CONVERGENCE THEOREMS FOR GENERALIZED EQUILIBRIUM PROBLEMS AND ASYMPTOTICALLY κ-STRICT PSEUDO-CONTRACTIONS IN HILBERT SPACES

Abstract

Abstract. In this paper, we introduce an iterative scheme for ndinga common element of the set of solutions of a generalized equilibriumproblem and the set of common xed points of a nite family of asymp-totically k-strict pseudo-contractions in Hilbert spaces. Weak and strongconvergence theorems are established for the iterative scheme. 1. IntroductionLet Hbe a real Hilbert space with inner product h;iand induced norm kk.Let Cbe a nonempty closed convex subset of H. Assume that a bifunctionF: C C!Rsatis es the following conditions:(A1) F(x;x) = 0;8x2C;(A2)Fis monotone, i.e., F(x;y) + F(y;x) 0;8x;y2C;(A3)lim t#0 F(tz+ (1 t)x;y) F(x;y);8x;y;z2C;(A4) for each x2C;y7!F(x;y) is convex and lower semicontinuous.Let A: C !H be a nonlinear mapping. Then, we consider the followinggeneralized equilibrium problem(GEP) which is to nd z2Csuch thatGEP:F(z;y) + hAz;y zi0;8y2C: (1:1)In the case of A0, this problem (1.1) reduces to the equilibrium problem(EP),which is to nd z2Csuch thatEP: F(z;y) 0;8y2C: (1:2)In the case of F 0, this problem (1.1) reduces to the variational inequalityproblem(VIP), which is to nd z2Csuch thatVIP: hAz;y zi0;8y2C: (1:3)