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

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 → ∞. Using these inequalities, it is also shown that (tm+k - t m) → ∞ as m → ∞, where 1 ≤ k ≤ n - 1 and (tm) is an increasing sequence of zeros of an oscillatory solution of Dny + yf(t, y)|y|p-2 = 0, t ≥ 0, provided that W{.,λ) ∈ Lσ([0, ∞), ℝ+), 1 ≤ σ ≤ ∞ and for all λ > 0. A criterion for disconjugacy of nonlinear homogeneous equation is obtained in an interval [a, b]. © 2009 Texas State University Published February 5, 2009.