Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 125, pp. 1-14.
Title: Kwong-Wong-type integral equation on time scales
Author: Baoguo Jia (Zhongshan Univ., Guangzhou, China)
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.