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.