Asymptotic formulae for the solutions of a linear delay difference equation
Consider the linear delay difference equation \[ x_{n+1}- x_n= a_n x_{n-k} ,\tag \(*\) \] where \(a_n\)'s are real numbers and \(k\) is a positive integer. The authors establish asymptotic formulae for the solutions of the equation \((*)\) assuming \(\sum^\infty_{n=0} |a_n |^p< \infty\) for some \(p\geq 1\). The asymptotic formulae are given by an appropriate recursive sequence. They present explicit formulae for \(p\in \{1, 2, 3, 4\}\). They also construct an example which shows that some comparison results obtained by \textit{J. Yan} and \textit{C. Qian} [J. Math. Anal. Appl. 165, No. 2, 346-360 (1992; Zbl 0818.39002)] are not correct.
scientific article; zbMATH DE number 861769