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

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.