Positive solutions of difference equations

Abstract:

Consider the difference equation \[ ({\text {E}})\quad {( - 1)^{m + 1}}{\Delta ^m}{A_n} + \sum \limits _{k = 0}^\infty {{p_k}{A_{n - {l_k}}} = 0,} \] where $m$ is a positive integer, ${({p_k})_{k \geq 0}}$ is a sequence of positive real numbers and ${({l_k})_{k \geq 0}}$ is a sequence of integers with $0 \leq {l_0} < {l_1} < {l_2} < \cdots$. The characteristic equation of (E) is \[ ( * )\quad - {(1 - \lambda )^m} + \sum \limits _{k = 0}^\infty {{p_k}{\lambda ^{ - {l_k}}} = 0.} \] We prove the following theorem. Theorem. (i) For $m$ even, (E) has a positive solution ${({A_n})_{n \in Z}}$ with $\lim {\text {su}}{{\text {p}}_{n \to \infty }}{A_n} < \infty$ if and only if (*) has a root in $(0,1)$. (ii) For $m$ odd, (E) has a positive solution ${({A_n})_{n \in Z}}$ if and only if (*) has a root in $(0,1)$.