Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 28, pp. 1-14.
Title: Liapunov-type integral inequalities for certain
higher-order differential equations
Author: Saroj Panigrahi (Univ. of Hyderabad, India)
Abstract:
In this paper, we obtain Liapunov-type integral inequalities
for certain nonlinear, nonhomogeneous differential equations
of higher order with without any restriction on the zeros of
their higher-order derivatives of the solutions by using elementary
analysis. As an applications of our results, we show that oscillatory
solutions of the equation converge to zero as $t\to \infty$.
Using these inequalities, it is also shown that
$(t_{m+ k} - t_{m}) \to \infty $ as
$m \to \infty$, where $1 \le k \le n-1$ and $\langle t_m \rangle $ is an
increasing sequence of zeros of an oscillatory solution of
$ D^n y + y f(t, y)|y|^{p-2} = 0$, $t \ge 0$, provided that
$W(., \lambda) \in L^{\sigma}([0, \infty), \mathbb{R}^{+})$,
$1 \le \sigma \le \infty$ and for all $\lambda > 0$. A criterion for
disconjugacy of nonlinear homogeneous equation is obtained in an
interval $[a, b]$.
Submitted October 19, 2008. Published February 05, 2009.
Math Subject Classifications: 34C10.
Key Words: Liapunov-type inequality; oscillatory solution; disconjugacy;
higher order differential equations.