PORTUGALIAE MATHEMATICA Vol. 63, No. 4, pp. 393400 (2006) 

Uniqueness properties of functionals with Lipschitzian derivativeBiagio RicceriDepartment of Mathematics, University of Catania,Viale A. Doria 6, 95125 Catania  ITALY Email: ricceri@dmi.unict.it 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 $x\to\frac{1}{2}\xx_0\^2\frac{1}{L}J(x)$ 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 $\x_0y_r\=\dist(x_0,J^{1}(r))$ and, for every $r>0$, the restriction of the functional $J$ to the sphere $\{x\in X:\xx_0\=r\}$ has a unique global maximum. Full text of the article:
Electronic version published on: 7 Mar 2008.
© 2006 Sociedade Portuguesa de Matemática
