Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point P
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Research Article Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping Atid Kangtunyakarn Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Correspondence should be addressed to Atid Kangtunyakarn, [email protected] Received 8 November 2010; Accepted 14 December 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 Atid Kangtunyakarn. 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. The purpose of this paper is to prove the strong convergence theorem for finding a common element of the set of fixed point problems of strictly pseudocontractive mapping in Hilbert spaces and two sets of generalized equilibrium problems by using the hybrid method.
1. Introduction Let C be a closed convex subset of a real Hilbert space H, and let F : C × C → R be a bifunction. Recall that the equilibrium problem for a bifunction F is to find x ∈ C such that F x, y ≥ 0,
∀y ∈ C.
1.1
The set of solutions of 1.1 is denoted by EPF. Given a mapping T : C → H, let Fx, y T x, y − x for all x, y ∈ C. Then, z ∈ EPF if and only if T z, y − z ≥ 0 for all y ∈ C; that is, z is a solution of the variational inequality. Let A : C → H be a nonlinear mapping. The variational inequality problem is to find a u ∈ C such that v − u, Au ≥ 0
1.2
2
Fixed Point Theory and Applications
for all v ∈ C. The set of solutions of the variational inequality is denoted by VIC, A. Now, we consider the following generalized equilibrium problem: Find z ∈ C such that F z, y Az, y − z ≥ 0,
∀y ∈ C.
1.3
The set of z ∈ C is denoted by EPF, A, that is, EPF, A z ∈ C : F z, y Az, y − z ≥ 0, ∀y ∈ C .
1.4
In the case of A ≡ 0, EPF, A is denoted by EPF. In the case of F ≡ 0, EPF, A is also denoted by VIC, A. Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics are reduced to find a solution of 1.3; see, for instance, 1–3. A mapping A of C into H is called inverse strongly monotone mapping, see 4, if there exists a positive real number α such that 2 x − y, Ax − Ay ≥ αAx − Ay
1.5
for all x, y ∈ C. The following definition is well known. Definition 1.1. A mapping T : C → C is said to be a κ-strict pseudocontraction if there exists κ ∈ 0, 1 such that T x − T y2 ≤ x − y2 κI − T x − I − T y2 ,
∀x, y ∈ C.
1.6
A mapping T is called nonexpansive if T x − T y ≤ x − y
1.7
for all x, y ∈ C. We know that κ-strict pseudocontraction includes a class of nonexpansive mappings. If κ 1, T is said to be a pseudocontractive mapping. T is strong pseudocontraction if there exists a positive constant λ ∈ 0, 1
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