About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 351372, 7 pages
http://dx.doi.org/10.1155/2014/351372
Research Article

Approximations for Equilibrium Problems and Nonexpansive Semigroups

Department of Mathematics, Beijing University of Technology, Beijing 100124, China

Received 29 November 2013; Accepted 4 February 2014; Published 16 March 2014

Academic Editor: Hassen Aydi

Copyright © 2014 Huan-chun Wu and Cao-zong Cheng. 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 method for finding a common element of the set of solutions of an equilibrium problem and the set of all common fixed points of a nonexpansive semigroup and prove the strong convergence theorem in Hilbert spaces. Our result extends the recent result of Zegeye and Shahzad (2013). In the last part of the paper, by the way, we point out that there is a slight flaw in the proof of the main result in Shehu's paper (2012) and perfect the proof.

1. Introduction

Let be a real Hilbert space, and let be a nonempty closed convex subset of . A mapping is called nonexpansive if for all . We denote the set of fixed points of by . It is known that is closed and convex. A family of mappings from into itself is called a nonexpansive semigroup on if it satisfies the following conditions:(i) for all ;(ii) for all ;(iii) for all and ;(iv)for all is continuous.We denote by the set of all common fixed points of ; that is, . It is clear that is a closed convex subset.

The equilibrium problem for is to find such that for all . The set of such solutions is denoted by . Numerous problems in physics, optimization, and economics can be reduced to find a solution of the equilibrium problem (for instance, see [1]).

For solving equilibrium problem, we 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 .

Several methods have been proposed to solve the equilibrium problem; see [17]. For finding common fixed points of a nonexpansive semigroup, Nakajo and Takahashi [8] introduced a convergent sequence for nonexpansive semigroup as follows:

Some authors have paid more attention to find an element . Buong and Duong [9] constructed the following iterative sequence and proved the weak convergence theorem for an equilibrium problem and a nonexpansive semigroup in Hilbert spaces:

In 2012, Shehu [10] studied iterative methods for fixed point problem, variational inequality, and generalized mixed equilibrium problem and introduced a new algorithm which does not involve the CQ algorithm and viscosity approximation method. However, we discover that there is a slight flaw in the proof of Theorem 3.1 in [10].

Motivated by Nakajo and Takahashi [8], Buong and Duong [9], and especially Shehu [10] and Zegeye and Shahzad [11], we present a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of all common fixed points of a nonexpansive semigroup and prove the strong convergence theorem in Hilbert spaces. Our result extends the recent result of [11]. In the last part of the paper, we perfect and simplify the proof of Theorem 3.1 in [10].

2. Preliminaries

Throughout this paper, let be a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . We write to indicate that the sequence converges strongly to . Similarly, will symbolize weak convergence. It is well known that satisfies Opial’s condition; that is, for any sequence with , we have For any , there exists a unique point such that is called the metric projection of onto . We know that is a nonexpansive mapping of onto and satisfies For and , we have

The following lemmas will be used in the proof of our results.

Lemma 1 (see [1]). Let be a nonempty closed convex subset of , and let be a bifunction from to satisfying (A1)–(A4). If and , then there exists such that

Lemma 2 (see [2]). For , define a mapping   as follows: Then the following hold:(i) is single valued;(ii) is firmly nonexpansive; that is, for any , ;(iii);(iv) is closed and convex.

Lemma 3 (see [12]). Suppose that (A1)–(A4) hold. If and , then

Lemma 4 (see [13]). Let be a nonempty bounded closed subset of , and let be a nonexpansive semigroup on . Then, for every ,

Lemma 5 (see [14]). 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 (see [15]). Let be a sequence of nonnegative real numbers satisfying , where(i), ;(ii).Then, .

3. Strong Convergence Theorems

In this section, we introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of all common fixed points of a nonexpansive semigroup and prove the strong convergence theorem in Hilbert spaces.

Theorem 7. Let be a nonempty closed convex subset of a real Hilbert space , and let be a bifunction from to satisfying (A1)–(A4). Suppose that is a nonexpansive semigroup on such that . For , let , , and be generated by where the real sequences , in and satisfy the following conditions:(1) and ;(2);(3);(4), and .Then, the sequence converges strongly to .

Proof. Note that the set is closed and convex since and are closed and convex. For simplicity, we write .
From Lemmas 1 and 2, we have , and, for any , Observe that It follows that From a simple inductive process, one has which yields that is bounded. So are and .
Set . For any , we have It follows from Lemma 3 that Hence, This implies that It follows from Lemma 5 that . Thus,
For any , we have Thus From the convexity of , it follows that Hence Since and , one has Observe that As , the following equality holds:
Now we show that In fact, we have Notice that For any , let . It is easy to see that is a bounded closed convex subset and is a subset of . Since the sequence is contained in . It follows from Lemma 4 that From (29), (30), and (32), the expression (28) is obtained.
Next we prove that where . In order to show this inequality, we can choose a subsequence of such that Due to the boundedness of , there exists a subsequence of such that . Without loss of generality, we assume that . From (27), we see that . Since and is closed and convex, we get .
We first show that . By , we have It follows from the monotonicity of that Replacing by , we obtain From (25), (27), and (A4), we have For , , set . We have and . Hence Dividing by , we see that Letting and from (A3), we get That is, .
Second, we prove that . Note that the equality (27) implies that . Suppose for contradiction that ; that is, Then from Opial’s condition and (28), we obtain which is a contradiction. Therefore, . Consequently, one gets .
From (34) and the property of metric projection, we have The inequality (33) arrives.
Finally we show that . From (11), we have It follows from (33) and Lemma 6 that converges strongly to .

Remark 8. Let and . Setting , we see that satisfies (A1)–(A4). For , let Thus, it follows that is a nonexpansive semigroup such that . If we take then all assumptions and conditions in Theorem 7 are satisfied.

Remark 9. Taking in Theorem 7, we obtain the iterative method for minimum-norm solution of an equilibrium problem and a nonexpansive semigroup.
As a direct consequence of Theorem 7, we obtain the following corollary.

Corollary 10. Let be a nonempty closed convex subset of a real Hilbert space , and assume that is a nonexpansive semigroup on such that . Let and be real sequences in , and let and be generated by Suppose that the following conditions are satisfied:(1) and ;(2);(3), , and .Then the sequence converges strongly to .

Proof. Letting for all , , and in Theorem 7, we get the result.

Remark 11. Corollary 10 extends the recent results of Zegeye and Shahzad [11, Corollaries 3.2 and 3.3] from finite family of nonexpansive mappings to a nonexpansive semigroup.

4. A Note on Shehu’s Paper “Iterative Method for Fixed Point Problem, Variational Inequality and Generalized Mixed Equilibrium Problems with Applications”

In 2012, Shehu [10] studied iterative methods for fixed point problem, variational inequality, and generalized mixed equilibrium problem and gave an interesting convergence theorem. However, there is a slight flaw in the proof of the main result (Theorem 3.1 in [10]).

Shehu obtained the following result (for more details, see [10]).

Theorem 12 (see [10]). Let be a closed convex subset of a real Hilbert space , let be a bifunction from satisfying (A1)–(A4), let be a proper lower semicontinuous and convex function with assumption (B1) or (B2), let A be a -Lipschitzian, relaxed -cocoercive mapping of into , and let be an -inverse, strongly monotone mapping of into . Suppose that is a nonexpansive mapping of into itself such that . Let and be two real sequences in and . Let , , and be generated by , Suppose that the following conditions are satisfied:(a) and ;(b);(c);(d).Then, the sequence converges strongly to an element of .

This theorem is proved in [10] by the following steps.

Step 1. The sequence is bounded.

Step 2. The following equalities hold:

Step 3. If is a weak limit of which is a subsequence of , then .

Step 4. The sequence converges strongly to .

In Step 4, in order to show that the sequence converges strongly to , the author shows the inequality by defining a mapping as follows: , where is a Banach limit. It is proved that the set and . An element of is taken arbitrarily and is denoted by . Of course, the element is not necessarily the weak sequential cluster point of . However, in Step 3, the symbol stands for the weak limit of which is a subsequence of . In the sequel, the author obtains The meaning of the element in (51) is ambiguous. It is difficult to ensure consistency.

Now, we perfect and simplify the proof of Step 4. According to the equality in Step 2, , for all , we see that the set contains only one element. Since is a relaxed -cocoercive mapping of into , that is, there exist such that it follows that the mapping is one-to-one. Therefore, the set is a singleton. By Step 3, the sequence possesses only one weak sequential cluster point. It follows from Lemma 2.38 in [16] that converges weakly to and so Since converges weakly to , it follows from Lemma 2.2 in [10] or Lemma 6 in this paper that converges strongly to .

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgment

The authors would like to thank referees and editors for their valuable comments and suggestions.

References

  1. E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1-4, pp. 123–145, 1994. View at Zentralblatt MATH · View at MathSciNet
  2. P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005. View at Zentralblatt MATH · View at MathSciNet
  3. Y. Liu, “A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4852–4861, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. D.-N. Qu and C.-Z. Cheng, “A strong convergence theorem on solving common solutions for generalized equilibrium problems and fixed-point problems in Banach space,” Fixed Point Theory and Applications, vol. 2011, article 17, 2011. View at Zentralblatt MATH · View at MathSciNet
  6. R. D. Chen, X. L. Shen, and S. J. Cui, “Weak and strong convergence theorems for equilibrium problems and countable strict pseudocontractions mappings in Hilbert space,” Journal of Inequalities and Applications, vol. 2010, Article ID 474813, 11 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. C. Jaiboon and P. Kumam, “Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities,” Journal of Inequalities and Applications, vol. 2010, Article ID 728028, 43 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. N. Buong and N. D. Duong, “Weak convergence theorem for an equilibrium problem and a nonexpansive semigroup in Hilbert spaces,” International Mathematical Forum, vol. 5, no. 53–56, pp. 2775–2786, 2010. View at Zentralblatt MATH · View at MathSciNet
  10. Y. Shehu, “Iterative method for fixed point problem, variational inequality and generalized mixed equilibrium problems with applications,” Journal of Global Optimization, vol. 52, no. 1, pp. 57–77, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. Zegeye and N. Shahzad, “Approximation of the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2013, article 1, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. F. Cianciaruso, G. Marino, and L. Muglia, “Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 146, no. 2, pp. 491–509, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, vol. 2005, Article ID 685918, 2005. View at Zentralblatt MATH · View at MathSciNet
  15. H.-K. Xu, “Another control condition in an iterative method for nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 65, no. 1, pp. 109–113, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, NY, USA, 2011. View at Publisher · View at Google Scholar · View at MathSciNet