Nonoscillation and eventual disconjugacy

Abstract

If every solution of an nth order linear differential equation has only a finite number of zeros in [ 0 , ∞ ) [0,\infty ) , it is not generally true that for sufficiently large c , c > 0 c,c > 0 , every solution has at most n − 1 n - 1 zeros in [ c , ∞ ) [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 {L_n}y + p(x)y = 0 , where L n {L_n} is an nth order disconjugate differential operator and p ( x ) p(x) is a continuous function of a fixed sign.