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Criteria of property \(A\) for third order superlinear differential equations. (English) Zbl 0776.34028
The paper deals with two superlinear differential equations, namely \(y'''+p(t)y'+q(t)| y|^ \alpha\text{sgn} y=0\) and \(y'''+p(t)y'+q(t)f(y)=0\), where \(\alpha>1\) is a quotient of odd positive numbers. Property \(A\) means that each solution \(u\) of the related equation is either oscillatory or there exists a point \(T\geq a\) such that \((-1)^ ju^{(j)}\text{sgn} u(t)>0\) for every \(t\geq T\), \(j=0,1,2\), and \(\lim_{t\to\infty}u^{(j)}(t)=0\), \(j=0,1,2\). Sufficient conditions for property \(A\) are obtained for both the equations above which extends and improves some related results cited in the references.
Reviewer: J.Andres (Olomouc)

34C11 Growth and boundedness of solutions to ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text: EuDML
[1] ELIAŠ J.: Properties of the nonoscillatory solution for a third order nonlinear differential equation. Mat. časopis 20 (1970), 249-253. · Zbl 0213.36602
[2] ERBE L.: Oscillation, nonoscilation and asymptotic behavior for third order nonlinear differential equations. Ann. Mat. Pura Appl. (4) 110 (1976), 373-391. · Zbl 0345.34023
[3] CECCHI M., MARINI M.: On the oscillatory behavior of a third order nonlinear differential equation. Nonlinear Anal. 15 (1990), 141-153. · Zbl 0707.34029
[4] GRACE S. R.: Oscillation of even order nonlinear functional differential equations with deviating arguments. Math. Slovaca 41 (1991), 189-204. · Zbl 0753.34047
[5] 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
[6] LIČKO I., ŠVEC M.: La caractère oscillatoire des solutions de l’èquation \(y^{(n)} + f(t)y^\alpha = 0\), \(n > 1\). Czechoslovak Math. J. 13(88) (1963), 481-491. · Zbl 0123.28202
[7] MIKUŠINSKI J.: On Fite’s oscillation theorems. Colloq. Math. 2 (1949), 34-39. · Zbl 0039.31302
[8] NELSON J. L.: A stability theorem for a third order nonlinear differential equation. Pacific J. Math. 24 (1968), 341-344. · Zbl 0155.13901
[9] ŠKERLÍK A.: Oscillation theorems for third order nonlinear differential equations. Math. Slovaca 42 (1992), 471-484. · Zbl 0760.34031
[10] ŠOLTÉS P.: A remark on the oscillatoriness of solutions of a nonlinear third order equation. Mat. Časopis 23 (1973), 326-332.
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