Jump to ContentJump to Main Navigation
Show Summary Details
In This Section

Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose

1 Issue per year


Mathematical Citation Quotient (MCQ) 2015: 0.33


Emerging Science

Open Access
Online
ISSN
2353-0626
See all formats and pricing
In This Section

Local attractivity in nonautonomous semilinear evolution equations

Joël Blot
  • Corresponding author
  • Laboratoire SAMM EA 4543, Université Paris 1 Panthéon-Sorbonne, centre P.M.F., 90 rue de Tolbiac, 75634 Paris cedex 13, France
  • Email:
/ Constantin Buşe
  • West University of Timisoara, Department of mathematics, Bd V. Parvan No. 4, 300223-Timisoara, România
  • Email:
/ Philippe Cieutat
  • Laboratoire de Mathématiques de Versailles, UMR-CNRS 8100, Université Versailles-Saint-Quentin-en-Yvelines, 45 avenue des États-Unis, 78035 Versailles cedex, France
  • Email:
Published Online: 2014-05-17 | DOI: https://doi.org/10.2478/msds-2014-0002

Abstract

We study the local attractivity of mild solutions of equations in the form u’(t) = A(t)u(t) + f (t, u(t)), where A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at infinity

Keywords: semilinear evolution equation; evolution family; exponential stability; attractivity; Nemytskii operator

References

  • [1] L. Amerio & G. Prouse, Almost-periodic functions and functional equations, Van Nostrand, New York, 1971.

  • [2] J.-B. Baillon, J. Blot, G.M. N’Guérékata & D. Pennequin, On C(n)-almost periodic solutions of some nonautonomous diferential equations in Banach spaces, Comment. Math., Prace Mat. XLVI(2) (2006), 263-273.

  • [3] J. Blot, P. Cieutat, G.M. N’Guérékata & D. Pennequin, Superposition operators between various spaces of almost periodic function spaces and applications, Commun. Math. Anal. 6(1) (2009), 42-70.

  • [4] J. Blot & B. Crettez, On the smoothness of optimal paths II: some local turnpike results, Decis. Econ. Finance 30(2) (2007), 137-150.

  • [5] H.S. Ding, W. Long & G.M. N’Guérékata, Almost automorphic solutions of nonautonomous evolution equations, Nonlinear Anal., 70(12) (2009), 4158-4164.

  • [6] S. Lang, Real and functional analysis, Third edition, Springer-Verlag, New York, Inc., 1993.

  • [7] N. V. Minh, F. Räbiger & R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotonomy of evolution equations on the half line, integr. Equ. Oper. Theory, 32 (1998), 332-353.

  • [8] G.M. N’Guérékata, Almost automorphic and almost periodic functions in abstract spaces, Kluwer Academic Publishers, New York, 2001.

  • [9] G.M. N’Guérékata, Topics in almost automorphy, Springer, New York, 2005.

  • [10] A. Pazy, Semigroups of linear operators and applications to partial diferential equations, Springer-Verlag New York, Inc., 1983.

  • [11] W. Rudin, Functional analysis, Second edition, McGraw-Hiil, Inc., New York, 1993.

  • [12] L. Schwartz, Cours d’analyse; tome 1, Hermann, Paris, 1967.

  • [13] T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Springer-Verlag, New York, 1975.

  • [14] S. Zaidman, Almost-periodic functions in abstract spaces, Pitman Publishong, Inc., Marshfeld, MA, 1985.

About the article

Received: 2014-01-17

Accepted: 2014-03-21

Published Online: 2014-05-17

Published in Print: 2014-01-01



Citation Information: Nonautonomous Dynamical Systems, ISSN (Online) 2353-0626, DOI: https://doi.org/10.2478/msds-2014-0002. Export Citation

© 2014 Joël Blot et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Comments (0)

Please log in or register to comment.
Log in