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Zbl 0826.34061
Colombo, Giovanni; Garay, Barnabas M.
Existence results for infinite dimensional differential equations without compactness. (English)
[J] Rend. Semin. Mat. Univ. Padova 92, 127-133 (1994). ISSN 0041-8994

A result of A. N. Godunov (1975) states that in the Banach space $E$ the Cauchy problem $x'= f(x)$, $x(0)= x$ admits solutions $(\forall)$ $x_0\in E$, $(\forall)$ $f\in C(E, E)$ (i.e. $f$ continuous) iff $E$ is finite-dimensional. However, the authors of this paper show that if $f$ is a limit of a sequence of bounded Lipschitz functions, uniformly on compact sets, then there exists a continuous extension $F: R\times E\to R\times E$ of $f$ such that the Cauchy problem $X'= F(X)$, $X(0)= X_0$ admits solutions in $[0, \infty)$ for all initial conditions in $[0, \infty)\times E$. They also prove an existence result valid for the particular case $f(x)= x/\sqrt{|x|}$, $x= 0$ and $f(0)= 0$ (so the set of solutions is not compact).
[Gheorghe Moroşanu (Iaşi)]

MSC 2000:: *34G20 Nonlinear ODE in abstract spaces
34A12 Initial value problems for ODE

Keywords: Banach space; Cauchy problem; existence

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