American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with $p$-Laplacian
  • DCDS-B Home
  • This Issue
  • Next Article
    Optimal control of the discrete-time fractional-order Cucker-Smale model
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 & 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.  Google Scholar

[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.  Google Scholar

[3]

A. Burton, Stability by Fixed Point Theory for Functional Differential Equations 1st edition, Dover Publications, New York, 2006.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. N. Raffoul and E. Yankson, Positive periodic solutions in neutral delay difference equations, Adv. Dyn. Syst. Appl., 5 (2010), 123-130.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[17]

W. Wang and X. Yang, Positive periodic solutions for neutral functional difference equations, Int. J. Difference Equ., 7 (2012), 99-109.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

A. Burton, Stability by Fixed Point Theory for Functional Differential Equations 1st edition, Dover Publications, New York, 2006.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. N. Raffoul and E. Yankson, Positive periodic solutions in neutral delay difference equations, Adv. Dyn. Syst. Appl., 5 (2010), 123-130.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[17]

W. Wang and X. Yang, Positive periodic solutions for neutral functional difference equations, Int. J. Difference Equ., 7 (2012), 99-109.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Sami Baraket, Soumaya Sâanouni, Nihed Trabelsi. Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1013-1063. doi: 10.3934/dcds.2020069
[2]

M.I. Gil’. Existence and stability of periodic solutions of semilinear neutral type systems. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 809-820. doi: 10.3934/dcds.2001.7.809
[3]

Robert Jankowski, Barbara Łupińska, Magdalena Nockowska-Rosiak, Ewa Schmeidel. Monotonic solutions of a higher-order neutral difference system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 253-261. doi: 10.3934/dcdsb.2018017
[4]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[5]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793
[6]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
[7]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations & Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032
[8]

Josef Diblík, Zdeněk Svoboda. Existence of strictly decreasing positive solutions of linear differential equations of neutral type. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 67-84. doi: 10.3934/dcdss.2020004
[9]

Syed M. Assad, Chjan C. Lim. Statistical equilibrium of the Coulomb/vortex gas on the unbounded 2-dimensional plane. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 1-14. doi: 10.3934/dcdsb.2005.5.1
[10]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2021, 16 (1) : 1-29. doi: 10.3934/nhm.2020031
[11]

Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315
[12]

Ewa Schmeidel, Karol Gajda, Tomasz Gronek. On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2681-2690. doi: 10.3934/dcdsb.2014.19.2681
[13]

Josef Diblík, Radoslav Chupáč, Miroslava Růžičková. Existence of unbounded solutions of a linear homogenous system of differential equations with two delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2447-2459. doi: 10.3934/dcdsb.2014.19.2447
[14]

Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437
[15]

Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134
[16]

Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko, J. Tomás Lázaro. Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4483-4507. doi: 10.3934/dcds.2018196
[17]

David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253
[18]

Daniel Núñez, Pedro J. Torres. Periodic solutions of twist type of an earth satellite equation. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 303-306. doi: 10.3934/dcds.2001.7.303
[19]

Anatoli F. Ivanov, Bernhard Lani-Wayda. Periodic solutions for three-dimensional non-monotone cyclic systems with time delays. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 667-692. doi: 10.3934/dcds.2004.11.667
[20]

Hernán R. Henríquez, Claudio Cuevas, Juan C. Pozo, Herme Soto. Existence of solutions for a class of abstract neutral differential equations. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2455-2482. doi: 10.3934/dcds.2017106

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (92)
  • HTML views (141)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy