Disfocality and Liapunov-type inequalities for third-order equations

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) not equal 0, t is an element of (a, b). If y(t) is a solution of (*) with y(t(1)) = 0 = y(t(2)) = y(t(3)) = (t(4)) (t(1) < t(2) < t(3) < t(4)) and y(t) not equal 0 for t is an element of U-i=1(3) (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]. (C) 2003 Elsevier Science Ltd. All rights reserved.