Hiroshima Mathematical Journal

New oscillation criteria for second-order neutral dynamic equations on time scales via Riccati substitution

S. H. Saker and Donal O’Regan

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Abstract

In this paper, we consider the second-order nonlinear neutral functional dynamic equation

$\left( p(t)\left( \left[ y(t)+r(t)y(\tau (t))\right] ^{\Delta }\right) ^{\gamma }\right) ^{\Delta }+f(t,y(\delta (t)))=0$

on a time scale $\mathbb{T}$ and establish some new sufficient conditions for oscillation. Our results improve oscillation results for neutral delay dynamic equations on time scales and are new when $\delta (t)>t$ and/or $% 0<\gamma <1.$ Furthermore our results can be applied on the time scales $% \mathbb{T=}h\mathbb{T}$, for $h>0$, $\mathbb{T=}q^{\mathbb{N}}=\{t:t=q^{k}$% \textbf{, }$k\in \mathbb{N}$, $q>1$, $\mathbb{T=N}^{2}=\{t^{2}:t\in \mathbb{N% }\},$ $\mathbb{T}_{2}\mathbb{=}\{\sqrt{n}:n\in \mathbb{N}_{0}\},$ $\mathbb{T}% _{3}\mathbb{=}\{\sqrt[3]{n}:n\in \mathbb{N}_{0}\},$ and when $\mathbb{T=T}% _{n}=\{t_{n}:n\in \mathbb{N}_{0}\}$ where $\{t_{n}\}$ is the set of harmonic numbers, etc.

Article information

Source
Hiroshima Math. J. Volume 42, Number 1 (2012), 77-98.

Dates
First available: 30 March 2012

Permanent link to this document
http://projecteuclid.org/euclid.hmj/1333113007

Zentralblatt MATH identifier
06040401

Mathematical Reviews number (MathSciNet)
MR2952073

Subjects
Primary: 34K11: Oscillation theory 39A10: Difference equations, additive 39A99: None of the above, but in this section

Keywords
Oscillation second-order neutral dynamic equation time scales

Citation

Saker, S. H.; O’Regan, Donal. New oscillation criteria for second-order neutral dynamic equations on time scales via Riccati substitution. Hiroshima Mathematical Journal 42 (2012), no. 1, 77--98. http://projecteuclid.org/euclid.hmj/1333113007.


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