A variational inequality index for condensing maps in Hilbert spaces and applications to semilinear elliptic inequalities

摘要

A variational inequality index for γ-condensing maps is established in Hilbert spaces. New results on existence of nonzero positive solutions of variational inequalities for such maps are proved by using the theory of variational inequality index. Applications of such a theory are given to existence of nonzero positive weak solutions for semilinear second order elliptic inequalities, where previous results of variational inequalities for S-contractive maps cannot be applied.