Electron. J. Diff. Equ., Vol. 2011 (2011), No. 125, pp. 1-14.

Kwong-Wong-type integral equation on time scales

Baoguo Jia

Abstract:
Consider the second-order nonlinear dynamic equation
$$ [r(t)x^\Delta(\rho(t))]^\Delta+p(t)f(x(t))=0, $$
where $p(t)$ is the backward jump operator. We obtain a Kwong-Wong-type integral equation, that is: If $x(t)$ is a nonoscillatory solution of the above equation on $[T_0,\infty)$, then the integral equation
$$ \frac{r^\sigma(t)x^\Delta(t)}{f(x^\sigma(t))} =P^\sigma(t)+\int^\infty_{\sigma(t)}\frac{r^\sigma(s) [\int^1_0f'(x_h(s))dh][x^\Delta(s)]^2}{f(x(s)) f(x^\sigma(s))}\Delta s $$
is satisfied for $t\geq T_0$, where $P^\sigma(t)=\int^\infty_{\sigma(t)}p(s)\Delta s$, and $x_h(s)=x(s)+h\mu(s)x^\Delta(s)$. As an application, we show that the superlinear dynamic equation
$$ [r(t)x^{\Delta}(\rho(t))]^\Delta+p(t)f(x(t))=0, $$
is oscillatory, under certain conditions.

Submitted July 21, 2011. Published September 29, 2011.
Math Subject Classifications: 34K11, 39A10, 39A99.
Key Words: Nonlinear dynamic equation; integral equation; nonoscillatory solution.

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Baoguo Jia
School of Mathematics and Computer Science
Zhongshan University, Guangzhou, 510275, China
email: mcsjbg@mail.sysu.edu.cn

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