Oscillation theorems for second-order half-linear neutral difference equations

Downloads

DOI:

https://doi.org/10.26637/mjm204/013

Abstract

In this article, some new oscillation criteria are established for the second order neutral difference equation of the form
$$
\Delta\left(a(n) \Delta(z(n))^\alpha\right)+q(n) x^\alpha(\sigma(n))=0, n \geq n_0
$$
where \(z(n)=x(n)+p(n) x(\tau(n))\). Our results improve and extend some known results in the literature. Some examples are also provided to show the importance of these results.

Keywords:

Half-linear, neutral, oscillation, difference equations

Mathematics Subject Classification:

39A10
  • R. Arul Department of Mathematics, Kandaswami Kandar’s College, Tamil Nadu - 638 182, India.
  • T.J. Raghupathi Department of Mathematics, Kandaswami Kandar’s College, Tamil Nadu - 638 182, India.
  • Pages: 460-471
  • Date Published: 01-10-2014
  • Vol. 2 No. 04 (2014): Malaya Journal of Matematik (MJM)

R. P. Agarwal, Difference Equations and Inequalities, Second Edition, Marcel Dekker, NewYork, 2000. DOI: https://doi.org/10.1201/9781420027020

R. P. Agarwal, M. Bohner and S. R. O'Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, New York, 2005. DOI: https://doi.org/10.1155/9789775945198

R. P. Agarwal, M. M. S. Manuel and E. Thandapani, Oscillatory and nonoscillatory behavior of second order neutral delay difference equations, Math. Comput. Model., 24 (1996), 5 -11. DOI: https://doi.org/10.1016/0895-7177(96)00076-3

D. D. Bainov and D. P. Mishev, Classification and existence of positive solutions of second order nonlinear neutral difference equations, Funk. Ekvae., 40(1997), 371-396.

Y. Bolat On the oscillation of higher order half-linear delay difference equation, Appl. Math. Inform Sci. 6(2012), 423-427.

J. Cheng, Kamanev-type oscillation criteria for delay difference equations, Acta Math. Sci., 27B(2007), 574-580. DOI: https://doi.org/10.1016/S0252-9602(07)60057-5

S. R. Grace and H. A. El-Morshedy, Oscillation criteria of comparison type for second order difference equations, J. Appl. Anal., 6(2000), 87-103. DOI: https://doi.org/10.1515/JAA.2000.87

B. S. Lalli and S. R. Grace, Oscillation theorems for second order delay and neutral difference equation, Utilitas Math., 45(1994), 197-212.

H. J. Li and C. C. Yeh Oscillation criteria for second order neutral delay difference equations, Comp.Math.Appl., 36(1998), 123-132. DOI: https://doi.org/10.1016/S0898-1221(98)80015-1

S. H. Saker, New oscillation criteria for second order nonlinear neutral delay difference equations, Appl. Math. Comput., 142(2003), 99-111. DOI: https://doi.org/10.1016/S0096-3003(02)00286-2

Y. G. Sun, S. H. Saker, Oscillation of second order nonlinear neutral delay difference equations, Appl. Math. Comput., 163(2005), 909 - 918. DOI: https://doi.org/10.1016/j.amc.2004.04.017

X. H. Tang and Y. Liu, Oscillation for nonlinear delay difference equations, Tamkang J. Math.,32(2001), 275-280. DOI: https://doi.org/10.5556/j.tkjm.32.2001.342

E.Thandapani, J. R. Greaf and P. W. Spikes, On the oscillation of solutions of second order quasilinear difference equations, Nonlin. World 3(1996), 545-565.

E. Thandapani, N. Kavitha and S. Pinelas Comparision and oscillation theorem for second order nonlinear neutral difference equations of mixed type, Dyn. Sys. Appl., 21(2012), 83-92.

E. Thandapani and P. Mohankumar, Oscillation and nonoscillation of nonlinear neutral delay difference equations, Tamkang J. Math., 38 (2007), 323-333. DOI: https://doi.org/10.5556/j.tkjm.38.2007.66

E. Thandapani and S. Selvarangam, Oscillation theorems for second order nonlinear neutral difference equations, J. Math. Comput. Sci., 2(2012), no.4, 866-879. DOI: https://doi.org/10.1186/1687-1847-2012-4

E. Thandapani, P. Sundaram and I. Gyori, Oscillaion of second order nonlinear neutral delay difference equations, Jour. Math. Phy. Sci., 31(1997), 121-132.

G. Zhang, Oscillation for nonlinear neutral difference equations, Appl. Math. E-Notes, 2(2002), 22-24.

  • NA

Metrics

Metrics Loading ...

Published

01-10-2014

How to Cite

R. Arul, and T.J. Raghupathi. “Oscillation Theorems for Second-Order Half-Linear Neutral Difference Equations”. Malaya Journal of Matematik, vol. 2, no. 04, Oct. 2014, pp. 460-71, doi:10.26637/mjm204/013.