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

Published online by Cambridge University Press:  20 November 2018

Jia Baoguo ,
Lynn Erbe  and
Allan Peterson
Jia Baoguo
Affiliation:
School of Mathematics and Computer Science, Zhongshan University, Guangzhou, China, 510275e-mail: mcsjbg@mail.sysu.edu.cn
Lynn Erbe
Affiliation:
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A.e-mail: erbe2@math.unl.edue-mail: apeterson1@math.unl.edu
Allan Peterson
Affiliation:
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A.e-mail: erbe2@math.unl.edue-mail: apeterson1@math.unl.edu
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Abstract

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Consider the second order superlinear dynamic equation

$$(*)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{x}^{\Delta \Delta }}(t)+p(t)f(x(\sigma (t)))=0$$

where $p\,\in \,C(\mathbb{T},\,\mathbb{R})$, $\mathbb{T}$ is a time scale, $f\,:\,\mathbb{R}\,\to \,\mathbb{R}$ is continuously differentiable and satisfies ${{f}^{'}}(x)>0$, and $x\,f\,(x)\,>\,0$ for $x\,\ne \,0$. 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 $\left( * \right)$ in the real case $\mathbb{T}\,=\,\mathbb{R}$, we obtain analogues in the difference equation and $q$-difference equation cases.

Keywords

34K1139A1039A99OscillationEmden–Fowler equationsuperlinear
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 4 , 01 December 2011 , pp. 580 - 592
Copyright
Copyright © Canadian Mathematical Society 2011

References

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