This article concerns oscillatory and asymptotic properties at infinity for differential equations of the type
with , , , and for all . In the linear case , will be denoted . Seven theorems give connections between oscillation and properties A or B of or , as defined by I. T. Kiguradze and Z. A. Chanturiya [Mathematics and its Applications, Soviet Series 89, Dordrecht: Kluwer Academic Publ. (1993; Zbl 0782.34002)]. The main theorems in the linear case state that has at least one nontrivial oscillatory solution if and only if it has property A [property B, respectively], extending results of M. Greguš [Third order linear differential equations. Mathematics and its Applications. D. Reidel Publ. Comp. (1987; Zbl 0602.34005)] and M. Gera [Acta Math. Univ. Comenianae 46/47, 189-203 (1985; Zbl 0612.34029)]. Corollaries provide sufficient conditions on , for equivalence of (i) property A for and property B for its adjoint equations; and (ii) property B for and property A for its adjoint.
In the second part of the paper these results are applied to generate sufficient conditions for to have property A and for to have property B. Related results of the authors are given in [Ann. Mat. Pura Appl., IV. Ser. 173, 373-389 (1997) (to appear) and Nonlinear Anal., Theory Methods Appl. 30, No. 3, 1583-1594 (1997; Zbl 0892.34032)].