Interpolation between consecutive conjugate points of an $n$th order linear differential equation

Abstract:

The interpolation problem ${x^{(n)}} + {P_{n - 1}}{x^{(n - 1)}} + \cdots + {P_0}x = 0$, ${x^{(i)}}({t_j}) = 0,i = 0, \cdots ,{k_j} - 1,j = 0, \cdots ,m$, is studied on the conjugate interval $[a,{\eta _1}(a)]$. The main result is that there exists an essentially unique nontrivial solution of the problem almost everywhere, provided ${k_1} + \cdots + {k_m} \geq n$, and cer tain other inequalities are satisfied, with $a = {t_0} < {t_1} < \cdots < {t_m} = {\eta _1}(a)$. In particular, this paper corrects the results of Azbelev and Caljuk (Mat. Sb. 51 (93) (1960), 475-486; English transl., Amer. Math. Soc. Transl. (2) 42 (1964), 233-245) on third order equations, and shows that their results are correct almost everywhere.