Relaxed and composite viscosity methods for variational inequalities, fixed points of nonexpansive mappings and zeros of accretive operators

Lu-Chuan Ceng,Qamrul Hasan Ansari,Abdullah Al-Otaibi,Abdul Latif

doi:10.1186/1687-1812-2014-29

Abstract

AbstractIn this paper, we present relaxed and composite viscosity methods for computing a common solution of a general systems of variational inequalities, common fixed points of infinitely many nonexpansive mappings and zeros of accretive operators in real smooth and uniformly convex Banach spaces. The relaxed and composite viscosity methods are based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for relaxed and composite viscosity algorithms not only in the setting of a uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results given in the literature.

Highlights

The theory of variational inequalities is well established and a tool to solve many problems arising from science, engineering, social sciences, etc., see, for example, [ – ] and the references therein
We mainly propose two different methods, namely, relaxed viscosity method and composite viscosity method, to find a common fixed point of an infinite family of nonexpansive mappings, a system of variational inequalities and zero of an accretive operator in the setting of a uniformly convex and -uniformly smooth Banach spaces
Motivated and inspired by the research going on in this area, we introduce some relaxed and composite viscosity methods for finding a zero of an accretive operator A ⊂ X × X such that D(A) ⊂ C ⊂ r> R(I + rA), solving system of two variational inequalities (SVI) ( . ) and the common fixed point problem of an infinite family {Tn} of nonexpansive self-mappings on C

Summary

Introduction

The theory of variational inequalities is well established and a tool to solve many problems arising from science, engineering, social sciences, etc., see, for example, [ – ] and the references therein. We mainly propose two different methods, namely, relaxed viscosity method and composite viscosity method, to find a common fixed point of an infinite family of nonexpansive mappings, a system of variational inequalities and zero of an accretive operator in the setting of a uniformly convex and -uniformly smooth Banach spaces These methods are based on Korpelevich’s extragradient method, viscosity approximation method and Mann iteration method. ] Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X and for each i = , , Bi : C → X be an αi-inverse strongly accretive mapping. ] Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X and ΠC be a sunny nonexpansive retraction from X onto C.

Jrn Gxn

Jrn xn

Wn is the

Note that