A system of nonlinear set valued variational inclusions.

Yong-Kun Tang,Salahuddin Salahuddin,Shih-Sen Chang

doi:10.1186/2193-1801-3-318

Abstract

In this paper, we studied the existence theorems and techniques for finding the solutions of a system of nonlinear set valued variational inclusions in Hilbert spaces. To overcome the difficulties, due to the presence of a proper convex lower semicontinuous function ϕ and a mapping g which appeared in the considered problems, we have used the resolvent operator technique to suggest an iterative algorithm to compute approximate solutions of the system of nonlinear set valued variational inclusions. The convergence of the iterative sequences generated by algorithm is also proved.AMS Mathematics subject classification49J40; 47H06

Highlights

It is well known that variational inequality theory and complementarity problems are very powerful tools of current mathematical technology
The classical variational inequality and complementarity problems have been extended and generalized to study a large variety of problems arising in economics, control problems, contact problems, mechanics, transportation, equilibrium problems, optimization theory, nonlinear programming, transportation equilibrium and engineering sciences, see (Aubin 1982; Baiocchi and Capelo 1984; Chang 1984; Giannessi and Maugeri 1995)
Inspired and motivated by the research work going on this field, in this works, the methods for finding the common solutions of a system of nonlinear set valued variational inclusions involving different nonlinear operators and fixed point problem are considered and studied, via proximal method in the framework of Hilbert spaces

Summary

Introduction

It is well known that variational inequality theory and complementarity problems are very powerful tools of current mathematical technology. Let Ni : H × H → H be a nonlinear function, gi : K → H be a nonlinear operator, Ai, Bi : K → CB(H) be the nonlinear set valued mappings and let ri be a fixed positive real number for each i = 1, 2, 3.

Results

Conclusion