Mathematica Bohemica, Vol. 143, No. 1, pp. 67-87, 2018


Oscillations of nonlinear difference equations with deviating arguments

George E. Chatzarakis, Julio G. Dix

Received May 14, 2016.  First published May 24, 2017.

Abstract:  This paper is concerned with the oscillatory behavior of first-order nonlinear difference equations with variable deviating arguments. The corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given.
Keywords:  infinite sum condition; retarded argument; advanced argument; oscillatory solution; nonoscillatory solution
Classification MSC:  39A10, 39A21


References:
[1] L. Berezansky, E. Braverman: On existence of positive solutions for linear difference equations with several delays. Adv. Dyn. Syst. Appl. 1 (2006), 29-47. MR 2287633 | Zbl 1124.39002
[2] G. E. Chatzarakis, R. Koplatadze, I. P. Stavroulakis: Optimal oscillation criteria for first order difference equations with delay argument. Pac. J. Math. 235 (2008), 15-33. DOI 10.2140/pjm.2008.235.15 | MR 2379767 | Zbl 1153.39010
[3] G. E. Chatzarakis, S. Pinelas, I. P. Stavroulakis: Oscillations of difference equations with several deviated arguments. Aequationes Math. 88 (2014), 105-123. DOI 10.1007/s00010-013-0238-2 | MR 3250787 | Zbl 1306.39007
[4] G. E. Chatzarakis, I. P. Stavroulakis: Oscillations of first order linear delay difference equations. Aust. J. Math. Anal. Appl. (electronic only) 3 (2006), Article ID 14, 11 pages. MR 2223018 | Zbl 1096.39003
[5] J. P. Dix, J. G. Dix: Oscillations of solutions to nonlinear first-order delay differential equations. Involve 9 (2016), 465-482. DOI 10.2140/involve.2016.9.465 | MR 3509339 | Zbl 06590119
[6] L. H. Erbe, Q. Kong, B. G. Zhang: Oscillation Theory for Functional-Differential Equations. Pure and Applied Mathematics 190. Marcel Dekker, New York (1994). MR 1309905 | Zbl 0821.34067
[7] L. H. Erbe, B. G. Zhang: Oscillation of discrete analogues of delay equations. Differ. Integral Equ. 2 (1989), 300-309. MR 0983682 | Zbl 0723.39004
[8] G. Ladas: Explicit conditions for the oscillation of difference equations. J. Math. Anal. Appl. 153 (1990), 276-287. DOI 10.1016/0022-247X(90)90278-N | MR 1080131 | Zbl 0718.39002
[9] G. Ladas, C. G. Philos, Y. G. Sficas: Sharp conditions for the oscillation of delay difference equations. J. Appl. Math. Simulation 2 (1989), 101-111. DOI 10.1155/S1048953389000080 | MR 1010549 | Zbl 0685.39004
[10] G. S. Ladde, V. Lakshmikantham, B. G. Zhang: Oscillation Theory of Differential Equations with Deviating Arguments. Pure and Applied Mathematics 110. Marcel Dekker, New York (1987). MR 1017244 | Zbl 0622.34071
[11] B. Li: Oscillation of first order delay differential equations. Proc. Am. Math. Soc. 124 (1996), 3729-3737. DOI 10.1090/S0002-9939-96-03674-X | MR 1363175 | Zbl 0865.34057
[12] X. Li, D. Zhu: Oscillation of advanced difference equations with variable coefficients. Ann. Differ. Equations 18 (2002), 254-263. MR 1940383 | Zbl 1010.39001
[13] X. N. Luo, Y. Zhou, C. F. Li: Oscillation of a nonlinear difference equation with several delays. Math. Bohem. 128 (2003), 309-317. MR 2012607 | Zbl 1055.39015
[14] X. H. Tang, J. S. Yu: Oscillation of delay difference equation. Comput. Math. Appl. 37 (1999), 11-20. DOI 10.1016/S0898-1221(99)00083-8 | MR 1688201 | Zbl 0937.39012
[15] X. H. Tang, R. Y. Zhang: New oscillation criteria for delay difference equations. Comput. Math. Appl. 42 (2001), 1319-1330. DOI 10.1016/S0898-1221(01)00243-7 | MR 1861531 | Zbl 1002.39022
[16] X. Wang: Oscillation of delay difference equations with several delays. J. Math. Anal. Appl. 286 (2003), 664-674. DOI 10.1016/S0022-247X(03)00508-0 | MR 2008855 | Zbl 1033.39017
[17] W. Yan, Q. Meng, J. Yan: Oscillation criteria for difference equation of variable delays. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13A (2006), Part 2, suppl., 641-647. MR 2219618
[18] B. G. Zhang, Y. Zhou: Oscillations of difference equations with several delays. Comput. Math. Appl. 44 (2002), 817-821. DOI 10.1016/S0898-1221(02)00193-1 | MR 1925823 | Zbl 1035.39010

Affiliations:   George E. Chatzarakis, Department of Electrical Engineering and Department of Electronic Engineering, School of Pedagogical and Technological Education (ASPETE) 14121, Heraklio, Athens, Greece, e-mail: geaxatz@otenet.gr, gea.xatz@aspete.gr; Julio G. Dix, Department of Mathematics, Texas State University, MCS583, Pickard St., San Marcos, TX78666, USA, e-mail: jd01@txstate.edu


 
PDF available at: