# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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

${y}^{\left(3\right)}-q\left(x\right){y}^{\text{'}}±r\left(x\right)f\left(y\right)=0,\phantom{\rule{1.em}{0ex}}x\in {ℝ}_{+},\phantom{\rule{2.em}{0ex}}\left({\mathrm{E}}_{±}\right)$

with $q,r\in C\left({ℝ}_{+},{ℝ}_{+}\right)$, $f\in C\left(ℝ,ℝ\right)$, $r\left(x\right)>0$, and $tf\left(t\right)>0$ for all $t\ne 0$. In the linear case $f\left(t\right)\equiv t$, $\left({\text{E}}_{±}\right)$ will be denoted $\left({\text{L}}_{±}\right)$. Seven theorems give connections between oscillation and properties A or B of $\left({\text{L}}_{±}\right)$ or $\left({\text{E}}_{±}\right)$, 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 $\left({\text{L}}_{+}\right)\left[\left({\text{L}}_{-}\right)\right]$ 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 $\left({\text{L}}_{+}\right)$ and property B for its adjoint equations; and (ii) property B for $\left({\text{L}}_{-}\right)$ and property A for its adjoint.

In the second part of the paper these results are applied to generate sufficient conditions for $\left({\text{E}}_{+}\right)$ to have property A and for $\left({\text{E}}_{-}\right)$ 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