A remark on the nonlinear complementarity problem

Abstract

It is shown that previous sufficient conditions for the solution of the nonlinear complementarity problem can be considerably weakened using a result of Rockafellar. Let E be a Banach space and K a closed convex cone in E. The polar K? of K is the set Ex* e E*lsup x*(K) O whenever x* e T(x), y* e T(y) and x, y E D(T) = ixlT(x) ;01. We say that T is a-monotone if there exists a strictly increasing function a: [0, -) -0, ) with a(O) = 0 and a(r) -, as r such that (x y xy*) > lix y c a(llx yfl) whenever x* e T(x) and y* e T(y) for each x, y e D(T) . If a(r) = kr for k > O, then T is said to be strongly monotone. If T is single valued and continuous from line segments in D(T) to E* with the weak* topology, then T is said to be hemicontinuous. We consider the following version (to multivalued maps) of the generalized complementarity problem (GCP) as formulated by Karamardian [ 2]: Let T: E 2E; find x e K and x* e T(x) satisfying x* E-K' and (x, x ) = 0. The (GCP) is important in that it is the form for many problems in mathematical programming, game theory, economics, etc. For this we refer the reader to [ 2] and [ 4] and the references given there. It is our purpose in this note to prove the following theorem which has as a corollary an extension of a recent theorem of Bazaraa, Goode, and Received by the editors September 5, 1973 and, in revised form, October 9, 1973 and January 7, 1974. AMS (MOS) subject classifications (1970)Y Primary 46N05, 47H05, 47H15; Secondary 90C30.