## 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