January 2018, 23(1): 253-261. doi: 10.3934/dcdsb.2018017

Monotonic solutions of a higher-order neutral difference system

1. 

Institute of Mathematics, University of Białystok, Ciolkowskiego 1M, 15-245 Bialystok, Poland

2. 

Institute of Mathematics, Lodz University of Technology, Wolczanska 215, 90-924 Lodz, Poland

* Corresponding author: Robert Jankowski

Received  August 2016 Revised  December 2016 Published  January 2018

A class of a higher-order nonlinear difference system with delayed arguments where the first equation of the system is of a neutral type is considered. A classification of non-oscillatory solutions is given and results on their boundedness or unboundedness are derived. The obtained results are illustrated by examples.

Citation: 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
References:
[1]

R. P. AgarwalS. R. Grace and E. Akin-Bohner, On the oscillation of higher order neutral difference equations of mixed type, Dynam. Systems Appl., 11 (2002), 459-469.

[2]

R. P. AgarwalE. Thandapani and P. J. Y. Wong, Oscillations of higher order neutral difference equations, Appl. Math. Lett., 10 (1997), 71-78. doi: 10.1016/S0893-9659(96)00114-0.

[3]

Y. Bolat, Oscillation of higher order neutral type nonlinear difference equations with forcing terms, Chaos, Solitons, Fractals, 42 (2009), 2973-2980. doi: 10.1016/j.chaos.2009.04.006.

[4]

J. DiblíkB. ŁupińskaM. Růžičková and J. Zonenberg, Boundedness and unboundedness of non-oscillatory solutions of a four-dimensional nonlinear neutral difference system, Boundedness and unboundedness of non-oscillatory solutions of a four-dimensional nonlinear neutral difference system, 2015 (2015), 1-11. doi: 10.1186/s13662-015-0662-9.

[5]

J. R. GraefA. MicianoP. SpikesP. Sundaram and E. Thandapani, Oscilatory and asymptotic behavior of solutions of nonlinear neutral-type difference equations, J. Austral. Math. Soc. Ser. B, 38 (1996), 163-171. doi: 10.1017/S0334270000000552.

[6]

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.

[7]

R. JankowskiE. Schmeidel and J. Zonenberg, Oscillatory properties of solutions of the fourth order difference equations with quasidifferences, Opuscula Math., 34 (2014), 789-797. doi: 10.7494/OpMath.2014.34.4.789.

[8]

W. T. Li and S. S. Cheng, Asymptotic trichotomy for positive solutions of a class of odd order nonlinear neutral difference equations, Comput. Math. Appl., 35 (1998), 101-108. doi: 10.1016/S0898-1221(98)00048-0.

[9]

M. Migda, On unstable neutral difference equations of higher order, Indian J. Pure Appl. Math., 36 (2005), 557-567.

[10]

M. Migda and J. Migda, Asymptotic properties of solutions of second-order neutral difference equations, Nonlinear Anal., 63 (2005), e789-e799. doi: 10.1016/j.na.2005.02.005.

[11]

M. Migda and J. Migda, Oscillatory and asymptotic properties of solutions of even order neutral difference equations, J. Difference Equ. Appl., 15 (2009), 1077-1084. doi: 10.1080/10236190903032708.

[12]

N. Parhi and A. K. Tripathy, Oscillation of a class of nonlinear neutral difference equations of higher order, J. Math. Anal. Appl., 284 (2003), 756-774. doi: 10.1016/S0022-247X(03)00298-1.

[13]

E. Schmeidel, Asymptotic trichotomy of solutions of a class of even order nonlinear neutral difference equations with quasidifferences, Proceedings of the International Conference on Difference Equations, Special Functions and Orthogonal Polynomials, World Scientific Publishing Co, (2007), 600-609. doi: 10.1142/9789812770752_0052.

[14]

A. Zafer, Oscillatory and asymptotic behavior of higher order difference equations, Math. Comput. Modelling, 21 (1995), 43-50. doi: 10.1016/0895-7177(95)00005-M.

[15]

Y. Zhou and B. G. Zhang, Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients, Comput. Math. Appl., 45 (2003), 991-1000. doi: 10.1016/S0898-1221(03)00074-9.

show all references

References:
[1]

R. P. AgarwalS. R. Grace and E. Akin-Bohner, On the oscillation of higher order neutral difference equations of mixed type, Dynam. Systems Appl., 11 (2002), 459-469.

[2]

R. P. AgarwalE. Thandapani and P. J. Y. Wong, Oscillations of higher order neutral difference equations, Appl. Math. Lett., 10 (1997), 71-78. doi: 10.1016/S0893-9659(96)00114-0.

[3]

Y. Bolat, Oscillation of higher order neutral type nonlinear difference equations with forcing terms, Chaos, Solitons, Fractals, 42 (2009), 2973-2980. doi: 10.1016/j.chaos.2009.04.006.

[4]

J. DiblíkB. ŁupińskaM. Růžičková and J. Zonenberg, Boundedness and unboundedness of non-oscillatory solutions of a four-dimensional nonlinear neutral difference system, Boundedness and unboundedness of non-oscillatory solutions of a four-dimensional nonlinear neutral difference system, 2015 (2015), 1-11. doi: 10.1186/s13662-015-0662-9.

[5]

J. R. GraefA. MicianoP. SpikesP. Sundaram and E. Thandapani, Oscilatory and asymptotic behavior of solutions of nonlinear neutral-type difference equations, J. Austral. Math. Soc. Ser. B, 38 (1996), 163-171. doi: 10.1017/S0334270000000552.

[6]

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.

[7]

R. JankowskiE. Schmeidel and J. Zonenberg, Oscillatory properties of solutions of the fourth order difference equations with quasidifferences, Opuscula Math., 34 (2014), 789-797. doi: 10.7494/OpMath.2014.34.4.789.

[8]

W. T. Li and S. S. Cheng, Asymptotic trichotomy for positive solutions of a class of odd order nonlinear neutral difference equations, Comput. Math. Appl., 35 (1998), 101-108. doi: 10.1016/S0898-1221(98)00048-0.

[9]

M. Migda, On unstable neutral difference equations of higher order, Indian J. Pure Appl. Math., 36 (2005), 557-567.

[10]

M. Migda and J. Migda, Asymptotic properties of solutions of second-order neutral difference equations, Nonlinear Anal., 63 (2005), e789-e799. doi: 10.1016/j.na.2005.02.005.

[11]

M. Migda and J. Migda, Oscillatory and asymptotic properties of solutions of even order neutral difference equations, J. Difference Equ. Appl., 15 (2009), 1077-1084. doi: 10.1080/10236190903032708.

[12]

N. Parhi and A. K. Tripathy, Oscillation of a class of nonlinear neutral difference equations of higher order, J. Math. Anal. Appl., 284 (2003), 756-774. doi: 10.1016/S0022-247X(03)00298-1.

[13]

E. Schmeidel, Asymptotic trichotomy of solutions of a class of even order nonlinear neutral difference equations with quasidifferences, Proceedings of the International Conference on Difference Equations, Special Functions and Orthogonal Polynomials, World Scientific Publishing Co, (2007), 600-609. doi: 10.1142/9789812770752_0052.

[14]

A. Zafer, Oscillatory and asymptotic behavior of higher order difference equations, Math. Comput. Modelling, 21 (1995), 43-50. doi: 10.1016/0895-7177(95)00005-M.

[15]

Y. Zhou and B. G. Zhang, Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients, Comput. Math. Appl., 45 (2003), 991-1000. doi: 10.1016/S0898-1221(03)00074-9.

[1]

Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431

[2]

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

[3]

Ewa Schmeidel, Robert Jankowski. Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2691-2696. doi: 10.3934/dcdsb.2014.19.2691

[4]

John R. Graef, R. Savithri, E. Thandapani. Oscillatory properties of third order neutral delay differential equations. Conference Publications, 2003, 2003 (Special) : 342-350. doi: 10.3934/proc.2003.2003.342

[5]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577

[6]

Xianyi Li, Deming Zhu. Comparison theorems of oscillation and nonoscillation for neutral difference equations with continuous arguments. Communications on Pure & Applied Analysis, 2003, 2 (4) : 579-589. doi: 10.3934/cpaa.2003.2.579

[7]

Mustafa Hasanbulli, Yuri V. Rogovchenko. Classification of nonoscillatory solutions of nonlinear neutral differential equations. Conference Publications, 2009, 2009 (Special) : 340-348. doi: 10.3934/proc.2009.2009.340

[8]

Emmanuel Frénod, Sever A. Hirstoaga, Eric Sonnendrücker. An exponential integrator for a highly oscillatory vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 169-183. doi: 10.3934/dcdss.2015.8.169

[9]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125

[10]

Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915

[11]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639

[12]

Chunhua Jin, Jingxue Yin. Periodic solutions of a non-divergent diffusion equation with nonlinear sources. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 101-126. doi: 10.3934/dcdsb.2012.17.101

[13]

Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1519-1537. doi: 10.3934/dcdss.2017078

[14]

Francesca Faraci, Alexandru Kristály. One-dimensional scalar field equations involving an oscillatory nonlinear term. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 107-120. doi: 10.3934/dcds.2007.18.107

[15]

Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640

[16]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133

[17]

Min Zou, An-Ping Liu, Zhimin Zhang. Oscillation theorems for impulsive parabolic differential system of neutral type. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2351-2363. doi: 10.3934/dcdsb.2017103

[18]

Saroj Panigrahi, Rakhee Basu. Oscillation results for second order nonlinear neutral differential equations with delay. Conference Publications, 2015, 2015 (special) : 906-912. doi: 10.3934/proc.2015.0906

[19]

Philippe Laurençot, Barbara Niethammer, Juan J.L. Velázquez. Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel. Kinetic & Related Models, 2018, 11 (4) : 933-952. doi: 10.3934/krm.2018037

[20]

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

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (16)
  • HTML views (54)
  • Cited by (0)

[Back to Top]