Criteria for disfocality and disconjugacy for third order differential equations

S. Panigrahi, University of Hyderabad, Hyderabad, India

E. J. Qualitative Theory of Diff. Equ., No. 23. (2009), pp. 1-17.

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) \ne 0, t \in (a,b) $ or (ii) $ y(a) = 0 = y'(a), y(b) = 0 = y'(b)$ with $ y(t)\ne 0, t\in (a,b)$. If y(t) is a solution of (*) with $ y(t_{i}) = 0, 1 \le i \le n, n\ge 4, (t_{1} <t_{2} < ...< t_{n} )$ and $ y(t) \neq 0, t \in \bigcup_{i = 1}^{i =n - 1} (t_{i}, t_{i+1})$, then lower bound for spacing $(t_{n}-t_{1})$ is obtained. A new criteria for disconjugacy is obtained for (*) in $[a,b]$.This papers improves many known bounds in the literature.

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