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 \mathbb{R}_+,\tag{E$$_\pm$$}$ with $$q, r\in C(\mathbb{R}_+,\mathbb{R}_+)$$, $$f\in C(\mathbb{R},\mathbb{R})$$, $$r(x)>0$$, and $$tf(t)>0$$ for all $$t\neq 0$$. 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 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 $$(\text{L}_+)[(\text{L}_-)]$$ has at least one nontrivial oscillatory solution if and only if it has property A [property B, respectively], extending results of 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 $$(\text{L}_+)$$ 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 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations
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References:

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