Pergamon Nonlinear Analysis, Theory, Methods & Applmtions, Vol. 26, No. 9, pp. 1573-1603, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 %15.OO+O.a) 0362-546X(94)00364-5 RECESSION MAPPINGS AND NONCOERCIVE VARIATIONAL INEQUALITIES SAMIR ADLY, DANIEL GOELEVEN? and MICHEL THBRA LACO, URA 1586, UniversitC de Limoges, F-87060 Limoges Cedex, France (Received 1 September 1994; received for publication 29 December 1994) Key words andphruses: Maximal monotone operator, subdifferential operator, recession function, non- coercive variational inequality, Navier-Stokes equations, puffed-up membrane, elastic plate, unilateral buckling. 1. INTRODUCTION Throughout the paper, unless specifically stated, X denotes a real reflexive Banach space, X* the topological dual of X and (. , *) the associated duality pairing. We will write “s”, “-” and 6,&P, to denote respectively, the strong (norm) convergence, the weak convergence and the convergeice with respect to a topology T. This study is concerned with the variational inequality ?‘./.I.@, A, f, a, K) : find u E K n D(L) such that (Lu + Au - f, u - u> + Q(v) - ‘B(u) 2 0, VU E K. (1.1) We suppose that the assumptions (H) described below are satisfied: (I) K is a nonempty closed and convex subset of X such that 0 E K; (II) L : D(L) c X -+ X* is a closed linear monotone operator, with domain D(L) dense in X; (III) f E x*; (IV) a,: X -+ IR is a convex and continuous function such that Q(O) = 0; (V) A : X + X* is pseudomonotone and bounded. Recall that A : X --t X* is pseudomonotone iff the following hold true. For each sequence (u, 1n E NJ such that then, u -u n and lim sup (Au,, u, - u) 5 0, Tl*+CC (Au, u - u) 5 lim inf (Au,, u, - v), VVEX. PI*+00 The theory of variational inequalities introduced by Stampacchia and Fichera in potential theory and mechanics and developed by the French and Italian schools, has emerged as an inter- esting branch of applied mathematics. This theory is a very useful and effective tool for study- ing a wide class of linear and nonlinear problems in a unified natural and general framework. Several theoretical results are well known when a coerciveness property holds for A (see for instance, the contributions by BrCzis [l], Browder [2-41 and Mosco [5]). However, the variational formulation of many engineering problems leads, generally, to variational inequalities which t Present address: FacuItCs Universitaires Notre Dame de la Paix, B-5000 Namur, Belgique. 1573