for a nonexpansive mapping sequence (1.9)(1.10) We will prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Moudafi's viscosity approximations with the weak contraction.

2. Preliminaries and Basic Results

Throughout this paper, a Banach space will always be over the real scalar field. We denote its norm by and its dual space by . The value of at is denoted by and the normalized duality mapping from into is defined by (2.1) Let denote the set of all fixed points for a mapping , that is, , and let denote the set of all positive integers. We write (resp. ) to indicate that the sequence weakly (resp. wea) converges to ; as usual will symbolize strong convergence.

In the proof of our main results, we need the following definitions and results. Let denote the unit sphere of a Banach space is said to have (i) a Gâteaux differentiable norm (we also say that is smooth ), if the limit (2.2) exists for each ; (ii) a uniformly Gâteaux differentiable norm , if for each in , the limit (2.2) is uniformly attained for ; (iii) a Fréchet differentiable norm , if for each , the limit (2.2) is attained uniformly for ; (iv) a uniformly Fréchet differentiable norm (we also say that is uniformly smooth ), if the limit (2.2) is attained uniformly for . A Banach space is said to be (v) strictly convex if (vi) uniformly convex if for all such that For more details on geometry of Banach spaces, see [21, 22].

If is a nonempty convex subset of a Banach space and is a nonempty subset of , then a mapping is called a retraction if is continuous with . A mapping is called sunny if whenever and . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto . We note that if is closed and convex of a Hilbert space , then the metric projection coincides with the sunny nonexpansive retraction from onto . The following lemma is well known which is given in [22, 23].Lemma 2.1 (see [22, Lemma  5.1.6]). Let be nonempty convex subset of a smooth Banach space the normalized duality mapping of , and a retraction. Then is both sunny and nonexpansive if and only if there holds the inequality: (2.3) Hence, there is at most one sunny nonexpansive retraction from onto .

In order to showing our main outcomes, we also need the following results. For completeness, we give a proof.

Proposition 2.2. Let be a convex subset of a smooth Banach space . Let be a subset of and let be the unique sunny nonexpansive retraction from onto . Suppose is a weak contraction with a function on and is a nonexpansive mapping. Then

(i) the composite mapping is a weak contraction on ; (ii) For each , a mapping is a weak contraction on . Moreover, defined by (2.4) is well definition: (2.4) (iii) if and only if is a unique solution of the following variational inequality: (2.5)

Proof. For any , we have (2.6) So, is a weakly contractive mapping with a function . For each fixed and , we have (2.7) Namely, is a weakly contractive mapping with a function . Thus, Theorem 1.2 guarantees that has a unique fixed point in , that is, satisfying (2.4) is uniquely defined for each . (i) and (ii) are proved.

Subsequently, we show (iii). Indeed, by Theorem 1.2, there exists a unique element such that . Such a fulfils (2.5) by Lemma 2.1. Next we show that the variational inequality (2.5) has a unique solution . In fact, suppose is another solution of (2.5). That is, (2.8) Adding up gets (2.9) Hence by the property of . This completes the proof.

Let be a sequence of nonexpansive mappings with on a closed convex subset of a Banach space and let be a sequence in with (C1). is said to have Browder's property if for each , a sequence defined by (2.10) for , converges strongly. Let be a sequence in with (C1) and (C2). Then is said to have Halpern's property if for each , a sequence defined by (2.11) for , converges strongly.

We know that if is a uniformly smooth Banach space or a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, is bounded, is a constant sequence , then has both Browder's and Halpern's property (see [7, 10, 11, 23], resp.). Lemma 2.3 (see [24, Proposition   4]). Let have Browder's property. For each , put , where is a sequence in defined by (2.10). Then is a nonexpansive mapping on .Lemma 2.4 (see [24, Proposition  5]). Let have Halpern's property. For each , put , where is a sequence in defined by (2.11). Then the following hold: (i) does not depend on the initial point . (ii) is a nonexpansive mapping on .Proposition 2.5. Let be a smooth Banach space, and have Browder's property. Then is a sunny nonexpansive retract of , and moreover, define a sunny nonexpansive retraction from to .Proof. For each , it is easy to see from (2.10) that (2.12)(2.13) This implies for any and some (2.14) The smoothness of implies the norm weak continuity of [22, Theorems   4.3.1,  4.3.2], so (2.15) Thus (2.16) By Lemma 2.1, is a sunny nonexpansive retraction from to . We will use the following facts concerning numerical recursive inequalities (see [25–27]). Lemma 2.6. Let and be two sequences of nonnegative real numbers, and a sequence of positive numbers satisfying the conditions and . Let the recursive inequality (2.17) be given where is a continuous and strict increasing function for all with . Then ( 1) converges to zero, as ; ( 2) there exists a subsequence such that (2.18)

3. Main Results

We first discuss Browder's type convergence. Theorem 3.1. Let have Browder's property. For each , put , where is a sequence in defined by (2.10). Let be a weak contraction with a function . Define a sequence in by (3.1) Then converges strongly to the unique point satisfying .Proof. We note that Proposition 2.2(ii) assures the existence and uniqueness of . It follows from Proposition 2.2(i) and Lemma 2.3 that is a weak contraction on , then by Theorem 1.2, there exists the unique element such that . Define a sequence in by (3.2) Then by the assumption, converges strongly to . For every , we have (3.3) then (3.4) Therefore, (3.5) Hence, (3.6) Consequently, converges strongly to . This completes the proof. We next discuss Halpern's type convergence. Theorem 3.2. Let have Halpern's property. For each , put , where is a sequence in defined by (2.11). Let be a weak contraction with a function . Define a sequence in by and (3.7) Then converges strongly to the unique point satisfying . Moreover, there exist a subsequence , and with such that (3.8)Proof. It follows from Proposition 2.2(i) and Lemma 2.4 that is a weak contraction on , then by Theorem 1.2, there exists a unique element such that . Thus we may define a sequence in by (3.9) Then by the assumption, as . For every , we have (3.10) Thus, we get for the following recursive inequality: (3.11) where , and . Thus by Lemma 2.6, (3.12) Hence, (3.13) Consequently, we obtain the strong convergence of to , and the remainder estimates now follow from Lemma 2.6.Theorem 3.3. Let be a Banach space whose norm is uniformly Gâteaux differentiable, and satisfies the condition (C2). Assume that have Browder's property and for every , where is a bounded sequence in defined by (2.10). then have Halpern's property.Proof. Define a sequence in by and (3.14) It follows from Proposition 2.5 and the assumption that is the unique sunny nonexpansive retraction from to Subsequently, we approved that (3.15) In fact, since , then we have (3.16) then (3.17) where is a constant such that by the boundedness of and Therefore, using , and , we get (3.18) On the other hand, since the duality map is norm topology to weak topology uniformly continuous in a Banach space with uniformly Gâteaux differentiable norm, we get that as (3.19) Therefore for any such that for all and all , we have that (3.20) Hence noting (3.18), we get that (3.21)

(3.15) is proved. From (2.10) and , we have for all ,

(3.22) Thus, (3.23) Consequently, we get for the following recursive inequality: (3.24) where and The strong convergence of to follows from Lemma 2.6. Namely, have Halpern's property.

4. Deduced Theorems

Using Theorems 3.1, 3.2, and 3.3, we can obtain many convergence theorems. We state some of them.

We now discuss convergence theorems for families of nonexpansive mappings. Let be a nonempty closed convex subset of a Banach space . A (one parameter) nonexpansive semigroups is a family of selfmappings of such that

(i) for (ii) for and ; (iii) for ; (iv) for each is nonexpansive, that is, (4.1) We will denote by the common fixed point set of , that is, (4.2) A continuous operator semigroup is said to be uniformly asymptotically regular (in short, u.a.r.) (see [28–31]) on if for all and any bounded subset of (4.3) Recently, Song and Xu [31] showed that have both Browder's and Halpern's property in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm whenever . As a direct consequence of Theorems 3.1, 3.2, and 3.3, we obtain the following. Theorem 4.1. Let be a real reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of , and a u.a.r. nonexpansive semigroup from into itself such that , and a weak contraction. Suppose that and satisfies the condition , and satisfies the conditions and . If and defined by (4.4) Then as , both and strongly converge to , where is a sunny nonexpansive retraction from to .

Let a sequence of positive real numbers divergent to , and for each and is the average given by (4.5) Recently, Chen and Song [32] showed that have both Browder's and Halpern's property in a uniformly convex Banach space with a uniformly Gâeaux differentiable norm whenever . Then we also have the following. Theorem 4.2. Let be a uniformly convex Banach space with uniformly Gâteaux differentiable norm, and let be as in Theorem 4.1. Suppose that a nonexpansive semigroups from into itself such that and defined by (4.6) where , and satisfies the condition , and satisfies the conditions and . Then as , both and strongly converge to , where is a sunny nonexpansive retraction from to .

Acknowledgments

The authors would like to thank the editors and the anonymous referee for his or her valuable suggestions which helped to improve this manuscript.

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