Electron. J. Diff. Eqns., Vol. 2009(2009), No. 28, pp. 1-14.

Liapunov-type integral inequalities for certain higher-order differential equations

Saroj Panigrahi

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 5, 2009.
Math Subject Classifications: 34C10.
Key Words: Liapunov-type inequality; oscillatory solution; disconjugacy; higher order differential equations.

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Saroj Panigrahi
Department of Mathematics and Statistics
University of Hyderabad, Hyderabad 500 046, India
email: spsm@uohyd.ernet.in, panigrahi2008@gmail.com

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