Split equality monotone variational inclusions and fixed point problem of set-valued operator

Abstract

In this paper, we are concerned with the split equality problem of finding an element in the zero point set of the sum of two monotone operators and in the common fixed point set of a finite family of quasi-nonexpansive set-valued mappings. Strong convergence theorems are established under suitable condition in an infinite dimensional Hilbert spaces. Some applications of the main results are also provided.

Split equality variational inequality problems for pseudomonotone mappings in Banach spaces

“…The SEVIP is quite general and it includes as special cases, split equality zero point problem (see, [18]), common solutions of the variational inequality problem [12], common zeros of mappings [16], split equality feasibility problem [27], has been studied extensively by many authors and applied to solving many real life problems such as in modelling intensity-modulated radiation therapy treatment planning [8,9], modelling of inverse problems arising from phase retrieval, and in sensor networks in computerised tomography and data compression [5,17]. If, in (1.1), we consider H 2 = H 3 , and B = I, the identity mapping on H 2 , the SEVIP reduces to the split varitional inequality problem (SVIP) that was recently introduced by Censor et al [10].…”

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.…”