# American Institute of Mathematical Sciences

January  2018, 23(1): 239-252. doi: 10.3934/dcdsb.2018016

## Qualitative properties of solutions of higher order difference equations with deviating arguments

 Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60-965 Poznań, Poland

* Corresponding author: Alina Gleska

Received  December 2016 Published  January 2018

In the paper the general higher order difference equation
 $(-1)^z Δ^m x(n)=f(n, x(σ_1(n)),x(σ_2(n)),..., x(σ_k(n)))$
with several deviating arguments is considered. According to the kind of the deviations
 $σ_i$
sufficient conditions for the equation to have property A and B are established.
Citation: Alina Gleska, Małgorzata Migda. Qualitative properties of solutions of higher order difference equations with deviating arguments. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 239-252. doi: 10.3934/dcdsb.2018016
##### References:
 [1] R. P. Agarwal, M. Bohner, S. R. Grace and D. O'Regan, Discrete Oscillation Theory Hindawi Publishing Corporation, New York, 2005. doi: 10.1155/9789775945198. [2] R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications Second Edition, Revised and Expanded, Marcel Dekker, New York, 2000. [3] J. Baštinec, L. Berezansky, J. Diblík and Z. šmarda, A final result on the oscillation of solutions of the linear discrete delayed equation $Δ x(n)=-p(n)x(n-k)$ with a positive coefficient Abstract and Applied Analysis 2011 (2011), Art. ID 586328, 28 pp. doi: 10.1155/2011/586328. [4] G. E. Chatzarakis, R. Koplatadze and I. P. Stavroulakis, Oscillation criteria of first order linear difference equations with delay argument, Nonlinear Anal., 68 (2008), 994-1005.  doi: 10.1016/j.na.2006.11.055. [5] G. E. Chatzarakis, M. Lafci and I. P. Stavroulakis, Oscillation results for difference equations with several oscillating coefficients, Applied Mathematics and Computation, 251 (2015), 81-91.  doi: 10.1016/j.amc.2014.11.045. [6] G. E. Chatzarakis, Ch. G. Philos and I. P. Stavroulakis, On the oscillation of the solutions to linear difference equations with variable delay, Electron. J. Differ. Equ., 50 (2008), 1-15. [7] G. E. Chatzarakis and Ö. Öcalan, Oscillation of difference equations with non-monotone retarded arguments, Applied Mathematics and Computation, 258 (2015), 60-66.  doi: 10.1016/j.amc.2015.01.110. [8] G. E. Chatzarakis, H. Péics, S. Pinelas and I. P. Stavroulakis, Oscillation results for difference equations with oscillating coefficients Advances in Difference Equations 2015 (2015), 17pp. doi: 10.1186/s13662-015-0391-0. [9] L. H. Erbe and B. G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations, 2 (1989), 300-309. [10] G. Grzegorczyk and J. Werbowski, On oscillatory solutions of certain difference equations, Opuscula Mathematica, 26 (2006), 317-326. [11] G. Grzegorczyk and J. Werbowski, Oscillation of higher-order linear difference equations, Comput. Math. Appl., 42 (2001), 711-717.  doi: 10.1016/S0898-1221(01)00190-0. [12] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications Clarendon Press, Oxford, New York, 1991. [13] W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications Second edition, Harcourt/Academic Press, San Diego, CA, 2001. [14] G. Ladas, Ch. G. Philos and Y. G. Sficas, Sharp conditions for the oscillation of delay difference equations, J. Appl. Math. Simulation, 2 (1989), 101-111.  doi: 10.1155/S1048953389000080. [15] J. Migda, Approximative solutions of difference equations, Electron. J. Qual. Theory Differ. Equ. 13 (2014), 26 pp. [16] M. Migda, Existence of nonoscillatory solutions of some higher order difference equations, Appl. Math. E-Notes, 4 (2004), 33-39. [17] M. Migda and J. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations, Nonlinear Analysis, 47 (2001), 4687-4695.  doi: 10.1016/S0362-546X(01)00581-8. [18] M. Migda, Asymptotic properties of nonoscillatory solutions of higher order neutral difference equations, Opuscula Mathematica, 26 (2006), 507-514. [19] W. Nowakowska and J. Werbowski, On connections between oscillatory solutions of functional, difference and differential equations, Fasc. Math., 44 (2010), 95-106. [20] Ch. G. Philos, I. K. Purnaras and I. P. Stavroulakis, Sufficient conditions for the oscillation of delay difference equations, J. Difference Equ. Appl., 10 (2004), 419-435.  doi: 10.1080/10236190410001648239. [21] J. Werbowski, Bounded oscillations of differential equations generated by deviating arguments, Utilitas Math., 31 (1987), 191-198. [22] A. Wyrwińska, Oscillation criteria of a higher order linear difference equation, Bull. Inst. Math. Acad. Sinica, 22 (1994), 259-266.

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##### References:
 [1] R. P. Agarwal, M. Bohner, S. R. Grace and D. O'Regan, Discrete Oscillation Theory Hindawi Publishing Corporation, New York, 2005. doi: 10.1155/9789775945198. [2] R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications Second Edition, Revised and Expanded, Marcel Dekker, New York, 2000. [3] J. Baštinec, L. Berezansky, J. Diblík and Z. šmarda, A final result on the oscillation of solutions of the linear discrete delayed equation $Δ x(n)=-p(n)x(n-k)$ with a positive coefficient Abstract and Applied Analysis 2011 (2011), Art. ID 586328, 28 pp. doi: 10.1155/2011/586328. [4] G. E. Chatzarakis, R. Koplatadze and I. P. Stavroulakis, Oscillation criteria of first order linear difference equations with delay argument, Nonlinear Anal., 68 (2008), 994-1005.  doi: 10.1016/j.na.2006.11.055. [5] G. E. Chatzarakis, M. Lafci and I. P. Stavroulakis, Oscillation results for difference equations with several oscillating coefficients, Applied Mathematics and Computation, 251 (2015), 81-91.  doi: 10.1016/j.amc.2014.11.045. [6] G. E. Chatzarakis, Ch. G. Philos and I. P. Stavroulakis, On the oscillation of the solutions to linear difference equations with variable delay, Electron. J. Differ. Equ., 50 (2008), 1-15. [7] G. E. Chatzarakis and Ö. Öcalan, Oscillation of difference equations with non-monotone retarded arguments, Applied Mathematics and Computation, 258 (2015), 60-66.  doi: 10.1016/j.amc.2015.01.110. [8] G. E. Chatzarakis, H. Péics, S. Pinelas and I. P. Stavroulakis, Oscillation results for difference equations with oscillating coefficients Advances in Difference Equations 2015 (2015), 17pp. doi: 10.1186/s13662-015-0391-0. [9] L. H. Erbe and B. G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations, 2 (1989), 300-309. [10] G. Grzegorczyk and J. Werbowski, On oscillatory solutions of certain difference equations, Opuscula Mathematica, 26 (2006), 317-326. [11] G. Grzegorczyk and J. Werbowski, Oscillation of higher-order linear difference equations, Comput. Math. Appl., 42 (2001), 711-717.  doi: 10.1016/S0898-1221(01)00190-0. [12] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications Clarendon Press, Oxford, New York, 1991. [13] W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications Second edition, Harcourt/Academic Press, San Diego, CA, 2001. [14] G. Ladas, Ch. G. Philos and Y. G. Sficas, Sharp conditions for the oscillation of delay difference equations, J. Appl. Math. Simulation, 2 (1989), 101-111.  doi: 10.1155/S1048953389000080. [15] J. Migda, Approximative solutions of difference equations, Electron. J. Qual. Theory Differ. Equ. 13 (2014), 26 pp. [16] M. Migda, Existence of nonoscillatory solutions of some higher order difference equations, Appl. Math. E-Notes, 4 (2004), 33-39. [17] M. Migda and J. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations, Nonlinear Analysis, 47 (2001), 4687-4695.  doi: 10.1016/S0362-546X(01)00581-8. [18] M. Migda, Asymptotic properties of nonoscillatory solutions of higher order neutral difference equations, Opuscula Mathematica, 26 (2006), 507-514. [19] W. Nowakowska and J. Werbowski, On connections between oscillatory solutions of functional, difference and differential equations, Fasc. Math., 44 (2010), 95-106. [20] Ch. G. Philos, I. K. Purnaras and I. P. Stavroulakis, Sufficient conditions for the oscillation of delay difference equations, J. Difference Equ. Appl., 10 (2004), 419-435.  doi: 10.1080/10236190410001648239. [21] J. Werbowski, Bounded oscillations of differential equations generated by deviating arguments, Utilitas Math., 31 (1987), 191-198. [22] A. Wyrwińska, Oscillation criteria of a higher order linear difference equation, Bull. Inst. Math. Acad. Sinica, 22 (1994), 259-266.
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