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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
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
MathSciNet review: 1024260
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Abstract: Consider the difference equation

$\displaystyle ({\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

$\displaystyle ( * )\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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1024260-5
PII: S 0002-9939(1990)1024260-5
Keywords: Difference equation, solution, positive solution
Article copyright: © Copyright 1990 American Mathematical Society