Existence and iterative approximation of solutions of some systems of variational inequalities and inclusions

Abstract

AbstractIn 1968, Brézis [Ann. Inst. Fourier (Grenoble), 18(1) (1968) 115‐175] initiated the study of the existence theory of a class of variational inequalities later known as variational inclusions, using proximal‐point mappings due to Moreau [Bull. Soc. Math. France, 93 (1965) 273‐299]. Variational inclusions include variational, quasi‐variational, variational‐like inequalities as special cases.In 1985, Pang [Math. Prog. 31 (1985) 206‐219] showed that a variety of equilibrium models can be uniformly modelled as a variational inequality defined on the product sets equivalent to a system of variational inequalities and discuss the convergence of method of decomposition for system of variational inequalities.Motivated by the recent research work in this directions, we consider some systems of variational (‐like) inequalities and inclusions; develop the iterative algorithms for finding the approximate solutions and discuss their convergence criteria. Further, we study the sensitivity analysis of solution of the system of variational inclusions. The techniques and results presented here improve the corresponding techniques and results for the variational inequalities and inclusions in the literature. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)