Communications in Contemporary Mathematics, Vol. 4, No. 1 (2002) 145–160 c  World Scientific Publishing Company STABILITY OF THE SOLUTION SET OF NON-COERCIVE VARIATIONAL INEQUALITIES SAMIR ADLY * and MICHEL TH ´ ERA † LACO, Universit´ e de Limoges, 123 Avenue A. Thomas, 87060 Limoges Cedex, France * adly@unilim.fr † michel.thera@unilim.fr EMIL ERNST Laboratoire de Modelisation en M´ ecanique et Thermodynamique (LMMT), Casse 322, Facult´ e de Sciences et Techniques de Saint J´ erome, Avenue Escadrille Normandie-Niemen 13397 Marseille Cedex 20, France Emil.Ernst@VMESA12.U-3MRS.fr Received 5 December 2000 Revised 15 May 2001 In this paper, we study the stability of the solution set of a non-coercive variational inequality with respect to small perturbations of the data involved in the problem. This research is done using well-known tools of convex analysis and the concept of well- positioned convex sets (which is defined and studied). Keywords : Convex analysis; barrier cone; support functional; recession analysis; semi- coercive functional. Mathematics Subject Classification 2000: 47H05, 52A41, 39B82 1. Introduction In this paper we are concerned with the variational inequality VI (A, f, Φ,K) : Find u ∈ K ∩ dom Φ such that 〈Au − f,v − u〉 + Φ(v) − Φ(u) ≥ 0 , ∀ v ∈ K, (1.1) where K is a closed convex set in a reflexive Banach space X , f is an element in X ∗ , the continuous dual of X , A is an operator from X to X ∗ ,Φ: X → R ∪{+∞} is a lower semi-continuous convex function that we assume to be bounded from below, and K ∩ dom Φ = ∅, where dom Φ := {x ∈ X ‖Φ(x) < +∞}. We denote by 〈·, ·〉 the duality pairing between X and X ∗ and by ‖·‖ and ‖·‖ ∗ the norm and the dual norm on X and X ∗ , respectively. The interplay of the different assumptions on A, K and Φ for the existence of a solution of the variational inequality VI (A, f, Φ,K) has been extensively studied in 145