Abstract and Applied Analysis

Volume 2013 (2013), Article ID 204948, 24 pages

http://dx.doi.org/10.1155/2013/204948

Research Article

A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

Haitao Che and Xintian Pan

School of Mathematics and Information Science, Weifang University, Weifang, Shandong 261061, China

Received 28 May 2013; Accepted 18 June 2013

Academic Editor: Sehie Park

Copyright © 2013 Haitao Che and Xintian Pan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.

1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed convex subset of and let be a bifunction. We consider the following equilibrium problem (EP) which is to find such that

The set of solutions of is denoted by .

Let be a finite family of bifunctions from into , where is the set of real numbers. The system of equilibrium problems for is to find a common element such that

We denote the set of solutions of (2) by , where is the set of solutions to the equilibrium problems, that is,

If , then the problem (2) is reduced to the equilibrium problems.

If and , then the problem (2) is reduced to the variational inequality problems of finding such that

The set of solutions of (4) is denoted by .

The equilibrium problem is very general in the sense that it includes, as special cases, fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems in noncooperative games, and numerous problems in physics, economics, and others. Some methods have been proposed to solve , , and ; see, for example, [1–29] and references therein. Formulations (2) extend this formulism to such problems, covering in particular various forms of feasibility problems [30, 31].

Definition 1. One-parameter family mapping from into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions:

(i) for all ,(ii) for all ,(iii)for each , the mapping is continuous,(iv) for all and .

Remark 2. We denote by the set of all common fixed points of , that is, .

Let be a nonlinear mapping. Now, we recall the following definitions.

(1) is said to be monotone if

(2) is called -Lipschitzian if there exists a positive constant such that

(3) is said to be -strongly monotone if there exists a positive constant such that

(4) is said to be nonexpansive if

 And denotes the set of fixed points of the mapping , that is, .(5) is said to be -strictly pseudocontractive mapping if there exists a constant such that

(6) is said to be -inverse-strongly monotone if there exists a constant such that

(7) is said to be relaxed -cocoercive if there exist positive real numbers such that

(8)A linear bounded operator is strong positive if there exists a constant with the property

(9)A set-valued mapping is called monotone if for all , and imply .(10)A monotone mapping is called maximal if the graph of is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping is maximal if and only if for , for every implies .

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. 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, is the fixed point set of a nonexpansive mapping , and is a given point in [16].

For finding a common element of the set of fixed points of nonexpansive mappings and the set of the variational inequalities, in 2006, Marino and Xu [16] introduced the general iterative method and proved that for a given , the sequence generated by the algorithm

where is a self-nonexpansive mapping on , is an -contraction of into itself i.e., and , satisfies certain conditions, and is strongly positive bounded linear operator on and converges strongly to fixed point of which is the unique solution to the following variational inequality:

which is the optimality condition for the minimization problem

where is a potential function for i.e., for .

Takahashi and Toyoda [32] introduced the following iterative scheme:

where is a -inverse-strongly monotone mapping, is a sequence in , and is a sequence in . They showed that if , then the sequence generated by (17) converges weakly to some .

Yamada [33] introduced the following iterative scheme called the hybrid steepest descent method:

where , , and let be a strongly monotone and Lipschitz continuous mapping and is a positive real number. He proved that the sequence generated by (18) converges strongly to the unique solution of .

Let be a nonempty closed convex subset of . Given the operators defined by

are called the resolvent of (see [19]). It is shown in [19] that under suitable hypotheses on (to be stated precisely in Section 2), is single-valued and firmly nonexpansive and satisfied .

For finding a common element of , S. Takahashi and W. Takahashi [23] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let be a nonexpansive mapping. Starting with arbitrary initial point , define sequences and recursively by

They proved that under certain appropriate conditions imposed on and , the sequences and converge strongly to , where .

In 2012, Chamnarnpan and Kumam [34] introduced the following explicit viscosity scheme with respect to -mappings for an infinite family of nonexpansive mappings

They prove that sequence and converge strongly to , where is an equilibrium point for and is the unique solution of the variational inequality

In 2012, Kangtunyakarn [35] modify the set of variational inequality to construct a new iterative scheme for finding a common element of the set of fixed point problems of infinite family of pseudocontractive mappings and the set of equilibrium problem and two sets of variational inequality problems. Let

Starting with arbitrary initial point , define sequences and recursively by

where is the sequence defined by (37), is and -inverse-strongly monotone mapping, respectively, , and . Under certain appropriate conditions they proved that the sequences and converge strongly to , where .

Let be a mapping, for . By modification of (4), for , we have

In this paper, motivated by the above results, we extend the nonexpansive mapping of hybrid steepest descent method (18) to nonexpansive semigroups and introduce a new iterative scheme for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of solutions of variational inequality problems for relaxed cocoercive mapping in a real Hilbert space by the hybrid steepest descent method. The results shown in this paper improve and extend the recent ones announced by many others.

2. Preliminaries

Throughout this paper, we always assume that is a nonempty closed convex subset of a Hilbert space . We write to indicate that the sequence converges weakly to . implies that converges strongly to . We denote by and the sets of positive integers and real numbers, respectively. For any , there exists a unique nearest point in , denoted by , such that

Such a is called the metric projection of onto . It is known that is nonexpansive. Furthermore, for and ,

It is widely known that satisfies Opial’s condition [8], that is, for any sequence with , the inequality

holds for every with .

In order to solve the equilibrium problem for a bifunction , we assume that satisfies the following conditions:

(A1),(A2) is monotone, that is, ,(A3),(A4) For each is convex and lower semicontinuous.

Let us recall the following lemmas which will be useful for our paper.

Lemma 3 (see [19]). Let be a bifunction from into satisfying (A1), (A2), (A3), and (A4). Then, for any and , there exists such that

Furthermore, if , then the following hold:

(1)  is single-valued,(2)  is firmly nonexpansive, that is,

(3) ,(4)  is closed and convex.

Lemma 4 (see [12]). Let be a nonempty bounded closed and convex subset of a real Hilbert space . Let from be a nonexpansive semigroup on C, then for all ,

Lemma 5 (see [13]). Let be a nonempty bounded closed and convex subset of a real Hilbert space , let be a sequence, and let from be a nonexpansive semigroup on C, if the following conditions are satisfied:

(i),(ii),

then, .

Lemma 6 (see [36]). In a Hilbert space , there holds the inequality

Lemma 7 (see [16]). Assume be a strongly positive linear bounded operator on with coefficient and , then .

Lemma 8 (see [37]). Let be a monotone mapping of into and let be the normal cone to at , that is,

and define a mapping on by

Then is maximal monotone and if and only if, for all .

Lemma 9 (see [27]). Let and be bounded sequences in a Banach space and be a sequence in satisfying the following condition:

Suppose that and . Then .

Lemma 10 (see [28]). Assume that is a sequence of nonnegative real numbers such that

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

(i),(ii) or .

Then, .

Let be a nonempty closed convex subset of a Hilbert space . Let be mapping of into self. For all , let where and . For every , we define the mapping as follows:

This mapping is called -mapping generated by and .

Lemma 11 (see [38]). Let be a nonempty closed convex subset of a Hilbert space . Let be a -strict pseudocontractive mapping of into self with and let where , , , and for all . For every , let and -mapping generated by and and , and , respectively. Then, for every and , the limit exists.

In view of the previous lemma, we will define the mapping as follows:

Remark 12 (see [38]). For each , is nonexpansive and for every bounded subset of .

Lemma 13 (see [38]). Let be a nonempty closed convex subset of a Hilbert space . Let be a -strict pseudocontractive mapping of into self such that with and let where , , , and for all . For every , let and -mapping generated by and , respectively. Then, .

3. Main Results

In this section, we will present our main results. To establish our results, we need the following technical lemmas.

Lemma 14. Let be a nonempty closed convex subset of a Hilbert space and let be -Lipschitz continuous and relaxed -cocoercive mappings with , for . If , then, for and ,

Proof. The proof is by induction. This holds for . In fact, for , it is obvious that

Next, we will show that .

Let

It follows that

Then, for every , one has

From and (43), one has

which means

On the other hand, from , we have

This together with (46) leads to

Furthermore, for every , from (46) and (48), we obtain

which implies

It follows from (45) and (42) that

It yields that

From and (52), one has

That is,

Therefore, for every , from (52) and (54), we obtain

which means

And hence,

Thus, we have

Thus,

Assume now that is true for some , and we show that it continues to hold for . For and , we have

By induction, holds for and this completes the proof.

Lemma 15. Let be a nonempty closed convex subset of a Hilbert space , let from be a nonexpansive semigroup on , and let be -Lipschitz continuous and relaxed -cocoercive mappings with ,  for . Assume that , for and . If , where is the sequence defined by (37) with , then is a nonexpansive mapping in . Furthermore, is a nonexpansive mapping in .

Proof. Since , for every , we have

Thus, we obtain that is a nonexpansive mapping.

Similarly, we can obtain that is a nonexpansive mapping in and this completes the proof.

The following main results follow from Lemmas 14 and 15.

Theorem 16. Let be a nonempty closed convex subset of a real Hilbert space , and let be a bifunction from satisfying (A1)–(A4). Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be -strict pseudo-contractive mappings of into self with and let , where , , and for all . For every , let and -mapping generated by and with . Let be -Lipschitz continuous and relaxed -cocoercive mappings with ,  for , let be a contraction of into itself with , and let be is a strongly positive linear bounded self-adjoint operator with the coefficients and . Assume that

Let be a sequence generated by and

where is the sequence defined by (37) and , . If , are two sequences in and and , for is a real sequence in satisfy the following conditions:

(i), , (ii) and ,(iii) and , for ,(iv) and ,(v).

Then converges strongly to , where is the unique solution of variational inequality

which is the optimality condition for the minimization problem

where is a potential function for i.e., for .

Proof. From the restrictions on control sequences, we may assume, without loss of generality, that for all . From Lemma 7, we know that if , then . We will assume that . Since is a strongly positive linear bounded self-adjoint operator on , we have

Note that

That is, is positive. Furthermore,

Next, We divide the proof of Theorem into five steps.

Step 1. We show that is bounded.

Take . Let , for and , for any . Since is nonexpansive for each and , we have

From Lemma 15 and (69), one has

It follows that

Furthermore,

By induction, we have

Hence, is bounded and we also obtain that and are all bounded.

Step 2. We claim that .

From the definition of and Lemma 15, for , we have

First, we will show that if is bounded, then

for .

From Step 2 of the proof in [4], we have for

For , notice that

It follows that

Therefore, from (76), we conclude (75).

Second, we estimate . From and , we obtain

Taking in (79) and in (80), we have

So, from (A2), one has

Furthermore,

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

Third, we estimate . It follows from (37) that

which means that

where is a constant such that , for all .

Next, we estimate . Substituting (84) and (86) into (74), one has

From (61), we have

Substitution (87) into (88) yields that

where is an appropriate constant such that

It follows from (89) that

Consequently, from (75) and the conditions in Theorem 16, we obtain

Hence, by Lemma 9, one has

Since , this shows that

Step 3. We claim that .

Observing , we obtain

which means that

This together with the conditions (i) and (ii) imply that

From (93) and (97), one has

For , we see that

It follows from (42) that

Substituting (99) into (100) yields that

Furthermore,

It follows that

From (94) and the condition (i), for , we have

Then, for and ,

On the other hand, one has

which means that

It follows that

Therefore, from (108) and (102), one has

Then,

From (94), (105), and condition (i), one has

Let and . Since is firmly nonexpansive, we obtain

It follows that

Consequently, from (108), one has

Then,

That is,

By condition (i) and (94), for , we obtain

Therefore, we have

From (117), one has

Notice that

Applying (119) and (93), we have

Since

this together with (94) yields that

Consequently, we obtain

Step 4. Letting , we show

We know that is a contraction. Indeed, for any , we have

and hence is a contraction due to . Thus, Banach’s Contraction Mapping Principle guarantees that has a unique fixed point, which implies .

We claim that . Since is bounded in , without loss of generality, we can assume that . Since is closed and convex, is weakly closed. Thus we have . For , notice that

It follows from (124) and Lemma 4 that

Thus, (128) and Lemma 5 assert that . Since is bounded in , without loss of generality, we can assume that . It follows from (94) that . Since is closed and convex, is weakly closed. Thus we have .

Let us show . For the sake of contradiction, suppose that , that is, . Since , by our assumption, we have and then . Hence . Therefore, by (124) and Opial condition, we have

which derives a contradiction. Thus, we obtain .

Next, we claim that . Since for , we obtain

From (A2), one has

Replacing by , we have

It follows from and that

for .

Put for all and . Then, we have and then . Hence, from (A1) and (A4), we have

which means . From (A3), we obtain for and then for , that is, .

Finally, we claim that .

We define the maximal monotone operator

Since is relaxed -cocoercive for , we have

which yields that is monotone. Thus, is maximal monotone. Let . Since and , we have

On the other hand, it follows from that

and hence

It follows that

which implies that

Since is maximal monotone, we obtain that . From Lemma 8, we obtain , that is, . Thus, .

Since , one has

Furthermore,

From (93) and (142), we have

Step 5. Finally, we show that converges strongly to . Indeed, from (61) and (70), we obtain

which implies that

It follows from (146) that

From condition (i) and (142), we know that

we can conclude from Lemma 10 that as . This completes the proof of Theorem 16.

Theorem 17. Let be a nonempty closed convex subset of a real Hilbert space , and let be bifunction from satisfying (A1)–(A4). Let be -strict pseudocontractive mappings of into self with and let , where , , , and for all . For every , let and be -mapping generated by and and , and , respectively. Let be -Lipschitz continuous and relaxed -cocoercive mappings with , let be a contraction of into itself with , and let be a strongly positive linear bounded self-adjoint operator with the coefficients and . Assume that

Let be a sequence generated by and

where is the sequence defined by (37). If , are two sequences in and and , for is a real sequence in satisfing the following conditions:

(i), , (ii) and ,(iii) and , for ,(iv) and .

Then converges strongly to , where is the unique solution of variational inequality

which is the optimality condition for the minimization problem

where is a potential function for i.e., for .

Proof. By Theorem 16, for , letting , we can obtain Theorem 17.

Theorem 18. Let be a nonempty closed convex subset of a real Hilbert space , and let be a bifunction from satisfying (A1)–(A4). Let be -strict pseudo-contractive mappings of into self with and let , where , , , and for all . For every , let and be -mapping generated by and and , and , respectively. Let be -Lipschitz continuous and relaxed -cocoercive mappings with , for , let be a contraction of into itself with , and let be is a strongly positive linear bounded self-adjoint operator with the coefficients and . Assume that

Let be a sequence generated by and

where is the sequence defined by (37) and , . If , are two sequences in and and , for is a real sequence in satisfing the following conditions:

(i), , (ii) and ,(iii) and ,(iv) and .

Then converges strongly to , where is the unique solution of variational inequality

which is the optimality condition for the minimization problem

where is a potential function for i.e., for .

Proof. By Theorem 16, letting for all , we can obtain Theorem 19.

Theorem 19. Let be a nonempty closed convex subset of a real Hilbert space , and let be a bifunction from satisfying (A1)–(A4). Let be -Lipschitz continuous and relaxed -cocoercive mappings with , and let be a contraction of into itself with , and let be is a strongly positive linear bounded self-adjoint operator with the coefficients and . Assume that

Let be a sequence generated by and

If , are two sequences in and and , for is a real sequence in satisfing the following conditions:

(i), , (ii) and ,(iii) and , for .(iv) and .

Then converges strongly to , where is the unique solution of variational inequality

which is the optimality condition for the minimization problem

where is a potential function for i.e., for .

Proof. By Theorem 17, letting for all , we can obtain Theorem 19.

Acknowledgments

The authors would like to thank the anonymous referees and the editor for their constructive comments and suggestions, which greatly improved this paper. This project is supported by the Natural Science Foundation of China (Grant nos. 11171180, 11171193, 11126233, and 10901096) Shandong Provincial Natural Science Foundation (Grant no. ZR2011AM016), and the Project of Science and Technology Program of Weifang (Grant no. 20121103).

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