Uniqueness properties of functionals with Lipschitzian derivative

Biagio Ricceri

Abstract: In this paper, we prove that if $X$ is a real Hilbert space and if $J:X\to\R$ is a $C^1$ functional whose derivative is Lipschitzian, with Lipschitz constant $L$, then, for every $x_0\in X$, with $J'(x_0)\neq 0$, the following alternative holds: either the functional $x\to\frac{1}{2}\|x-x_0\|^2-\frac{1}{L}J(x)$ has a global minimum in $X$, or, for every $r>J(x_0)$, there exists a unique $y_r\in J^{-1}(r)$ such that $\|x_0-y_r\|=\dist(x_0,J^{-1}(r))$ and, for every $r>0$, the restriction of the functional $J$ to the sphere $\{x\in X:\|x-x_0\|=r\}$ has a unique global maximum.

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