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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 $$y^{(3)}- q(x)y'\pm r(x)f(y)= 0,\quad x\in \bbfR_+,\tag{E_\pm}$$ with $q, r\in C(\bbfR_+,\bbfR_+)$, $f\in C(\bbfR,\bbfR)$, $r(x)>0$, and $tf(t)>0$ for all $t\ne 0$. In the linear case $f(t)\equiv t$, $(\text{E}_\pm)$ will be denoted $(\text{L}_\pm)$. Seven theorems give connections between oscillation and properties A or B of $(\text{L}_\pm)$ or $(\text{E}_\pm)$, as defined by {\it I. T. Kiguradze} and {\it 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 $(\text{L}_+)[(\text{L}_-)]$ has at least one nontrivial oscillatory solution if and only if it has property A [property B, respectively], extending results of {\it M. Greguš} [Third order linear differential equations. Mathematics and its Applications. D. Reidel Publ. Comp. (1987; Zbl 0602.34005)] and {\it 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 $(\text{L}_+)$ and property B for its adjoint equations; and (ii) property B for $(\text{L}_-)$ and property A for its adjoint. In the second part of the paper these results are applied to generate sufficient conditions for $(\text{E}_+)$ to have property A and for $(\text{E}_-)$ 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)].

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34C11 Qualitative theory of solutions of ODE: growth, boundedness
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##### References:
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