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Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales

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Published:2011-03-08
Printed: Dec 2011
• Jia Baoguo,
School of Mathematics and Computer Science, Zhongshan University, Guangzhou, China, 510275
• Lynn Erbe,
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A.
• Allan Peterson,
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A.
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Abstract

Consider the second order superlinear dynamic equation \begin{equation*} (*)\qquad x^{\Delta\Delta}(t)+p(t)f(x(\sigma(t)))=0\tag{$*$} \end{equation*} where $p\in C(\mathbb{T},\mathbb{R})$, $\mathbb{T}$ is a time scale, $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is continuously differentiable and satisfies $f'(x)>0$, and $xf(x)>0$ for $x\neq 0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which includes the nonlinear function $f(x)=x^\alpha$ with $\alpha>1$, commonly known as the Emden--Fowler case. Here the coefficient function $p(t)$ is allowed to be negative for arbitrarily large values of $t$. In addition to extending the result of Kiguradze for \eqref{star1} in the real case $\mathbb{T}=\mathbb{R}$, we obtain analogues in the difference equation and $q$-difference equation cases.
 Keywords: Oscillation, Emden-Fowler equation, superlinear
 MSC Classifications: 34K11 - Oscillation theory 39A10 - Difference equations, additive 39A99 - None of the above, but in this section

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