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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Athanassios G. Kartsatos
Journal: Proc. Amer. Math. Soc. 125 (1997), 1469-1478
MSC (1991): Primary 47H17; Secondary 47H05, 47H10
MathSciNet review: 1371130
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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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Additional Information

Athanassios G. Kartsatos
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
Email: hermes@gauss.math.usf.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03739-8
PII: S 0002-9939(97)03739-8
Keywords: Maximal monotone operator, pathwise connected set, invariance of domain, compact perturbation, existence of zeros
Received by editor(s): September 12, 1995
Received by editor(s) in revised form: November 27, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society