Exponential separation, exponential dichotomy, and almost periodicity of linear difference equations

On Exponential Dichotomy of Variational Difference Equations

Exponential Stability and Exponential Dichotomy of Semigroups of Linear Operators

The aim of this paper is to establish necessary and sufficient conditions for exponential stability of semigroups of linear operators and to show how this conditions can be applied in order to characterize the exponential dichotomy. First, we prove that an exponentially bounded semigroup is exponentially stable if and only if it is (l(N, X), l∞(N, X))-stable, where p ∈ (1,∞). After that this re...

Exponential Stability, Exponential Expansiveness, and Exponential Dichotomy of Evolution Equations on the Half-line

For an evolution family on the half-line U = (U(t; s)) ts0 of bounded linear operators on a Banach space X we introduce operators G 0 ; G X and I X on certain spaces of X-valued continuous functions connected with the integral equation u(t) = U(t; s)u(s) + R t s U(t;)f()dd: We characterize exponential stability, exponential expansiveness and exponential dichotomy of U by properties of G 0 ; G X...

Sufficient Conditions for Exponential Stability and Dichotomy of Evolution Equations

We present several suucient conditions for exponential stability and di-chotomy of solutions of the evolution equation u 0 (t) = A(t)u(t) () on a Banach space X. Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform constants and A() has a suuciently small HH older constant, then () has exponential dichotomy. We further stud...

On Exponential Stability of Variational Difference Equations

We prove that a general system of variational difference equations is uniformly exponentially stable if and only if certain associated sets are of the second category. We also deduce necessary and sufficient conditions for uniform exponential stability of systems with uniformly bounded coefficients. We apply our results for the study of exponential stability of linear skew-product flows, genera...