Characterization of non-disconjugacy for a one parameter family of n-th order linear differential equations

Abstract

The aim of this paper is to obtain different criteria which allow us to affirm that the one parameter family of $n^{\mathrm{th}}-$order linear differential equations, given by the following expression \[ T_n[M]\,u(t) \equiv u^{(n)}(t)+a_1(t)\, u^{(n-1)}(t)+\cdots +a_{n-1}(t)\, u'(t)+(a_{n}(t)+M)\,u(t)=0 \,,\quad t\in I\equiv[a,b]\,, \] is not disconjugate for every $M\in \mathbb{R}$. Three different sufficient criteria, which ensure that such property holds, are presented. Moreover, a characterization of this property is given. To finish the paper, three examples, where the different criteria are applied, are shown.