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

Anteneh Getachew Gebrie,Rabian Wangkeeree

doi:10.1186/s13663-018-0630-7

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.

Highlights

1 Introduction In 1994 Censor and Elfving [1] introduced a notion of the split feasibility problem, which is to find an element of a closed convex subset of the Euclidean space whose image under a linear operator is an element of another closed convex subset of a Euclidean space
In 2009 Censor and Segal [2] introduced the split common fixed point problem (SCFPP) where split feasibility problem becomes a special case of SCFPP
Dinh, Son, Jiao and Kim [16] proposed the linesearch algorithm which combines the extragradient method incorporated with the Armijo linesearch rule for solving the problem (FPSCSEP) in real Hilbert spaces under the assumptions that the first bifunction is pseudomonotone with respect to its solution set, the second bifunction is monotone, and fixed point mappings are nonexpansive

Summary

Introduction

In 1994 Censor and Elfving [1] introduced a notion of the split feasibility problem, which is to find an element of a closed convex subset of the Euclidean space whose image under a linear operator is an element of another closed convex subset of a Euclidean space. They introduced the iterative scheme which converges strongly to a common solution of the split equilibrium problem, the variational inequality problem and the fixed point problem for a nonexpansive mapping. Dinh, Son, and Anh [15] considered the following fixed point set-constrained split equilibrium problems (FPSCSEPs): C such that. Dinh, Son, Jiao and Kim [16] proposed the linesearch algorithm which combines the extragradient method incorporated with the Armijo linesearch rule for solving the problem (FPSCSEP) in real Hilbert spaces under the assumptions that the first bifunction is pseudomonotone with respect to its solution set, the second bifunction is monotone, and fixed point mappings are nonexpansive. In the last section we will see applications supported by an example and numerical results

Preliminary

Modified projected subgradient-proximal algorithm

Conclusion