Ordinary differential equations with a continuous right-hand side in Fréchet spaces
type: Item
instance of: scholarly article
title: Ordinary differential equations with a continuous right-hand side in Fréchet spaces
zbMATH Open document ID: 0803.34061
DOI: 10.1007/BF01210224
author: S. G. Lobanov
published in: Mathematical Notes
publication date: 1994-07-13T00:00:00Z
review text: Considering the differential equation (1) \(\dot x= f(t,x)\) with a continuous righthand side \(f\in C(\mathbb{R}\times E,E)\), \(E\) a nonnormed Fréchet space the author shows in particular, that for arbitrary nonnormed Fréchet spaces not even the theorem of Peano and Kneser are satisfied in the weakened formulation: uniqueness of solutions does not necessarily imply continuity in the dependence of solutions on the initial data; an example similar to the second of \textit{Dieudonné's} example [Acta Sci. Math., Szeged 12 B, L. Fejér et F. Riesz LXX annos natis dedic., 38-40 (1950; Zbl 0037.060)]\ is possible iff there exists a bounded nonprecompact set in \(E\). If \(S_ t(f,(t_ 0,x_ 0))= \{\varphi(t)\), \(\varphi\) solution of (1) with the initial condition \(\varphi(t_ 0)= x_ 0\}\), the basic results are given by the following two theorems: Theorem 1. For an arbitrary nonnormed Fréchet space \(E\) and an arbitrary positive integer \(n\) mappings of \(f_ 0\), \(f_ n\), \(f_ \infty\),\(f\in C(\mathbb{R}\times E,E)\) may be found such that (1) \(S_ t(f_ 0(t_ 0,x_ 0))\) is an empty set for all \((t,t_ 0,x_ 0)\in \mathbb{R}\times E\times E\), (2) \(\forall t\neq 0\), \(S_ t(f_ n(0,0))\) contains exactly \(n\) points, (3) \(\forall t\neq 0\), \(S_ t(f_ \infty(0,0))\) is a denumerable nonclosed unbounded set, (4) the equation (1) is uniquely solvable for all initial conditions on the whole real line, but, for \(t\neq 0\), dependence of the solution with initial condition \(x(0)= x_ 0\) on \(x_ 0\) is not continuous at the point \(x_ 0= 0\). Theorem 2. If in a Fréchet space \(E\) bounded sets are relatively compact, then every solution of (1), bounded on the interval \((a,b)\), has a limit as \(t\to a\) and \(t\to b\). In the contrary case, an \(f\in C(\mathbb{R}\times E,E)\) may be found such that a solution of (1) is bounded on \([0,1)\) but its limit as \(t\to 1\) does not exist.
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 567918
zbMATH Keywords: nonnormed Fréchet space
zbMATH Keywords: continuity in the dependence of solutions on the initial data
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 567918