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November  2008, 21(4): 1025-1046. doi: 10.3934/dcds.2008.21.1025

Characterization of stable manifolds for nonuniform exponential dichotomies

Luis Barreira 1, and  Claudia Valls 2,

1. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa
2. 

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received  August 2007 Revised  March 2008 Published  May 2008

We establish the existence of smooth stable manifolds for nonautonomous differential equations $v'=A(t)v+f(t,v)$ in a Banach space, obtained from sufficiently small perturbations of a linear equation $v'=A(t)v$ admitting a nonuniform exponential dichotomy. In addition to the exponential decay of the flow on the stable manifold we also obtain the exponential decay of its derivative with respect to the initial condition. Furthermore, we give a characterization of the stable manifold in terms of the exponential growth rate of the solutions.
Keywords: Nonuniform exponential dichotomies, stable manifolds..
Mathematics Subject Classification: Primary: 37D10, 34D9.
Citation: Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025
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