Strong Convergence of Iterative Algorithm for a New System of GeneralizedH·,·-η-Cocoercive Operator Inclusions in Banach Spaces

Saud M Alsulami,Eskandar Naraghirad,Nawab Hussain

doi:10.1155/2013/540108

Saud M Alsulami, Eskandar Naraghirad + Show 1 more

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https://doi.org/10.1155/2013/540108

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Abstract

We introduce and study a new system of generalizedH·,·-η-cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated withH·,·-η-cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.

Highlights

The resolvent operator technique is a powerful tool to study the approximation solvability of nonlinear variational inequalities and variational inclusions, which have been applied widely to optimization and control, mechanics and physics, economics and transportation equilibrium, and engineering sciences, see, for example, [1,2,3,4] and the references therein.In a series of papers [5,6,7,8], the authors investigated (A, η)-accretive and H(⋅, ⋅)-accretive operators for solving variational inclusions in Banach spaces
Motivated and inspired by the research works mentioned above, in this paper, we introduce and study a new system of H(⋅, ⋅) − η-cocoercive mapping inclusions in Banach spaces
Using the resolvent operator technique associated with H(⋅, ⋅) − η-cocoercive operators, we propose a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method

Summary

Introduction

The resolvent operator technique is a powerful tool to study the approximation solvability of nonlinear variational inequalities and variational inclusions, which have been applied widely to optimization and control, mechanics and physics, economics and transportation equilibrium, and engineering sciences, see, for example, [1,2,3,4] and the references therein. In a series of papers [5,6,7,8], the authors investigated (A, η)-accretive and H(⋅, ⋅)-accretive operators for solving variational inclusions in Banach spaces. Some results on H((⋅, ⋅), η)accretive operators and application for solving set-valued variational inclusions in Banach spaces have been proved in [7]. Motivated and inspired by the research works mentioned above, in this paper, we introduce and study a new system of H(⋅, ⋅) − η-cocoercive mapping inclusions in Banach spaces. Using the resolvent operator associated with H(⋅, ⋅) − η-cocoercive mapping, we suggest and analyze a new general algorithm and establish the existence and uniqueness of solutions for this system of H(⋅, ⋅) − η-cocoercive mappings

Preliminaries

Strong Convergence Theorem