Advances in Differential Equations

Degree theories and invariance of domain for perturbed maximal monotone operators in Banach spaces

Athanassios G. Kartsatos and Igor V. Skrypnik

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. Differential Equations Volume 12, Number 11 (2007), 1275-1320.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867415

Mathematical Reviews number (MathSciNet)
MR2372240

Zentralblatt MATH identifier
1160.47044

Subjects
Primary: 47H11: Degree theory [See also 55M25, 58C30]
Secondary: 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H14: Perturbations of nonlinear operators [See also 47A55, 58J37, 70H09, 70K60, 81Q15]

Citation

Kartsatos, Athanassios G.; Skrypnik, Igor V. Degree theories and invariance of domain for perturbed maximal monotone operators in Banach spaces. Adv. Differential Equations 12 (2007), no. 11, 1275--1320. https://projecteuclid.org/euclid.ade/1355867415.


Export citation