On multivalued nonlinear variational inclusion problems with (A,η)-accretive mappings in Banach spaces

Heng-You Lan

doi:10.1155/jia/2006/59836

Abstract

Based on the notion of -accretive mappings and the resolvent operators associated with -accretive mappings due to Lan et al., we study a new class of multivalued nonlinear variational inclusion problems with -accretive mappings in Banach spaces and construct some new iterative algorithms to approximate the solutions of the nonlinear variational inclusion problems involving -accretive mappings. We also prove the existence of solutions and the convergence of the sequences generated by the algorithms in -uniformly smooth Banach spaces.

Highlights

In order to study extensively variational inequalities and variational inclusions, which are providing mathematical models to some problems arising in economics, mechanics, and engineering science, Ding [1], Huang and Fang [10], Fang and Huang [3], Verma [14, 15], Fang and Huang [4, 5], Huang and Fang [9], Fang et al [2] have introduced the concepts of η-subdifferential operators, maximal η-monotone operators, generalized monotone operators, A-monotone operators, (H,η)-monotone operators in Hilbert spaces, H-accretive operators, generalized maccretive mappings and (H,η)-accretive operators in Banach spaces, and their resolvent operators, respectively
The iterative algorithms for the variational inclusions with H-accretive operators can be found in the paper [6]
Lan et al [12] introduced and studied some new iterative algorithms for solving a class of nonlinear variational inequalities with multivalued mappings in Hilbert spaces, and gave some convergence analysis of iterative sequences generated by the algorithms

Summary

Introduction

In order to study extensively variational inequalities and variational inclusions, which are providing mathematical models to some problems arising in economics, mechanics, and engineering science, Ding [1], Huang and Fang [10], Fang and Huang [3], Verma [14, 15], Fang and Huang [4, 5], Huang and Fang [9], Fang et al [2] have introduced the concepts of η-subdifferential operators, maximal η-monotone operators, generalized monotone operators (named H-monotone operators), A-monotone operators, (H,η)-monotone operators in Hilbert spaces, H-accretive operators, generalized maccretive mappings and (H,η)-accretive operators in Banach spaces, and their resolvent operators, respectively. Lan et al [11] introduced a new concept of (A,η)-accretive mappings, which generalizes the existing monotone or accretive operators, studied some properties of (A,η)-accretive mappings, and defined resolvent operators associated with (A,η)-accretive mappings. Lan et al [12] introduced and studied some new iterative algorithms for solving a class of nonlinear variational inequalities with multivalued mappings in Hilbert spaces, and gave some convergence analysis of iterative sequences generated by the algorithms. Motivated and inspired by the above works, the purpose of this paper is to introduce the notion of (A,η)-accretive mappings and the resolvent operators associated with (A,η)-accretive mappings due to Lan et al, to study a new class of multivalued nonlinear variational inclusion problems with (A,η)-accretive mappings in Banach spaces, and to construct some new iterative algorithms to approximate the solutions of the nonlinear variational inclusion problems involving (A,η)-accretive mappings. We prove the existence of solutions and the convergence of the sequences generated by the algorithms in q-uniformly smooth Banach spaces

Preliminaries

Iterative algorithms and convergence