Approximate Equilibrium Problems and Fixed Points

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.


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 �xed 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 �rst 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.
roughout the paper, let be a real Hilbert space; and let be a nonempty, closed, bounded, and convex subset of .
Condition 1. e following condition appears implicitly in [1]. We assume that the map satis�es the following conditions: for all ; (ii) is monotone, that is, for all ; (iii) for all , (iv) for each �xed , the function is convex and lower semicontinuous.

Condition 2.
Assume that the map for satis�es the following conditions. 1 ) for all .
( 2 ) is approximate monotone, that is, for all .
( 3 ) For all , ( 4 ) For each �xed , the function is convex and lower semicontinuous. �e�nition 1 (see [1]). We say that * is an equilibrium point of if there exists a * , such that * ≥ ∀ ; the set of such * is denoted by EP ; that is, �e�nition 2. Suppose , we say that * ∈ is an approximate equilibrium point of if there exists a * ∈ , such that * , ≥ ∈ (5) In this paper, the set of such an * ∈ is denoted by AEP( , that is, and we set

Preliminaries
In the following we will present a known lemma which is needed in the proof of some results (see [2]).
en (a) is single valued.
(b) is �rmly nonexpansive� that is, International Journal of Analysis 3 (c) Take . en (d) At last, we claim that AEP( ) is closed and convex. Indeed, since is �rmly nonexpansive, is also nonexpansive, and since the �xed-point set of a nonexpansive operator is closed and convex [3, proposition 1.5.3]. erefore follows from (b), (c).
In the following we will present a known theorem which is needed in the proof of some results (see [4]).

eorem 6. Let be a map satis�es Condition 1, and let be a nonexpentive mapping such that ( ) EP ( )
. Let { } and { } be sequences generated initially by an arbitrary element 1 and then by where { } and { } satisfy the following conditions: In the following we will present a theorem which is extended eorem 6. eorem 8. Let be a map satis�es Condition 2, and let be a map such that ( ) AEP ( ) . Let { } and { } be sequences generated initially by an arbitrary element 1 and then by where { } and { } satisfy the following conditions: en, the sequences { } and { } converge weakly to an element of ( ) AEP ( ).