zbMATH — the first resource for mathematics

Oscillation theorems for a class of linear fourth order differential equations. (English) Zbl 0542.34030
Consider the equation (*) \(y^{(4)}+P(t)y''+R(t)y'+Q(t)y=0\), where P(t), R(t), Q(t) are real-valued continuous functions on \(I=[a,\infty)\), \(- \infty<a<\infty\). Assume that (A): P(t)\(\leq 0\), R(t)\(\leq 0\), \(R^ 2(t)\leq 2P(t)Q(t)\) for all \(t\in I\) and Q(t) is not identical to zero in any interval of I. (B): \(\int^{\infty}_{\tau_ 0}t^{2+\alpha}Q(t)dt=-\infty, \tau_ 0<\max \{a,0\}\), for some \(0\leq \alpha<1\). The author proves the following theorem: Suppose that \(\int^{\infty}_{\tau_ 0}tP(t)dt>-\infty,\) Q(t)\(\leq R(t)\) for all \(t\geq \tau_ 0\) and that (A) and (B) hold, or (B) holds and \(\int^{\infty}_{\tau_ 0}tP(t)dt>-\infty,\) P(t)\(\leq R(t)\leq 0\), 2Q(t)\(\leq R(t)\) for all \(t\in I\). Then equation (*) is oscillatory and there exists a fundamental system of solutions of (*) such that two solutions of this system are oscillatory, other solutions of this system are nonoscillatory and one of them tends monotonically to \(\infty\) as \(t\to \infty\) and the other tends to zero if \(\int^{\infty}_{\tau_ 0}s^{2+\alpha}R(s)ds>-\infty.\)
Reviewer: T.S.Liu

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text: EuDML
[1] Ahmad Shair: On the oscillation of solutions of a class of linear fourth order differential equations. Pac. J. Math. 34, 1970. · Zbl 0176.20402
[2] Heidel J. W.: Qualitative behavior of solutions of a third order nonlinear differential equation. Pac. J. Math., 27, 1968. · Zbl 0172.11703
[3] Leighton W., Nehari Z.: On the oscillation of solutions of self-adjoint linear differential equations of the fourth order. Trans. Amer. Math. Soc, 89, 1958. · Zbl 0084.08104
[4] Regenda J.: Oscillation and nonoscillation properties of the solutions of the differential equation \(y^{(4)}+P(t)y^{\prime\prime}+Q(t)y=0\). Math. Slov., 28, 1978. · Zbl 0406.34041
[5] Regenda J.: Oscillation criteria for fourth order linear differential equations. Math. Slov., 29, 1979. · Zbl 0408.34032
[6] Regenda J.: On the oscillation of solutions of a class of linear fourth order differential equations. Czech. Math. J., to appear in 1983. · Zbl 0547.34023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.