Variational relation problems: existence of solutions and fixed points of contraction mappings

Abdul Latif,Dinh The Luc

doi:10.1186/1687-1812-2013-315

Abstract

We propose a new approach to study variational relation problems. Namely, we apply Mizoguchi and Takahashi’s fixed point theorem of contraction mappings and an error bound of a system of linear inequalities to establish existence conditions for a variational relation problem in which the variational relation linearly depends on the decision variable. Then we develop an algorithm to compute a solution of a linear variational relation problem. MSC:49J52, 47H10.

Highlights

( ) xis a fixed point of S, that is, x ∈ S(x), ( ) R(x, y) holds for every y ∈ T(x), where X and Y are nonempty sets, S is a set-valued mapping from X to itself, T is a setvalued mapping from X to Y, and R(x, y) is a relation linking x ∈ X and y ∈ Y
Existence conditions of solutions to variational relation problems were analyzed in great generality, the stability of solutions of a parametric variational relation was studied with respect to the continuity of set-valued mappings, and very recently a numerical method was developed to solve variational relation problems when the data are linear [ ]
As far as we know, all conditions established for existence of solutions of variational relation problems in the above cited papers utilize intersection theorems or fixed point theorems involving the KKM property of set-valued mappings in one or another form [ ]

Summary

Introduction

We consider the following variational relation problem: find x ∈ X such that ( ) xis a fixed point of S, that is, x ∈ S(x), ( ) R(x, y) holds for every y ∈ T(x), where X and Y are nonempty sets, S is a set-valued mapping from X to itself, T is a setvalued mapping from X to Y , and R(x, y) is a relation linking x ∈ X and y ∈ Y. Given a real-valued function φ on X × Y , a variational relation can be defined by any of the following equality and inequalities: φ(x, y) = , φ(x, y) = , φ(x, y) > or φ(x, y) ≥ .

Results

Conclusion