Nonoscillation and eventual disconjugacy

Abstract:

If every solution of an nth order linear differential equation has only a finite number of zeros in $[0,\infty )$, it is not generally true that for sufficiently large $c,c > 0$, every solution has at most $n - 1$ zeros in $[c,\infty )$. Settling a known conjecture, we show that for any n, the above implication does hold for a special type of equation, ${L_n}y + p(x)y = 0$, where ${L_n}$ is an nth order disconjugate differential operator and $p(x)$ is a continuous function of a fixed sign.