Two-Step Viscosity Approximation Scheme for Variational Inequality in Banach Spaces

Abstract

This paper introduces and analyzes a viscosity iterative algorithm for an infinite family of nonexpansive mappings in the framework of a strictly convex and uniformly smooth Banach space. It is shown that the proposed iterative method converges strongly to a common fixed point of , which solves specific variational inequalities. Necessary and sufficient convergence conditions of the iterative algorithm for an infinite family of nonexpansive mappings are given. Results shown in this paper represent an extension and refinement of the previously known results in this area.

1. Introduction

The variational inequality problem was first introduced by Hartman and Stampacchia [1]. This problem has achieved increasing attention in many research fields, such as mathematical programming, constrained linear and nonlinear optimization, automatic control, manufacturing system design, signal and image processing, and complementarity problem in economics and pattern recognition (see [2–4] and the references therein). Nowadays, the theory of variational inequalities and fixed point theory are two important and dynamic areas in nonlinear analysis and optimization.

One promising approach to handle these problems is to develop iterative schemes to compute the approximate solutions of variational inequalities and to find a common fixed point of a given family of operators. There is a variety of techniques to suggest and analyze various iterative algorithms for solving variational inequalities and the related optimization problems. The fixed point theory has played an important role in the development of various algorithms for solving variational inequalities.

In this paper, the purpose is to develop a new iterative method for solving a specific variational inequality.

Let be a real Banach space and a nonempty closed convex subset of . Recall that a mapping is said to be a contraction on if there is a constant such that for all . We use to denote the collection of all contractions on . That is, is acontraction with constant . A mapping is said to be nonexpansive if for all . We denote by the set of fixed points of mapping ; that is, .

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems (see [5–8] and the references therein). A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is a linear bounded operator defined on , is the fixed point set of the nonexpansive mapping , and is a potential function for (i.e., for all ).

Let be a nonexpansive mapping. For given and define a contraction mapping by It follows from Banach’s contraction principle that it yields a unique fixed point of ; that is, is the unique solution of the following equation:

Moudafi [9] first proposed the viscosity approximation method and proved that if is a real Hilbert space, then the sequence converges strongly to a fixed point of in which is the unique solution to the following variational inequality:

In 2004, Xu [10] extended Moudafi’s results [9] to the framework of uniformly smooth Banach spaces and proved the strong convergence of both the continuous scheme and iterative scheme. Very recently, Yao et al. [11] introduced the following iteration scheme: where the sequences and . By using the viscosity approximation method, they proved that the approximate solutions converge strongly to a solution of a variational inequality under some mild conditions.

Let be an infinite family of nonexpansive mappings and let be real numbers such that for every (the set of positive integers). Let be the identity operator on a real Banach space . For any , the mapping is defined by Such a mapping is called the -mapping generated by and (see [12]). Nonexpansivity of each ensures the nonexpansivity of .

Shimoji and Takahashi [12] first introduced an iterative algorithm given by an infinite family of nonexpansive mappings. Furthermore, they considered the feasibility problem of finding a solution of infinite convex inequalities and the problem of finding a common fixed point of infinite nonexpansive mappings. Bauschke and Borwein [13] pointed out that the well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings. The problem of finding an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [14]). A simple algorithmic solution to the problem of minimizing a quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation (see [14, 15]). It is now one of the main tools in studying convergence of iterative methods for approaching a common fixed point of an infinite family of nonlinear mappings.

Cho et al. [16] proposed the following iterative scheme: where is defined by (6) and the sequences and are in . Under some conditions, they proved the strong convergence of the sequence defined by (7) and extended the results of [11].

Motivated and inspired by the earlier methods proposed in the literature and their convergence, we consider the following two-step viscosity approximation method for finding common fixed point of an infinite family of nonexpansive mappings : where , , , and and . By using viscosity approximation methods, the purpose of this paper is to study necessary and sufficient conditions for the convergence of the iterative algorithm (8) for finding approximate common fixed points of an infinite countable family of nonexpansive mappings . The results presented in this paper extend and improve some recent results.

2. Preliminaries

Let be a Banach space with dimension and let be its dual. The modulus of convexity of is the function defined by A Banach space is uniformly convex if and only if for all . A Banach space is said to be strictly convex if

Suppose that is a sequence in ; then (resp., ) will denote strong (resp., weak) convergence of the sequence to .

Let the value of at be denoted by . The normalized duality mapping from into is defined by where denotes the dual space of a real Banach space .

Let . The norm of is said to be Gâteaux differentiable (and is said to be smooth) if the limit exists for all . The norm is said to be uniformly Gâteaux differentiable if, for all , the limit is attained uniformly for each . The norm of is said to be Fréchet differentiable if, for all , the limit exists uniformly for each . The norm of is said to be uniformly Fréchet differentiable (or is said to be uniformly smooth) if the limit is attained uniform for all . It is well known that (uniform) Fréchet differentiability of the norm implies (uniform) Gâteaux differentiability of norm . It is known (see [17]) that if is smooth, then the normalized duality mapping is single-valued and norm to weak star continuous. And we know that if the norm of is uniformly Gâteaux differentiable, then the normalized duality mapping is norm to weak star uniformly continuous on each bounded subset of .

Let and be nonempty subsets of a Banach space such that is nonempty closed convex and ; then a mapping is said to be a retraction if for all . A retraction is said to be sunny [18] if for all and with . A sunny nonexpansive retraction is a sunny retraction, which is also a nonexpansive mapping. In a smooth Banach space , it is well known [18] that is a sunny nonexpansive retraction from to if and only if the following inequality holds:

Concerning , the next lemmas play a crucial role for proving our main results.

Lemma 1 (cf. [12]). Let be a nonempty, closed, and convex subset of a strictly convex Banach space . Let be nonexpansive mappings of into itself such that is nonempty and let be real numbers such that for any . Then, for any and , the limit exists.

Using Lemma 1, we can define the mapping of into itself as follows: Such a mapping is said to be the -mapping generated by and . Throughout this paper, we will assume that for all .

Lemma 2 (cf. [12]). Let be a nonempty, closed, and convex subset of a strictly convex Banach space . Let be nonexpansive mappings of into itself such that is nonempty and let be real numbers such that for any . Then .

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

Lemma 3. Let be a real Banach space and let be the normalized duality mapping; then for any the following inequality holds:

Lemma 4 (cf. [19], Lemma 2.5). Let be a sequence of nonnegative real numbers satisfying the following relation: where (i) , ; (ii) ; (iii) , . Then converges to zero as .

Lemma 5 (cf. [20]). Let , be two bounded sequences in a Banach space and with . Suppose that for all integers and . Then .

It follows from [10, Theorem 4.1] that we have the following results.

Lemma 6 (cf. [10]). Let be a uniformly smooth Banach space. Let be a nonempty, closed, and convex subset of , and let be a nonexpansive mapping with and . Then the sequence defined by converges strongly to a fixed point of as . If we define by then solves the following variational inequality: In particular, if is a constant, then (19) is reduced to the sunny nonexpansive retraction from onto :

3. Main Results

In the sequel, denotes the set of common fixed points for a family of nonexpansive mappings .

Lemma 7. Let be a real strictly convex and uniformly smooth Banach space. Let be a nonempty, closed, and convex subset of and let be a nonexpansive mapping from into itself for . Assume that and . Suppose that the sequences , , , and in satisfy the following conditions: (1), and ;(2) and .Let the two-step viscosity approximation iterative scheme be defined by (8). Then (i)the sequence is bounded;(ii);(iii).

Proof. (i) We should prove that for all and given and so , , , , , and are bounded.
Indeed, take a given . It follows from (8) that It follows from (8) and (21) that By mathematical induction, we obtain that for all . Hence, is bounded and so are , , , , , and .
(ii) Putting , we have Then we have Since and are nonexpansive, from (6), we obtain where is a constant such that for all . Similarly, we have where is a constant such that for all . Combining (25), (26), and (27), we have From conditions (1), (2), and , we get It follows from Lemma 5 that . Noting (24), we obtain Thus, we get that holds.
(iii) Observe that which implies that Since and , there exists an integer such that for some constants . Hence we conclude that, for all , Since and and , , and are bounded sequences, we have On the other hand, we have Since for any and for any , there exists a positive integer such that for all and for all . In particular, for all . Thus we have that This together with (36) implies This completes the proof.

Theorem 8. Let be a real strictly convex and uniformly smooth Banach space. Let be a nonempty, closed, and convex subset of and let be a nonexpansive mapping from into itself for . Assume that and . Suppose that the sequences , , , and in satisfy the following conditions: (1), and ;(2) and .Then the two-step viscosity approximation iterative scheme defined by (8) converges strongly to , where is the unique solution of the following variational inequality:

Proof. It follows from Lemma 6 that there exists a solution of a variational inequality: That is, , where is defined by (17).
We first show that , where with being the fixed point of the contraction: Then, we can write Suppose that a subsequence is such that and for some . It follows from (38) that . Putting it follows from (42), Lemma 3, that The last inequality implies that Letting and noting (44) yield that where is a constant such that for all and . Taking in (47) and noticing the fact that the two limits are interchangeable due to the fact that is uniformly continuous on bounded subsets of from the strong topology of to the weak* topology of , we have Indeed, letting , from (47) we have Thus, for arbitrary , there exists a positive number such that, for any , we have Since as , the set is bounded and the duality mapping is norm-to-norm uniformly continuous on bounded subset of ; there exists such that, for any , Choose ; we have, for all and , which implies that This together with (50) implies that Since is arbitrary, we have that .
Next, we claim that . Indeed, notice that (35) implies that It follows from (8) and Lemma 3 that This implies that where . Set It follows from condition (1), (48), and (55) that , , and . Then, (57) reduces to From Lemma 4 with , we see that . This completes the proof.

If is a constant in Theorem 8, then we have the following result.

Corollary 9. Let be a real strictly convex and uniformly smooth Banach space. Let be a nonempty, closed, and convex subset of and let be a nonexpansive mapping from into itself for . Assume that and . Suppose that the sequences , , , and in satisfy the following conditions: (1), and ;(2) and .Let be the sequence of successive approximations for the iterative method defined by where is a mapping defined by (6). Then converges strongly to , where and is the unique sunny nonexpansive retraction; that is, satisfies the following property:

Theorem 10. Let be a real strictly convex and uniformly smooth Banach space. Let be a nonempty, closed, and convex subset of and let be a nonexpansive mapping from into itself for . Assume that and . For given , let be the sequence of successive approximations for the iterative method defined by where is a mapping defined by (6), , are sequences in , and is sequence in with . Then (1) converges strongly to some common fixed point if and only if and ;(2)if converges strongly to some common fixed point , for given , , then is the unique solution of the following variational inequality in :

Proof. (1) The sufficiency is obvious. Indeed, if we set for all in (8), it follows from Theorem 8 that converges strongly to some common fixed point .
Now we prove necessity. Assume that converges strongly to some common fixed point . If we set in (62), we have Therefore, we obtain that This implies that Notice that in general is not equal to . However, since is an arbitrary contraction, we get . Thus, .
On the other hand, let us set , , , and in (62) for all , where is the identity operator. Clearly, is the unique common fixed point of and for all . Moreover, we have Because , we obtain Therefore, we get that ; equivalently, .
(2) If , for all , we define a contraction by Banach’s contraction principle guarantees that has a unique fixed point in . It follows from Lemmas 2 and 6 that and is the unique solution of the following variational inequality in :

If we set in (62), we have the following result.

Corollary 11. Let be a real strictly convex and uniformly smooth Banach space. Let be a nonempty, closed, and convex subset of and let be a nonexpansive mapping from into itself for . Assume that and . For given , let be the sequence of successive approximations for the iterative method defined by where is a mapping defined by (6) and is sequence in . Then (1) converges strongly to some common fixed point if and only if and ;(2)if converges strongly to some common fixed point , for given , , then is the unique solution of the following variational inequality in :

Remark 12. Corollary 11 improves Theorem 4.2 of [10] from a single nonexpansive mapping to an infinite countable family of nonexpansive mappings and generalizes the corresponding results by Xu [19], Halpern [21], Lions [22], and Wittmann [23] to the viscosity methods. And our iterative method presented in this paper can be reviewed as a refinement and modification of the iterative methods in the literature. Moreover we show necessary and sufficient conditions for the convergence of the viscosity iterative algorithm for finding approximate common fixed points of an infinite family of nonexpansive mappings.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (Grant no. 61374081) and the Natural Science Foundation of Guangdong Province (S2013010013034).