A variational inequality theory in reflexive smooth Banach spaces and applications to pp-Laplacian elliptic inequalities

We develop a theory of variational inequality for demicontinuous SS-contractive maps in reflexive smooth Banach spaces. Variational inequality theorems are proved and applied to study existence of nonzero positive weak solutions for pp-Laplacian elliptic inequalities and a population model of one species arising in mathematical biology.