Hyers–Ulam stability and discrete dichotomy

Discrete Methods and Exponential Dichotomy of Semigroups

The aim of this paper is to characterize the uniform exponential dichotomy of semigroups of linear operators in terms of the solvability of discrete-time equations over N. We give necessary and sufficient conditions for uniform exponential dichotomy of a semigroup on a Banach space X in terms of the admissibility of the pair (l∞(N, X), c00(N, X)). As an application we deduce that a C0-semigroup...

Discrete and Continuous: A Fundamental Dichotomy in Mathematics

STABILITY AND DICHOTOMY OF POSITIVE SEMIGROUPS ON Lp

A new proof of a result of Lutz Weis is given, that states that the stability of a positive strongly continuous semigroup (e)t≥0 on Lp may be determined by the quantity s(A). We also give an example to show that the dichotomy of the semigroup may not always be determined by the spectrum σ(A). Consider a strongly continuous semigroup (e)t≥0 acting on a Banach space X with unbounded generator A. ...

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...

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...