Mann-Type Extragradient Methods for General Systems of Variational Inequalities with Multivalued Variational Inclusion Constraints in Banach Spaces

Lu-Chuan Ceng,Saleh A Al-Mezel,Abdul Latif

doi:10.1155/2013/616549

Abstract

We introduce Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

Highlights

Let X be a real Banach space whose dual space is denoted by X∗
We introduce Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces
Motivated and inspired by the research going on this area, we introduce Mann-type extragradient methods for finding solutions of the general system of variational inequalities (GSVI) (9) which are ones of the multivalued variational inclusion (MVVI) (15) and common fixed points of a countable family of nonexpansive mappings

Summary

Introduction

Let X be a real Banach space whose dual space is denoted by X∗. The normalized duality mapping J : X → 2X∗ is defined byJ (x) = {x∗ ∈ X∗ : ⟨x, x∗⟩ = ‖x‖2 = 󵄩󵄩󵄩󵄩x∗󵄩󵄩󵄩󵄩2} , ∀x ∈ X, (1)where ⟨⋅, ⋅⟩ denotes the generalized duality pairing.

Results

Conclusion