TAILIEUCHUNG - Existence of a solution of the quasi-variational inequality with semicontinuous operator
The paper considers quasi-variational inequalities with point to set operator. The existence of a solution, in the case when the operator of the quasi-variational inequality is semi-continuous and the feasible set is convex and compact, is proved. | Yugoslav Journal of Operations Research 16 (2006), Number 2, 147-152 EXISTENCE OF A SOLUTION OF THE QUASIVARIATIONAL INEQUALITY WITH SEMICONTINUOUS OPERATOR Djurica S. JOVANOV Faculty of Organizational Sciences University of Belgrade, Serbia Received: April 2004 / Accepted: June 2006 Abstract. The paper considers quasi-variational inequalities with point to set operator. The existence of a solution, in the case when the operator of the quasi-variational inequality is semi-continuous and the feasible set is convex and compact, is proved. Keywords: Quasi-variational inequality, existence of a solution, semi continuous operator. 1. INTRODUCTION Let X be a real Hilbert space, U ⊂ X convex closed subset of the space X , Q :U 2U point to set mapping from U to its subsets. The quasi-variational inequality QVI ( F ,U , Q) is the problem: Find u ∈ U such that there exists y ∈ F (u ) satisfying u ∈ Q(u ) and (∀v ∈ Q(u )) ≥ 0 . There are many problems which can be formulated as quasi-variational inequalities, for example: equilibrium problems in economics, impulse control problems, etc. Existence of a solution of the quasi-variational inequality is considered in [1], [3]. In this paper we prove existence of a solution of the quasi-variational inequality QVI ( F ,U , Q) with semi-continuous operator F and continuous mapping Q . If Q(u ) = U the quasi-variational inequality is variational inequality VI ( F ,U ) . In Section 2 some properties of the point to set mapping are considered. In Section 3 some existence theorems are proved. 148 Dj. Jovanov / Existence of a Solution of the Quasi-Variational Inequality 2. DEFINITIONS, NOTATIONS, PRELIMINARIES Let X be a real Hilbert space, U ⊂ X convex, closed subset of the space X , Q : U → 2U point to set mapping. Definition . We say that F : U → 2U is upper semi continuous at u0 if for any open set N such that F (u0 ) ∈ N there exists a neighborhood M of u0 such that F ( M ) ⊂ N . Definition . We say that F : U → 2U is .
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