Lipschitzian solutions of the implicit Cauchy problem \(g(x')=f(t,x)\), \(x(0)=0\), with \(f\) discontinuous in \(x\).
type: Item
instance of: scholarly article
title: Lipschitzian solutions of the implicit Cauchy problem \(g(x')=f(t,x)\), \(x(0)=0\), with \(f\) discontinuous in \(x\).
zbMATH Open document ID: 0578.34002
DOI: 10.1007/BF02844891
author: Biagio Ricceri
publication date: 1985-01-01T00:00:00Z
review text: Let a be a positive real number, n a positive integer, \(S=[0,a]\times {\mathbb{R}}^ n\). Denote by \({\mathcal I}\) the family of all subsets of S for which at least one of the projections on the axes is a set of Lebesgue measure zero. Moreover, let \({\mathcal G}\) be the family of all subsets Y of \({\mathbb{R}}^ n\) such that, for each \(i=1,2,...,n\), the supremum and the infimum of the projection of \(\overline{conv}(Y)\) on the ith axis are both positive or both negative. Our main result is the following theorem. Let Y be a non-empty arcwise connected subset of \({\mathbb{R}}^ n\) belonging to \({\mathcal G}\); g a continuous real function on Y; f a real function on S; A a subset of S belonging to \({\mathcal F}\). Assume that: (i) for each \(r\in {\mathbb{R}}\), the boundary of \(f^{-1}(r)\) in S belongs to \({\mathcal F}\); (ii) the restriction of f to \(S\setminus A\) is continuous and bounded and the closure in \({\mathbb{R}}\) of its range is contained in the range of g. Under such hypotheses, there exist a set \(E\subseteq [0,a]\), of measure zero, and a Lipschitzian function h: [0,a]\(\to {\mathbb{R}}^ n\) such that \(g(h'(t))=f(t,h(t))\) a.e. in [0,a], \(h(0)=0\), and the restriction of h' to [0,a]\(\setminus E\) is continuous. Several counterexamples show that none of the hypotheses of the theorem above can be dropped. In particular, assumption (i) cannot be replaced with the more general one: ''for each \(r\in {\mathbb{R}}\), the boundary of \(f^{-1}(r)\) in S is a set of Lebesgue measure zero in \({\mathbb{R}}^{n+1}\)''. To see this, an example is presented.
Mathematics Subject Classification ID: 34A99
zbMATH DE Number: 3925294
zbMATH Keywords: Lipschitzian solutions
zbMATH Keywords: implicit Cauchy problem
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02844891
OpenAlex ID: W2052597193
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 3925294