Ordinary differential equations with a continuous right-hand side in Fréchet spaces
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.
continuity in the dependence of solutions on the initial data
scientific article; zbMATH DE number 567918
scientific article