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Some properties of third order differential operators

Authors:	M. Cecchi Z. Došlá M. Marini

Affiliation:	(1) Depart. of Electr, Eng., University of Florence, Via S. Marta 3, 50139 Firenze, Italy;(2) Depart. of Mathematics, Masaryk University, Janákovo nám. 2a, 662 95 Brno, Czech Republic

Abstract:	Consider the third order differential operator L given by $Lleft(cdotright) equiv frac{1}{{a_3 (t)}}frac{d}{{dt}}frac{1}{{a_2 (t)}}frac{d}{{dt}}frac{1}{{a_1 (t)}}frac{d}{{d(t)}}left(cdotright)$ and the related linear differential equation L(x)(t) + x(t) = 0. We study the relations between L, its adjoint operator, the canonical representation of L, the operator obtained by a cyclic permutation of coefficients a_i, i = 1,2,3, in L and the relations between the corresponding equations.We give the commutative diagrams for such equations and show some applications (oscillation, property A).

Keywords:	Differential operators linear differential equation of third order canonical forms adjoint equation cyclic permutation oscillatory solution Kneser solution property A
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