Oscillation criteria for two dimensional linear neutral delay difference systems

Tripathy, Arun Kumar

Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 447-460

Abstract (VO)

In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form $$ \Delta \left [\begin{matrix} x(n)+p(n)x(n-m)\\ y(n)+p(n)y(n-m) \end{matrix} \right ]= \left [\begin{matrix} a(n) b(n) \\ c(n) d(n) \end{matrix} \right ]\left [\begin{matrix} x(n-\alpha )\\ y(n-\beta ) \end{matrix} \right ] $$ are established, where $m>0$, $\alpha \geq 0$, $\beta \geq 0$ are integers and $a(n)$, $b(n)$, $c(n)$, $d(n)$, $p(n)$ are sequences of real numbers.

Détail

DOI : 10.21136/MB.2022.0048-21

Classification : 34C10, 34K11, 39A13
Keywords: oscillation; nonoscillation; system of neutral equations; Krasnoselskii's fixed point theorem

Bibliographie

[1] Agarwal, R. P.: Difference Equations and Inequalities: Theory, Methods, and Applications. Pure and Applied Mathematics, Marcel Dekker 228. Marcel Dekker, New York (2000). | DOI | MR | JFM

[2] Agarwal, R. P., Bohner, M., Grace, S. R., O'Regan, D.: Discrete Oscillation Theory. Hindawi Publishing, New York (2005). | DOI | MR | JFM

[3] Agarwal, R. P., Wong, P. J. Y.: Advanced Topics in Difference Equations. Mathematics and Its Applications (Dordrecht) 404. Kluwer Academic, Dordrecht (1997). | DOI | MR | JFM

[4] Chatzarakis, G. E., Groumpas, E. I.: Oscillations in systems of difference equations. Far East J. Dyn. Syst. 17 (2011), 17-31. | MR | JFM

[5] Diblík, J., Łupińska, B., Růžičková, M., Zonenberg, J.: Bounded and unbounded non-oscillatory solutions of a four-dimensional nonlinear neutral difference systems. Adv. Difference Equ. 2015 (2015), Article ID 319, 11 pages. | DOI | MR | JFM

[6] Elaydi, S. N.: An Introduction to Difference Equations. Undergraduate Texts in Mathematics. Springer, New York (1996). | DOI | MR | JFM

[7] Graef, J. R., Thandapani, E.: Oscillation of two-dimensional difference systems. Comput. Math. Appl. 38 (1999), 157-165. | DOI | MR | JFM

[8] Jiang, J., Tang, X.: Oscillation and asymptotic behaviour of two-dimensional difference systems. Comput. Math. Appl. 54 (2007), 1240-1249. | DOI | MR | JFM

[9] Li, W.-T.: Classification schemes for nonoscillatory solutons of two-dimensional nonlinear difference systems. Comput. Math. Appl. 42 (2001), 341-355. | DOI | MR | JFM

[10] Migda, M., Schmeidel, E., Zdanowicz, M.: Periodic solutions of a 2-dimensional system of neutral difference equations. Discrete Contin. Dyn. Syst., Ser. B 23 (2018), 359-367. | DOI | MR | JFM

[11] Parhi, N., Tripathy, A. K.: Oscillatory behavior of second order difference equations. Commun. Appl. Nonlinear Anal. 6 (1999), 79-100. | MR | JFM

[12] Parhi, N., Tripathy, A. K.: Oscillation of a class of neutral difference equations of first order. J. Difference Equ. Appl. 9 (2003), 933-946. | DOI | MR | JFM

[13] Parhi, N., Tripathy, A. K.: Oscillation of forced nonlinear neutral delay difference equations of first order. Czech. Math. J. 53 (2003), 83-101. | DOI | MR | JFM

[14] Schmeidel, E.: Oscillation of nonlinear three-dimensional difference systems with delays. Math. Bohem. 135 (2010), 163-170. | DOI | MR | JFM

[15] Schmeidel, E., Zdanowicz, M.: Existence of the asymptotically periodic solution to the system of nonlinear neutral difference equations. Tatra Mt. Math. Publ. 79 (2021), 149-162. | DOI | MR | JFM

[16] Stević, S., Diblík, J., Iričanin, J., Šmarda, B. Z.: On a third-order system of difference equations with variable coefficients. Abstr. Appl. Anal. 2012 (2012), Article ID 508523, 22 pages. | DOI | MR | JFM

[17] Tripathy, A. K.: Oscillation criteria for first-order systems of linear difference equations. Electron. J. Differ. Equ. 2009 (2009), Article ID 29, 11 pages. | MR | JFM