TECHNICAL NOTE On Quasimonotone Variational Inequalities 1 D. AUSSEL 2 AND N. HADJISAVVAS 3 Communicated by S. Schaible Abstract. The purpose of this paper is to prove the existence of solutions of the Stampacchia variational inequality for a quasimonotone multivalued operator without any assumption on the existence of inner points. Moreover, the operator is not supposed to be bounded valued. The result strengthens a variety of other results in the literature. Key Words. Variational inequalities, quasimonotone operators, generalized monotonicity, existence results. 1. Introduction and Definitions Given a Banach space X with topological dual X*, a subset K of X, and a multivalued operator T: K ! 2 X  , the Stampacchia variational inequality problem is to find x 2 K such that 8y 2 K; 9x  2 TðxÞ : hx  ; y  xi 0: ð1Þ Existence of solutions of (1) under a generalized monotonicity assumption for T has been investigated intensively in recent years. In most cases, T was assumed to be pseudomonotone (in the sense of Karamardian); see e.g. Refs. 1–2. Extension of these results to the broader class of quasimonotone operators has been established also, but only at the cost of restrictive 1 This work was prepared while the second author was visiting the Mathematics Department of the University of Perpignan, Perpignan, France. The author wishes to thank the Mathematics Department for its hospitality. 2 Associate Professor, De´ partement de Mathe´ matiques, Universite´ de Perpignan, Perpignan, France. 3 Professor, Department of Product and Systems Design Engineering, University of the Aegean, Hermoupolis, Syros, Greece. JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 121, No. 2, pp. 445–450, May 2004 (Ó 2004) 445 0022-3239/04/0500-0445/0 Ó 2004 Plenum Publishing Corporation