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Kiguradzetype Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales


Published:20110308
Printed: Dec 2011
Jia Baoguo,
School of Mathematics and Computer Science, Zhongshan University, Guangzhou, China, 510275
Lynn Erbe,
Department of Mathematics, University of NebraskaLincoln, Lincoln, NE 685880130, U.S.A.
Allan Peterson,
Department of Mathematics, University of NebraskaLincoln, Lincoln, NE 685880130, U.S.A.
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 EmdenFowler 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.