A Viscosity Approximation Method for Finding Common Solutions of Variational Inclusions, Equilibrium Problems, and Fixed Point Problems in Hilbert Spaces

Abstract

We introduce an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a variational inclusion with set-valued maximal monotone mapping, and inverse strongly monotone mappings and the set of solutions of an equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of J. W. Peng et al. (2008) and many others.

1. Introduction

Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of , and let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that

(1.1)

The set of solutions of (1.1) is denoted by Recently, Combettes and Hirstoaga [1] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. Let be a nonlinear map. The classical variational inequality which is denoted by is to find such that

(1.2)

The variational inequality has been extensively studied in literature. See, for example, [2, 3] and the references therein. Recall that a mapping of into itself is called nonexpansive if

(1.3)

A mapping is called contractive if there exists a constant such that

(1.4)

We denote by the set of fixed points of .

Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping; see, for instance, [3–6] and the references therein. Recently, Plubtieng and Punpaeng [6] introduced the following iterative scheme. Let and let , and be sequences generated by

(1.5)

They proved that if the sequences and of parameters satisfy appropriate conditions, then the sequences and both converge strongly to the unique solution of the variational inequality

(1.6)

which is the optimality condition for the minimization problem

(1.7)

where is a potential function for

Let be a single-valued nonlinear mapping, and let be a set-valued mapping. We consider the following variational inclusion, which is to find a point such that

(1.8)

where is the zero vector in . The set of solutions of problem (1.8) is denoted by . If , then problem (1.8) becomes the inclusion problem introduced by Rockafellar [7]. If , where is a nonempty closed convex subset of and is the indicator function of C, that is,

(1.9)

then the variational inclusion problem (1.8) is equivalent to variational inequality problem (1.2). It is known that (1.8) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity, variational inequalities, optimal control, mathematical economics, equilibria, game theory. Also various types of variational inclusions problems have been extended and generalized (see [8] and the references therein.)

Very recently, Peng et al. [9] introduced the following iterative scheme for finding a common element of the set of solutions to the problem (1.8), the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in Hilbert space. Starting with , define sequence, , , and by

(1.10)

for all , where , and . They proved that under certain appropriate conditions imposed on and , the sequences , , and generated by (1.10) converge strongly to , where .

Motivated and inspired by Plubtieng and Punpaeng [6], Peng et al. [9] and Aoyama et al. [10], we introduce an iterative scheme for finding a common element of the set of solutions of the variational inclusion problem (1.8) with multi-valued maximal monotone mapping and inverse-strongly monotone mappings, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Starting with an arbitrary define sequences , and by

(1.11)

for all , where , , and let ; be a strongly bounded linear operator on , and is a sequence of nonexpansive mappings on . Under suitable conditions, some strong convergence theorems for approximating to this common elements are proved. Our results extend and improve some corresponding results in [3, 9] and the references therein.

2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section.

Let be a real Hilbert space with inner product and norm , respectively.

It is wellknown that for all and , there holds

(2.1)

Let be a nonempty closed convex subset of . Then, for any , there exists a unique nearest point of , denoted by , such that for all . Such a is called the metric projection from into . We know that is nonexpansive. It is also known that, and

(2.2)

It is easy to see that (2.2) is equivalent to

(2.3)

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:

(A1)  for all

(A2) is monotone, that is,

(A3)for each

(2.4)

(A4)for each is convex and lower semicontinuous.

The following lemma appears implicitly in [11] and [1].

Lemma 2.1 (See [1, 11]).

Let be a nonempty closed convex subset of and let be a bifunction of in to satisfying (A1)–(A4). Let and . Then, there exists such that

(2.5)

Define a mapping as follows:

(2.6)

for all . Then, the following hold:

(1) is single-valued;

(2) is firmly nonexpansive, that is, for any ,

(2.7)

(3)

(4) is closed and convex.

We also need the following lemmas for proving our main result.

Lemma 2.2 (See [12]).

Let be a Hilbert space, a nonempty closed convex subset of , a contraction with coefficient , and a strongly positive linear bounded operator with coefficient . Then :

(1)if   , then

(2)if   , then .

Lemma (See [13]).

Assume is a sequence of nonnegative real numbers such that

(2.8)

where is a sequence in and is a sequence in such that

(1)

(2) or

Then

Recall that a mapping is called -inverse-strongly monotone, if there exists a positive number such that

(2.9)

Let be the identity mapping on . It is well known that if is -inverse-strongly monotone, then is -Lipschitz continuous and monotone mapping. In addition, if , then is a nonexpansive mapping.

A set-valued is called monotone if for all and imply . A monotone mapping is maximal if its graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for for every implies .

Let the set-valued mapping be maximal monotone. We define the resolvent operator associated with and as follows:

(2.10)

where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive and -inverse-strongly monotone, see for example [14] and that a solution of problem (1.8) is a fixed point of the operator for all , see for instance [15].

Lemma 2.4 (See [14]).

Let be a maximal monotone mapping and be a Lipschitz-continuous mapping. Then the mapping is a maximal monotone mapping.

Remark (See [9]).

Lemma 2.4 implies that is closed and convex if is a maximal monotone mapping and be an inverse strongly monotone mapping.

Lemma 2.6 (See [10]).

Let be a nonempty closed subset of a Banach space and let a sequence of mappings of into itself. Suppose that . Then, for each , converges strongly to some point of . Moreover, let be a mapping from into itself defined by

(2.11)

Then .

3. Main Results

We begin this section by proving a strong convergence theorem of an implicit iterative sequence obtained by the viscosity approximation method for finding a common element of the set of solutions of the variational inclusion, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.

Throughout the rest of this paper, we always assume that is a contraction of into itself with coefficient , and is a strongly positive bounded linear operator with coefficient and . Let be a nonexpansive mapping of into . Let be an -inverse-strongly monotone mapping, be a maximal monotone mapping and let be defined as in (2.10). Let be a sequence of mappings defined as Lemma 2.1. Consider a sequence of mappings on defined by

(3.1)

where By Lemma 2.2, we note that is a contraction. Therefore, by the Banach contraction principle, has a unique fixed point such that

(3.2)

Theorem 3.1.

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be a nonexpansive mapping on . Let be an -inverse-strongly monotone mapping, be a maximal monotone mapping such that Let be a contraction of into itself with a constant and let be a strongly bounded linear operator on with coefficient and . Let , and be sequences generated by and

(3.3)

where , and satisfy and . Then, , and converges strongly to a point in which solves the variational inequality:

(3.4)

Equivalently, we have

Proof.

First, we assume that . By Lemma 2.2, we obtain . Let Since we have

(3.5)

We note from that . As is nonexpansive, we have

(3.6)

for all Thus, we have

(3.7)

It follows that Hence is bounded and we also obtain that ,, , and are bounded. Next, we show that . Since , we note that

(3.8)

Moreover, it follows from Lemma 2.1 that

(3.9)

and hence Therefore, we have

(3.10)

and hence

(3.11)

Since is bounded and it follows that as

Put . From (3.10), it follows by the nonexpansive of and the inverse strongly monotonicity of that

(3.12)

which implies that

(3.13)

Since , we have as . Since is –inverse-strongly monotone and is nonexpansive, we have

(3.14)

Thus, we have

(3.15)

From (3.5), (3.10), and (3.15), we have

(3.16)

Thus, we get

(3.17)

Since , as , we have as . It follows from the inequality that as . Moreover, we have as .

Put . Since both and are nonexpansive, we have is a nonexpansive mapping on and then we have for all . It follows by Theorem 3.1 of Plubtieng and Punpaeng [6] that converges strongly to , where and , for all . We will show that . Since converges strongly to , we also have . Let us show Assume Since and , we have Since it follows by the Opial's condition that

(3.18)

This is a contradiction. Hence We now show that . In fact, since is –inverse-strongly monotone, is an -Lipschitz continuous monotone mapping and . It follows from Lemma 2.4 that is maximal monotone. Let , that is, . Again since , we have that is,

(3.19)

By the maximal monotonicity of , we have

(3.20)

and so

(3.21)

It follows from , and that

(3.22)

Since is maximal monotone, this implies that that is, . Hence, . Since , we have . It implies that is the unique solution of the variational inequality (3.4).

Now we prove the following theorem which is the main result of this paper.

Theorem 3.2.

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be a sequence of nonexpansive mappings on . Let be an -inverse-strongly monotone mapping, be a maximal monotone mapping such that Let be a contraction of into itself with a constant and let be a strongly bounded linear operator on with coefficient and . Let , and be sequences generated by and

(3.23)

for all , where , and satisfy

(3.24)

Suppose that for any bounded subset of . Let be a mapping of into itself defined by and suppose that . Then, , and converges strongly to , where is a unique solution of the variational inequalities (3.4).

Proof.

Since , we may assume that for all . First we will prove that is bonded. Let Then, we have

(3.25)

It follows from (3.25) and induction that

(3.26)

Hence is bounded and therefore , and are also bounded. Next, we show that . Since is nonexpansive, it follows that

(3.27)

Then, we have

(3.28)

where . On the other hand, we note that

(3.29)

(3.30)

Putting in (3.29) and in (3.30), and By (A2), we have

(3.31)

and hence

(3.32)

Since we assume that there exists a real number such that for all Thus, we have

(3.33)

and hence

(3.34)

where . From (3.28), we have

(3.35)

Since is bounded, it follows that . Hence, by Lemma 2.3, we have as . From (3.34) and , we have . Moreover, we have from (3.27) that .

We note from (3.23) that . Then, we have

(3.36)

Since , and , we get . From the proof of Theorem 3.1, we have

(3.37)

for all . Therefore, we have

(3.38)

and hence

(3.39)

Since is bounded, and , it follows that as

Put . It follows from (3.38), the nonexpansive of and the inverse strongly monotonicity of that

(3.40)

This implies that

(3.41)

Since and , we have as .From (3.5), (3.15) and (3.38), we have

(3.42)

Thus, we obtain

(3.43)

Since , and , we have as . It follows from the inequality that as . Moreover, we note that as . Since

(3.44)

for all , it follows that . Next, we show that

(3.45)

where is a unique solution of the variational inequality (3.4). To show this inequality, we choose a subsequence of such that

(3.46)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that From we obtain . Let us show . It follows by (3.23) and (A2) that

(3.47)

and hence

(3.48)

Since and it follows by (A4) that for all For with and let Since and we have and hence So, from (A1) and (A4) we have

(3.49)

and hence . From (A3), we have for all and hence By the same argument as in proof of Theorem 3.1, we have and hence . This implies that

(3.50)

Finally we prove that . From (3.23), we have

(3.51)

This implies that

(3.52)

where and . It easily verified that , and . Hence, by Lemma 2.1, the sequence converges strongly to .

As in [10, Theorem  4.1], we can generate a sequence of nonexpansive mappings satisfying condition . for any bounded of by using convex combination of a general sequence of nonexpansive mappings with a common fixed point.

Corollary 3.3.

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be an -inverse-strongly monotone mapping, be a maximal monotone mapping. Let be a contraction of into itself with a constant and let be a strongly bounded linear operator on with coefficient and . Let be a family of nonnegative numbers with indices with such that

(i)

(ii)

(iii)

Let be a sequence of nonexpansive mappings on with and let , and be sequences generated by and

(3.53)

for all , where , and satisfy

(3.54)

Then, , and converges strongly to in which solves the variational inequality:

(3.55)

If and in Theorem 3.2, we obtain the following corollary.

Corollary 3.4 (see Peng et al. [9]).

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be a nonexpansive mapping on . Let be an -inverse-strongly monotone mapping, be a maximal monotone mapping such that Let be a contraction of into itself with a constant . Let , and be sequences generated by and

(3.56)

for all , where , and satisfy

(3.57)

Then, , and converges strongly to , where .

If and in Theorem 3.2, we obtain the following corollary.

Corollary 3.5 (see S. Plubtieng and R. Punpaeng [6]).

Let be a real Hilbert space, let be a bifunction from satisfying (A1)–(A4) and let be a nonexpansive mapping on such that Let be a contraction of into itself with a constant and let be a strongly bounded linear operator on with coefficient and . Let , and be sequences generated by and

(3.58)

for all , where and satisfy

(3.59)

Then, and converges strongly to a point in which solves the variational inequality:

(3.60)