COMPARISON THEOREMS ON THE OSCILLATION OF A CLASS OF NEUTRAL DIFFERENCE EQUATIONS WITH CONTINUOUS VARIABLES

Title & Authors
COMPARISON THEOREMS ON THE OSCILLATION OF A CLASS OF NEUTRAL DIFFERENCE EQUATIONS WITH CONTINUOUS VARIABLES
Karpuz, Basak; Ocalan, Ozkan;

Abstract
In this paper, we introduce an iterative method to study oscillatory properties of delay difference equations of the following form $\small{{\nabla}_{\alpha}\;[x(t)\;-\;r(t)x(t\;-\;k)]\;+\;p(t)x(t\;-\;{\tau})\;-\;q(t)x(t\;-\;{\sigma})\;=\;0}$, $\small{t\;{\geq}\;t_0}$, where $\small{t_0\;{\in}\;\mathbb{R}}$, t varies in the real interval ($\small{t_0,\;{\infty}}$), $\small{\alpha}$ > 0, $\small{\kappa}$, $\small{\tau}$, $\small{{\sigma}\;{\geq}\;0}$, $\small{r\;{\in}\;C\;([t_0-{\alpha},\;{\infty}),\;\mathbb{R}^+}$, p, $\small{q\;{\in}\;C\;([t_0,\;{\infty}),\;\mathbb{R}^+)}$ and $\small{{\nabla}_{\alpha}x(t)\;=\;x(t)\;-\;x(t\;-\;{\alpha})}$ for $\small{t\;{\geq}\;t_0}$.
Keywords
continuous variable;neutral difference equations;oscillation;positive and negative coefficients;
Language
English
Cited by
References
1.
I. Gyori and G. Ladas, Oscillation Theory of Delay Difference Equations with Applications, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991.

2.
X. Li and D. Zhu, Comparison theorems of oscillation and nonoscillation for neutral difference equations with continuous arguments, Commun. Pure Appl. Anal. 2 (2003), no. 4, 579–589.

3.
C. G. Philos, Oscillation for first order linear delay differential equations with variable coefficients, Funkcial. Ekvac. 35 (1992), no. 2, 307–319.

4.
J. H. Shen, Comparison theorems for oscillations of difference equations with continuous variables, Kexue Tongbao (Chinese) 41 (1996), no. 16, 1441–1444.

5.
B. G. Zhang, Oscillations of a class of difference equations with continuous arguments, Appl. Math. Lett. 14 (2001), no. 5, 557–561.

6.
B. G. Zhang and B. M. Liu, Linearized oscillation theorems for certain nonlinear difference equations with continuous arguments, Math. Comput. Modelling 30 (1999), no. 1-2, 89–96.

7.
B. G. Zhang, J. Yan, and S. K. Choi, Oscillation for difference equations with continuous variable, Comput. Math. Appl. 36 (1998), no. 9, 11–18.

8.
Y. Zhang and J. Yan, Oscillation criteria for difference equations with continuous arguments, Acta Math. Sinica 38 (1995), no. 3, 406–411.

9.
Y. Zhou, Oscillation of difference equations with variable coefficients and continuous arguments, Economic Math. 13 (1996), no. 1, 86–89.