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. 2011
Received: 8 November 2010
Accepted: 14 December 2010
Published: 20 December 2010
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.
In the case of , is denoted by . In the case of , is also denoted by . 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].
The class of -strict pseudocontractions falls into the one between classes of nonexpansive mappings, and the pseudocontraction mappings, and the class of strong pseudocontraction mappings is independent of the class of -strict pseudocontraction.
converges weakly to a fixed point of provided the control sequence satisfies the conditions that for all and . In 1974, S. Ishikawa proved the following strong convergence theorem of pseudocontractive mapping.
Theorem 1.2 (see ).
In order to prove a strong convergence theorem of Mann algorithm (1.12) associated with strictly pseudocontractive mapping, in 2006, Marino and Xu  proved the following theorem for strict pseudocontractive mapping in Hilbert space by using method.
Theorem 1.3 (see ).
Many authors study the problem for finding a common element of the set of fixed point problem and the set of equilibrium problem in Hilbert spaces, for instance, [2, 3, 11–15]. The motivation of (1.14), (1.15), and the research in this direction, we prove the strong convergence theorem for finding solution of the set of fixed points of strictly pseudocontractive mapping and two sets of generalized equilibrium problems by using the hybrid method.
Lemma 2.1 (see ).
The following lemma is well known.
Lemma 2.3 ((demiclosedness principle) (see ).
The following lemma appears implicitly in .
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Lemma 2.7 (see ).
3. Main Result
By using our main result, we have the following results in Hilbert spaces.
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