Hybrid methods for accretive variational inequalities involving pseudocontractions in Banach spaces

Rudong Chen,Yaqin Wang

doi:10.1186/1687-1812-2011-63

Abstract

We use strongly pseudocontractions to regularize a class of accretive variational inequalities in Banach spaces, where the accretive operators are complements of pseudocontractions and the solutions are sought in the set of fixed points of another pseudocontraction. In this paper, we consider an implicit scheme that can be used to find a solution of a class of accretive variational inequalities.Our results improve and generalize some recent results of Yao et al. (Fixed Point Theory Appl, doi:10.1155/2011/180534, 2011) and Lu et al. (Nonlinear Anal, 71(3-4), 1032-1041, 2009).2000 Mathematics subject classification 47H05; 47H09; 65J15

Highlights

Throughout this paper, we always assume that E is a real Banach space, 〈·, ·〉 is the dual pair between E and E*, and 2E denotes the family of all the nonempty subsets of E
Let f : C ® C be a Lipschitz strongly pseudocontraction, S : C ® C be a Lipschitz pseudocontraction, and T : C ® C be a continuous pseudocontraction with Fix(T) ≠ ∅
Since f is strongly pseudocontractive, it is easy to see that the solution of the variational inequality (3.2) is unique

Summary

Introduction

Throughout this paper, we always assume that E is a real Banach space, 〈· , ·〉 is the dual pair between E and E*, and 2E denotes the family of all the nonempty subsets of E. In [2], Lu et al considered the following type of monotone variational inequality problem in Hilbert spaces(denoted by VI(1.2)) We consider the following variational inequality problem in Banach spaces (denoted by VI(1.3))

Results

Conclusion