Convergence of the Solutions for a Neutral Difference Equation with Negative Coefficients

Chatzarakis, George E.; Miliaras, George N.

doi:10.2478/tmmp-2013-0003

Abstract

In this paper, we investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form

where τj (n), j = 1, . . . , w are general retarded arguments, σ(n) is a general deviated argument, [###] is a sequence of positive real numbers such that p(n) ≥ p, p ∈ R+, and Δ denotes the forward difference operator Δx(n) = x(n + 1) − x(n).

References

[1] AGARWAL, R. P.-BOHNER, M.-GRACE, S. R.-O’REGAN, D.: Discrete OscillationTheory. Hindawi Publishing Corporation, New York, 2005.10.1155/9789775945198
Search in Google Scholar
[2] BAŠTINEC, J.-DIBLÍK, J.-ŠMARDA, Z.: Oscillation of solutions of a linear second--order discrete-delayed equation, Adv. Difference Equ. Vol. 2010, Article ID 693867, 12 p., 2010.
Search in Google Scholar
[3] BAŠTINEC, J.-DIBLÍK, J.-ŠMARDA, Z.: Existence of positive solutions of discretelinear equations with a single delay, J. Difference Equ. Appl. 16 (2010), 1047-1056.10.1080/10236190902718026
Search in Google Scholar
[4] BELLMAN, R.-COOKE, K. L.: Differential-Difference Equations. Academic Press, New York, 1963.10.1016/B978-0-12-395651-4.50022-2
Search in Google Scholar
[5] BRAYTON, R. K.-WILLOUGHBY, R. A.: On the numerical integration of a symmetricsystem of difference-differential equations of neutral type, J. Math. Anal. Appl. 18 (1967), 182-189.10.1016/0022-247X(67)90191-6
Search in Google Scholar
[6] BRUMLEY, W. E.: On the asymptotic behavior of solutions of differential-differenceequations of neutral type, J. Difference Equ. 7 (1970), 175-188.10.1016/0022-0396(70)90131-2
Search in Google Scholar
[7] CHATZARAKIS, G. E.-KARAKOSTAS, G. L.-STAVROULAKIS, I. P.: Convergenceof the positive solutions of a nonlinear neutral difference equation, Nonlinear Oscill. 14 (2011), 407-418.
Search in Google Scholar
[8] CHATZARAKIS, G. E.-MILIARAS, G. N.: Convergence and divergence of the solutionsof a neutral difference equation, J. Appl. Math., Vol. 2011, Article ID 262316, 18 p., 2011.
Search in Google Scholar
[9] DIBLÍK, J.-RUŽIČKOVÁ, M.-ŠMARDA, Z.-ŠUTÁ, Z.: Asymptotic convergence ofthe solutions of a dynamic equation on discrete time scales, Abstr. Appl. Anal., Vol. 2012, Article ID 580750, 20 p., 2012.
Search in Google Scholar
[10] GEORGIOU, D. A.-GROVE, E. A.-LADAS, G.: Oscillation of neutral difference equationswith variable coefficients, in: Differential Equations, Stability and Control, Lecture Notes in Pure and Appl. Math., Vol. 127, Dekker, New York, 1991, pp. 165-173.
Search in Google Scholar
[11] GY˝ORI, I.-HORVÁTH, L.: Asymptotic constancy in linear difference equations: limitformulae and sharp conditions, Adv. Difference Equ., Vol. 2010, Article ID 789302, 20 p., 2010.
Search in Google Scholar
[12] GY˝ORI, I.-LADAS, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford, 1991.
Search in Google Scholar
[13] HUNG, D. C.: Oscillation and convergence for a neutral difference equation, J. Sci., Math.-Phys. 24 (2008), 133-143.
Search in Google Scholar
[14] JIA, J.-ZHONG, X.-GONG, X.-QUYANG, R.-HAN, H.: Nonoscillation of first--order neutral difference equation, Mod. Appl. Sci. 3 (2009), 30-33.10.5539/mas.v3n11p30
Search in Google Scholar
[15] LALLI, B. S.-ZHANG, B. G.: On existence of positive solutions and bounded oscillationsfor neutral difference equations, J. Math. Anal. Appl. 166 (1992), 272-287.10.1016/0022-247X(92)90342-B
Search in Google Scholar
[16] LALLI, B. S.-ZHANG, B. G.-LI, J. Z.: On the oscillation of solutions of neutral differenceequations, J. Math. Anal. Appl. 158 (1991), 213-233.10.1016/0022-247X(91)90278-8
Search in Google Scholar
[17] MIGDA, M.-ZHANG, G.: On unstable neutral difference equations with maxima, Math. Slovaca 56 (2006), 451-463.
Search in Google Scholar
[18] PEICS, H.: Positive solutions of neutral delay difference equation, Novi Sad J. Math. 35 (2005), 111-122.
Search in Google Scholar
[19] SNOW, W.: Existence, uniqueness and stability for nonlinear differential-difference equationsin the neutral case, N. Y. U. Courant, Inst. Math. Sci., IMM-NYU 328 (February 1965).
Search in Google Scholar
[20] TANG, X. H.: Asymptotic behavior of solutions for neutral difference equations, Comput. Math. Appl. 44 (2002), 301-315.10.1016/S0898-1221(02)00149-9
Search in Google Scholar
[21] TANG, X. H.-CHENG, S. S.: Positive solutions of a neutral difference equation withpositive and negative coefficients, Georgian Math. J. 11 (2004), 177-185.10.1515/GMJ.2004.177
Search in Google Scholar
[22] THANDAPANI, E.-KUMAR, P. M.: Oscillation and nonoscillation of nonlinear neutraldelay difference equations, Tamkang J. Math. 38 (2007), 323-333.10.5556/j.tkjm.38.2007.66
Search in Google Scholar
[23] THANDAPANI, E.-ARUL, R.-RAJA, P. S.: The asymptotic behavior of nonoscillatorysolutions of nonlinear neutral type difference equations, Math. Comput. Modelling 39 (2004), 1457-1465.10.1016/j.mcm.2004.07.004
Search in Google Scholar
[24] THANDAPANI, E.-MARIAN, S. L.-GRAEF, J. R.: Asymptotic behavior of nonoscillatory solutions of neutral difference equations, Comput. Math. Appl. 45 (2003), 1461-1468.10.1016/S0898-1221(03)00107-X
Search in Google Scholar
[25] WANG, X.: Asymptotic behavior of solutions for neutral difference equations, Comput. Math. Appl. 52 (2006), 1595-1602.10.1016/j.camwa.2005.08.037
Search in Google Scholar
[26] WEI, J.: Asymptotic behavior results for nonlinear neutral delay difference equations, Appl. Math. Comput. 217 (2011), 7184-7190.
Search in Google Scholar
[27] YU, J. S.-WANG, Z. C.: Asymptotic behavior and oscillation in neutral delay differenceequations, Funkcial. Ekvac. 37 (1994), 241-248.
Search in Google Scholar

Convergence of the Solutions for a Neutral Difference Equation with Negative Coefficients

Abstract

Paradigm

My account