## Advances in Differential Equations

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

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

Athanassios G. Kartsatos and Igor V. Skrypnik

#### 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