Perturbed Iterative Approximation of Solutions for Nonlinear General -Monotone Operator Equations in Banach Spaces

Xing Wei,Heng-You Lan,Xian-Jun Zhang

doi:10.1155/2009/290713

Abstract

We introduce and study a new class of nonlinear general -monotone operator equations with multivalued operator. By using Alber's inequalities, Nalder's results, and the new proximal mapping technique, we construct some new perturbed iterative algorithms with mixed errors for solving the nonlinear general -monotone operator equations and study the approximation-solvability of the nonlinear operator equations in Banach spaces. The results presented in this paper improve and generalize the corresponding results on strongly monotone quasivariational inclusions and nonlinear implicit quasivariational inclusions.

Highlights

Let X be a real Banach space with the topological dual space of X∗, let x, y be the pairing between x ∈ X∗ and y ∈ X, let 2X∗ denote the family of all subsets of X∗, and let CB X denote the family of all nonempty closed bounded subsets of X
We note that for appropriate and suitable choices of, g, A, M, f, T, and X, it is easy to see that the problem 1.1 includes a number of quasivariational inclusions, generalized quasivariational inclusions, quasivariational inequalities, implicit quasivariational inequalities, complementarity problems, and equilibrium problems studied by many authors as special cases; see, for example, 4–7 and the references therein
Motivated and inspired by the works of Xia and Huang 2, Cui et al 1 introduced first a new class of general A-monotone operators in Banach spaces, studied some properties of general Amonotone operator, and defined a new proximal mapping associated with the general Amonotone operator

Summary

Introduction

Let X be a real Banach space with the topological dual space of X∗, let x, y be the pairing between x ∈ X∗ and y ∈ X, let 2X∗ denote the family of all subsets of X∗, and let CB X denote the family of all nonempty closed bounded subsets of X. We note that for appropriate and suitable choices of , g, A, M, f, T , and X, it is easy to see that the problem 1.1 includes a number of quasivariational inclusions, generalized quasivariational inclusions, quasivariational inequalities, implicit quasivariational inequalities, complementarity problems, and equilibrium problems studied by many authors as special cases; see, for example, 4–7 and the references therein. Motivated and inspired by the works of Xia and Huang 2 , Cui et al 1 introduced first a new class of general A-monotone operators in Banach spaces, studied some properties of general Amonotone operator, and defined a new proximal mapping associated with the general Amonotone operator. The results presented in this paper improve and extend some corresponding results in recent literature

Preliminaries

Perturbed Algorithms and Convergence