Variational inequalities and fixed point theorems for PM-maps

We study maps T for which J - T is pseudo-monotone; we call such T a PM-map. This includes compact maps in suitable spaces and pseudo-contractive maps in Hilbert spaces. We study variational inequalities in a situation which was previously done only when T is compact. We show that our variational inequalities are well suited to treating existence of fixed points for generalized inward tin particular, weakly inward) PM-maps in Hilbert spaces. Our new results on existence of fixed points generalizes many earlier results obtained by using other methods. An application to homogeneous integral equations is provided. (C) 1998 Academic Press.