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Generalized Sturm-Liouville equations. II. (English) Zbl 0715.34045

We deal with a system of generalized differential equations of the form (1) \(dv=zdR\), \(dz=vdP\) which is closely connected to the classical Sturm- Liouville equation of the second order. For a more detailed description and motivation see the second author [(*) Arch. Math., Brno 23, No.2, 97- 107 (1987; Zbl 0632.34004)] and K. Kreith [Colloq. Math. Soc. Janos Bolyai 47, 573-589 (1987; Zbl 0617.34025)]. In the case of the generalized differential equations of the form (1) the solutions in general exhibit discontinuities, and by (1), systems with strong impulses can be described in the sense of K. Kreith (loc. cit.).
The main goal in (*) was to derive some generalized version of the Sturmian comparison theorem. To this aim a certain identity was derived and we obtained a comparison theorem in which the distribution of “zeroes” of a solution was described for two equations of the form (1) with the same coefficient R and different coefficients \(P_ 1\) and \(P_ 2\). The method used in (*) was of “variational” nature and, moreover, the proof of the results was based on integration without mentioning anything which would correspond to differentiation.
Here we derive a more general and complete result for systems of the form (1). First we derive a Sturm type comparison theorem for classical systems of the form \(\dot x=r(s)y\), \(\dot y=p(s)x\) with locally integrable coefficients r,p such that \(r\geq 0\) almost everywhere in the interval of definition of the system. The second part of the paper is devoted to the concept of prolongation of a function (of bounded variation) along a given increasing function. This concept then enables us to transfer results for classical systems of ordinary differential equations to systems of the form (1); this technique is also used for deriving the desired comparison theorem in the last part of the present paper.

MSC:

34B24 Sturm-Liouville theory
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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References:

[1] G. Aumann: Reelle Funktionen. Springer-Verlag Berlin, Heidelberg, New York 1969. · Zbl 0181.05801
[2] D. Franková: Continuous dependence on a parameter of solutions of generalized differential equations. Časopis pěst. mat.
[3] Ch. S. Hönig: Volterra-Stieltjes Integral Equations. North-Holland Amsterdam 1975.
[4] Ph. Hartman: Ordinary Differential Equations. John Wiley & Sons New York, London, Sydney 1964. · Zbl 0125.32102
[5] K. Kreith: Sturm-Liouville oscillations in the presence of strong impulses. Colloquia Mathematica Soc. J. Bolyai, 47, Differential Equations: Qualitative Theory, Szeged (1984), 573-589.
[6] Š. Schwabik: Generalized Differential Equations. Fundamental results. Rozpravy ČSAV, Academia Praha 1985. · Zbl 0594.34002
[7] Š. Schwabik: Generalized Sturm-Liouville equations. Archivum Mathematicum (Brno), Vol. 23, No. 2 (1987), 95-107. · Zbl 0632.34004
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