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Scalar and matrix complex nonoscillation criteria

Published online by Cambridge University Press:  20 January 2009

H. C. Howard
H. C. Howard
Affiliation:
University of Kentucky, Lexington, Kentucky 40506, U.S.A.
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In his recent book on ordinary differential equations Hille (3) devotes a chapter to complex oscillation theory. Drawing upon his own work in this area and the work of Nehari, Schwarz, Taam, and others, he gives a variety of oscil-lation and nonoscillation theorems for solutions of the differential equation

where z is a complex variable and p is regular in some appropriate domain. There are a number of results for (1.1) with an arbitrary coefficient o and some discussions for special cases of classical interest, such as the Bessel and Mathieu equations. There is a bibliography at the end of the chapter. For other recent work in this area attention is directed to papers by Herold (1, 2) Kim (4, 5) and Lavie (6) where other references are given.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 18 , Issue 3 , June 1973 , pp. 173 - 189
Copyright
Copyright © Edinburgh Mathematical Society 1973

References

REFERENCES

(1) Herold, H., Nullstellen bei Lösungen komplexer linearer Differentialgleichungen 2. Ordnung, Monatsh. Math. 74 (1970), 41–49.CrossRefGoogle Scholar
(2) Herold, H., Nullstellen und kritische Stellender Lösunger von DirFerentialgleichungen 2. Ordnung im Komplexen, Math. Z. 108 (1969), 269–284.Google Scholar
(3) Hille, E., Lectures on Ordinary Differential Equations (Addison-Wesley, 1969).Google Scholar
(4) Kim, W. J., Disfocality of Differential Systems, J. Math. Anal. Appl. 26 (1969), 9–19.CrossRefGoogle Scholar
(5) Kim, W. J., On the Zeros of y (n)+py=0, J. Math. Anal. Appl. 25 (1969), 189–208.Google Scholar
(6) Lavie, M., On Disconjugacy and Interpolation in the Complex Domain, J. Math. Anal. Appl. 32 (1970), 246–263.Google Scholar
(7) Barrett, J., Fourth order boundary value problems and comparison theorems, Canad. J. Math. 13 (1961), 625–638.CrossRefGoogle Scholar
(8) Howard, H. C., Some oscillation criteria for general self-adjoint differential equations, J. London Math. Soc. 43 (1968), 401–406CrossRefGoogle Scholar
Howard, H. C., Some oscillation criteria for general self-adjoint differential equations, J. London Math. Soc. (2) 1(1969), 660.Google Scholar