Criteria for Disfocality and Disconjugacy for Third Order Differential Equations

Abstract

In this paper, lower bounds for the spacing (b - a) of the zeros of the solutions and the zeros of the derivative of the solutions of third order differential equations of the form y′″ + q(t)y′ + p(t)y =0 (*) are derived under the some assumptions on p and q. The concept of disfocality is introduced for third order differential equations (*). This helps to improve the Liapunov-type inequality, when y(t) is a solution of (*) with (i) y{a) =0 =y′(b) or y′(a) =0 =y(b) with y(t) ≠ 0, t ∈ (a,b) or (ii) y(a) =0 =y′(a), y(b) =0 =y′(b) with y(t) ≠0, t ∈ (a, b). If y(t) is a solution of (*) with y{ti) =0, I ≤ i ≤ n, n ≥ 4, (t1 < t2 < ... < tn) and y(t) ≠ 0, t ∈ Ui=n-1i=1(t i, ti+ i), then lower bound for spacing (tn - t1) is obtained. A new criteria for disconjugacy is obtained for (*) in [a, b]. This papers improves many known bounds in the literature.