Uniqueness properties of functionals with Lipschitzian derivative
type: Item
instance of: scholarly article
title: Uniqueness properties of functionals with Lipschitzian derivative
zbMATH Open document ID: 1131.49005
author: Biagio Ricceri
published in: Portugaliae Mathematica. Nova Série
publication date: 2007-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0504019
full work available at URL: https://eudml.org/doc/53795
review text: Summary: We prove that if \(X\) is a real Hilbert space and if \(J:X\to\mathbb R\) 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 \(x\to\frac12\|x-x_0\|^2- \frac1L J(x)\) 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 \(\|x_0-y_r\|= \text{dist}(x_0,J^{-1}(r))\) 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.
Mathematics Subject Classification ID: 49J27
zbMATH DE Number: 5187462
zbMATH Keywords: global minima and maxima uniqueness
zbMATH Keywords: Lipschitz differentiable functionals
MaRDI profile type: Publication
arXiv ID: math/0504019
description: scientific article