Implicit Iteration Scheme with Perturbed Mapping for Equilibrium Problems and Fixed Point Problems of Finitely Many Nonexpansive Mappings.

We introduce an implicit iteration scheme with a perturbed mapping for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space. Then, we establish some convergence theorems for this implicit iteration scheme which are connected with results by Xu and Ori (Numer. Funct. Analysis Optim. 22:767–772, ), Zeng and Yao (Nonlinear Analysis, Theory, Methods Appl. 64:2507–2515, ) and Takahashi and Takahashi (J. Math. Analysis Appl. 331:506–515, ). In particular, necessary and sufficient conditions for strong convergence of this implicit iteration scheme are obtained. [ABSTRACT FROM AUTHOR]