# 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)
Integral criteria of oscillation for a third order linear differential equation. (English) Zbl 0855.34038
The differential equation (1) $y''' + p(t)y' + q(t)y = 0$ is considered, where either $p(t) \le 0$, $q(t) > 0$ or $p(t) \le 0$, $p'(t) - q(t) > 0$ on $I = (a, \infty) \subset (0, \infty)$. New criteria for (1) to be oscillatory on $I$ (i.e. (1) has at least one oscillatory solution on $I)$ are presented in integral form. These criteria extend and improve some oscillation ones for (1) and may be applied to the Euler equation.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A30 Linear ODE and systems, general
Full Text:
##### References:
 [1] CECCHI I., MARINI M.: On the oscillatory behavior of a third order nonlinear differential equation. Nonlinear Anal. 15 (1990), 141-153. · Zbl 0707.34029 · doi:10.1016/0362-546X(90)90118-Z [2] CHANTURIYA T. A.: Integral conditions of oscillation of solutions of higher-order linear differential equations. (Russian), Differentsiaľnye Uravneniya 16 (1980), 470-482. · Zbl 0479.34014 [3] DŽURINA J.: Comparison theorems for nonlinear ODE’s. Math. Slovaca 42 (1992), 299-315. · Zbl 0760.34030 · eudml:32117 [4] ELIAŠ J.: Properties of the nonoscillatory solutions for a third order nonlinear differential equation. Mat. Časopis 20 (1970), 249-253. · Zbl 0213.36602 · eudml:29703 [5] ERBE L.: Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations. Pacific J. Math. 64 (1976), 369-385. · Zbl 0339.34030 · doi:10.2140/pjm.1976.64.369 [6] ERBE L.: Oscillation, nonoscillation and asymptotic behavior for third order nonlinear differential equations. Ann. Mat. Pura Appl. (4) 110 (1976), 373-391. · Zbl 0345.34023 · doi:10.1007/BF02418014 [7] HANAN M.: Oscillation criteria for a third order linear differential equations. Pacific J. Math. 11 (1961), 919-944. · Zbl 0104.30901 · doi:10.2140/pjm.1961.11.919 [8] JONES G. D.: An asymptotic property of solutions $y''' + p(x)y' + q(x)y = 0$. Pacific J. Math. 17 (1973), 135-138. · Zbl 0264.34040 · doi:10.2140/pjm.1973.47.135 [9] KHVEDELIDZE N. N., CHANTURIYA T. A.: Oscillation of solutions of third-order linear ODE’s. (Russian), Differentsial’nye Uravneniya 27 (1991), 452-460, 611-618. · Zbl 0731.34033 [10] LAZER A. C : The behavior of solutions of the differential equation $y''' + p(x)y' + q(x)y = 0$. Pacific J. Math. 17 (1966), 435-466. · Zbl 0143.31501 · doi:10.2140/pjm.1966.17.435 [11] LIČKO I., ŠVEC M.: La caractère oscillatoire des solutions de ľequations $y^{(n)} +f(t)y^\alpha = 0$, $n > 1$. Czechoslovak Math. J. 13(88) (1963), 481-491. [12] NELSON J. L.: A stability theorem for a third order nonlinear differential equation. Pacific J. Math. 24 (1968), 341-344. · Zbl 0155.13901 · doi:10.2140/pjm.1968.24.341 [13] OLÁH R.: Integral conditions of oscillation of a linear differential equation. Math. Slovaca 39 (1989), 323-329. · Zbl 0685.34029 · eudml:32432 [14] ROVDER J.: Oscillation criteria for third-order linear differential equations. Mat. Časopis 25 (1975), 231-244. · Zbl 0309.34028 · eudml:29549 [15] ŠKERLÍK A.: Criteria of property A for third order superlinear differential equations. Math. Slovaca 45 (1993), 171-183. · Zbl 0776.34028 · eudml:31894 [16] ŠOLTÉS P.: A remark on the oscillatoriness of solutions of a nonlinear third order equation. Mat. Časopis 23 (1973), 326-332. · Zbl 0297.34029 · eudml:29733