Existence results for infinite dimensional differential equations without compactness (Q1891522)

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).