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