Strong Convergence Theorems by an Extragradient Method for Solving Variational Inequalities and Equilibrium Problems in a Hilbert Space

In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for monotone, Lipschitz-continuous mappings. The iterative process is based on the so-called extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parametric controlling conditions. This main theorem extends a recent result of Yao, Liou and Yao [Y. Yao, Y. C. Liou and J.-C. Yao, ''An Extragradient Method for Fixed Point Problems and Variational Inequality Problems,'' Journal of Inequalities and Applications Volume 2007, Article ID 38752, 12 pages doi:10.1155/2007/38752] and many others.

Anahtar Kelimeler:

Nonexpansive mapping; Equilibrium problem; Fixed point; Lipschitz-continuous mappings; Variational inequality; Extragradient method

Strong Convergence Theorems by an Extragradient Method for Solving Variational Inequalities and Equilibrium Problems in a Hilbert Space

Keywords:

Nonexpansive mapping; Equilibrium problem; Fixed point; Lipschitz-continuous mappings; Variational inequality; Extragradient method,

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Poom KUMAM Received 14.04.2008
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, (KMUTT) Bang Mod, Bangkok 10140, THAILAND e—mail: poom.kum@kmutt.ac.th