   ISSN 1088-6826(online) ISSN 0002-9939(print)

Positive solutions of difference equations

Authors: Ch. G. Philos and Y. G. Sficas
Journal: Proc. Amer. Math. Soc. 108 (1990), 107-115
MSC: Primary 39A10
DOI: https://doi.org/10.1090/S0002-9939-1990-1024260-5
MathSciNet review: 1024260
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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 $(0,1)$.

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