Annals of Functional Analysis

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

Raffaele Chiappinelli

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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 F(x),x—where 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 ω(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

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

Received: 17 November 2017
Accepted: 29 January 2018
First available in Project Euclid: 14 July 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.

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


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.

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