# American Institute of Mathematical Sciences

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. Agarwal, S. 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. Agarwal, E. 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. Diblík, B. Łupińska, M. 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. Graef, A. Miciano, P. Spikes, P. 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. Jankowski, E. 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.

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##### References:
 [1] R. P. Agarwal, S. 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. Agarwal, E. 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. Diblík, B. Łupińska, M. 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. Graef, A. Miciano, P. Spikes, P. 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. Jankowski, E. 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.
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