Weak Convergence Theorem for Finding Fixed Points and Solution of Split Feasibility and Systems of Equilibrium Problems

Abstract

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of fixed points of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge weakly to a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems under mild conditions. Our main result improves and extends the recent ones announced by Ceng et al. (2012) and many others.

Convergence Theorems for Common Solutions of Split Variational Inclusion and Systems of Equilibrium Problems

“…In the process of studying equilibrium problems and split inverse problems, not only techniques and methods for solving the respective problems have been proposed (see, for example, CQ-algorithm in Byrne [37,38], relaxed CQ-algorithm in Yang [39] and Gibali et al [40], self-adaptive algorithm in López et al [41], Moudafi and Thukur [42], and Gibali [43]), but also the common solution of equilibrium problems, split inverse problems, and other problems have been considered in many works (see, for example, Plubtieng and Sombut [44] considered the common solution of equilibrium problems and nonspreading mappings; Sobumt and Plubtieng [45] studied a common solution of equilibrium problems and split feasibility problems in Hilbert spaces; Sitthithakerngkiet et al [46] investigated a common solution of split monotone variational inclusion problems and fixed points problem of nonlinear operators; Eslamian and Fakhri [47] considered split equality monotone variational inclusion problems and fixed point problem of set-valued operators; Censor and Segal [48], Plubtieng and Sriprad [49] explored split common fixed point problems for directed operators). In particular, some applications to mathematical models for studying a common solution of convex optimizations and compressed sensing whose constraints can be presented as equilibrium problems and split variational inclusion problems, which stimulated our research on this kind of problem.…”