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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation
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by Jitsuro Sugie and Masahiko Tanaka PDF
Proc. Amer. Math. Soc. 145 (2017), 2059-2073 Request permission

Abstract:

A nonoscillation problem is dealt with the second-order linear difference equation\[ c_nx_{n+1} + c_{n-1}x_{n-1} = b_nx_n, \] 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 \[ q_n = \frac {c_{n-1}}{b_n}\frac {c_n}{b_{n+1}}\frac {c_n}{c_{n-1}} = \frac {c_n^2}{b_nb_{n+1}}. \] 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.
References
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Additional Information
  • Jitsuro Sugie
  • Affiliation: Department of Mathematics, Shimane University, Matsue 690-8504, Japan
  • MR Author ID: 168705
  • Email: jsugie@riko.shimane-u.ac.jp
  • Masahiko Tanaka
  • Affiliation: Department of Mathematics, Shimane University, Matsue 690-8504, Japan
  • Email: qut4527@yahoo.co.jp
  • 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
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2059-2073
  • MSC (2010): Primary 39A06, 39A21; Secondary 39A10
  • DOI: https://doi.org/10.1090/proc/13338
  • MathSciNet review: 3611320