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4.—Comparison Theorems for a Class of Second-order Non-linear Ordinary Differential Equations

Published online by Cambridge University Press:  14 February 2012

Donald C. Benson
Donald C. Benson
Affiliation:
Department of Mathematics, University of California, Davis
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Synopsis

Integral inequalities are used to obtain comparison theorems for a class of second-order differential equations which includes the Emden-Fowler equation, certain Liénard equations, and linear equations of the form d2y/dx2+f(x)y = 0. For these linear equations the results below imply Sturm's classical comparison theorem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 73 , 1975 , pp. 77 - 84
Copyright
Copyright © Royal Society of Edinburgh 1975

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References

1Bellman, R., Stability theory of ordinary differential equations. New York: McGraw-Hill, 1953.Google Scholar
2Benson, D. C, Comparison and oscillation theory for Liénard's equation with positive damping. SIAMJ. Appl. Math., 24, 251–271, 1973.CrossRefGoogle Scholar
3Benson, D. C, Inequalities involving integrals of functions and their derivatives, J.Math.Anal. Appl., 17, 292–308, 1967.CrossRefGoogle Scholar
4Burton, T. A. and Townsend, C. G., On the generalized Liénard equation with forcing function. J.Differential Equations, 4, 620–633, 1968.CrossRefGoogle Scholar
5Coffman, C. V. and Wong, J. S. W., Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations. Trans. Amer. Math. Soc, 167, 399–134, 1972.CrossRefGoogle Scholar
6Fowler, R. H., Further studies of Emden's and similar differential equations. Quart. J. Math. Oxford Ser., 2, 259–288, 1931.CrossRefGoogle Scholar
7Gelfand, I. M. and Fomin, S. V., Calculus of variations. Englewood Cliffs: Prentice Hall, 1963.Google Scholar
8Graef, J. R., On the generalized Liénard equation with negative damping. J.Differential Equations, 12, 34–62, 1972.CrossRefGoogle Scholar
9Leighton, W., Comparison theorems for linear differential equations of the second order. Proc. Amer. Math. Soc, 13, 603–610, 1962.CrossRefGoogle Scholar