# Article

 Title: Oscillation properties for a scalar linear difference equation of mixed type (English) Author: Berezansky, Leonid Author: Pinelas, Sandra Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 141 Issue: 2 Year: 2016 Pages: 169-182 Summary lang: English . Category: math . Summary: The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type $$\Delta x(n)+\sum _{k=-p}^{q}a_{k}(n)x(n+k)=0,\quad n>n_{0},$$ where $\Delta x(n)=x(n+1)-x(n)$ is the difference operator and $\{a_{k}(n)\}$ are sequences of real numbers for $k=-p,\ldots ,q$, and $p>0$, $q\geq 0$. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced. (English) Keyword: oscillation Keyword: difference equation Keyword: mixed type Keyword: asymptotic behavior MSC: 39A21 MSC: 39A99 idZBL: Zbl 06587861 idMR: MR3499783 DOI: 10.21136/MB.2016.14 . Date available: 2016-05-19T09:04:52Z Last updated: 2020-07-01 Stable URL: http://hdl.handle.net/10338.dmlcz/145711 . Reference: [1] Agarwal, R. P.: Difference Equations and Inequalities: Theory, Methods, and Applications.Pure and Applied Mathematics 228 Marcel Dekker, New York (2000). Zbl 0952.39001, MR 1740241 Reference: [2] Asada, T., Yoshida, H.: Stability, instability and complex behavior in macrodynamic models with policy lag.Discrete Dyn. Nat. Soc. 5 (2001), 281-295. Zbl 0980.34070, 10.1155/S1026022600000583 Reference: [3] Berezansky, L., Braverman, E.: Some oscillation problems for a second order linear delay differential equation.J. Math. Anal. Appl. 220 (1998), 719-740. Zbl 0915.34064, MR 1614948, 10.1006/jmaa.1997.5879 Reference: [4] k, J. Diblí, Janglajew, K., ková, M. Kúdelčí: An explicit coefficient criterion for the existence of positive solutions to the linear advanced equation.Discrete Contin. Dyn. Syst. Ser. B 19 (2014), 2461-2467. MR 3275005, 10.3934/dcdsb.2014.19.2461 Reference: [5] k, J. Diblí, ková, M. Kúdelčí: New explicit integral criteria for the existence of positive solutions to the linear advanced equation $y'=c(t)y(t+\tau)$.Appl. Math. Lett. 38 (2014), 144-148. MR 3258218, 10.1016/j.aml.2014.06.020 Reference: [6] Dubois, D., Stecke, K. E.: Dynamic analysis of repetitive decision-free discrete-event processes: applications to production systems.Ann. Oper. Res. 26 (1990), 323-347. MR 1087827, 10.1007/BF02248590 Reference: [7] Ferreira, J. M., Pinelas, S.: Oscillatory mixed difference systems.Adv. Difference Equ. (2006), 1-18. Zbl 1139.39011, MR 2238984 Reference: [8] Frisch, R., Holme, H.: The characteristic solutions of a mixed difference and differential equation occurring in economic dynamics.Econometrica 3 (1935), 225-239. 10.2307/1907258 Reference: [9] Gandolfo, G.: Economic Dynamics.Berlin Springer (2010). Zbl 1177.91094, MR 2841165 Reference: [10] Iakovleva, V., Vanegas, C. J.: On the solution of differential equations with delayed and advanced arguments.Electron. J. Differ. Equ. 13 (2005), 57-63. Zbl 1092.34549, MR 2312906 Reference: [11] James, R. W., Belz, M. H.: The significance of the characteristic solutions of mixed difference and differential equations.Econometrica 6 (1938), 326-343. 10.2307/1905410 Reference: [12] Krisztin, T.: Nonoscillation for functional differential equations of mixed type.J. Math. Anal. Appl. 245 (2000), 326-345. Zbl 0955.34054, MR 1758543, 10.1006/jmaa.2000.6735 Reference: [13] Ladde, G. S., Lakshmikantham, V., Zhang, B. G.: Oscillation Theory of Differential Equations with Deviating Arguments.Pure and Applied Mathematics 110 Marcel Dekker, New York (1987). Zbl 0832.34071, MR 1017244 Reference: [14] Rogovchenko, Y. V.: Oscillation criteria for certain nonlinear differential equations.J. Math. Anal. Appl. 229 (1999), 399-416. Zbl 0921.34034, MR 1666412, 10.1006/jmaa.1998.6148 .

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