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Differential inequalities and extension of Lyapunov's method

Published online by Cambridge University Press:  24 October 2008

V. Lakshmikantham
V. Lakshmikantham
Affiliation:
RIAS, Baltimore, U.S.A.
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Extract

One of the most important techniques in the theory of non-linear differential equations is the direct method of Lyapunov and its extensions. It depends basically on the fact that a function satisfying the inequality

is majorized by the maximal solution of the equation

Using this comparison principle and the concept of Lyapunov's function various properties of solutions of differential equations have been considered (1–11).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 4 , October 1964 , pp. 891 - 895
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Antosiewiez, H. A.An inequality for approximate solutions of ordinary differential equations. Math. Z. 78 (1962), 44–52.CrossRefGoogle Scholar
(2)Brauer, F.Global behaviour of solutions of ordinary differential equations. Jour. Math. Anal. Appl. 2 (1961), 145–158.CrossRefGoogle Scholar
(3)Lakshmikantham, V.On the boundedness of solutions of non-linear differential equations. Proc. American Math. Soc. 8 (1957), 1044–1048.Google Scholar
(4)Lakshmikantham, V.Notes on a variety of problems of differential systems. Arch. Rational Mech. Anal. 10 (1962), 119–126.CrossRefGoogle Scholar
(5)Lakshmikantham, V.Differential systems and extension of Lyapunov's method. Michigan J. Math. 9 (1962), 311–320.CrossRefGoogle Scholar
(6)Lakshmikantham, V.Lyapunov function and a basic inequality in delay-differential equations. Arch. Rational Mech. Anal. 10 (1962), 305–310.CrossRefGoogle Scholar
(7)Lakshmikantham, V.Upper and lower bounds on the norm of solutions of differential equations. Proc. American Math. Soc. 13 (1962), 615–616.CrossRefGoogle Scholar
(8)Lakshmikantham, V.Properties of solutions of abstract differential inequalities. Proc.London Math. Soc. (to appear).Google Scholar
(9)Lakshmikantham, V.Upper and lower bounds on the norm of solutions of differential equations II. Proc. American Math. Soc. 14 (1963), 509–513.CrossRefGoogle Scholar
(10)Lakshmikantham, V.Differential equations in Banach spaces and extensions of Lia- punov's method. Proc. Cambridge Philos. Soc. 59 (1963), 373–381.CrossRefGoogle Scholar
(11)Lakshmikantham, V.On the stability and boundedness of differential systems. Proc. Cambridge Philos. Soc. 58 (1962), 492–496.CrossRefGoogle Scholar