Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $G.$

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47H17, 47H05, 47H10

Retrieve articles in all journals with MSC (1991): 47H17, 47H05, 47H10

Additional Information

Athanassios G. Kartsatos
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

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