Cent. Eur. J. Math. • 6(4) • 2008 • 526-536 DOI: 10.2478/s11533-008-0049-1 Central European Journal of Mathematics Generalized variational-like inequalities for pseudo-monotone type III operators Research Article Mohammad S.R. Chowdhury 1* , Kok-Keong Tan 2 1 Department of Mathematics, Lahore University of Management Sciences (LUMS), Phase II, Opposite Sector U, DHA, Lahore Cantt., Lahore - 54792, Pakistan 2 Department of Mathematics & Stattistics, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 3J5 Received 3 April 2008; accepted 31 July 2008 Abstract: Our aim in this paper is mainly to prove some existence results for solutions of generalized variational-like inequal- ities with (η )-pseudo-monotone type III operators defined on non-compact sets in topological vector spaces. Keywords: generalized variational-like inequalities • 0-diagonally concave relation • pseudo-monotone type III operators © Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction Browder [4] and Hartman and Stampacchia [20] first introduced variational inequalities and Minty [26, 27] first introduced the theory of monotone nonlinear operators. Since then, there have been many generalizations on the theory of monotone nonlinear operators as well as in variational inequality problems. In 1996, Chowdhury and Tan first obtained generalized variational inequalities for pseudo-monotone type I operators [8]. Moreover, Chowdhury and Tan’s (1997) results on generalized variational inequalities for quasi-monotone operators were obtained in [9]. The recent results of Chowdhury and Tarafdar on generalized variational inequalities for pseudo-monotone type III operators were established in [10]. There are wide applications of variational inequalities in nonlinear elliptic boundary value problems, obstacle problems, complementarity problems, mathematical programming, mathematical economics and in many other areas. There are too many citations of papers concerning variational inequalities. Some references are [20], [23] and [31] where citations of many further references can be found. The topic on monotone variational inequalities is discussed in [35, 36]. In [22], Karamardian showed that the complementarity problems can be reduced to the variational inequality problems. In [25], Mancino and Stampacchia showed the relationship between mathematical programming and the variational inequalities. * E-mail: msrchowdhury@yahoo.com.au 526