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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 \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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