Disconjugacy and the limit-point condition for fourth-order operators

Abstract

SynopsisLetLdenote the ordinary differential operator given byLf= (pf″)″ + (qf′)′ +rf, withp″,q′ andrcontinuous functions on [0,∞), and withp>0,q≦ 0, andr≧ 0. It is proved that if the equationLg= 0 possesses a non-oscillatory solution, then any non-trivial solutionftoLf= 0 such thatf(0) =f′(0) = 0 is eventually bounded away from zero.This theorem is used to prove that, for a general class of functionsqandrcontaining the polynomials as a very special case, the equationLg= 0 has at most two linearly independent square integrable solutions, whenpis identically one,q≦ 0 andr≧ 0.Finally, the main theorem is applied to show that certain sixth-order operators are limit-3.