Hybrid projected subgradient-proximal algorithms for solving split equilibrium problems and split common fixed point problems of nonexpansive mappings in Hilbert spaces

Abstract

Abstract In this paper, we propose two strongly convergent algorithms which combines diagonal subgradient method, projection method and proximal method to solve split equilibrium problems and split common fixed point problems of nonexpansive mappings in a real Hilbert space: fixed point set constrained split equilibrium problems (FPSCSEPs) in real Hilbert spaces. The computations of first algorthim requires prior knowledge of operator norm. To estimate the norm of an operator is not always easy, and if it is not easy to estimate the norm of an operator, we purpose another iterative algorithm with a way of selecting the step-sizes such that the implementation of the algorithm does not need any prior information as regards the operator norm. The strong convergence properties of the algorithms are established under mild assumptions on equilibrium bifunctions. We also report some applications and numerical results to compare and illustrate the convergence of the proposed algorithms.

Keywords

• nonexpansive mappings
• common fixed point problem
• equilibrium problem
• split equilibrium problem
• monotone bifunction
• pseudomonotone bifunction