An invariance of domain result for multi-valued maximal monotone operators whose domains do not necessarily contain any open sets

Kartsatos, Athanassios

doi:10.1090/S0002-9939-97-03739-8

An invariance of domain result for multi-valued maximal monotone operators whose domains do not necessarily contain any open sets
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by Athanassios G. Kartsatos PDF

Proc. Amer. Math. Soc. 125 (1997), 1469-1478 Request permission

Abstract:

Let $X$ be a real, reflexive, locally uniformly convex Banach space with $X^{*}$ locally uniformly convex. Let $T:X\supset D(T)\to 2^{X^{*}}$ be a maximal monotone operator and $G\subset X$ open and bounded. Assume that $M\subset X^{*}$ is pathwise connected and such that $T(D(T)\cap G)\cap M \not = \emptyset$ and $\overline {T(D(T)\cap \partial G)}\cap M = \emptyset .$ Then $M\subset T(D(T)\cap G).$ If, moreover, $T$ is of type ($S$) on $\partial G,$ then $\overline {T(D(T)\cap \partial G)}$ may be replaced above by $T(D(T)\cap \partial G).$ The significance of this result lies in the fact that it holds for multi-valued mappings $T$ which do not have to satisfy $\text {int}D(T) \not = \emptyset .$ It has also been used in this paper in order to establish a general “invariance of domain” result for maximal monotone operators, and may be applied to a greater variety of problems involving partial differential equations. No degree theory has been used. In addition to the above, necessary and sufficient conditions are given for the existence of a zero (in an open and bounded set $G$) of a completely continuous perturbation $T+C$ of a maximal monotone operator $T$ such that $T+C$ is locally monotone on $G.$

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