Some results concerning the solution mappings of mixed variational inequality problems

Qi-Qing Song,Hui Yang

doi:10.1186/s13660-016-1176-z

Abstract

This paper shows some continuities of mappings between the space of mixed variational inequality problems and the graph space of their solution mappings. The space of mixed variational inequality problems is homeomorphic to the graph of a continuous mapping. These generalize the results in the corresponding references.

Highlights

Introduction and preliminariesVariational inequalities have become important methods to analyze many linear and nonlinear problems; see [ ] and [ ], such as linear complementarity problems in [ ], convex optimization in [ ], imaging problems in [ ], etc.Classical variational inequalities were introduced by Hartman and Stampacchia in the s; see [ ] and [ ]
The spaces consisting of nonlinear problems may have some interesting topological properties: some nonlinear problem spaces are homeomorphic to the graph spaces of their solution mappings, such as bimatrix games in [ ], normal game problems in [, ], game trees in [ ], classical variational inequality problems [ ], etc
Duvaut and Lions considered a kind of mixed variational inequality, see [ ], which added a function to a classical variational inequality

Summary

Introduction

Introduction and preliminariesVariational inequalities have become important methods to analyze many linear and nonlinear problems; see [ ] and [ ], such as linear complementarity problems in [ ], convex optimization in [ ], imaging problems in [ ], etc.Classical variational inequalities were introduced by Hartman and Stampacchia in the s; see [ ] and [ ]. Let K be a compact convex subset of Rn. We consider the following mixed variational inequality problem: to find a point u∗ ∈ K such that u∗ satisfies θ (u) – θ u∗ + u – u∗ T f u∗ ≥ , ∀u ∈ K , ( ) Denote by M the set as follows: θ : K → R is continuous and convex; M = (θ , f ): f : K → Rn is continuous and monotone .

Results

Conclusion