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Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation


Authors: Jitsuro Sugie and Masahiko Tanaka
Journal: Proc. Amer. Math. Soc. 145 (2017), 2059-2073
MSC (2010): Primary 39A06, 39A21; Secondary 39A10
DOI: https://doi.org/10.1090/proc/13338
Published electronically: January 11, 2017
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Abstract: A nonoscillation problem is dealt with the second-order linear difference equation

$\displaystyle c_nx_{n+1} + c_{n-1}x_{n-1} = b_nx_n, $

where $ \{b_n\}$ and $ \{c_n\}$ are positive sequences. For all sufficiently large $ n \in \mathbb{N}$, the ratios $ c_n/c_{n-1}$ and $ c_{n-1}/b_n$ play an important role in the results obtained. To be precise, our nonoscillation criteria are described in terms of the sequence

$\displaystyle q_n = \frac {c_{n-1}}{b_n}\frac {c_n}{b_{n+1}}\frac {c_n}{c_{n-1}} = \frac {c_n^2}{b_nb_{n+1}}. $

These criteria are compared with those that have been reported in previous researches by using some specific examples. Figures are attached to facilitate understanding of the concrete examples.

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Additional Information

Jitsuro Sugie
Affiliation: Department of Mathematics, Shimane University, Matsue 690-8504, Japan
Email: jsugie@riko.shimane-u.ac.jp

Masahiko Tanaka
Affiliation: Department of Mathematics, Shimane University, Matsue 690-8504, Japan
Email: qut4527@yahoo.co.jp

DOI: https://doi.org/10.1090/proc/13338
Keywords: Linear difference equations, nonoscillation, Riccati transformation, Sturm's separation theorem
Received by editor(s): February 20, 2016
Received by editor(s) in revised form: June 19, 2016
Published electronically: January 11, 2017
Additional Notes: The first author’s work was supported in part by Grant-in-Aid for Scientific Research No.25400165 from the Japan Society for the Promotion of Science
Communicated by: Mourad Ismail
Article copyright: © Copyright 2017 American Mathematical Society