A new topological degree theory for densely defined quasibounded $({\widetilde{S}}_{+})$-perturbations of multivalued maximal monotone operators in reflexive Banach spaces

Abstract

Let $X$ be an infinite-dimensional real reflexive Banach space with dual space $X^{\ast }$ and $G\subset X$ open and bounded. Assume that $X$ and $X^{\ast }$ are locally uniformly convex. Let $T:X\supset D\left(T\right)\to 2^{X^{\ast }}$ be maximal monotone and $C:X\supset D\left(C\right)\to X^{\ast }$ quasibounded and of type $\left({\tilde{S}}_{+}\right)$. Assume that $L\subset D\left(C\right)$, where $L$ is a dense subspace of $X$, and $0\in T\left(0\right)$. 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.