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**On the focal points of solutions of a certain fourth order differential equation.**
*(English)*
Zbl 0643.34040

This paper is focussed on the existence and multiplicity of zeros of the first derivative of solutions (the so-called accompanying points) related to the fourth order differential equation, obtained by iteration of the second order oscillatory linear homogeneous differential equation \(y''(t)+q(t)y(t)=0,\) where \(q(t)\in C^ 2(-\infty,+\infty)\), \(q(t)>0\) on the interval \((-\infty,+\infty).\)

For this study they are distinguished the particular bundles, to which the space of all solutions related to the differential equation under consideration can be dissociated. This investigation has the immediate connection to the foregoing author papers concerning the distribution of zeros of the solutions (the so called conjugate points) related to the fourth order differential equation.

For this study they are distinguished the particular bundles, to which the space of all solutions related to the differential equation under consideration can be dissociated. This investigation has the immediate connection to the foregoing author papers concerning the distribution of zeros of the solutions (the so called conjugate points) related to the fourth order differential equation.

Reviewer: V.Vlček

### MSC:

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

### Keywords:

multiplicity of zeros; accompanying points; fourth order differential equation; iteration; second order oscillatory linear homogeneous differential equation; distribution of zeros of the solutions; conjugate points
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\textit{V. Vlček}, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 25, 97--115 (1986; Zbl 0643.34040)

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### References:

[1] | Sherman T. L.: Properties of solutions of \(N^{\text th}\) order linear differential equations. Pacific J. Math, Vol. 15, No. 3, 1965 |

[2] | Borůvka O.: Lineare Differentialtransformationen 2. Ordnung. VEB Deutscher Verlag der Wissenschaften, Berlin, 1967 · Zbl 0153.11201 |

[3] | Vlček V.: On a distribution of zeros of solutions of a fourth-order iterated linear differential equation. Acta UP Olom., F. R. N., Tom 57, 1978 · Zbl 0422.34037 |

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