Disfocality and Liapunov-type inequalities for third-order equations

Abstract

The concept of disfocality is introduced for third-order differential equations y ‴ + p(t)y = 0. This helps to improve the Liapunov inequality when y(t) is a solution of (*) with y(a) = 0 = y′(a), y(b) = 0 = y′(b), and y(t) ≠ 0, t ε (a, b). If y(t) is a solution of (*) with y(t 1) = 0 = y (t 2) = y(t 3) = y(t 4) (t 1 < t 2 < t 3 < t 4) and y(t) ≠ 0 for t ε ∪ 3i=1(t i,t i+1), then the lower bound for (t 4-t 1) is obtained. A new criteria is obtained for disconjugacy of (*) in [a, b]. © 2003 Elsevier Science Ltd. All rights reserved.