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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 |
$\left\{ \begin{align}&Δ≤(x_1(n)-p_1(n)\,x_1(n-τ_1))=a_1(n)\,f_1(x_1(n-σ_1),x_2(n-σ_2))\\&Δ≤(x_2(n)-p_2(n)\,x_2(n-τ_2))=a_2(n)\,f_2(x_1(n-σ_3),x_2(n-σ_4)),\end{align} \right.$ |
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 1st 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 1st 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 1st 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 1st 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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