Iterative Algorithms for Finding Common Solutions to Variational Inclusion Equilibrium and Fixed Point Problems

Abstract

The main purpose of this paper is to introduce an explicit iterative algorithm to study the existence problem and the approximation problem of solution to the quadratic minimization problem. Under suitable conditions, some strong convergence theorems for a family of nonexpansive mappings are proved. The results presented in the paper improve and extend the corresponding results announced by some authors.

1. Introduction

Throughout this paper, we assume that is a real Hilbert space with inner product and norm , is a nonempty closed convex subset of , and is the set of fixed points of mapping .

A mapping is called nonexpansive if

(1.1)

Let be a single-valued nonlenear mapping and be a multivalued mapping. The so-called quasivariational inclusion problem (see [1–3]) is to find such that

(1.2)

The set of solutions to quasivariational inclusion problem (1.2) is denoted by .

Special Cases

• (I) If , where is a proper convex lower semi-continuous function and is the subdifferential of , then the quasivariational inclusion problem (1.2) is equivalent to finding such that

  

  (1.3)

which is called the mixed quasivariational inequality (see [4]).

• (II) If , where is a nonempty closed convex subset of and is the indicator function of , that is,

  

  (1.4)

then the quasivariational inclusion problem (1.2) is equivalent to finding such that

(1.5)

This problem is called the Hartman-Stampacchia variational inequality (see [5]). The set of solutions to variational inequality (1.5) is denoted by .

Let be a nonlinear mapping and be a bifunction. The so-called generalized equilibrium problem is to find a point such that

(1.6)

The set of solutions to (1.6) is denoted by GEP (see [5, 6]). If , then (1.6) reduces to the following equilibrium problem: to find such that

(1.7)

The set of solutions to (1.7) is denoted by EP.

Iterative methods for nonexpansive mappings and equilibrium problems have been applied to solve convex minimization problems (see [7–9]). 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 :

(1.8)

where is the fixed point set of a nonexpansive mapping on .

In 2010, Zhang et al. (see [10]) proposed the following iteration method for variational inclusion problem (1.5) and equilibrium problem (1.6) in a Hilbert space :

(1.9)

Under suitable conditions, they proved the sequence generated by (1.9) converges strongly to the fixed point , which solves the quadratic minimization problem (1.8).

Motivated and inspired by the researches going on in this direction, especially inspired by Zhang et al. [10], the purpose of this paper is to introduce an explicit iterative algorithm to studying the existence problem and the approximation problem of the solution to the quadratic minimization problem (1.8) and prove some strong convergence theorems for a family of nonexpansive mappings in the setting of Hilbert spaces.

2. Preliminaries

Let be a real Hilbert space, and be a nonempty closed convex subset of . For any , there exists a unique nearest point in , denoted by , such that

(2.1)

Such a mapping from onto is called the metric projection. It is well-known that the metric projection is nonexpansive.

In the sequel, we use and to denote the weak convergence and the strong convergence of the sequence , respectively.

Definition 2.1.

A mapping is called -inverse strongly monotone if there exists an such that

(2.2)

A multivalued mapping is called monotone if ,

(2.3)

A multivalued mapping is called maximal monotone if it is monotone and for any , when

(2.4)

then .

Proposition 2.2 (see [11]).

Let be an -inverse strongly monotone mapping. Then, the following statements hold:

(i) is an -Lipschitz continuous and monotone mapping;

(ii)if is any constant in , then the mapping is nonexpansive, where is the identity mapping on .

Lemma 2.3 (see [12]).

Let be a strictly convex Banach space, be a closed convex subset of , and be a sequence of nonexpansive mappings. Suppose . Let be a sequence of positive numbers with . Then the mapping defined by

(2.5)

is well defined. And it is nonexpansive and

(2.6)

Definition 2.4.

Let be a Hilbert space and be a multivalued maximal monotone mapping. Then, the single-valued mapping defined by

(2.7)

is called the resolvent operator associated with, where is any positive number and is the identity mapping.

Proposition 2.5 (see [11]).

• (i) The resolvent operator associated with is single-valued and nonexpansive for all , that is,

  

  (2.8)

 (ii) The resolvent operator is 1-inverse strongly monotone, that is,

(2.9)

Definition 2.6.

A single-valued mapping is said to be hemicontinuous if for any , function is continuous at 0.

It is well-known that every continuous mapping must be hemicontinuous.

Lemma 2.7 (see [13]).

Let and be bounded sequences in a Banach space . Let be a sequence in with

(2.10)

Suppose that

(2.11)

Then,

(2.12)

Lemma 2.8 (see [14]).

Let be a real Banach space, be the dual space of , be a maximal monotone mapping, and be a hemicontinuous bound monotone mapping with . Then, the mapping is a maximal monotone mapping.

Lemma 2.9 (see [15]).

Let be a uniformly convex Banach space, let be a nonempty closed convex subset of , and be a nonexpansive mapping with a fixed point. Then, is demiclosed in the sense that if is a sequence in satisfying

(2.13)

then

(2.14)

Throughout this paper, we assume that the bifunction satisfies the following conditions:

for all ;

is monotone, that is,

(2.15)

for each ,

(2.16)

for each , is convex and lower semi-continuous.

Lemma 2.10 (see [16]).

Let be a real Hilbert space, be a nonempty closed convex subset of , and be a bifunction satisfying the conditions . Let and . Then, there exists a point such that

(2.17)

Moreover, if is a mapping defined by

(2.18)

then the following results hold:

(i) is single-valued and firmly nonexpansive, that is, for any ,

(2.19)

(ii)EP is closed and convex, and .

Lemma 2.11.

• (i) (see [11])   is a solution of variational inclusion (1.2) if and only if

  

  (2.20)

that is,

(2.21)

• (ii) (see [10])   is a solution of generalized equilibrium problem (1.6) if and only if

  

  (2.22)

that is,

(2.23)

 (iii) (see [10]) Let be an -inverse strongly monotone mapping and be a -inverse strongly monotone mapping. If and , then is a closed convex subset in and GEP is a closed convex subset in .

Lemma 2.12 (see [17]).

Assume that is a sequence of nonnegative real numbers such that

(2.24)

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

(i);

(ii) or .

Then, .

3. Main Results

Theorem 3.1.

Let be a real Hilbert space, be a nonempty closed convex subset of , be an -inverse strongly monotone mapping and be a -inverse strongly monotone mapping. Let be a maximal monotone mapping, be a sequence of nonexpansive mappings with , be the nonexpansive mapping defined by (2.5), and be a bifunction satisfying conditions . Let be the sequence defined by

(3.1)

where the mapping is defined by (2.18), and are two constants with , and

(3.2)

If

(3.3)

where and GEP is the set of solutions of variational inclusion (1.2) and generalized equilibrium problem (1.6), respectively, then the sequence defined by (3.1) converges strongly to , which is the unique solution of the following quadratic minimization problem:

(3.4)

Proof.

We divide the proof of Theorem 3.1 into four steps.

Step 1 (The sequence is bounded).

Set

(3.5)

Taking , then it follows from Lemma 2.11 that

(3.6)

Since both and are nonexpansive, and are -inverse strongly monotone and -inverse strongly monotone, respectively, from Proposition 2.2, we have

(3.7)

(3.8)

This implies that

(3.9)

It follows from (3.1) and (3.9) that

(3.10)

where . This shows that is bounded. Hence, it follows from (3.9) that the sequence and are also bounded.

It follows from (3.5), (3.6), and (3.9) that

(3.11)

This shows that is bounded.

Step 2.

Now, we prove that

(3.12)

Since is nonexpansive, from (3.5) and (3.9), we have that

(3.13)

(3.14)

Let in (3.14), in view of condition , we have

(3.15)

By virtue of Lemma 2.7, we have

(3.16)

This implies that

(3.17)

We derive from (3.17) that

(3.18)

From (3.1) and (3.8), we have

(3.19)

where

(3.20)

that is,

(3.21)

Let , noting the assumptions that , , from (3.2) and (3.18), we have

(3.22)

By virtue of Lemma 2.10(i) and (3.1), we have

(3.23)

Simplifying it, we have

(3.24)

Similarly, in view of Proposition 2.5(ii) and (3.1), we have

(3.25)

Simplifying it, from (3.24), we have

(3.26)

From (3.19) and (3.26), we have

(3.27)

Let nd in view of (3.18) and (3.22), we have

(3.28)

This shows that

(3.29)

Then, we have

(3.30)

Step 3 (sequence converges strongly to ).

Because is bounded, without loss of generality, we can assume that . In view of (3.12), it yields that and . From Lemma 2.9 and (3.30), we know that .

Next, we prove that .

Since , we have

(3.31)

It follows from condition that

(3.32)

Therefore,

(3.33)

For any and , then . From (3.33), we have

(3.34)

Since is -inverse strongly monotone, from Proposition 2.2(i) and (3.12), we have

(3.35)

Let in (3.34), in view of condition and , we have

(3.36)

It follows from conditions , and (3.36) that

(3.37)

that is,

(3.38)

Let to 0 in (3.38), we have

(3.39)

This shows that .

Step 4 (now, we prove that ).

Since is -inverse strongly monotone, from Proposition 2.2 (i), we know that is an -Lipschitz continuous and monotone mapping and , where is the domain of . It follows from Lemma 2.8 that is maximal monotone. Let , that is, . Since , we have , that is, . By virtue of the maximal monotonicity of , we have

(3.40)

Therefore we have

(3.41)

Since is monotone, this implies that

(3.42)

Since

(3.43)

from (3.42), we have

(3.44)

Since is maximal monotone, , that is, .

Summing up the above arguments, we have proved that

(3.45)

On the other hand, for any , we have

(3.46)

and so we have

(3.47)

Put in (3.47), we have

(3.48)

where and . Since , it is easy to see that and . By Lemma 2.12, we conclude that as , where is the unique solution of the following quadratic minimization problem:

(3.49)

This completes the proof of Theorem 3.1.

In Theorem 3.1, if , then the following corollary can be obtained immediately.

Corollary 3.2.

Let be a real Hilbert space, be a nonempty closed convex subset of , be an -inverse strongly monotone mapping and be a -inverse strongly monotone mapping. Let be a maximal monotone mapping, be a nonexpansive mappings with . Let be a bifunction satisfying conditions . Let be the sequence defined by

(3.50)

where the mapping is defined by (2.18), and are two constants with , and

(3.51)

If

(3.52)

where and GEP are the sets of solutions of variational inclusion (1.2) and generalized equilibrium problem (1.6), then the sequence defined by (3.50) converges strongly to , which is the unique solution of the following quadratic minimization problem:

(3.53)

In Theorem 3.1, if , where is the indicator function of , then the variational inclusion problem (1.2) is equivalent to variational inequality (1.5), that is, to find such that , for all . Since . Consequently, we have the following corollary.

Corollary 3.3.

Let be a real Hilbert space, be a nonempty closed convex subset of , be an -inverse strongly monotone mapping and be a -inverse strongly monotone mapping. Let and be a nonexpansive mappings with . Let be a bifunction satisfying conditions . Let be the sequence defined by

(3.54)

where the mapping is defined by (2.18), and are two constants with , and

(3.55)

If

(3.56)

where and GEP are the sets of solutions of variational inclusion (1.5) and generalized equilibrium problem (1.6), then the sequence defined by (3.54) converges strongly to , which is the unique solution of the following quadratic minimization problem:

(3.57)