General Nonlinear Random Equations with Random Multivalued Operator in Banach Spaces

Heng-You Lan,Wei Xie,Yeolje Cho

doi:10.1155/2009/865093

Abstract

We introduce and study a new class of general nonlinear random multivalued operator equations involving generalized -accretive mappings in Banach spaces. By using the Chang's lemma and the resolvent operator technique for generalized -accretive mapping due to Huang and Fang (2001), we also prove the existence theorems of the solution and convergence theorems of the generalized random iterative procedures with errors for this nonlinear random multivalued operator equations in -uniformly smooth Banach spaces. The results presented in this paper improve and generalize some known corresponding results in iterature.

Highlights

Introduction and PreliminariesThe variational principle has been one of the major branches of mathematical sciences for more than two centuries
The variational principles have played a fundamental role in the development of the general theory of relativity, gauge field theory in modern particle physics and soliton theory
These principles have been enriched by the discovery of the variational inequality theory, which is mainly due to Hartman and Stampacchia 1

Summary

Introduction and Preliminaries

The variational principle has been one of the major branches of mathematical sciences for more than two centuries. Huang et al 6, 7 , Huang 8 , Jin and Liu 9 and Lan et al 11 introduced and studied some new classes of nonlinear variational inclusions involving generalized m-accretive mappings in Banach spaces. Inspired and motivated by recent works in these fields see 3, 11, 12, 16, 25– 28 , in this paper, we introduce and study a new class of general nonlinear random multivalued operator equations involving generalized m-accretive mappings in Banach spaces. By using the Chang’s lemma and the resolvent operator technique for generalized m-accretive mapping due to Huang and Fang 15 , we prove the existence theorems of the solution and convergence theorems of the generalized random iterative procedures with errors for this nonlinear random multivalued operator equations in q-uniformly smooth Banach spaces. The problem 1.2 is called the general nonlinear random equation with multivalued operator involving generalized m-accretive mapping in Banach spaces.

Random Iterative Algorithms

Existence and Convergence Theorems