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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A remark on a stability theorem of M. Marachkoff


Author: John R. Haddock
Journal: Proc. Amer. Math. Soc. 31 (1972), 209-212
DOI: https://doi.org/10.1090/S0002-9939-1972-0288370-1
MathSciNet review: 0288370
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Abstract: By placing certain conditions on $ f(t,x)$ for the system of ordinary differential equations (1)

$\displaystyle x' = f(t,x),\;f(t,0) \equiv 0,$

Marachkoff weakened the conditions on the Liapunov function of the classical asymptotic stability theorem of Liapunov theory and obtained asymptotic stability of the zero solution of (1). Later, Massera gave a shorter proof of Marachkoff's result. In this note we show that Marachkoff's theorem can be proven without the use of one of the conditions.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0288370-1
Keywords: Zero solution, stability, asymptotic stability, Liapunov function, positive definite, negative definite, decrescent, total derivative
Article copyright: © Copyright 1972 American Mathematical Society