Nonlinear Analysis 37 (1999) 933 – 951 The study of maximal elements, xed points for L S -majorized mappings and their applications to minimax and variational inequalities in product topological spaces 1 Paul Deguire a , Kok Keong Tan b , George Xian-Zhi Yuan c;* a Department of Mathematics and Statistics, The University of Moncton, Moncton, New Brunswick, Canada E1A 3E9 b Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5 c Department of Mathematics, The University of Queensland, Brisbane 4072 QLD, Australia Received 7 March 1996; accepted 24 October 1997 Dedicated to Professor Ky Fan Keywords: LS mapping; LS -majorized function; Maximal element; Fan–Browder xed point; Minimax inequality and variational inequality 1. Introduction Recently, by establishing the existence of selection functions for set-valued mappings with open bers in product spaces, Deguire and Lassonde [10] gave some xed-point theorems in product spaces for both compact and non-compact domains. It is well- known that each xed-point theorem has an equivalent version of a maximal element (we recall that a point x ∈ X is said to be a maximal element of a mapping F : X → 2 Y if F (x)= ∅, where X and Y are both topological spaces). The existence of maximal elements for mappings in topological (vector) spaces and its important applications to mathematical economies have been studied by many authors in both mathematics and economies, for example; see [1, 2, 10, 11, 26, 27, 29–31]. In this paper, it is our purpose, * Corresponding author. E-mail: xzy@maths.uq.edu.au. 1 This project is supported in part by Australian Research Council. 0362-546X/99/$ – see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S0362-546X(98)00084-4