# Hybrid Approximate Proximal Point Algorithms for Variational Inequalities in Banach Spaces

- L. C. Ceng
^{1, 2}, - S. M. Guu
^{3}and - J. C. Yao
^{4}Email author

**2009**:275208

**DOI: **10.1155/2009/275208

© L. C. Ceng et al. 2009

**Received: **10 March 2009

**Accepted: **7 June 2009

**Published: **14 July 2009

## Abstract

Let be a nonempty closed convex subset of a Banach space with the dual , let be a continuous mapping, and let be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator we study the variational inequality (for short, VI( ): find such that for all , where is a given element. By combining the approximate proximal point scheme both with the modified Ishikawa iteration and with the modified Halpern iteration for relatively nonexpansive mappings, respectively, we propose two modified versions of the approximate proximal point scheme L. C. Ceng and J. C. Yao (2008) for finding approximate solutions of the VI( ). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of the VI( ), which is also a fixed point of .

## 1. Introduction

where is the normalized duality mapping on and is the generalized projection operator which assigns to an arbitrary point the minimum point of the functional with respect to . In [1, Theorem?8.2], Alber proved that the above sequence converges strongly to the solution , that is, as , if the following conditions hold:

where is a continuous strictly increasing function for all with ;

where is a continuous nondecreasing function for all with . Note that solution methods for the problem (1.1) has also been studied in [2–10].

Let be a nonempty closed convex subset of a real Banach space with the dual . Assume that is a continuous mapping on and is a relatively nonexpansive mapping such that . The purpose of this paper is to introduce and study two new iterative algorithms (1.5) and (1.6) in a uniformly convex and uniformly smooth Banach space .

Algorithm 1.1.

where are sequences in , is a bounded sequence in , and is assumed to exist for each ,

Algorithm 1.2.

where is a sequence in , is a bounded sequence in , and is assumed to exist for each , .

In this paper, strong convergence results on these two iterative algorithms are established; that is, under appropriate conditions, both the sequence generated by algorithm (1.5) and the sequence generated by algorithm (1.6) converge strongly to the same point , which is a solution of the . Our results represent the improvement, generalization, and development of the previously known results in the literature including Li [8], Zeng and Yao [9], Ceng and Yao [10], and Qin and Su [11].

Notation 1.

## 2. Preliminaries

where denotes the generalized duality pairing. It is well known that if is smooth, then is single-valued and if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of . We will still denote the single-valued duality mapping by .

Recall that if is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. In this connection, Alber [1] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

It is clear that in a Hilbert space , (2.2) reduces to .

The existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, e.g., [13]). In a Hilbert space, .

Let be a closed convex subset of , and let be a mapping from into itself. A point in is called an asymptotically fixed point of [14] if contains a sequence which converges weakly to such that . The set of asymptotical fixed points of will be denoted by . A mapping from into itself is called relatively nonexpansive [15–17] if and for all and .

exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for . Recall also that if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of . A Banach space is said to have the Kadec-Klee property if for any sequence , whenever and , we have . It is known that if is uniformly convex, then has the Kadec-Klee property; see [18, 19] for more details.

Remark 2.1 ([11]).

If is a reflexive, strictly convex, and smooth Banach space, then for any , if and only if . It is sufficient to show that if , then . From (2.4), we have . This implies that . From the definition of , we have . Therefore, we have ; see [18, 19] for more details.

We need the following lemmas and proposition for the proof of our main results.

Lemma 2.2 (Kamimura and Takahashi [20]).

Let be a uniformly convex and smooth Banach space and let and be two sequences of . If and either or is bounded, then .

Lemma 2.3 (Alber [1]).

Lemma 2.4 (Alber [1]).

Lemma 2.5 (Matsushita and Takahashi [21]).

Let be a strictly convex and smooth Banach space, let be a closed convex subset of , and let be a relatively nonexpansive mapping from into itself. Then is closed and convex.

Lemma 2.6 (Chang [7]).

## 3. Main Results

Now we are in a position to prove the main theorems of this paper.

Theorem 3.1.

where is assumed to exist for each , If is uniformly continuous and , then converges strongly to , which is a solution of the (1.1).

Proof.

As by the induction assumption, the last inequality holds, in particular, for all . This together with the definition of implies that . Hence (3.7) holds for all . This implies that is well defined.

On the other hand, it follows from the definition of that . Since , we have

and hence is bounded. Again from we know that is also bounded.

On account of the boundedness and nondecreasing property of we deduce that exists. From Lemma 2.4, we derive

Since is uniformly continuous, it follows from (3.26), (3.30) and that .

Finally, let us show that converges strongly to , which is a solution of the (1.1). Indeed, assume that is a subsequence of such that . Then . Next let us show that and convergence is strong. Put . From and , we have . Now from weakly lower semicontinuity of the norm, we derive

This shows that is a solution of the (1.1). This completes the proof.

Corollary 3.2 ([11, Theorem?2.1]).

where is the single-valued duality mapping on . If is uniformly continuous, then converges strongly to .

Proof.

for all . Thus algorithm (3.1) reduces to algorithm (3.39). By Theorem 3.1 we obtain the desired result.

Theorem 3.3.

where is assumed to exist for each , If is uniformly continuous and , then converges strongly to , which is a solution of the (1.1).

Proof.

Since is uniformly continuous, it follows from (3.50) and (3.54) that .

Finally, let us show that converges strongly to , which is a solution of the (1.1). Indeed, assume that is a subsequence of such that . Then . Next let us show that and convergence is strong. Put . From and , we have . Now from weakly lower semicontinuity of the norm, we derive

This shows that is a solution of the (1.1). This completes the proof.

Corollary 3.4 ([11, Theorem?2.2]).

where is the single-valued duality mapping on . If is nonempty, then converges strongly to .

Proof.

for all . Thus algorithm (3.42) reduces to algorithm (3.61). Thus under the lack of the uniform continuity of it follows from (3.55) that . By the careful analysis of the proof of Theorem 3.3, we can obtain the desired result.

## Declarations

### Aknowledgments

The first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Shanghai Leading Academic Discipline Project (S30405) and Innovation Program of Shanghai Municipal Education Commission (09ZZ133). The third author was partially supported by the Grant NSF 97-2115-M-110-001.

## Authors’ Affiliations

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