A remark on a stability theorem of M. Marachkoff

Haddock, John R.

doi:10.1090/S0002-9939-1972-0288370-1

A remark on a stability theorem of M. Marachkoff
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by John R. Haddock

Proc. Amer. Math. Soc. 31 (1972), 209-212

DOI: https://doi.org/10.1090/S0002-9939-1972-0288370-1

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Abstract:

By placing certain conditions on $f(t,x)$ for the system of ordinary differential equations (1) \[ x’ = f(t,x),\;f(t,0) \equiv 0,\] 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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Bibliographic Information

Journal: Proc. Amer. Math. Soc. 31 (1972), 209-212
DOI: https://doi.org/10.1090/S0002-9939-1972-0288370-1
MathSciNet review: 0288370