About certain fourth-order ordinary differential operators - oscillation and discreteness of their spectrum. (English) Zbl 0685.34033

Fiedler, Frank

Math. Nachr. 142, 235-250 (1989).

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

Keywords:

discreteness; boundedness; weighted Hilbert space

References:

[1]	Ahlbrandt, J. Diff. Equs. 9 pp 420– (1971)
[2]	Ahlbrandt, Canad. J. Math. 33 pp 229– (1981) · Zbl 0462.47030
[3]	and , Linear operators II, New York – London 1963
[4]	Fiedler, J. Diff. Equs. 42 pp 155– (1981)
[5]	Fiedler, Proc. Roy. Soc. Edinburgh 109 A pp 127– (1988) · Zbl 0657.34036
[6]	Oscillation and discreteness criteria of certain fourth-order ordinary differential operators, prepared for publication
[7]	Fiedler, Math. Nachr. 107 pp 187– (1982)
[8]	Direct methods of qualitative spectral analysis of singular differential operators, Jerusalem 1965
[9]	Kalyabin, Diff. Equs. 9 pp 951– (1973)
[10]	Lewis, Pac. J. Math. 51 pp 221– (1974) · Zbl 0281.34027
[11]	Lewis, Proc. Amer. Math. Soc. 42/2 pp 480– (1974)
[12]	Lewis, Canad. J. Math. 27 pp 138– (1975) · Zbl 0287.34027
[13]	Moore, J. Math. 5 pp 125– (1955)
[14]	Spectral theory of ordinary differential operators, Ellis Hoorwood, Chichester 1981
[15]	Müller-Pfeiffer, J. Diff. Equs. 46/2 pp 194– (1982)
[16]	Reid, Pac. J. Math. 13 pp 665– (1963) · Zbl 0119.07401
[17]	Ordinary differential equations, New York, John Wiley and Sons 1971