Strong Convergence Theorems for Strict Pseudocontractions in Uniformly Convex Banach Spaces

Abstract

The viscosity approximation methods are employed to establish strong convergence theorems of the modified Mann iteration scheme to -strict pseudocontractions in -uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm. The main result improves and extends many nice results existing in the current literature.

1. Introduction

Let be a real Banach space, and let be a nonempty closed convex subset . We denote by the normalized duality map from to defined by

(1.1)

A mapping is said to be a -strictly pseudocontractive mapping (see, e.g., [1]) if there exists a constant such that

(1.2)

for all . We note that the class of -strict pseudocontractions strictly includes the class of nonexpansive mappings which are mapping on such that

(1.3)

for all . Obviously, is nonexpansive if and only if is a -strict pseudocontraction. A mapping is said to be a -strictly pseudocontractive mapping with respect to if, for all , there exists a constant such that

(1.4)

A mapping is called -contraction if there exists a constant such that

(1.5)

We denote by the set of fixed point of , that is, .

Recall the definition of Mann's iteration; letbe a nonempty convex subsetand letbe a self-mapping of. For any, the sequenceis defined by

(1.6)

where is a real sequence in .

In the last ten years or so, there have been many nice papers in the literature dealing with the iteration approximating fixed points of Lipschitz strongly pseudocontractive mappings by utilizing the Mann iteration process. Results which had been known only for Hilbert spaces and Lipschitz mappings have been extended to more general Banach spaces and more general class of mappings; see, for example, [1–6] and the references therein for more information about this problem.

In 2007, Marino and Xu [2] showed that the Mann iterative sequence converges weakly to a fixed point of -strict pseudocontractions in Hilbert spaces. Meanwhile, they have proposed an open question; that is,is the result of[2, Theorem ]true in uniformly convex Banach spaces with Fréchet differentiable norm? In other words, can Reich's theorem [7, Theorem ], with respect to nonexpansive mappings, be extended to -strict pseudocontractions in uniformly convex Banach spaces?

In 2008, using the Mann iteration and the modified Ishikawa iteration, Zhou [3] obtained some weak and strong convergence theorems for -strict pseudocontractions in Hilbert spaces which extend the corresponding results in [2].

Recently, Hu and Wang [4] obtained that the Mann iterative sequence converges weakly to a fixed point of -strict pseudocontractions with respect to in -uniformly convex Banach spaces.

In this paper, we first introduce the modified Mann iterative sequence.Letbe a nonempty closed convex subset ofand letbe a-contraction. For any, the sequenceis defined by

(1.7)

where, for all, , andin. The iterative sequence (1.7) is a natural generalization of the Mann iterative sequences (1.6). If we take , for all , in (1.7), then (1.7) is reduced to the Mann iteration.

The purpose in this paper is to show strong convergence theorems of the modified Mann iteration scheme for -strict pseudocontractions with respect to in -uniformly convex Banach spaces with uniformly differentiable norm by using viscosity approximation methods. Our theorems improve and extend the comparable results in the following four aspects: in contrast to weak convergence results in [2–4], strong convergence theorems of the modified Mann iterative sequence are obtained in -uniformly convex Banach spaces; in contrast to the results in [7, 8], these results with respect to nonexpansive mappings are extended to -strict pseudocontractions with respect to ; the restrictions and in [8, Theorem ] are removed; our results partially answer the open question.

2. Preliminaries

The modulus of convexity of is the function defined by

(2.1)

  is uniformly convex if and only if, for all such that . is said to be -uniformly convex if there exists a constant such that . Hilbert spaces, (or ) spaces () and Sobolev spaces () are -uniformly convex. Hilbert spaces are -uniformly convex, while

(2.2)

A Banach space is said to have differentiable norm if the limit

(2.3)

exists for each , where . The norm of is a uniformlydifferentiable if for each , the limit is attained uniformly for . It is well known that if is a uniformly differentiable norm, then the duality mapping is single valued and norm-to-weak uniformly continuous on each bounded subset of .

Lemma 2.1 (see [4]).

Let be a real -uniformly convex Banach space, and let be a nonempty closed convex subset of . Let be a -strict pseudocontraction with respect to , and let be a real sequence in . If is defined by , for all , then for all , the inequality holds

(2.4)

where is a constant in [9, Theorem ]. In addition, if , , and , then , for all .

Lemma 2.2 (see [10]).

Let and be bounded sequences in a Banach space such that

(2.5)

where is a sequence in such that . Assuming

(2.6)

then .

Lemma 2.3.

Let be a real Banach space. Then, for all and , the following inequality holds:

(2.7)

Lemma 2.4 (see [11]).

Let be a sequence of nonnegative real number such that

(2.8)

where is a sequence in and is a sequence in satisfying the following conditions: ; or . Then, .

3. Main Results

Theorem 3.1.

Let be a real -uniformly convex Banach space with a uniformly Gteaux differentiable norm, and let be a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a -strict pseudocontraction with respect to , and . Let be a -contraction with . Assume that real sequences , , and in satisfy the following conditions:

,

and

and , where .

For any , the sequence is generated by

(3.1)

where , for all . Then, the sequence converges strongly to a fixed point of .

Proof.

Equation (3.1) can be expressed as follows:

(3.2)

where

(3.3)

Taking , we obtain from Lemma 2.1

(3.4)

Therefore, the sequence is bounded, and so are the sequences , , and . Since and the condition we know that is bounded. We estimate from (3.3) that

(3.5)

Since , where is the identity mapping, we have

(3.6)

and imply from (3.5) and (3.6) that

(3.7)

Hence, by Lemma 2.2, we obtain

(3.8)

From (3.3), we get

(3.9)

and so it follows from (3.8) and (3.9) that . Since and , we have

(3.10)

For any , defining , we have

(3.11)

Since is a nonexpansive mapping, we have from [12, Theorem ] that the net generated by converges strongly to , as . Clearly,

(3.12)

In view of Lemma 2.3, we find

(3.13)

and hence

(3.14)

Since the sequences , , and are bounded and , we obtain

(3.15)

where . We also know that

(3.16)

From the facts that , as , is bounded, and the duality mapping is norm-to-weak uniformly continuous on bounded subset of , it follows that

(3.17)

Combining (3.15), (3.16), and the two results mentioned above, we get

(3.18)

From (3.9) and the fact that the duality mapping is norm-to-weak uniformly continuous on bounded subset of , it follows that

(3.19)

Writing

(3.20)

and from Lemma 2.3, we find

(3.21)

where

(3.22)

From (3.18), (3.19), and the conditions it follows that and . Consequently, applying Lemma 2.4 to (3.21), we conclude that .

Corollary 3.2.

Let , , , , , and be as in Theorem 3.1. For any , the sequence is generated by

(3.23)

where , for all . Then the sequence converges strongly to a fixed point of .

Remark 3.3.

Theorem 3.1 and Corollary 3.2 improve and extend the corresponding results in [2–4, 7, 8] essentially since the following facts hold.

Theorem 3.1 and Corollary 3.2 give strong convergence results in -uniformly convex Banach spaces for the modification of Mann iteration scheme in contrast to the weak convergence result in [2, Theorem ], [3, Theorem and Corollary ], and [4, Theorems and ].

In contrast to the results in [7, Theorem ], and [8, Theorem ], these results with respect to nonexpansive mappings are extended to -strict pseudocontraction in -uniformly convex Banach spaces.

In contrast to the results in [8, Theorem ], the restrictions and are removed.