Regional topology and approximative solutions of dierence and dierential equations

Abstract

We introduce a topology, which we call the regional topology, on the space of
all real functions on a given locally compact metric space. Next we obtain a new
versions of Schauder's fixed point theorem and Ascoli's theorem. We use these
theorems and the properties of the iterated remainder operator to establish con-
ditions under which there exist solutions, with prescribed asymptotic behavior, of
some dierence and dierential equations.