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On generic asymptotic stability of differential equations in Banach space


Authors: F. S. De Blasi and J. Myjak
Journal: Proc. Amer. Math. Soc. 65 (1977), 47-51
MSC: Primary 34G05; Secondary 58F10
MathSciNet review: 0447730
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Abstract: The asymptotic stability of the zero solution of the differential equation $ ( \ast )\;x' = Ax + f(x)$ is studied, when the pertubation f is in a given complete metric space $ \mathfrak{M}$. It is known that the zero solution of $ ( \ast )$ is asymptotically stable whenever f is in a certain proper subset $ \mathfrak{N} \subset \mathfrak{M}$. It is shown that, while $ \mathfrak{N}$ is of Baire first category in $ \mathfrak{M}$, on the contrary the set $ {\mathfrak{M}_0}$ of all those f for which the zero solution of $ ( \ast )$ is asymptotically stable is a proper residual subset of $ \mathfrak{M}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0447730-0
Keywords: Differential equations, Banach space, Asymptotically stable, generically asymptotically stable, Baire first category, residual set
Article copyright: © Copyright 1977 American Mathematical Society