Table of Contents Author Guidelines Submit a Manuscript
International Journal of Analysis
Volume 2013 (2013), Article ID 659493, 4 pages
http://dx.doi.org/10.1155/2013/659493
Research Article

Approximate Equilibrium Problems and Fixed Points

1Faculty of Mathematics, Yazd University, Yazd, Iran
2Faculty of Mathematics, Vali-e-Asr University, Rafsanjan, Iran

Received 16 September 2012; Revised 21 November 2012; Accepted 12 December 2012

Academic Editor: Malte Braack

Copyright © 2013 H. Mazaheri and S. A. M. Mohsenalhosseini. 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 find a common element of the set of fixed points of a map and the set of solutions of an approximate equilibrium problem in a Hilbert space. Then, we show that one of the sequences weakly converges. Also we obtain some theorems about equilibrium problems and fixed points.

1. Introduction

Equilibrium is “everywhere:” in economics, physics, engineering, chemistry, biology, and so forth. From the mathematical modelling point of view equilibrium can be described in fixed point theorems, optimisation problems, variational inequalities, complementarity problems, and so forth. Equilibrium systems can be studied from several points of view: existence of solutions; existence of nontrivial solutions; number of solutions; properties of the solution set; and the numerical approximation of solutions. In the first a short description of what is a mathematical equilibrium system in general, we will present several particular classes of approximate equilibrium systems and the relations between then.

closed, bounded, and convex subset of .

Condition 1. The following condition appears implicitly in [1].
We assume that the map satisfies the following conditions:(i); (ii); (iii), (iv), .

Condition 2. Assume that the map for satisfies the following conditions.). (). (), (), .

Definition 1 (see [1]). We say that is an equilibrium point of if there exists a , such that the set of such is denoted by ; that is,

Definition 2. Suppose , we say that is an approximate equilibrium point of if there exists a , such that In this paper, the set of such an is denoted by , that is,
and we set

2. Preliminaries

In the following we will present a known lemma which is needed in the proof of some results (see [2]).

Lemma 3. Let be a map satisfies Condition 1. Let , and let . Then there exists a , such that

Lemma 4. Let be a map satisfies Condition 2. Let , , and let . Then there exists a , such that

Proof. Consider the map by . The map satisfies Condition 1. Then by Lemma 3, there exists a , such that
Thus

Lemma 5. Let be a map satisfies Condition 2. For , , we defined , such that
Then(a) is single valued.(b) is firmly nonexpansive; that is, (c). (d) is nonempty, closed, and convex.

Proof. (a) For and , let . Then, Then Since is approximate monotone, Now, since and , we have So, we have .
(b) Now we claim that is a firmly nonexpansive. Indeed, if , Adding the two inequalities, we have With , we have Now since and , then so (c) Take . Then (d) At last, we claim that is closed and convex. Indeed, since is firmly nonexpansive, is also nonexpansive, and since the fixed-point set of a nonexpansive operator is closed and convex [3, proposition ]. Therefore follows from (b), (c).

In the following we will present a known theorem which is needed in the proof of some results (see [4]).

Theorem 6. Let be a map satisfies Condition 1, and let be a nonexpentive mapping such that . Let and be sequences generated initially by an arbitrary element and then by where and satisfy the following conditions:(i) for some ;(ii) and .
Then, the sequences and converge weakly to an element of .

Lemma 7. Let be a map by that satisfies Condition 1, and let be a nonexpentive mapping, then if and only if .

In the following we will present a theorem which is extended Theorem 6.

Theorem 8. Let be a map satisfies Condition 2, and let be a map such that . Let and be sequences generated initially by an arbitrary element and then by where and satisfy the following conditions:(i) for some ;(ii) and .Then, the sequences and converge weakly to an element of .

Proof. Suppose the nap by that satisfies Condition 1. Since by , Lemma 9 implies that and holds conditions theorem (2.8), therefore the sequences and converge weakly to an element of . By Lemma 9 the sequences and converge weakly to .

Lemma 9. Let be a map satisfies Condition 1. For , define a mapping such that is contraction. If   for all , then there exists .

Proof. Since is a nonempty closed, bounded, and convex subset of , and is continues, then by Browde’s theorem there exists a . Since  for all  , thus   for all  . So .

Lemma 10. Let be a map satisfies Condition 1. For , define a mapping such that is contraction. If ,  for all , then .

Proof. Since is a nonempty closed, bounded, and convex subset of , and is contraction, now by Theorem 2.1 of [4], since converges to fixed point , then there exists a . Since for all , , thus  for all  . So there exists a , such that .

Lemma 11. Let be a real-valued function, let be a map satisfies Condition 2, and let be a nonlinear onto mapping and satisfying If   , then .

Proof. If , then Since , there exists , such that . Therefore , and so It follows that and .

Lemma 12. Let the map satisfy Condition 2. For , define a mapping such that is contraction. If ,  for all  , then there exists a .

Proof. Since is contraction by Theorem 2.1. of [1] , , and since for all , therefore for all and for some , ,  for all   . Thus.

References

  1. L.-C. Ceng, S. Al-Homidan, Q. H. Ansari, and J.-C. Yao, “An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 967–974, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. 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 Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984. View at MathSciNet
  4. L. C. Ceng, N. C. Wong, and J. C. Yao, “Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings,” Journal of Inequalities and Applications, vol. 2006, Article ID 65983, 11 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet