Polynomial stability of evolution operators in Banach spaces
The paper considers three concepts of polynomial stability for linear evolution operators which are defined in a general Banach space and whose norms can increase not faster than exponentially. Our approach is based on the extension of techniques for exponential stability to the case of polynomial stability. Some illustrating examples clarify the relations between the stability concepts considered in paper. The obtained results are generalizations of well-known theorems about the uniform and nonuniform exponential stability.
ON NONUNIFORM POLYNOMIAL DICHOTOMY OF EVOLUTION OPERATORS IN BANACH SPACES
In this paper we study two nonuniform polynomial dichotomy concepts for evolution operators in Banach spaces. Our main objective is to give integral characterizations for nonuniform polynomial dichotomies. As for applications we obtain characterizations of these concepts in terms of Lyapunov functions. Some illustrating examples are given. AMS Subject Classification: 34D05, 34E05
Smooth stable invariant manifolds and arbitrary growth rates
Abstract For nonautonomous linear equations v ′ = A ( t ) v with a generalized exponential dichotomy, we show that there is a smooth stable invariant manifold for the perturbed equation v ′ = A ( t ) v + f ( t , v ) provided that f is sufficiently small. The generalized exponential dichotomies may exhibit stable and unstable behaviors with respect to arbitrary growth rates e c ρ ( t ) for some function ρ ( t ) . We consider the general case of nonuniform exponential dichotomies, and the result is obtained in Banach spaces. Moreover, we show that for an equivariant system, the dynamics on the stable manifold in a certain class of graphs is also equivariant. We emphasize that this result cannot be obtained by averaging over the symmetry.