Liapunov-type integral inequalities for higher order dynamic equations on time scales

Pages: 629 - 641, Issue special, November 2013

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Saroj Panigrahi - Department of Mathematics and Statistics, University of Hyderabad, Hyderabad-500 046, India (email)

Abstract: In this paper, we obtain Liapunov-type integral inequalities for certain nonlinear, nonhomogeneous dynamic equations of higher order without any restriction on the zeros of their higher-order delta derivatives of solutions by using time scale 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 generalized zeros of an oscillatory solution of $ D^n y + y f(t, y(\sigma(t)))|y(\sigma(t))|^{p-2} = 0$, $t \ge 0$, provided that $W(., \lambda) \in L^{\mu}([0, \infty)_{\mathbb{T}}, \mathbb{R}^{+})$, $1 \le \mu \le \infty$ and for all $\lambda > 0$. A criterion for disconjugacy of nonlinear homogeneous dynamic equation is obtained in an interval $[a, \sigma(b)]_{\mathbb{T}}$.

Keywords:  Liapunov-type inequality, disconjugacy, oscillatory solutions, dynamic equations, time scales.
Mathematics Subject Classification:  34C10, 34N05, 26E70.

Received: August 2012;      Revised: November 2012;      Published: November 2013.