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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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  • [1] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff Int. Publ., Leyden (The Netherlands), 1975. MR 52:11666
  • [2] H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl Math. 23 (1970), 123-144. MR 41:2454
  • [3] F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure Appl. Math., 18, Part 2, Providence, 1976.
  • [4] F. Browder, Degree of mapping and its generalizations, Contemp. Math. 21 (1983), 15-40. MR 85e:47086
  • [5] J. Dugundji, Topology, Allyn & Bacon, Boston, 1970. MR 33:1824
  • [6] Z. Guan, Ranges of operators of monotone type in Banach spaces, J. Math. Anal. Appl. 174 (1993), 256-264. MR 95b:47068
  • [7] Z. Guan and A. G. Kartsatos, Ranges of perturbed maximal monotone and $m$-accretive operators in Banach spaces, Trans. Amer. Math. Soc. 347 (1995), 2403-2435. MR 95i:47096
  • [8] A. G. Kartsatos, Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, 1992, vol. III, Walter De Gruyter, New York, 1995, pp. (2197-2222). CMP 96:12
  • [9] A. G. Kartsatos, On compact perturbations and compact resolvents of nonlinear m-accretive operators in Banach spaces, Proc. Amer. Math. Soc. 119 (1993), 1189-1199. MR 94c:47076
  • [10] A. G. Kartsatos, Sets in the ranges of sums for perturbations of nonlinear $m$-accretive operators in Banach spaces, Proc. Amer. Math. Soc. 123 (1995), 145-156. MR 95c:47072
  • [11] A. G. Kartsatos, Sets in the ranges of nonlinear accretive operators in Banach spaces, Studia Math. 114 (1995), 261-273. CMP 95:14
  • [12] A. G. Kartsatos, Degree theoretic solvability of inclusions involving perturbations of nonlinear $m$-accretive operators in Banach spaces, Yokohama Math. J. 42 (1994), 171-181. MR 96c:47098
  • [13] A. G. Kartsatos, On the connection between the existence of zeros and the asymptotic behaviour of resolvents of maximal monotone operators in reflexive Banach spaces, Trans. Amer. Math. Soc. (to appear).
  • [14] A. G. Kartsatos, New results in the perturbation theory of maximal monotone and $m$-accretive operators, Trans. Amer. Math. Soc. 348 (1996), 1663-1707. MR 96j:47050
  • [15] J. A. Park, Invariance of domain theorem for demicontinuous mappings of type (S$_{+}$), Bull. Korean Math. Soc. 29 (1992), 81-87. MR 93d:47134
  • [16] D. Pascali and S. Sburlan, Nonlinear mappings of monotone type, Sijthoff and Noordhoff Intern. Publ., Bucure\c{s}ti, Romania, 1978.
  • [17] E. H. Rothe, Introduction to Various Aspects of Degree Theory in Banach Spaces, Math. Surveys and Monographs, No. 23, A.M.S., Providence, 1986. MR 87m:47145
  • [18] G. H. Yang, The ranges of nonlinear mappings of monotone type, J. Math. Anal. Appl. 173 (1993), 165-172. MR 94b:47069
  • [19] E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B, Springer-Verlag, New York, 1990. MR 91b:47002

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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

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