Abstract and Applied Analysis

Volume 2012 (2012), Article ID 414718, 26 pages

doi:10.1155/2012/414718

Research Article

A New General System of Generalized Nonlinear Mixed Composite-Type Equilibria and Fixed Point Problems with an Application to Minimization 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 9 August 2012; Accepted 24 September 2012

Academic Editor: RuDong Chen

Copyright © 2012 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 general system of generalized nonlinear mixed composite-type equilibria and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. Finally, we also give a numerical example which supports our results. The results obtained in this paper extend the recent ones announced by 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 and convex subset of . Let be a real-valued function, where is the set of real numbers. Let be two nonlinear mappings and be an equilibrium-like function, that is, for all . We consider the following new generalized equilibrium problem: find such that

The set of solutions of the problem (1.1) is denoted by . As special cases of the problem (1.1), we have the following results.

(1)If , then problem (1.1) reduces to the following generalized equilibrium problem: find such that

which was considered by Cho et al. [1] for more details. The set of solutions of the problem (1.1) is denoted by .(2)If and , where is an equilibrium bifunction, then problem (1.1) reduces to the following mixed equilibrium problem: find such that

which was considered by Ceng and Yao [2] for more details. The set of solutions of the problem (1.3) is denoted by .(3)If , and where is an equilibrium bifunction, then problem (1.1) reduces to the following equilibrium problem: find such that

The set of solutions of problem (1.4) is denoted by .(4)If , , then problem (1.1) reduces to the following classical variational inequality problem: find such that

The set of solutions of the problem (1.5) is denoted by .

In brief, for an appropriate choice of the mapping , the function , and the convex set , one can obtain a number of the various classes of equilibrium problems as special cases. In particular, the equilibrium problems (1.4) which were introduced by Blum and Oettli [3] and Noor and Oettli [4] in 1994 have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. In [3, 4], it has been shown that equilibrium problems include variational inequalities, fixed point, minimax problems, Nash equilibrium problems in noncooperative games, and others as special cases. This means that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. Hence collectively, equilibrium problems cover a vast range of applications. Related to the equilibrium problems, we also have the problems of finding the fixed points of the nonlinear mappings, which is the subject of current interest in functional analysis. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of the set of solutions of the equilibrium problems and the set of fixed points of nonlinear mappings (e.g., see [5–18] and the references therein).

Recall the following definitions.

Definition 1.1. The mapping is said to be

(1)nonexpansive if

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

(3)-strict pseudocontraction [19] if there exists a constant such that

(4)pseudocontractive if

Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. It is easy to see that (1.8) is equivalent to

that is, is -inverse-strongly monotone. From [19], we know that if is a -strictly pseudocontractive mapping, then is Lipschitz continuous with constant , that is, , for all .

In this paper, we use to denote the set of fixed points of .

Definition 1.2. A countable family of mapping is called a family of -strict pseudocontraction mappings if there exists a constant such that

On the other hand, let be a nonempty closed and convex subset of a real Hilbert space . Let be three bifunctions and let be six nonlinear mappings and let be three functions. We consider the following problem of finding such that

which is called a new general system of generalized nonlinear mixed composite-type equilibria, where for all . Next, we present some special cases of problem (1.12) as follows.

(1)If , , , and for all , then the problem (1.12) reduces to the following new general system of generalized nonlinear mixed composite-type equilibria: find such that

where for all .(2)If , , , and , then the problem (1.12) reduces to the following general system of generalized nonlinear mixed composite-type equilibria: find such that

which was introduced and considered by Ceng et al. [20], where for all .(3)If , , , and for all and , then the problem (1.12) reduces to the following a general system of generalized equilibria: find such that

which was introduced and considered by Ceng and Yao [21], where for all .(4)If , , and , , for all , then the problem (1.12) reduces to the following generalized mixed equilibrium problem with perturbed mapping: find such that

which was introduced and considered by Hu and Ceng [22].(5)If , , and for all , then the problem (1.12) reduces to the following general system of variational inequalities: find such that

which was introduced and considered by Kumam et al. [23], where for all . (6)If , , for all , and , then the problem (1.12) reduces to the following general system of variational inequalities: find such that

which was introduced and considered by Ceng et al. [24], where for all .

In 2010, Cho et al. [1] introduced an iterative method for finding a common element of the set of solutions of generalized equilibrium problems (1.2), the set of solutions for a systems of nonlinear variational inequalities problems (1.18), and the set of fixed points of nonexpansive mappings in Hilbert spaces. Ceng and Yao [21] introduced and considered a relaxed extragradient-like method for finding a common element of the set of solutions of a system of generalized equilibria, the set of fixed points of a strictly pseudocontractive mapping, and the set of solutions of a equilibrium problem in a real Hilbert space and obtained a strong convergence theorem. The result of Ceng and Yao [21] included, as special cases, the corresponding ones of S. Takahashi and W. Takahashi [10], Ceng et al. [24], Peng and Yao [25], and Yao et al. [26].

Motivated and inspired by the works in the literature, we introduce a new general system of generalized nonlinear mixed composite-type equilibria (1.12) and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. The results presented in this paper extend the recent results of Cho et al. [1], Ceng and Yao [21], Ceng et al.[20], and many authors.

2. Preliminaries

A bounded linear operator is said to be strongly positive, if there exists a constant such that

Recall that, a mapping is said to be contractive if there exists a constant such that

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

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

is called the metric projection of onto . It is well known that is nonexpansive (see [27]) and for ,

Let be a real-valued function, be a mapping and be an equilibrium-like function. Let be a positive real number. For all , we consider the following problem. Find such that

which is known as the auxiliary generalized equilibrium problem.

Let be the mapping such that, for all , is the solution set of the auxiliary problem (2.6), that is,

Then, we will assume the Condition [28] as follows:

(a) is single-valued;(b) is nonexpansive;(c).

Notice that the examples of showing the sufficient conditions for the existence of the condition can be found in [6].

Throughout this paper, we assume that a bifunction and is a lower semicontinuous and convex function satisfy the following conditions:

(H1), ;(H2) is monotone, that is, , ;(H3)for all ,   is weakly upper semicontinuous;(H4)for all ,   is convex and lower semicontinuous;(A1)for all and , there exist a bounded subset and such that for all ,

(A2) is a bounded set.

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

Lemma 2.1 (see [29]). Let be a nonempty closed and convex subset of a real Hilbert sapce . Let be a bifunction satisfying condition – and let be a lower semicontinuous and convex function. For and define a mapping follows

Assume that either or holds, then the following statements hold

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

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

Remark 2.2. If , then is rewritten as (see [21, Lemma 2.1] for more details).

Lemma 2.3 (see [30]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .

Lemma 2.4 (see [31]). Let be a real Hilbert space. Then the following inequalities hold.

(i),  and .(ii).

Definition 2.5 (see [32]). Let be a sequence of mappings from a subset of a real Hilbert space into itself. We say that satisfies the condition if

where , for all .

Lemma 2.6 (see [32]). Suppose that satisfies the -condition such that

(i)for each , is converse strongly to some point in (ii)let the mapping defined by for all .

Then, .

Lemma 2.7 (see [33]). Let be a closed and convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose is nonempty. Let be a sequence of positive numbers with . Then a mapping on defined by for all is well defined, nonexpansive, and holds.

Lemma 2.8 (see [19]). Let be a -strict pseudocontraction. Define by for each . Then, as , S is nonexpansive such that .

Lemma 2.9 (see [34]). Let be a closed and convex subset of a real Hilbert space and let be a nonexpansive mapping. then, the mapping is demiclosed. That is, if is a sequence in such that and , then .

Lemma 2.10 (see [35]). 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, .

Lemma 2.11. Let be a nonempty closed and convex subset of a real Hilbert space . Let mappings be -inverse-strongly monotone and -inverse-strongly monotone, respectively. Then, we have

where . In particular, if , then is nonexpansive.

Proof. From Lemma 2.4, for all , we have

It is clear that, if , then is nonexpansive. This completes the proof.

Lemma 2.12. Let be a nonempty closed and convex subset of a real Hilbert space . Let mappings be -inverse-strongly monotone and -inverse-strongly monotone, respectively. Let be the mapping defined by

If , then is nonexpansive.

Proof. From Lemma 2.11, for all , we have

which implies that is nonexpansive. This completes the proof.

Lemma 2.13. Let be a nonempty closed and convex subset of a real Hilbert space . Let    be a bifunction satisfying conditions – and let be a nonlinear mapping. Suppose that be a real positive number. Let be a lower semicontinuous and convex function. Assume that either condition or holds. Then, for is a solution of the problem (1.12) if and only if , and , where is the mapping defined as in Lemma 2.12.

Proof. Let be a solution of the problem (1.12). Then, we have

This completes the proof.

Corollary 2.14 (see [20]). Let be a nonempty closed and convex subset of a real Hilbert space . Let be a bifunction satisfying conditions – and let be a nonlinear mapping. Suppose that be a real positive number. Let be a lower semicontinuous and convex function. Assume that either condition or holds. Then, for is a solution of the problem (1.14) if and only if , , where is the mapping defined by

Corollary 2.15 (see [21]). Let be a nonempty closed and convex subset of a real Hilbert space . Let be a bifunction satisfying conditions – and let be a nonlinear mapping. Suppose that is a real positive number. Assume that either condition or holds. Then, for is a solution of the problem (1.15) if and only if , , where is the mapping defined by

Corollary 2.16 (see [23]). Let be a nonempty closed and convex subset of a real Hilbert space . For given is a solution of the problem (1.17) if and only if , and , where is the mapping defined by

Corollary 2.17 (see [24]). Let be a nonempty closed and convex subset of a real Hilbert space . For given is a solution of the problem (1.18) if and only if , , where is the mapping defined by

3. Main Results

We are now in a position to prove the main result of this paper.

Theorem 3.1. Let be a nonempty closed and convex subset of a real Hilbert space such that . Let be lower semicontinuous and convex functionals, be an equilibrium-like function, be a mapping, and be a bifunction satisfying conditions –. Assume that either condition or holds. Let be -inverse-strongly monotone, be -inverse-strongly monotone and -inverse-strongly monotone, respectively. Let be a family of -strict pseudocontraction mappings. Define a mapping , for all , and . Assume that the condition is satisfied and , where is defined as in Lemma 2.13. Let , , and be three constants. Let be a contraction mapping with a coefficient and let be a strongly positive bounded linear operator on with a coefficient such that . For , let the sequence defined by

where such that , , , , and are two sequences in . Suppose that satisfies the -condition. Let be the mapping defined by for all and suppose that . Assume the following conditions are satisfied:

(C1) and , (C2).

Then the sequence defined by (3.1) converges strongly to , where is the unique solution of the variational inequality

or equivalently, , where is a metric projection mapping from onto , and is a solution of the problem (1.12), where and .

Proof. Note that from the conditions and , we may assume, without loss of generality, that for all . Since is a linear bounded self-adjoint operator on , by (2.2), we have

Observe that

This show that is positive. It follows that

First, we show that is bounded. Taking , it follows from Lemma 2.13 that

Putting and , we obtain . Notice that . Since is nonexpansive and is -inverse-strongly monotone, we have

and hence

We observe that

Setting . By Lemma 2.8, we have is a nonexpansive mapping such that for all . Then, we have

It follows that

By induction, we have

Hence, is bounded, so are , , , and . From definition of and for all , it follows that

By our assumption, satisfies the PT-condition, we obtain that

that is satisfies the PT-condition.

Next, we show that . Since , we have

It follows from (3.15) that

Let for all . Then, we have

Combining (3.16) and (3.17), we have

Since satisfies the PT-condition, we can define a mapping by

We observe that

Be Lemma 2.6, we have

Consequently, it follows from the conditions , , and (3.18) that

Hence, by Lemma 2.3, we obtain that

Consequently, we have

On the other hand, we observe that

It follows that

From the conditions , , and (3.24), we obtain that

Next, we show that

To show this, we take a subsequence of such that

Since is bounded, without loss of generality, we can assume that . So, we get

Next, we show that . Define a mapping by

From (3.27), we have

For all , it follows that

Since satisfies the PT-condition, we obtain that

that is satisfies the PT-condition. Define a mapping by

By Lemma 2.6, we obtain that

From Lemma 2.7, we see that is nonexpansive and

Notice that

From (3.32) and (3.39), we obtain that

Thus, by Lemma 2.9, we obtain that .

Finally, we show that as . From Lemma 2.4, we compute

It follows that

where . Put and

Then, the (3.41) reduces to the formula

It is easily seen that and (using (3.30)), we get

Hence, by Lemma 2.10, we conclude that as . This completes the proof.

Remark 3.2. Theorem 3.1 improves and generalizes [1, Theorem 2.1] in the following ways.

(i)From a one nonexpansive mapping to a countable family of strict pseudocontraction mappings.(ii)From a general system of variational inequalities to a general system of generalized nonlinear mixed composite-type equilibria.(iii)Theorem 3.1 for finding an element ( is defined as in Lemma 2.13) is more general the one of finding elements of ( is defined as in Lemma 2.17) in [1, Theorem 2.1].

Furthermore, our method of the proof is very different from that in [1, Theorem 2.1] because (3.1) involves the countable family of strict pseudocontraction mappings and strongly positive bounded linear operator.

4. Application to Minimization Problems

Let be a nonempty closed and convex subset of a real Hilbert space and be a strongly positive linear bounded operator with a constant . In this section, we will utilize the results presented in Section 3 to study the following minimization problem:

where is a nonempty closed and convex subsets of , is some constant and is a potential function for (i.e., for all ), where is a contraction mapping with a constant . Note that this kind of minimization problems has been studied extensively by many authors (e.g., see [18, 36–39]). We can apply Theorem 3.1 to solve the above minimization problem in the framework of Hilbert spaces as follows.

Theorem 4.1. Let be a nonempty closed and convex subset of a real Hilbert space such that . Let be a nonexpansive mappings such that . Let and be two constants. Let be a contraction mapping with a coefficient and be a strongly positive bounded linear operator on with a coefficient such that . Let be a sequence defined by and

where are two sequences in . Assume the following conditions are satisfied:

(C1) and , (C2).

Suppose that is a compact subset of . Then the sequence defined by (4.2) converges strongly to which solves the minimization problem (4.1).

Proof. Taking and in Theorem 3.1. Hence, from Theorem 3.1, we know that the sequence defined by (4.2) converges strongly to , where is the unique solution of the variational inequality

Since is nonexpansive, then is convex. Again by the assumption that is compact, then it is a compact and convex subset of , and

is a continuous mapping. By virtue of the well-known Weierstrass's theorem, there exists a point which is a minimal point of minimization problem (4.1). As is known to all, (4.3) is the optimality necessary condition [18] for the minimization problem (4.1). Therefore we also have

Since is the unique solution of (4.3), we have . This completes the proof.

5. A Numerical Example

In this section, we give a real example in which the conditions satisfy the ones of Theorem 3.1 and some numerical experiment results to explain the main result Theorem 3.1 as follows.

Example 5.1. Let , , , , , , , , , , and , with a constant , ,  , ,  , , and . For all , let define by , , we see that, is a family of -strictly pseudocontractive with . Then, is the sequence defined by

and as , where is the unique solution of the minimization problem

Proof. Step 1. We show that

where

Since , due to the definition of , in (2.7), we have

Also by the equivalent property (2.5) of the nearest projection from to , we obtain this conclusion. When we take , then . By the condition , we have . In a similar way, for all , we can get

and . Hence

Step 2. We show that is satisfies the -condition. Since , , and . For all , we have

that is satisfies the -condition.

Step 3. We show that

where is the unique solution of the minimization problem:

Due to (5.3) and (5.4), we can obtain a special sequence of (3.1) in Theorem 3.1 as follows:

Since , combining with (5.7), we have

By Lemma 2.10, we obtain that , where is the unique solution of the minimization problem , where is a constant number.

5.1. Numerical Experiment Results

Next, we show the numerical experiment results using software MATLAB 7.0 and we obtain the results shown in Tables 1 and 2 and Figure 1, which show that the iteration process of the sequence as initial points and , respectively.

Table 1: This table shows the value of sequence on each iteration step (initial value ).

1
1.000000000000000

2
0.400000000000000

3
0.236825396825397

4
0.156844963280237

5
0.109791474296166

6
0.079448383160412

7
0.058780740964773

8
0.044187178532649

9
0.033618033450111

10
0.000000000000000

Table 2: This table shows the value of sequence on each iteration step (initial value ).

1
0.500000000000000

2
0.225000000000000

3
0.135089285714286

4
0.089721577233324

5
0.062805104063327

6
0.045409812093980

7
0.033562589425687

8
0.025205072875411

9
0.019159476038106

10
0.000000000000000

(a)

(b)

Figure 1: These figures show the iteration comparison chart of different initial values (a)   and (b)  , respectively.

Acknowledgment

The authors were supported by the Higher Education Research Promotion and the National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC no. 55000613).

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