Article
| Title: | Oscillation criteria for first-order neutral $h$-difference equations with oscillatory periodic coefficients (English) |
| Author: | Gökteke, Reyhan |
| Author: | Bolat, Yaşar |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 2 |
| Year: | 2026 |
| Pages: | 459-468 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We consider the first-order neutral h-difference equations $\Delta (x_{n}+p_{n}x_{n-k})+q_{n}x_{n-l}=0,$ where $\Delta x_{n}=x_{n+h}-x_{n}$, $h$, $k$ and $l$ are positive integers, $p_{n}$ is an oscilatory periodic sequence with period $h$ and $q_{n}$ is an increasing positive sequence. The fact that the sequence $p$ is an oscillatory periodic sequence is considered for the first time in this study. Moreover, we consider $p_{n}$ can be, quickly oscillatory sequence, and we obtain some new oscillatory criteria. Also we give some example equations, and recurrence formulas of these example equations were obtained, and calculations were made using the Python program and graphs were created to prove that our results are true. (English) |
| Keyword: | first-order linear difference equation |
| Keyword: | delayed difference equation |
| Keyword: | neutral difference equation |
| Keyword: | oscillation |
| Keyword: | oscillatory |
| Keyword: | periodic coefficient |
| MSC: | 34C10 |
| MSC: | 39A06 |
| MSC: | 39A10 |
| MSC: | 39A21 |
| DOI: | 10.21136/CMJ.2026.0175-25 |
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| Date available: | 2026-05-22T11:18:48Z |
| Last updated: | 2026-05-25 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153642 |
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| Reference: | [1] Agarwal, R. P.: Difference Equations and Inequalities: Theory, Methods and Applications.Pure and Applied Mathematics, Marcel Dekker 228. Marcel Dekker, New York (2000). Zbl 0952.39001, MR 1740241 |
| Reference: | [2] Bolat, Y., Gokteke, R.: New oscillation criteria for $h$-difference equation with oscillatory periodic coefficients.Math. Methods Appl. Sci. 48 (2025), 8696-8701. Zbl 08081898, MR 4902214, 10.1002/mma.10746 |
| Reference: | [3] Chen, M.-P., Lalli, B. S., Yu, J. S.: Oscillation in neutral delay difference equations with variable coefficients.Comput. Math. Appl. 29 (1995), 5-11. Zbl 0819.39004, MR 1326277, 10.1016/0898-1221(94)00223-8 |
| Reference: | [4] Georgiou, D. A., Grove, E. A., Ladas, G.: Oscillations of neutral difference equations.Appl. Anal. 33 (1989), 243-253. Zbl 0685.39003, MR 1030111, 10.1080/00036818908839876 |
| Reference: | [5] Kordonis, I.-G. E., Philos, C. G.: Oscillation of neutral difference equations with periodic coefficients.Comput. Math. Appl. 33 (1997), 11-27. Zbl 0880.39015, MR 1442802, 10.1016/S0898-1221(97)00040-0 |
| Reference: | [6] Lalli, B. S., Zhang, B. G.: Oscillation and comparison theorems for certain neutral difference equations.J. Aust. Math. Soc., Ser. B 34 (1992), 245-256. Zbl 0759.39002, MR 1181576, 10.1017/S0334270000008754 |
| Reference: | [7] Lalli, B. S., Zhang, B. G., Li, J. Z.: On the oscillation of solutions and existence of positive solutions of neutral difference equations.J. Math. Anal. Appl. 158 (1991), 213-233. Zbl 0732.39002, MR 1113411, 10.1016/0022-247X(91)90278-8 |
| Reference: | [8] Thandapani, E., Arul, R., Raja, P. S.: Oscillation of first order neutral delay difference equations.Appl. Math. E-Notes 3 (2003), 88-94. Zbl 1027.39003, MR 1980570 |
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