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

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 → ∞. Using these inequalities, it is also shown that (tm+κ-t m) → ∞ as m → ∞, where 1 ≤ κ ≤ n - 1 and < tm > is an increasing sequence of generalized zeros of an oscillatory solution of Dny+yf(t, y(σ(t)))|y(σ(t))| p-2 = 0, t ≥ 0, provided that W(., λ) ∈ L μ ([0,∞)T,ℝ +), 1 ≤ μ ≤ ∞ and for all λ > 0. A criterion for disconjugacy of nonlinear homogeneous dynamic equation is obtained in an interval [a, σ(b)] T.