We study the asymptotic behavior of solutions of the higher-order neutral difference equation
where τ (n) is a general retarded argument, σ(n) is a general deviated argument, c ∈ \( \mathbb{R} \); (p(n)) n ≥ 0 is a sequence of real numbers, ∆ denotes the forward difference operator ∆x(n) = x(n+1) - x(n); and ∆j denotes the jth forward difference operator ∆j (x(n) = ∆ (∆j-1(x(n))) for j = 2, 3,…,m. Examples illustrating the results are also given.
Similar content being viewed by others
References
R. P. Agarwal, M. Bochner, S. R. Grace, and D. O’Regan, Discrete Oscillation Theory, Hindawi Publ. Co., New York (2005).
R. P. Agarwal and S. R. Grace, “Oscillations of higher-order nonlinear difference equations of neutral type,” Appl. Math. Lett., 12, 77–83 (1999).
R. P. Agarwal, E. Thandapani, and P. J. Y. Wong, “Oscillations of higher-order neutral difference equations,” Appl. Math. Lett., 10, 71–78 (1997).
G. E. Chatzarakis, G. L. Karakostas, and I. P. Stavroulakis, “Convergence of the positive solutions of a nonlinear neutral difference equation,” Nonlinear Oscillations, 14, No. 3, 407–418 (2011).
G. E. Chatzarakis and G. N. Miliaras, “Convergence and divergence of the solutions of a neutral difference equation,” J. Appl. Math., Art. ID 262316 (2011), 18 p.
G. E. Chatzarakis, G. N. Miliaras, I. P. Stavroulakis, and E. Thandapani, “Asymptotic behavior of first-order neutral difference equations with general arguments,” Panamer. Math. J., 23, 111–129 (2013).
D. A. Georgiou, E. A. Grove, and G. Ladas, “Oscillation of neutral difference equations,” Appl. Anal., 33, 243–253 (1989).
S. R. Grace and B. S. Lalli, “Oscillation theorems for second order delay and neutral difference equations,” Util. Math., 45, 197–211 (1994).
I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford (1991).
X. Gong, X. Zhong, J. Jia, and R. Quyang H. Han, “Oscillation of first-order neutral difference equation,” Modern Appl. Sci., 3, No. 8, 90–94 (2009).
D. C. Huong, “Oscillation and convergence for a neutral difference equation,” VNU J. Sci., Math.-Phys., 24, 133–143 (2008).
V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Acad. Publ., Dordrecht (1993).
G. Ladas and G. Qian, “Comparison results and linearized oscillations for higher order difference equations,” Int. J. Math. Math. Sci., 15, 129–142 (1992).
B. S. Lalli and B. G. Zhang, “On existence of positive solutions and bounded oscillations for neutral difference equations,” J. Math. Anal. and Appl., 166, 272–287 (1992).
W.-T. Li and R. P. Agarwal, “Positive solutions of higher-order nonlinear delay difference equations,” Comp. Math. Appl., 45, 1203–1211 (2003).
A. M. Samoilenko, “On a problem of the investigation of global solutions of linear differential equations with deviating argument,” Ukr. Mat. Zh., 55, No. 5, 631–640 (2003); English transl.: Ukr. Math. J., 55, No. 5, 761–772 (2003).
A. M. Samoilenko and A. G. Pelyukh, “Existence and uniqueness of an analytic solution of a nonlinear differential-functional equation of neutral type,” Nonlinear Oscillations, 2, No. 2, 225–230 (1999).
B. Szmanda, “Properties of solutions of higher order difference equations,” Math. Comput. Modelling, 28, 95–101 (1998).
M. K. Yildiz and Ö. Öcalan, “Oscillation results for higher order nonlinear neutral delay difference equations,” Appl. Math. Lett., 20, 13–17 (2007).
J. S. Yu and Z. C.Wang, “Asymptotic behavior and oscillation in neutral delay difference equations,” Funkc. Ekvacioj., 37, 243–247 (1994).
A. Zafer, “Oscillatory and asymptotic behavior of higher order difference equations,” Math. Comput. Modelling, 21, 43–50 (1995).
G. Zhang and S. S. Cheng, “Oscillation criteria for a neutral difference equation with delay,” Appl. Math. Lett., 8, 13–17 (1995).
X. Zhou, “Oscillatory and asymptotic properties of higher order nonlinear neutral difference equations with oscillating coefficients,” Appl. Math. Lett., 21, 1142–1148 (2008).
Y. Zhou and B. G. Zhang, “Existence and nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients,” Comput. Math. Appl., 45, 991–1000 (2003).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 3, pp. 430–450, March, 2013.
Rights and permissions
About this article
Cite this article
Chatzarakis, G.E., Khatibzadeh, H., Miliaras, G.N. et al. Asymptotic behavior of higher-order neutral difference equations with general arguments. Ukr Math J 65, 478–499 (2013). https://doi.org/10.1007/s11253-013-0790-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0790-x