Mariam Manjikashvili, Sulkhan Mukhigulashvili: Necessary and sufficient conditions of disconjugacy for the fourth order linear ordinary differential equations, 341-353

Abstract:

We study the disconjugacy of the fourth order linear ordinary differential equation

$$u^{(4)}(t)=p(t)u(t),$$

on the interval $[a,\,b].$ We find necessary and sufficient conditions for the disconjugacy on $[a,\,b]$, which have the comparison theorems character. Our results complete Kondrat'ev's second comparison theorem for the case of the fourth order ODE. The above mentioned conditions significantly improve Coppel's well-known condition which guarantees the disconjugacy of our equation for not necessarily constant sign coefficient $p,$ and generalise some optimal disconjugacy results proved for constant-coefficient equations.

Key Words: Disconjugacy, necessary and sufficient conditions, comparison theorem, $4$th order ordinary differential equations.

2010 Mathematics Subject Classification: Primary 34C10; Secondary 34B05.