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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces
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by Shui-Nee Chow and Hugo Leiva PDF
Proc. Amer. Math. Soc. 124 (1996), 1071-1081 Request permission

Abstract:

In this paper we introduce a concept of exponential dichotomy for linear skew-product semiflows (LSPS) in infinite dimensional Banach spaces, which is an extension of the classical concept of exponential dichotomy for time dependent linear differential equations in Banach spaces. We prove that the concept of exponential dichotomy used by Sacker-Sell and Magalhães in recent years is stronger than this one, but they are equivalent under suitable conditions. Using this concept we where able to find a formula for all the bounded negative continuations. After that, we characterize the stable and unstable subbundles in terms of the boundedness of the corresponding projector along (forward/backward) the LSPS and in terms of the exponential decay of the semiflow. The linear theory presented here provides a foundation for studying the nonlinear theory. Also, this concept can be used to study the existence of exponential dichotomy and the roughness property for LSPS.
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Additional Information
  • Shui-Nee Chow
  • Affiliation: CDSNS Georgia Tech, Atlanta, Georgia 30332
  • Email: chow@math.gatech.edu
  • Hugo Leiva
  • Affiliation: CDSNS Georgia Tech, Atlanta, Georgia 30332 and ULA-Venezuela
  • Email: leiva@math.gatech.edu
  • Received by editor(s): April 14, 1994
  • Additional Notes: This research was partially supported by NSF grant DMS-9306265.
  • Communicated by: Hal L. Smith
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 1071-1081
  • MSC (1991): Primary 34G10; Secondary 35B40
  • DOI: https://doi.org/10.1090/S0002-9939-96-03433-8
  • MathSciNet review: 1340377