Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces

Ravi P Agarwal,Narin Petrot,Yeol Je Cho

doi:10.1186/1687-1812-2011-31

Abstract

Abstract In this paper, the existing theorems and methods for finding solutions of systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces are studied. To overcome the difficulties, due to the presence of a proper convex lower semi-continuous function, φ and a mapping g, which appeared in the considered problem, we have used some applications of the resolvent operator technique. We would like to point out that although many authors have proved results for finding solutions of the systems of nonlinear set-valued (mixed) variational inequalities problems, it is clear that it cannot be directly applied to the problems that we have considered in this paper because of φ and g. 2000 AMS Subject Classification: 47H05; 47H09; 47J25; 65J15.

Highlights

Introduction and preliminaries LetH be a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉, and ||·||, respectively
Let H be a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉, and ||·||, respectively
A set-valued mapping A : H ® 2H is said to be ν-strongly monotone if there exists a constant ν >0, such that, w1 − w2, u1 − u2 ≥ ν||u1 − u2||2, ∀u1, u2 ∈ H, w1 ∈ Au1, w2 ∈ Au2

Summary

Introduction

Introduction and preliminaries LetH be a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉, and ||·||, respectively. For each fixed positive real numbers, r and h, we consider the following so-called system of general nonlinear set-valued mixed variational inequalities problems: Find x*, y*Î H, u* Î Ay*, v* Î Bx*, such that ρu∗ + x∗ − g(y∗), g(x) − x∗ + φ(g(x)) − φ(x∗) ≥ 0, ∀x ∈ H, g(x) ∈ H, ηv∗ + y∗ − g(x∗), g(x) − y∗ + φ(g(x)) − φ(y∗) ≥ 0, ∀x ∈ H, g(x) ∈ H.

Results

Conclusion