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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

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

##### References:
  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.  Google Scholar  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.  Google Scholar  A. Burton, Stability by Fixed Point Theory for Functional Differential Equations 1st edition, Dover Publications, New York, 2006. Google Scholar  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.  Google Scholar  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. Google Scholar  K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics 1st edition, Kluwer Academic Publishers, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  Y. N. Raffoul and E. Yankson, Positive periodic solutions in neutral delay difference equations, Adv. Dyn. Syst. Appl., 5 (2010), 123-130. Google Scholar  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.  Google Scholar  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. Google Scholar  W. Wang and X. Yang, Positive periodic solutions for neutral functional difference equations, Int. J. Difference Equ., 7 (2012), 99-109. Google Scholar  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.  Google Scholar  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.  Google Scholar
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