Let $\mathbb H$ a Hilbert space and $N:\mathbb H\to \mathbb H$ a continuous nonlinear mapping. In Fonda and Mawhin (Iterative and variational methods for the solvability of some semilinear equations in Hilbert spaces) they estimate from below and from above the quantity $(Nu_1-Nu_2,u_1-u_2)$. Then they say that from that estimation and "classical results" we have that $N$ is a gradient operator: there is $\eta:\mathbb H\to \mathbb R$ such that $N=\nabla \eta$.

I would like to know how to prove that assertion or where I can find those classical results.

  • $\begingroup$ In general the estimate from below means that $N$ is monotone and monotone operators are not necessarily gradients. The paper of Fonda and Mawhin assumes much more than that: $Nu(t,x)=V'(t,x,u(t,x))$ (whatever it means) and it is not possible to answer our question without reading their paper. $\endgroup$ Apr 16 '18 at 2:08

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