On Generalized Mixed Equilibrium Problems and Fixed Point Problems with Applications

In this paper, we introduce and investigate two new generalized mixed equilibrium problems and explore the relationship between them and the properties of their solutions in Banach spaces. Based on the generalized f -projection, we construct hybrid algorithms to find common fixed points of a countable family of quasi-φnonexpansive mappings in Banach spaces, a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points for quasi-φ-nonexpansive mappings and, further, prove some strong convergence theorems for these hybrid algorithms under some suitable assumptions. As some applications of the main results, the strong convergence theorems for the general H-monotone mappings and equilibrium problems are also proven. Key–Words: Generalized mixed equilibrium problem, Quasi-φ-nonexpansive mapping, Strong convergence theorem, Fixed point, Generalized f -projection.