A new topological degree theory for densely defined quasibounded (S˜+)-perturbations of multivalued maximal monotone operators in reflexive Banach spaces

Abstract

Let X be an infinite-dimensional real reflexive Banach space with dual space X∗ and G⊂X open and bounded. Assume that X and X∗ are locally uniformly convex. Let T:X⊃D(T)→2X∗ be maximal monotone and C:X⊃D(C)→X∗ quasibounded and of type (S˜+). Assume that L⊂D(C), where L is a dense subspace of X, and 0∈T(0). A new topological degree theory is introduced for the sum T+C. Browder's degree theory has thus been extended to densely defined perturbations of maximal monotone operators while results of Browder and Hess have been extended to various classes of single-valued densely defined generalized pseudomonotone perturbations C. Although the main results are of theoretical nature, possible applications of the new degree theory are given for several other theoretical problems in nonlinear functional analysis.