zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 $$y^{(3)}- q(x)y'\pm r(x)f(y)= 0,\quad x\in \bbfR_+,\tag{E$_\pm$}$$ 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)].

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34C11Qualitative theory of solutions of ODE: growth, boundedness
Full Text: DOI
[1] Cecchi, M.: Oscillation criteria for a class of third order linear differential equations. Boll. un. Mat. ital. 6, 297-306 (1983) · Zbl 0523.34029
[2] Cecchi, M.; Došlá, Z.; Marini, M.; Villari, G.: On the qualitative behavior of solutions of third order differential equations. J. math. Anal. appl. 197, 749-766 (1996) · Zbl 0856.34034
[3] Cecchi, M.; Došlá, Z.; Marini, M.: An equivalence theorem on propertiesab. Ann. mat. Pura appl. 173, 373-389 (1997)
[4] Cecchi, M.; Dožlá, Z.; Marini, M.: On nonlinear oscillations for equations associated to disconjugate operators. Nonlinear analysis, T., M. Appl. 30, 1583-1594 (1997) · Zbl 0892.34032
[5] Elias, U.: Oscillation theory of two-term differential equations. (1997) · Zbl 0878.34022
[6] Erbe, L.: Oscillation, nonoscillation and asymptotic behavior for third order nonlinear differential equations. Ann. mat. Pura appl. 110, 373-391 (1976) · Zbl 0345.34023
[7] Gera, M.: Über das verhalten der lösungen der gleichungxatxbtxctxct. Acta math. Univ. comenianae, 189-203 (1985) · Zbl 0612.34029
[8] Greguš, M.: Third order linear differential equation. (1987)
[9] Greguš, M.; Jr., M. Greguš: Asymptotic properties of solutions of a certain nonautonomous nonlinear differential equation of the third order. Boll. U.M.I. 7-A, 341-350 (1993)
[10] Greguš, M.; Jr., M. Greguš: Remark concerning oscillatory properties of solutions of a certain nonlinear equation of the third order. Achivum math. (Brno) 29, 51-55 (1992)
[11] Hanan, M.: Oscillation criteria for third-order linear differential equation. Pacific J. Math. 11, 919-944 (1961) · Zbl 0104.30901
[12] Hartman, P.: Ordinary differential equations. (1982) · Zbl 0476.34002
[13] Heidel, J. W.: Qualitative behavior of solutions of a third order nonlinear differential equation. Pacific J. Math. 27, 507-526 (1968) · Zbl 0172.11703
[14] Kiguradze, I. T.; Chanturia, T. A.: Asymptotic properties of solutions of nonautonomous ordinary differential equations. (1993)
[15] Lazer, A. C.: The behavior of solutions of the differential equationypxyqxy. Pacific J. Math. 17, 435-466 (1966) · Zbl 0143.31501
[16] Marini, M.; Zezza, P. L.: On the asymptotic behavior of the solutions of a class of second-order linear differential equations. J. diff. Equations 28, 1-17 (1978) · Zbl 0371.34032
[17] Parhi, N.; Das, P.: Oscillation criteria for a class of nonlinear differential equations of third order. Ann. polon. Math. 3, 219-229 (1992) · Zbl 0771.34024
[18] Villari, G.: Contributi allo studio asintotico dell’equazionextptxt. Ann. mat. Pura appl. 51, 301-328 (1960) · Zbl 0095.06903