Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 64, pp. 1-15.

Title: Necessary and sufficient conditions for the oscillatory 
       and asymptotic behaviour of solutions to neutral delay 
       dynamic equations
   
Authors: Basak Karpuz (Afyon Kocatepe Univ., Afyonkarahisar, Turkey)
         Ozkan Ocalan (Afyon Kocatepe Univ., Afyonkarahisar, Turkey)
	 Radhanath Rath (Veer Surendra Sai Univ., Orissa, India)

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.