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On third order differential equations with property A and B. (English) Zbl 0926.34025

This article concerns oscillatory and asymptotic properties at infinity for differential equations of the type

y (3) -q(x)y ' ±r(x)f(y)=0,x + ,(E ± )

with q,rC( + , + ), fC(,), r(x)>0, and tf(t)>0 for all t0. In the linear case f(t)t, (E ± ) will be denoted (L ± ). Seven theorems give connections between oscillation and properties A or B of (L ± ) or (E ± ), 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 (L + )[(L - )] 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 q, r for equivalence of (i) property A for (L + ) and property B for its adjoint equations; and (ii) property B for (L - ) and property A for its adjoint.

In the second part of the paper these results are applied to generate sufficient conditions for (E + ) to have property A and for (E - ) 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)].

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34C11Qualitative theory of solutions of ODE: growth, boundedness