Journal of Applied Mathematics

Volume 2013 (2013), Article ID 230392, 17 pages

http://dx.doi.org/10.1155/2013/230392

Research Article

A New Computational Technique for Common Solutions between Systems of Generalized Mixed Equilibrium and Fixed Point Problems

Pongsakorn Sunthrayuth and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bang Mod, Bangkok 10140, Thailand

Received 18 February 2013; Accepted 1 April 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Pongsakorn Sunthrayuth and Poom Kumam. 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.

We introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of nonexpansive mappings, the set solutions of a system of a general system of generalized equilibria in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common element of the above three sets under some suitable conditions. As applications, at the end of the paper, we apply our results to find the minimum-norm solutions which solve some quadratic minimization problems. The results obtained in this paper extend and improve many recent ones announced by many others.

1. Introduction

Throughout this paper, we denoted by the set of all real numbers. We always assume that is a real Hilbert space with inner product and induced norm and is a nonempty, closed, and convex subset of . denotes the metric projection of onto . A mapping is said to be -Lipschitzian if there exists a constant such that

If , then is a contraction, and if , then is a nonexpansive mapping. We denote by the set of all fixed points set of the mapping ; that is, .

A mapping is said to be monotone if

A mapping is said to be strongly monotone if there exists such that

Let be a real-valued function, an equilibrium bifunction, and a nonlinear mapping. The generalized mixed equilibrium problem is to find such that

which was introduced and studied by Peng and Yao [1]. The set of solutions of problem (4) is denoted by . As special cases of problem (4), we have the following results.

(1)If , then problem (4) reduces to mixed equilibrium problem. Find such that

which was considered by Ceng and Yao [2]. The set of solutions of problem (5) is denoted by . (2)If , then problem (4) reduces to generalized equilibrium problem. Find such that

which was considered by S. Takahashi and W. Takahashi [3]. The set of solutions of problem (6) is denoted by .(3)If , then problem (4) reduces to equilibrium problem. Find such that

The set of solutions of problem (7) is denoted by .(4)If , then problem (4) reduces to classical variational inequality problem. Find such that

The set of solutions of problem (8) is denoted by . It is known that is a solution of the problem (8) if and only if is a fixed point of the mapping , where is a constant and is the identity mapping.

The problem (4) is very general in the sense that it includes several problems, namely, fixed point problems, optimization problems, saddle point problems, complementarity problems, variational inequality problems, minimax problems, Nash equilibrium problems in noncooperative games, and others as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of problem (4) (see, e.g., [4–9]). Several iterative methods to solve the fixed point problems, variational inequality problems, and equilibrium problems are proposed in the literature (see, e.g., [1–3, 10–18]) and the references therein.

Let be two mappings. Ceng and Yao [12] considered the following problem of finding such that

which is called a general system of generalized equilibria, where and are two constants. In particular, if and , then problem (9) reduces to the following problem of finding such that

which is called a new system of generalized equilibria, where and are two constants.

If , , and , then problem (9) reduces to problem (7).

If , then problem (9) reduces to a general system of variational inequalities. Find such that

where and are two constants, which is introduced and considered by Ceng et al. [19].

In 2010, Ceng and Yao [12] proposed the following relaxed extragradient-like method for finding a common solution of generalized mixed equilibrium problems, a system of generalized equilibria (9), and a fixed point problem of a -strictly pseudocontractive self-mapping on as follows:

where are -inverse strongly monotone, -inverse strongly monotone, and -inverse strongly monotone, respectively. They proved strong convergence of the related extragradient-like algorithm (12) under some appropriate conditions , , , and satisfying, for all , to , where , with the mapping defined by

Very recently, Ceng et al. [11] introduced an iterative method for finding fixed points of a nonexpansive mapping on a nonempty, closed, and convex subset in a real Hilbert space as follows:

where is a metric projection from onto , is an -Lipschitzian mapping with a constant , and is a -Lipschitzian and -strongly monotone operator with constants and . Then, they proved that the sequences generated by (14) converge strongly to a unique solution of variational inequality as follows:

In this paper, motivated and inspired by the previous facts, we first introduce the following problem of finding such that

which is called a more general system of generalized equilibria in Hilbert spaces, where for all . In particular, if , and , then problem (16) reduces to problem (9). Finally, by combining the relaxed extragradient-like algorithm (12) with the general iterative algorithm (14), we introduce a new iterative method for finding a common element of a fixed point problem of a nonexpansive semigroup, the set solutions of a general system of generalized equilibria in a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to a common element of the previous three sets under some suitable conditions. Furthermore, we apply our results to finding the minimum-norm solutions which solve some quadratic minimization problem. The main result extends various results existing in the current literature.

2. Preliminaries

Let be a semigroup. We denote by the Banach space of all bounded real-valued functionals on with supremum norm. For each , we define the left and right translation operators and on by

for each and , respectively. Let be a subspace of containing . An element in the dual space of is said to be a mean on if . It is well known that is a mean on if and only if

for each . We often write instead of for and .

Let be a translation invariant subspace of (i.e., and for each ) containing . Then, a mean on is said to be left invariant (resp., right invariant) if (resp., ) for each and . A mean on is said to be invariant if is both left and right invariant [20–22]. is said to be left (resp., right) amenable if has a left (resp., right) invariant mean. is a amenable if is left and right amenable. In this case, also has an invariant mean. As is well known, is amenable when is commutative semigroup; see [23]. A net of means on is said to be left regular if

for each , where is the adjoint operator of .

Let be a nonempty, closed, and convex subset of . A family is called a nonexpansive semigroup on if for each , the mapping is nonexpansive and for each . We denote by the set of common fixed point of ; that is,

Throughout this paper, the open ball of radius centered at is denoted by , and for a subset of by , we denote the closed convex hull of . For and a mapping , the set of -approximate fixed point of is denoted by ; that is, .

In order to prove our main results, we need the following lemmas.

Lemma 1 (see [23–25]). Let be a function of a semigroup into a Banach space such that the weak closure of is weakly compact, and let be a subspace of containing all the functions with . Then, for any , there exists a unique element in such that

for all . Moreover, if is a mean on , then

One can write by .

Lemma 2 (see [23–25]). Let be a closed and convex subset of a Hilbert space , a nonexpansive semigroup from into such that , and a subspace of containing , the mapping an element of for each and , and a mean on .

If one writes instead of , then the following hold:

(i) is nonexpansive mapping from into ;(ii) for each ;(iii) for each ;(iv)if is left invariant, then is a nonexpansive retraction from onto .

Let be a real Hilbert space with inner product and norm , and let be a nonempty, closed, and convex subset of . We denote the strong convergence and the weak convergence of to by and , respectively. Also, a mapping denotes the identity mapping. For every point , there exists a unique nearest point of , denoted by , such that

Such a projection is called the metric projection of onto . We know that is a firmly nonexpansive mapping of onto ; that is,

It is known that

In a real Hilbert space , it is well known that

for all and .

If is -inverse strongly monotone, then it is obvious that is -Lipschitz continuous. We also have that, for all and ,

In particular, if , then is a nonexpansive mapping from to .

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

(A1)  for all ;(A2)  is monotone, that is, for each ;(A3)  is upper semicontinuous, that is, for each ,

(A4)  is convex and weakly lower semicontinuous for each ;(B1) for each and , there exists a bounded subset and such that for all ,

(B2)  is a bounded set.

Lemma 3 (see [1]). Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a bifunction satisfying conditions , and let be a lower semicontinuous and convex function. For and , define a mapping as follows:

Assume that either or holds. Then, the following hold:

(i) for all and is single valued;(ii) is firmly nonexpansive, that is, for all ,

(iii);(iv) is closed and convex.

Remark 4. If , then is rewritten as (see [12, Lemma 2.1] for more details).

Lemma 5 (see [26]). Let and be bounded sequences in a Banach space , and let be a sequence in with . Suppose that  for all integers and . Then, .

Lemma 6 (Demiclosedness Principle [27]). Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping with . If is a sequence in that converges weakly to and if converges strongly to , then ; in particular, if , then .

Lemma 7 (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, .

The following lemma can be found in [29, 30]. For the sake of the completeness, we include its proof in a real Hilbert space version.

Lemma 8. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a -Lipschitzian and -strongly monotone operator. Let and . Then, for each , the mapping defined by is a contraction with constant .

Proof. Since and , this implies that . For all , we have

It follows that

Hence, we have that is a contraction with constant . This completes the proof.

Lemma 9. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let    be a finite family of -inverse strongly monotone operator. Let be a mapping defined by

If for all , then is nonexpansive.

Proof. Put for and . Then, . For all , it follows from (28) that

which implies that is nonexpansive. This completes the proof.

Lemma 10. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let    be a nonlinear mapping. For given , where , , for , and . Then, is a solution of the problem (16) if and only if is a fixed point of the mapping defined as in Lemma 9.

Proof. Let be a solution of the problem (16). Then, we have

This completes the proof.

3. Main Results

Theorem 11. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let    a finite family of bifunctions which satisfy ,    a finite family of lower semicontinuous and convex functions, and    a finite family of a -inverse strongly monotone mapping and    a finite family of an -inverse strongly monotone mapping. Let be a semigroup, and let be a nonexpansive semigroup on such that . Let be a left invariant subspace of such that and the function is an element of for and . Let be a left regular sequence of means on X such that . Let be a -Lipschitzian and -strongly monotone operator with constants , and let be an -Lipschitzian mapping with a constant . Let and , where . Assume that , where is defined as in Lemma 9. For given , let be a sequence defined by

where , are sequences in , and is a sequence such that satisfying the following conditions:

  and ; ;  and for all .

Then, the sequence defined by (39) converges strongly to as , where solves uniquely the variational inequality

Equivalently, one has .

Proof. Note that from condition , we may assume, without loss of generality, that for all . First, we show that is bounded. Set

. Then, we have and . From Lemmas 3 and 9, we have that and are nonexpansive. Take ; we have

By Lemma 10, we have . It follows from (42) that

Set

Then, we can rewrite (39) as . From Lemma 8 and (43), we have

It follows from (45) that

By induction, we have

Hence, is bounded, and so are and .

Next, we show that

Observe that

Indeed,

Since is bounded and , then (49) holds. We observe that

Let be a bounded sequence in . Now, we show that

For the previous purpose, put , and we first show that

In fact, since and , we have

Substituting in (54) and in (55), then add these two inequalities, and using , we obtain

Hence,

we derive from (57) that

which implies that

Noticing that condition implies that (53) holds, from the definition of and the nonexpansiveness of , we have

for which (52) follows by (53). Since and , we have

Put a constant such that

From definition of , we note that

It follows from (51), (61), and (63) that

From condition , (49), and (52), we have

Hence, by Lemma 5, we obtain

Consequently,

From condition , we have

From (66) and (68), we have

Set , where . From (25), we have

It follows that

By the convexity of and (71), we have

where is an appropriate constant such that .

Next, we show that

From (28), we have

From (42), for all , we note that

From (72) and (75), we have

Substituting (74) into (72), we have

which in turn implies that

Since , , for all , from and (67), we obtain that

On the other hand, from Lemma 3 and (26), we have

which in turn implies that

Substituting (81) into (76), we have

which in turn implies that

Since , from, (67), and (79), we obtain that (73) holds. Consequently,

Next, we show that

From (28), we have

By induction, we have

From (72) and (75), we have

Substituting (87) into (88), we have

which in turn implies that

Since , from and (67), we obtain that (85) holds.

On the other hand, from (24) and (26), we have

which in turn implies that

By induction, we have

Substituting (93) into (88), we have

which in turn implies that

Since , from, (67), and (85), we obtain that

Consequently,

It follows from (84) and (97) that

Next, we show that

Put

Set . We remark that is nonempty, bounded, closed, and convex set, and , , and are in . We will show that

To complete our proof, we follow the proof line as in [31] (see also [23, 32, 33]). Let . By [34, Theorem 1.2], there exists such that

Also by [34, Corollary 1.1], there exists a natural number such that

for all and . Let . Since is strongly left regular, there exists such that for all and . Then, we have

On the other hand, by Lemma 2, we have

Combining (103)–(105), we have

for all and . Therefore,

Since is arbitrary, we obtain that (101) holds. Let and . Then, there exists satisfying (102). From (101) and condition , there exists such that and for all . From (69), there exists such that for all . Then, from (102) and (106), we have

for all . Hence, . Since is arbitrary, we obtain that (99) holds.

Next, we show that

where . To show this, we choose a subsequence of such that

Since is bounded, there exists a subsequence of such that . Now, we show that .

(i)We first show that . From (99), we have as for all . Then, from Demiclosedness Principle 2.6, we get .(ii)We show that , where is defined as in Lemma 9. Then, from (97), we have

and from (98), we also have . By Demiclosedness Principle 2.6, we get .(iii)We show that . Note that , for all . Then, we have

Replacing by in the last inequality and using , we have

Let for all and . This implies that . Then, we have

From (73), we have as . Furthermore, by the monotonicity of , we obtain . Then, from , we obtain

Using , , and (115), we also obtain

and, hence,

Letting and using , we have, for each ,

This implies that . Hence, . Therefore,

From (66) and (110), we obtain

Finally, we show that as . Notice that , where . Then, from (25), we have

It follows from (121) that

Put and . Then, (122) reduces to formula

It is easily seen that , and (using (120))

Hence, by Lemma 7, we conclude that as . This completes the proof.

Using the results proved in [35] (see also [32]), we obtain the following results.

Corollary 12. Let , , , , , , , and be the same as in Theorem 11. Let and be nonexpansive mappings on with . Assume that , where is defined as in Lemma 9. Let , , and be sequences satisfying . Then, the sequence defined by

converges strongly to , where solves uniquely the variational inequality (40).

Corollary 13. Let , , , , , , , and be the same as in Theorem 11. Let be a strongly continuous nonexpansive semigroup on . Assume that , where is defined as in Lemma 9. Let , , and be sequences satisfying . Then, the sequence defined by

where is an increasing sequence in with , converges strongly to , where solves uniquely the variational inequality (40).

Corollary 14. Let , , , , , , , and be the same as in Theorem 11. Let be a strongly continuous nonexpansive semigroup on . Assume that , where is defined as in Lemma 9. Let , , and be sequences satisfying . Then, the sequence defined by

where is a decreasing sequence in with , converges strongly to , where solves uniquely the variational inequality (40).

4. Some Applications

In this section, as applications, we will apply Theorem 11 to find minimum-norm solutions of some variational inequalities. Namely, find a point which solves uniquely the following quadratic minimization problem:

Minimum-norm solutions have been applied widely in several branches of pure and applied sciences, for example, defining the pseudoinverse of a bounded linear operator, signal processing, and many other problems in a convex polyhedron and a hyperplane (see [36, 37]).

Recently, some iterative methods have been studied to find the minimum-norm fixed point of nonexpansive mappings and their generalizations (see, e.g. [38–49] and the references therein).

Using Theorem 11 and Corollaries 12, 13, and 14, we immediately have the following results, respectively.

Theorem 15. Let and be the same as in Theorem 11. Let be a nonexpansive semigroup on such that . Let and be sequences satisfying . Then, the sequence defined by

converges strongly to , where is the minimum-norm fixed point of , where solves uniquely the quadratic minimization problem (128).

Theorem 16. Let and be the same as in Corollary 12. Let and be nonexpansive mappings on with such that . Let and be sequences satisfying . Then, the sequence defined by

converges strongly to , where is the minimum-norm fixed point of , where solves uniquely the quadratic minimization problem (128).

Theorem 17. Let and be the same as in Corollary 13. Let be a strongly continuous nonexpansive semigroup on such that . Let and be sequences satisfying . Then, the sequence defined by

where is an increasing sequence in with , converges strongly to , where is the minimum-norm fixed point of , where solves uniquely the quadratic minimization problem (128).

Theorem 18. Let and be the same as in Corollary 14. Let be a nonexpansive semigroup on such that . Let and be sequences satisfying . Then, the sequence defined by

where is a decreasing sequence in with , converges strongly to , where is the minimum-norm fixed point of , where solves uniquely the quadratic minimization problem (128).

Acknowledgment

The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (Grant no. NRU56000508).

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