A General System of Generalized Nonlinear Mixed Composite-Type Equilibria in Hilbert Spaces

Abstract

Very recently, Ceng and Yao [L. C. Ceng, J. C. Yao, A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem, Nonlinear Anal., 72 (2009), 1922-1937, suggested and analyzed a relaxed extragradient-like method for finding a common solution of a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem of a strict pseudocontractive mapping in a Hilbert space. In this paper, based on the authors' iterative method, we introduce a modification of the relaxed extragradient-like method for finding a common solution of a generalized mixed equilibrium problem with perturbed mapping, a general system of generalized nonlinear mixed composite-type equilibria and a fixed point problem of a strict pseudocontractive mapping in a Hilbert space, and then obtain a strong convergence theorem. Utilizing this theorem, we establish some new strong convergence results in fixed point problems, variational inequalities, mixed equilibrium problems and systems of generalized nonlinear mixed composite-type equilibria in Hilbert spaces.