## Annals of Functional Analysis

- Ann. Funct. Anal.
- Volume 10, Number 2 (2019), 170-179.

### Surjectivity of coercive gradient operators in Hilbert space and nonlinear spectral theory

#### Abstract

We consider continuous gradient operators $F$ acting in a real Hilbert space $H$, and we study their surjectivity under the basic assumption that the corresponding functional $\langle F\left(x\right),x\rangle $—where $\langle \cdot \rangle $ is the scalar product in $H$—is coercive. While this condition is sufficient in the case of a linear operator (where one in fact deals with a bounded self-adjoint operator), in the general case we supplement it with a compactness condition involving the number $\omega \left(F\right)$ introduced by Furi, Martelli, and Vignoli, whose positivity indeed guarantees that $F$ is proper on closed bounded sets of $H$. We then use Ekeland’s variational principle to reach the desired conclusion. In the second part of this article, we apply the surjectivity result to give a perspective on the spectrum of these kinds of operators—ones not considered by Feng or the above authors—when they are further assumed to be sublinear and positively homogeneous.

#### Article information

**Source**

Ann. Funct. Anal., Volume 10, Number 2 (2019), 170-179.

**Dates**

Received: 17 November 2017

Accepted: 29 January 2018

First available in Project Euclid: 14 July 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1531533617

**Digital Object Identifier**

doi:10.1215/20088752-2018-0003

**Mathematical Reviews number (MathSciNet)**

MR3941379

**Zentralblatt MATH identifier**

07083886

**Subjects**

Primary: 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05]

Secondary: 47H05: Monotone operators and generalizations 47H08: Measures of noncompactness and condensing mappings, K-set contractions, etc.

**Keywords**

boundedly invertible operator measure of noncompactness Ekeland’s variational principle positively homogeneous operator

#### Citation

Chiappinelli, Raffaele. Surjectivity of coercive gradient operators in Hilbert space and nonlinear spectral theory. Ann. Funct. Anal. 10 (2019), no. 2, 170--179. doi:10.1215/20088752-2018-0003. https://projecteuclid.org/euclid.afa/1531533617