proceedings of the american mathematical society Volume 123, Number 5, May 1995 OSCILLATION AND NONOSCILLATION CRITERIA FOR DELAY DIFFERENTIALEQUATIONS Á. ELBERT AND I. P. STAVROULAKIS (Communicated by Hal L. Smith) Abstract. Oscillation and nonoscillation criteria for the first-order delay dif- ferential equation x'{t)+p(t)x(T(t)) = 0, t > t0, T(0 < t, are established in the case where 1 / p(s) ds > - and lim / p(s) ds ■■ Jz(t) e í->°o yT(/) 1. Introduction The qualitative properties of the solutions of the delay differential equation (1) x'(t)+p(t)x(x(t)) = 0, t>t0, where r(t) < t, have been the subject of many investigations. The first system- atic study was made by Myshkis [6]. Among others he has shown [5] that all solutions of (1) oscillate if p(t)>0, limsup[í - t(/)] < oo, liminf[t-T(t)]-liminfp(t) > -. í-»oo '-*00 t-y°° e Later these conditions were improved, by Ladas [4] and Koplatadze and Chan- turija [3], to /" 1 (2) liminf / p(s)ds > -. '^°° Jx(t) e Concerning the constant £ in (2) we mention that if the inequality e p(s)ds < - 'r(t) f Jrli holds, then, according to a result in [3], (1) has a nonoscillatory solution. To the best of our knowledge there is no result in the case when we have f 1 f' 1 (3) / p(s)ds>- and lim / p(s)ds--. Jx(t) e '^°°Jx(t) e Received by the editors August 26, 1993. 1991 Mathematics Subject Classification. Primary 34K15; Secondary 34C10. Key words and phrases. Oscillation, nonoscillation, delay differential equations. ©1995 American Mathematical Society 0002-9939/95 $1.00+ $.25 per page 1503 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use