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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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  • [1] Francesco S. De Blasi and Józef Myjak, Sur la convergence des approximations successives pour les contractions non linéaires dans un espace de Banach, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 4, Aiii, A185–A187 (French, with English summary). MR 0425703
  • [2] -, Generic properties of hyperbolic partial differential equations, J. London Math. Soc. (to appear).
  • [3] G. E. Ladas and V. Lakshmikantham, Differential equations in abstract spaces, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 85. MR 0460832
  • [4] W. Orlicz, Zur Theorie der Differentialgleichung $ y' = f(x,y)$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. A 8/9 (1932), 221-228.
  • [5] G. Sansone and R. Conti, Non-linear differential equations, Revised edition. Translated from the Italian by Ainsley H. Diamond. International Series of Monographs in Pure and Applied Mathematics, Vol. 67, A Pergamon Press Book. The Macmillan Co., New York, 1964. MR 0177153
  • [6] F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83–93. MR 0011702,
  • [7] Giovanni Vidossich, Most of the successive approximations do converge, J. Math. Anal. Appl. 45 (1974), 127–131. MR 0335908,

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Keywords: Differential equations, Banach space, Asymptotically stable, generically asymptotically stable, Baire first category, residual set
Article copyright: © Copyright 1977 American Mathematical Society

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