Stability of Nonautonomous Differential Equations

Luis Barreira, Claudia Valls (auth.)3540747745, 978-3-540-74774-1

Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the construction and regularity of topological conjugacies, the study of center manifolds, as well as their reversibility and equivariance properties. Most results are obtained in the infinite-dimensional setting of Banach spaces. Furthermore, the linear variational equations are always assumed to possess a nonuniform exponential behavior, given either by the existence of a nonuniform exponential contraction or a nonuniform exponential dichotomy. The presentation is self-contained and has unified character. The volume contributes towards a rigorous mathematical foundation of the theory in the infinite-dimension setting, and may lead to further developments in the field. The exposition is directed to researchers as well as graduate students interested in differential equations and dynamical systems, particularly in stability theory.

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Table of contents :
Front Matter….Pages I-XIV
Front Matter….Pages 17-17
Introduction….Pages 1-16
Exponential dichotomies and basic properties….Pages 19-25
Robustness of nonuniform exponential dichotomies….Pages 27-51
Front Matter….Pages 53-53
Lipschitz stable manifolds….Pages 55-73
Smooth stable manifolds in Rn….Pages 75-117
Smooth stable manifolds in Banach spaces….Pages 119-143
A nonautonomous Grobman–Hartman theorem….Pages 145-167
Front Matter….Pages 169-169
Center manifolds in Banach spaces….Pages 171-196
Reversibility and equivariance in center manifolds….Pages 197-215
Front Matter….Pages 217-217
Lyapunov regularity and exponential dichotomies….Pages 219-248
Lyapunov regularity in Hilbert spaces….Pages 249-263
Stability of nonautonomous equations in Hilbert spaces….Pages 265-276
Back Matter….Pages 277-290