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The problem of minimizing locally a $ C^2$ functional around non-critical points is well-posed


Author: Biagio Ricceri
Journal: Proc. Amer. Math. Soc. 135 (2007), 2187-2191
MSC (2000): Primary 49K40, 90C26, 90C30; Secondary 49J35
DOI: https://doi.org/10.1090/S0002-9939-07-08789-8
Published electronically: March 1, 2007
MathSciNet review: 2299496
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove the following general result: Let $ X$ be a real Hilbert space and $ J:X\to {\bf R}$ a $ C^1$ functional, with locally Lipschitzian derivative.

Then, for each $ x_0\in X$ with $ J'(x_0)\neq 0$, there exists $ \delta>0$ such that, for every $ r\in ]0,\delta[$, the restriction of $ J$ to the sphere $ \{x\in X : \Vert x-x_0\Vert=r\}$ has a unique global minimum toward which every minimizing sequence strongly converges.


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Additional Information

Biagio Ricceri
Affiliation: Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy
Email: ricceri@dmi.unict.it

DOI: https://doi.org/10.1090/S0002-9939-07-08789-8
Keywords: Minimization, well-posedness, Hilbert spaces, non-critical points, locally Lipschitzian derivative, saddle points.
Received by editor(s): March 22, 2006
Published electronically: March 1, 2007
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.