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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Keywords:
Difference equation,
solution,
positive solution

Article copyright:
© Copyright 1990
American Mathematical Society