Ascoli's theorem for functions vanishing at infinity and selected applications. (Q1426989)

The aim of the paper is to prove a generalized version of Ascoli's theorem for functions vanishing at infinity. Let \(E\) be a complete metric space with distance \(d\) and let \(C_b({\mathbb R}^N;E)\) be the space of \(E\)-valued bounded continuous functions on \({\mathbb R}^N\). Let \(Z\subset E\) be a nonempty closed subset and \(C_Z({\mathbb R}^N;E)\) be the closed subspace of \(C_b({\mathbb R}^N;E)\) consisting of those functions that tend to \(Z\) at infinity. Sufficient conditions for a subset \({\mathcal H}\subset C_Z({\mathbb R}^N;E)\) to be relatively compact are described. The conditions are similar to those in the classical Ascoli theorem. The above characterization is used to prove some results concerning compactness of the Sobolev embeddings. Namely, the sufficient conditions for compactness of a bounded subset \(\mathcal H\) of Sobolev space \(W^{m,p}({\mathbb R}^n)\) regarded as a subspace of \(C_{\{0\}}({\mathbb R}^N;{\mathbb R})\) are described. The generalized Ascoli theorem is used also to study the properties of the differential operator \(u\mapsto \frac{d u}{dt} - F(u)\), \(F\) being the Nemytskii operator.