A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem

Abstract

Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space $X^{⁎}$. Let $T:X\supseteq D(T)\to {\mathrm{2}}^{X^{⁎}}$ be maximal monotone, $S:X\to {\mathrm{2}}^{X^{⁎}}$ be bounded and of type $(S_{+}),$ and $C:D(C)\to X^{⁎}$ be compact with $D(T)\subseteq D(C)$ such that $C$ lies in ${\mathrm{\Gamma }}_{\sigma }^{\tau }$ (i.e., there exist $\sigma \ge \mathrm{0}$ and $\tau \ge \mathrm{0}$ such that $‖Cx‖\le \tau ‖x‖+\sigma$ for all $x\in D(C)$). A new topological degree theory is developed for operators of the type $T+S+C$. The theory is essential because no degree theory and/or existence result is available to address solvability of operator inclusions involving operators of the type $T+S+C$, where $C$ is not defined everywhere. Consequently, new existence theorems are provided. The existence theorem due to Asfaw and Kartsatos is improved. The theory is applied to prove existence of weak solution (s) for a nonlinear parabolic problem in appropriate Sobolev spaces.