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2007 Degree theories and invariance of domain for perturbed maximal monotone operators in Banach spaces
Athanassios G. Kartsatos, Igor V. Skrypnik
Adv. Differential Equations 12(11): 1275-1320 (2007).

Abstract

Let $X$ be a real reflexive Banach space with dual $X^*.$ Let $T:X\supset D(T)\to 2^{X^*}$ be maximal monotone, and $C:X\supset D(C)\to X^*.$ A theory of domain invariance is developed, in which various conditions are given for a nonlinear operator of the type $T+C:D(T)\cap D(C)\to 2^{X^*}$ to map a given relatively open set onto an open set. The well-known invariance of domain theorem of Schauder about injective operators of the type $I+C,$ with $C$ compact, is extended to operators $T+C.$ Here, $T$ is a possibly densely defined operator with compact resolvents and $C$ is continuous and bounded, or $T$ is just maximal monotone and $C$ compact. The case of completely continuous resolvents of $T$ and demicontinuous operators $C$ is also covered. Another invariance of domain result is given for demicontinuous, bounded, and $(S_+)$-perturbations $C.$ This result makes use of the topological degrees of Browder and Skrypnik. Finally, three invariance of domain theorems are given for the case where $T$ is single-valued and both operators $T,~C$ are densely defined. These results make use of the topological degree theory that was recently developed by the authors for the sum $T+C,$ where $C$ satisfies conditions like quasiboundedness and $(S_+)$ with respect to $T.$ Applications to elliptic and parabolic problems are included.

Citation

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Athanassios G. Kartsatos. Igor V. Skrypnik. "Degree theories and invariance of domain for perturbed maximal monotone operators in Banach spaces." Adv. Differential Equations 12 (11) 1275 - 1320, 2007.

Information

Published: 2007
First available in Project Euclid: 18 December 2012

zbMATH: 1160.47044
MathSciNet: MR2372240

Subjects:
Primary: 47H11
Secondary: 47H07, 47H14

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.12 • No. 11 • 2007
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