Approximation solutions of the sum of monotone mapping inclusion, split equality, fixed point and variation inequality problems

Abstract

Many of the most important problems arising in nonlinear analysis reduce to solv ing inclusions of monotone mappings, split equilibrium, variational inequality or fixed point problems. Analytical methods for finding exact solutions of many non linear equations are rare or unknown. Therefore, methods of approximating the solutions of nonlinear inclusion problems are of interest where solutions are known to exist. In this thesis, we combine Douglas-Rachford and Viscosity methods and construct an iterative scheme which converges strongly to a zero of the sum of a finite fam ily of maximal monotone mappings, under suitable conditions, in the setting of Hilbert spaces. Moreover, we construct a Viscosity method to approximate a zero of the sum of maximal monotone mappings and a solution of the split equality monotone inclusion problem for the sum of two maximal monotone mappings in Hilbert spaces. We also establish forward-backward type and Halpern type al gorithms and prove strong convergent theorems to zeros of the sum of maximal monotone mappings in the setting of real reflexive Banach spaces. Furthermore, we introduce a new class of mappings called f-pseudocontractive mappings and construct an algorithm for finding common f-fixed points of those mappings and discuss its strong convergence in reflexive Banach spaces. Finally, we introduce the concept of a Bregman relatively f-nonexpansive map ping and investigate an algorithm for approximating common element of the set of solutions of variational inequality problems for Lipschitz monotone mappings and the set of f-fixed points of Bregman relatively f-nonexpansive mapping in the setting of Banach spaces.