Asymptotic properties of solutions to difference equations of Sturm–Liouville type

Abstract

We consider the discrete Sturm–Liouville type equation of the formΔ(rnΔxn)=anf(xσ(n))+bn.Assume s is a given nonpositive real number. We present sufficient conditions for the existence of solution x with the asymptotic behaviorxn=c(r1−1+⋯+rn−1−1)+d+o(ns)where c, d are given real numbers. Moreover, we establish conditions under which for a given solution x there exist real numbers c, d such that x has the above asymptotic behavior.