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Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces


Authors: Shui-Nee Chow and Hugo Leiva
Journal: Proc. Amer. Math. Soc. 124 (1996), 1071-1081
MSC (1991): Primary 34G10; Secondary 35B40
MathSciNet review: 1340377
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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

DOI: https://doi.org/10.1090/S0002-9939-96-03433-8
Keywords: Skew-product semiflow, exponential dichotomy, stable and unstable manifolds
Received by editor(s): April 14, 1994
Additional Notes: This research was partially supported by NSF grant DMS-9306265.
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society