Ascoli’s theorem for functions vanishing at infinity and selected applications

We give a new form of the Ascoli theorem for functions on RN tending to some given closed subset Z of a complete metric space E at infinity. For instance, when E is a normed space and Z = {0}, the usual uniform decay requirement is replaced by the assumption that the 0 function is the only continuous function produced by some limiting process. This formulation, which has significant practical value in concrete applications, is described in its general form, but with emphasis on the case when Z is totally disconnected. Variants in Sobolev spaces and the properness of nonlinear ordinary differential operators are discussed.  2003 Elsevier Inc. All rights reserved