Disfocality and Liapunov-type inequalities for third-order equations Original Research Article

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, tϵ (a, b). If y(t) is a solution of (∗) with y(t1) = 0 = y (t2) = y(t3) = y(t4 ) (t1 < t2 < t3 < t4) and y(t) ≠ 0 for tϵ ∪3i=1(ti,ti+1), then the lower bound for (t4 − t1) is obtained. A new criteria is obtained for disconjugacy of (∗) in [a, b].