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 be a real, reflexive, locally uniformly convex Banach space with locally uniformly convex. Let be a maximal monotone operator and open and bounded. Assume that is pathwise connected and such that and Then If, moreover, is of type () on then may be replaced above by The significance of this result lies in the fact that it holds for multi-valued mappings which do not have to satisfy 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 ) of a completely continuous perturbation of a maximal monotone operator such that is locally monotone on

**[1]**Viorel Barbu,*Nonlinear semigroups and differential equations in Banach spaces*, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR**0390843****[2]**H. Brezis, M. G. Crandall, and A. Pazy,*Perturbations of nonlinear maximal monotone sets in Banach space*, Comm. Pure Appl. Math.**23**(1970), 123–144. MR**0257805****[3]**F. Browder,*Nonlinear operators and nonlinear equations of evolution in Banach spaces*, Proc. Symp. Pure Appl. Math.,**18**, Part 2, Providence, 1976.**[4]**Felix E. Browder,*The degree of mapping, and its generalizations*, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982), Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 15–40. MR**729503**, 10.1090/conm/021/729503**[5]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[6]**Zhengyuan Guan,*Ranges of operators of monotone type in Banach space*, J. Math. Anal. Appl.**174**(1993), no. 1, 256–264. MR**1212931**, 10.1006/jmaa.1993.1115**[7]**Zhengyuan Guan and Athanassios G. Kartsatos,*Ranges of perturbed maximal monotone and 𝑚-accretive operators in Banach spaces*, Trans. Amer. Math. Soc.**347**(1995), no. 7, 2403–2435. MR**1297527**, 10.1090/S0002-9947-1995-1297527-2**[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]**Athanassios G. Kartsatos,*On compact perturbations and compact resolvents of nonlinear 𝑚-accretive operators in Banach spaces*, Proc. Amer. Math. Soc.**119**(1993), no. 4, 1189–1199. MR**1216817**, 10.1090/S0002-9939-1993-1216817-6**[10]**Athanassios G. Kartsatos,*Sets in the ranges of sums for perturbations of nonlinear 𝑚-accretive operators in Banach spaces*, Proc. Amer. Math. Soc.**123**(1995), no. 1, 145–156. MR**1213863**, 10.1090/S0002-9939-1995-1213863-5**[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]**Athanassios G. Kartsatos,*Degree-theoretic solvability of inclusions involving perturbations of nonlinear 𝑚-accretive operators in Banach spaces*, Yokohama Math. J.**42**(1994), no. 2, 171–182. MR**1332006****[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]**Athanassios G. Kartsatos,*New results in the perturbation theory of maximal monotone and 𝑚-accretive operators in Banach spaces*, Trans. Amer. Math. Soc.**348**(1996), no. 5, 1663–1707. MR**1357397**, 10.1090/S0002-9947-96-01654-6**[15]**Jong An Park,*Invariance of domain theorem for demicontinuous mappings of type (𝑆₊)*, Bull. Korean Math. Soc.**29**(1992), no. 1, 81–87. MR**1157248****[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*, Mathematical Surveys and Monographs, vol. 23, American Mathematical Society, Providence, RI, 1986. MR**852987****[18]**Guang Hong Yang,*The ranges of nonlinear mappings of monotone type*, J. Math. Anal. Appl.**173**(1993), no. 1, 165–172. MR**1205916**, 10.1006/jmaa.1993.1059**[19]**Eberhard Zeidler,*Nonlinear functional analysis and its applications. II/B*, Springer-Verlag, New York, 1990. Nonlinear monotone operators; Translated from the German by the author and Leo F. Boron. MR**1033498**

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

Email:
hermes@gauss.math.usf.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03739-8

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