Periodic solutions of a $2$-dimensional system of neutral difference equations

Małgorzata Migda; Ewa Schmeidel; Małgorzata Zdanowicz

doi:10.3934/dcdsb.2018024

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January 2018, 23(1): 359-367. doi: 10.3934/dcdsb.2018024

Periodic solutions of a $2$-dimensional system of neutral difference equations

Małgorzata Migda ^1, , Ewa Schmeidel ^2, and Małgorzata Zdanowicz ^2,,

1.	Poznan University of Technology, Piotrowo 3A, 60-965 Poznań, Poland
2.	University of Bialystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland

* Corresponding author: Ma lgorzata Zdanowicz

Received September 2016 Revised May 2017 Published January 2018

The 2-dimensional system of neutral type nonlinear difference equations with delays in the following form

$\left\{ \begin{align}&Δ≤(x_1(n)-p_1(n)\,x_1(n-τ_1))=a_1(n)\,f_1(x_1(n-σ_1),x_2(n-σ_2))\\&Δ≤(x_2(n)-p_2(n)\,x_2(n-τ_2))=a_2(n)\,f_2(x_1(n-σ_3),x_2(n-σ_4)),\end{align} \right.$

is considered. In this paper we use Schauder's fixed point theorem to study the existence of periodic solutions of the above system.

Keywords: System of difference equation, delays, 2-dimensional, neutral type, periodic solutions, existence.

Mathematics Subject Classification: Primary: 39A10; Secondary: 39A23.

Citation: Małgorzata Migda, Ewa Schmeidel, Małgorzata Zdanowicz. Periodic solutions of a $2$-dimensional system of neutral difference equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 359-367. doi: 10.3934/dcdsb.2018024

References:

[1]	A. Bellen, N. Guglielmi and A. E. Ruehli, Methods for linear systems of circuit delay differential equations of neutral type, IEEE Transactions on Circuits and Systems I, 46 (1999), 212-216. doi: 10.1109/81.739268.
[2]	R. K. Brayton and R. A. Willoughby, On the numerical integration of a symmetric system of difference-differential equations of neutral type, J. Math. Anal. Appl., 18 (1967), 182-189. doi: 10.1016/0022-247X(67)90191-6.
[3]	A. Burton, Stability by Fixed Point Theory for Functional Differential Equations 1^st edition, Dover Publications, New York, 2006.
[4]	G. E. Chatzarakis and G. N. Miliaras, Convergence and divergence of the solutions of a neutral difference equation J. Appl. Math. 2011 (2011), Art. ID 262316, 18 pp. doi: 10.1155/2011/262316.
[5]	M. Galewski, R. Jankowski, M. Nockowska-Rosiak and E. Schmeidel, On the existence of bounded solutions for nonlinear second-order neutral difference equations, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 1-12.
[6]	K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics 1^st edition, Kluwer Academic Publishers, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.
[7]	Z. Guo and M. Liu, Existence of non-oscillatory solutions for a higher-order nonlinear neutral difference equation, Electron. J. Differential Equations, 146 (2010), 1-7. doi: 10.1016/S0022-247X(03)00017-9.
[8]	R. Jankowski and E. Schmeidel, Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences, Discrete Contin. Dyn. Syst. (B), 19 (2014), 2691-2696. doi: 10.3934/dcdsb.2014.19.2691.
[9]	Z. Liu, Y. Xu and S. M. Kang, Global solvability for a second order nonlinear neutral delay difference equation, Comput. Math. Appl., 57 (2009), 587-595.
[10]	J. Migda, Asymptotically polynomial solutions to difference equations of neutral type, Appl. Math. Comput., 279 (2016), 16-27. doi: 10.1016/j.amc.2016.01.001.
[11]	M. Migda and J. Migda, A class of first-order nonlinear difference equations of neutral type, Math. Comput. Modelling, 40 (2004), 297-306. doi: 10.1016/j.mcm.2003.12.006.
[12]	M. Migda, E. Schmeidel and M. Zdanowicz, Bounded solutions of k-dimensional system of nonlinear difference equations of neutral type, Electron. J. Qual. Theory Differ. Equ., 80 (2015), 1-17. doi: 10.14232/ejqtde.2015.1.80.
[13]	M. Migda and G. Zhang, Monotone solutions of neutral difference equations of odd order, J. Difference Equ. Appl., 10 (2004), 691-703. doi: 10.1080/10236190410001702490.
[14]	Y. N. Raffoul and E. Yankson, Positive periodic solutions in neutral delay difference equations, Adv. Dyn. Syst. Appl., 5 (2010), 123-130.
[15]	X. H. Tang and S. S. Cheng, Positive solutions of a neutral difference equation with positive and negative coefficients, Georgian Math. J., 11 (2004), 177-185. doi: 10.1515/GMJ.2004.177.
[16]	E. Thandapani, R. Karunakaran and I. M. Arockiasamy, Bounded nonoscillatory solutions of neutral type difference systems, Electron. J. Qual. Theory Differ Equ. Spec. Ed. I, 25 (2009), 1-8.
[17]	W. Wang and X. Yang, Positive periodic solutions for neutral functional difference equations, Int. J. Difference Equ., 7 (2012), 99-109.
[18]	Z. Wang and J. Sun, Asymptotic behavior of solutions of nonlinear higher-order neutral type difference equations, J. Differ. Equ. Appl., 12 (2006), 419-432. doi: 10.1080/10236190500539352.
[19]	J. Wu, Two periodic solutions of $n$-dimensional neutral functional difference systems, J. Math. Anal. Appl., 334 (2007), 738-752. doi: 10.1016/j.jmaa.2007.01.009.

show all references

References:

[1]	A. Bellen, N. Guglielmi and A. E. Ruehli, Methods for linear systems of circuit delay differential equations of neutral type, IEEE Transactions on Circuits and Systems I, 46 (1999), 212-216. doi: 10.1109/81.739268.
[2]	R. K. Brayton and R. A. Willoughby, On the numerical integration of a symmetric system of difference-differential equations of neutral type, J. Math. Anal. Appl., 18 (1967), 182-189. doi: 10.1016/0022-247X(67)90191-6.
[3]	A. Burton, Stability by Fixed Point Theory for Functional Differential Equations 1^st edition, Dover Publications, New York, 2006.
[4]	G. E. Chatzarakis and G. N. Miliaras, Convergence and divergence of the solutions of a neutral difference equation J. Appl. Math. 2011 (2011), Art. ID 262316, 18 pp. doi: 10.1155/2011/262316.
[5]	M. Galewski, R. Jankowski, M. Nockowska-Rosiak and E. Schmeidel, On the existence of bounded solutions for nonlinear second-order neutral difference equations, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 1-12.
[6]	K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics 1^st edition, Kluwer Academic Publishers, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.
[7]	Z. Guo and M. Liu, Existence of non-oscillatory solutions for a higher-order nonlinear neutral difference equation, Electron. J. Differential Equations, 146 (2010), 1-7. doi: 10.1016/S0022-247X(03)00017-9.
[8]	R. Jankowski and E. Schmeidel, Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences, Discrete Contin. Dyn. Syst. (B), 19 (2014), 2691-2696. doi: 10.3934/dcdsb.2014.19.2691.
[9]	Z. Liu, Y. Xu and S. M. Kang, Global solvability for a second order nonlinear neutral delay difference equation, Comput. Math. Appl., 57 (2009), 587-595.
[10]	J. Migda, Asymptotically polynomial solutions to difference equations of neutral type, Appl. Math. Comput., 279 (2016), 16-27. doi: 10.1016/j.amc.2016.01.001.
[11]	M. Migda and J. Migda, A class of first-order nonlinear difference equations of neutral type, Math. Comput. Modelling, 40 (2004), 297-306. doi: 10.1016/j.mcm.2003.12.006.
[12]	M. Migda, E. Schmeidel and M. Zdanowicz, Bounded solutions of k-dimensional system of nonlinear difference equations of neutral type, Electron. J. Qual. Theory Differ. Equ., 80 (2015), 1-17. doi: 10.14232/ejqtde.2015.1.80.
[13]	M. Migda and G. Zhang, Monotone solutions of neutral difference equations of odd order, J. Difference Equ. Appl., 10 (2004), 691-703. doi: 10.1080/10236190410001702490.
[14]	Y. N. Raffoul and E. Yankson, Positive periodic solutions in neutral delay difference equations, Adv. Dyn. Syst. Appl., 5 (2010), 123-130.
[15]	X. H. Tang and S. S. Cheng, Positive solutions of a neutral difference equation with positive and negative coefficients, Georgian Math. J., 11 (2004), 177-185. doi: 10.1515/GMJ.2004.177.
[16]	E. Thandapani, R. Karunakaran and I. M. Arockiasamy, Bounded nonoscillatory solutions of neutral type difference systems, Electron. J. Qual. Theory Differ Equ. Spec. Ed. I, 25 (2009), 1-8.
[17]	W. Wang and X. Yang, Positive periodic solutions for neutral functional difference equations, Int. J. Difference Equ., 7 (2012), 99-109.
[18]	Z. Wang and J. Sun, Asymptotic behavior of solutions of nonlinear higher-order neutral type difference equations, J. Differ. Equ. Appl., 12 (2006), 419-432. doi: 10.1080/10236190500539352.
[19]	J. Wu, Two periodic solutions of $n$-dimensional neutral functional difference systems, J. Math. Anal. Appl., 334 (2007), 738-752. doi: 10.1016/j.jmaa.2007.01.009.

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