Auxiliary Principle for a System of Generalized Set-Valued Nonlinear Mixed Variational-Like Inequalities in Banach Spaces

Yangqing Qiu,Luchuan Ceng,Xiaolin Zhan

doi:10.1155/2019/5353748

Abstract

In this paper, the auxiliary principle technique is extended to study a system of generalized nonlinear mixed variational-like inequalities problem for set-valued mappings without compact values in Banach spaces with p-uniformly convex bidual spaces. First, the existence of the solutions of the related auxiliary problem is proved. Then, a new iterative algorithm based on the system of auxiliary variational inequalities is constructed. Finally, both the existence of the solutions of the original problem and the convergence of the iterative sequences generated by the algorithm are proved. And we also present a numerical example to demonstrate the result. Our results improve and extend some known results.

Highlights

We present a numerical example to demonstrate the result
It follows from Assumption 5(1) that ηi(u, V) = −ηi(V, u) and ηi(u, u) = 0, ∀u, V ∈ Ei
R be a function with E∗i 󳨀→ E∗i and Fi : Assumption 5 holds, the system of auxiliary variational inequalities problem (20) has a unique solution (x, y) ∈ E1 × E2

Summary

Introduction

In the system of the generalized set-valued nonlinear mixed variational-like inequalities problem (1), bi is a nonlinear mapping, so the projection method cannot be applied to it. Ding et al [3] extended it to study the existence and algorithm of solutions of generalized strongly nonlinear mixed variationallike inequalities in Banach spaces when N(T(⋅), A(⋅)) is a set-valued mapping. Noor [6] put forward that extending the projection methods and its variant forms for generalized set-valued mixed nonlinear variational inequalities involving the nonlinear form bi(⋅, ⋅) satisfying properties (i), (ii), and (iii) is still an open problem, and this needs further research efforts. The auxiliary principle technique is extended to study a system of generalized nonlinear mixed variationallike inequalities problem (1) for set-valued mappings without compact values in Banach spaces with p-uniformly convex bidual spaces.

Preliminaries

Auxiliary Problem and Algorithm

Existence and Convergence Theorem

Conflicts of Interest