Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}$.

References [Enhancements On Off] (What's this?)

  • [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,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34G05, 58F10

Retrieve articles in all journals with MSC: 34G05, 58F10

Additional Information

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