On nonoscillatory linear differential equations of second order

For the equationy" + q(t)y = 0, we derive necessary conditions for nonoscillation in terms of "general means" of Q (t) f q ds. 1. Main theorem. The object of this note is to obtain necessary conditions for (1.1) y" + q(t)y = 0, q E CO[ 0, w), to be nonoscillatory at t = X ( 0 for which the arguments of [2] are applicable. Roughly, the conditions imposed on K will include the following three types: first, a generalization of the condition that the map Q i-+ KQ in (1.2) is regular in the Received by the editors September 16, 1976. AMS (MOS) subject classifications (1970). Primary 34C10. 'This study was supported by NSF Grant MPS75-15733. C American Mathematical Society 1977