Degree theories and invariance of domain for perturbed maximal monotone operators in Banach spaces

Abstract

Let $X$ be a real reflexive Banach space with dual $X^*.$ Let $T:X\supset D(T)\to 2^{X^*}$ be maximal monotone, and $C:X\supset D(C)\to X^*.$ A theory of domain invariance is developed, in which various conditions are given for a nonlinear operator of the type $T+C:D(T)\cap D(C)\to 2^{X^*}$ to map a given relatively open set onto an open set. The well-known invariance of domain theorem of Schauder about injective operators of the type $I+C,$ with $C$ compact, is extended to operators $T+C.$ Here, $T$ is a possibly densely defined operator with compact resolvents and $C$ is continuous and bounded, or $T$ is just maximal monotone and $C$ compact. The case of completely continuous resolvents of $T$ and demicontinuous operators $C$ is also covered. Another invariance of domain result is given for demicontinuous, bounded, and $(S_+)$-perturbations $C.$ This result makes use of the topological degrees of Browder and Skrypnik. Finally, three invariance of domain theorems are given for the case where $T$ is single-valued and both operators $T,~C$ are densely defined. These results make use of the topological degree theory that was recently developed by the authors for the sum $T+C,$ where $C$ satisfies conditions like quasiboundedness and $(S_+)$ with respect to $T.$ Applications to elliptic and parabolic problems are included.