Electron. J. Diff. Eqns., Vol. 2009(2009), No. 64, pp. 1-15.

Necessary and sufficient conditions for the oscillatory and asymptotic behaviour of solutions to neutral delay dynamic equations

Basak Karpuz, Ozkan Ocalan, Radhanath Rath

Abstract:
This article concerns the asymptotic behaviour of solutions to nonlinear first-order neutral delay dynamic equations involving coefficients with opposite signs. We present necessary and sufficient conditions for the solutions to oscillate or to converge to zero. The coefficient associated with the neutral part is considered in three distinct ranges, in one of which the coefficient is allowed to oscillate. Illustrative examples show that the existing results do not apply to these examples and hence they show the significance of our results. The results of this article are also new for the particular choices of the time scale $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{Z}$.

Submitted March 10, 2009. Published May 12, 2009.
Math Subject Classifications: 39A10, 39A11.
Key Words: Asymptotic behavior; neutral dynamic equations; nonoscillation; oscillation; time scale.

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Basak Karpuz
Department of Mathematics, Facaulty of Science and arts
ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey
email: bkarpuz@gmail.com http://www2.aku.edu.tr/~bkarpuz
Ozkan Ocalan
Department of Mathematics, Faculty of Science and Arts
ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey
email: ozkan@aku.edu.tr http://www2.aku.edu.tr/~ozkan
Radhanath Rath
Department of Mathematics
Veer Surendra Sai University of Technology
Burla, Sambalpur, 768018 Orissa, India
email: radhanathmath@yahoo.co.in

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