## Annals of Functional Analysis

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

Raffaele Chiappinelli

#### 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 $\langle F(x),x\rangle$—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 $\omega (F)$ 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

#### 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

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