Variational Inequalities and Fixed Point Theorems for PM-MapsU

We study maps T for which JyT 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 in 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