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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 139123, 8 pages
http://dx.doi.org/10.1155/2013/139123
Research Article

New Hybrid Steepest Descent Algorithms for Equilibrium Problem and Infinitely Many Strict Pseudo-Contractions in Hilbert Spaces

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 4 May 2013; Accepted 17 June 2013

Academic Editor: Gue Myung Lee

Copyright © 2013 Peichao Duan. 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 propose an explicit iterative scheme for finding a common element of the set of fixed points of infinitely many strict pseudo-contractive mappings and the set of solutions of an equilibrium problem by the general iterative method, which solves the variational inequality. In the setting of real Hilbert spaces, strong convergence theorems are proved. The results presented in this paper improve and extend the corresponding results reported by some authors recently. Furthermore, two numerical examples are given to demonstrate the effectiveness of our iterative scheme.

1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed convex subset of .

Let be a nonlinear mapping; we consider the problem of finding such that It is known as the variational inequality problem (denoted by ).

Generally, is assumed to be Lipschitzian and strongly monotone. The relative definitions are listed as follows.(i)is called -Lipschitzian on , if there exists a constant such that (ii) is said to be -strongly monotone on , if there exists a constant such that (iii)A mapping of is said to be a -strict pseudo-contraction if there exists a constant such that for all ; see [1]. (iv)A mapping of is said to be a nonexpansive mapping if it is strictly pseudo-contractive with constant .

Obviously, the class of strict pseudo-contractions strictly includes the class of nonexpansive mappings. We denote the set of fixed points of by (i.e., ).

Let be a bifunction from to , where is the set of real numbers.

The equilibrium problem for is to determine its equilibrium points, that is, the set The set of such solutions is denoted by .

Many problems in applied sciences such as physics, optimization, and economics reduce into finding some element of . Some methods have been proposed to solve the equilibrium problem (5); see, for instance, [26]. In particular, Combettes and Hirstoaga [7] proposed several methods for solving the equilibrium problem. On the other hand, Mann [8] and Shimoji and Takahashi [9] considered iterative schemes for finding a fixed point of a nonexpansive mapping. Further, Acedo and Xu [10] projected new iterative methods for finding a fixed point of strict pseudo-contractions.

In 2006, Marino and Xu [5] proposed a general iterative method and proved that the algorithm converged strongly. Recently, Tian [11] revealed the inner contact of Yamada’s algorithm [12] and viscosity iterative algorithm and then introduced a new general iterative algorithm combining a -Lipschitzian and -strong monotone operator. On this basis, Wang [13] considered a general composite iterative method for infinitely many strict pseudo-contractions in 2010. However, the -mapping used in Wang’s paper requires many composite operations. Very recently, He and Sun [14] proposed a new operator to replace the -mapping for infinite family nonexpansive mappings.

The mapping is defined as follows: where such that , , and are infinite nonexpansive mappings. Because it does not contain many composite computations, it is more simple and easy to realize.

In this paper, we combine the operator and the general iterative algorithm to propose a new explicit iterative scheme involving equilibrium problem (5) and an infinite family of strict pseudo-contractions. Under certain assumptions, we will prove that the sequence converges strongly. Further an example will be given to demonstrate the effectiveness of our iterative scheme and another will be given to compare numerical results and convergence rate of the algorithm in this paper and [15].

2. Preliminaries

In the sequel, we will make use of the following lemmas in a real Hilbert space .

Lemma 1. Let be a real Hilbert space. There hold the following identities:(i)(ii)

Lemma 2 (see [13]). Let be a -Lipschitzian and -strongly monotone operator on a Hilbert space with , , , and . Then is a contraction with contractive coefficient and .

Lemma 3 (see [1]). Let be a -strict pseudo-contraction. Define by for each . Then, as , is a nonexpansive mapping such that .

Lemma 4. Let be an -Lipschitz mapping with coefficient and a -Lipschitzian continuous operator and -strongly monotone operator with , . Then, for , That is, is strongly monotone with coefficient .

Proof. Since is -Lipschitz and -strongly monotone, it is easy to get

Lemma 5 (see [16]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that(i)(ii)Then, .

Let be a sequence of -strict pseudo-contractions. Define , . Then, by Lemma 3, is nonexpansive. In order to find the common fixed point set of infinite mappings, -mapping is often used; see [9, 13, 15, 17, 18] and references therein. The mapping is defined by where are real numbers such that . Such a mapping is called a -mapping generated by and . As we have seen, -mapping contains many composite computation of , and it is complicated and needs a large number of complex operations. In [14], He and Sun proposed a new hybrid steepest descent method for solving fixed point problem defined on the common fixed point set of infinite nonexpansive mappings.

Lemma 6 (see [14]). Let be a real Hilbert and all nonexpansive mappings with . Let , where such that . Then is a nonexpansive mapping with .

Lemma 7 (see [14]). Let be a real Hilbert and all nonexpansive mappings with . Let , where such that . Assume , where . Then uniformly converges to in each bounded subset in .

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions:(A1)  for all ;(A2)  is monotone; that is, for any ;(A3) for each ;(A4)  is convex and lower semicontinuous for each .

We recall some lemmas which will be needed in the rest of this paper.

Lemma 8 (see [2]). Let be a nonempty closed convex subset of , let be bifunction from to satisfying (A1)–(A4), and let and . Then there exists such that

Lemma 9 (see [7]). For , , define a mapping as follows: for all . Then, the following statements hold:(i) is single valued; (ii) is firmly nonexpansive; that is, for any , (iii); (iv) is closed and convex.

Lemma 10 (see [19]). Let and be bounded sequences in a Banach space and a sequence of real numbers such that for all Suppose that for all and . Then .

Lemma 11 (see [6]). Let , , , and be as in Lemma 9. Then the following holds: for all and .

Lemma 12 (see [13]). Let be a Hilbert space, a nonempty closed convex subset of , and a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then .

We adopt the following notations:(1) stands for the weak convergence of to ,(2) stands for the strong convergence of to .

3. Main Result

Recall that, given a nonempty closed convex subset of a real Hilbert space , for any , there exists a unique nearest point in , denoted by , such that for all . Such a is called the metric (or the nearest point) projection of onto . As we all know, if and only if there holds the following relation:

Throughout the rest of this paper, we always assume that is an -Lipschitzian mapping of into itself with coefficient and is a -Lipschitzian continuous operator and -strongly monotone on with , . Assume that and .

Define a mapping . Since both and are nonexpansive, it is easy to get that is also nonexpansive. Consider the mapping on defined by where . By Lemmas 2 and 9, we have Since , it follows that is a contraction. Therefore, by the Banach contraction principle, has a unique fixed point such that

For simplicity, we will write for provided no confusion occurs. Next we prove the sequence converges strongly to a which solves the variational inequality By the property of the projection, we can get equivalently.

Theorem 13. Let be a nonempty closed convex subset of a real Hilbert space and a bifunction from to satisfying (A1)–(A4). Let be family -strict pseudo-contractions for some . Assume the set . Let be an -Lipschitzian mapping of into itself with , and let be a -Lipschitzian continuous operator and -strongly monotone on with , ,  , and . For every , let be the mapping generated by and with according to (6). Given , let and be sequences generated by the following algorithm: If , and satisfy the following conditions:(i),  and ;(ii); (iii) and , then, converges strongly to , which solves the variational inequality (24).

Proof. The proof is divided into several steps.
Step  1. Show first that is bounded.
Taking any , by Lemma 9, we have It follows from (25) that Further we get By induction, we obtain . Hence, is bounded, so are and . It follows from the Lipschitz continuity of and that , , and are also bounded. From the nonexpansivity of , it follows that is also bounded.
Step  2. Show that Suppose , then .
Hence, we have Observe that By the definition of , we have where
It follows from (30) and (32) that where .
Hence we get . Since is convergent, it is easy to see that is also convergent. Thus we have .
From conditions (i) and (iii) and Lemma 11, we obtain By Lemma 10, we have . Thus
By Lemma 11 and (30) and (29), we obtain
Step  3. Show that where .
Observe that From condition (i) and (25), we can obtain It follows from condition (ii) that By Lemma 9, we get This implies that By nonexpansivity of , we have It follows from (25) that This implies that From conditions (i) and (ii) and (29), we have Thus, we get
On the other hand, we have Combining (47) and Lemma 7, we obtain (37).
Step  4. Show that where is a unique solution of the variational inequality (24). Indeed, take a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume . From (37), we obtain .
By the same argument as in the proof of Theorem 13, we have . Since , it follows that
Step  5. Show that
Since It follows from (29) and (51) that This implies that where , . Put , . It is easy to see that . Hence, by Lemma 5, the sequence converges strongly to .

Remark 14. If we extend the equilibrium problem to be system of equilibrium problems, we still obtain the desired result by the similar proof of Theorem 13.

4. Numerical Result

In this section, we consider the following two simple examples to demonstrate the effectiveness, realization, and convergence of the algorithm in Theorem 13. Further, we compare convergence rates of the algorithm in this paper and [15].

First, we give an example as follows.

Example 15. In Theorem 13, let , , , for all . Define , and let , . Take with Lipschitz constant and strongly monotone constant , , for all with Lipschitz coefficient . Give the parameters , for every , and fix and . Then is the sequence generated by As , we have .

Let , ; then we have . Take the initial guess , using software MATLAB R2012, we obtain the numerical experiment results in Table 1.

tab1
Table 1: .

Let be the two-dimensional Euclidean space with usual inner product and induced norm .

Next, we consider another simple example.

Example 16. In Theorem 13, let , ,  , for all . Give , and let , , . Take with Lipschitz constant and strongly monotone constant , , , for all with contraction coefficient . Give the parameters , for every , and fix and . Then is the sequence generated by As , we have .

For analysis of the rate of convergence, we use the concept introduced by Rhoades [20] as follows.

Definition 17. Let be a closed interval on the real line and a continuous function. Suppose that and are two iterations which converge to the fixed point of . Then, is said to converge faster than if

Now we turn to numerical simulation using the algorithm (57). Take the initial guess and , respectively. All the numerical experiment results are given in Tables 2(a) and 3(a). Then we realize the algorithm in [15], and the -mapping is used in the paper. Further we obtain the corresponding numerical results which can be found in Tables 2(b) and 3(b).

tab2
Table 2: (a) . (b) .
tab3
Table 3: (a) . (b) .

It is easy to see that the approximation values obtained by the algorithm (25) in this paper are more close to the common fixed point at the same iterative number. And from the computer programming point of view, the algorithm is easier to implement in this paper.

Acknowledgments

The author would like to thank the referee for valuable suggestions to improve the paper and the Fundamental Research Funds for the Central Universities (Grant ZXH2012K001).

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