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About certain fourth-order ordinary differential operators - oscillation and discreteness of their spectrum. (English) Zbl 0685.34033
Following differential equation is considered: \[ (p(x)u''(x))''+q(x)u(x)=0,\quad 0<p\in C^ 2(0,\infty),\quad q\in C(0,\infty). \] New sufficient conditions are given and proved for the oscillation of the solutions to this differential equation, which include the known results as special case \(p(x)=x^{\alpha}\), where \(\alpha\) is real. With the aid of these conditions necessary conditions are derived for the discreteness and boundedness from below of the spectrum of the differential operator \[ Bu(x)=1/w(x)(p(x)u''(x))'',\quad D(B)=C^{\infty}_ 0(1,\infty),\quad 0<w\in C(1,\infty),\quad 0<p\in C^ 2(1,\infty) \] in the weighted Hilbert space \(L_{2,w}(1,\infty)\), in the most general case. In an Appendix some auxiliary results are given.
Reviewer: Á.Bosznay

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