Nonuniform Exponential Unstability of Evolution Operators in Banach Spaces

In this paper we consider a nonuniform unstability concept for evolution operators in Banach spaces. The relationship between this concept and the Perron condition is studied. Generalizations to the nonuniform case of some results of Van Minh, Räbiger and Schnaubelt are obtained. The theory we present here is applicable for general time varying linear equations in Banach spaces. 1. Evolution operators with exponential growth Let X be a real or complex Banach space. The norm on X and on the Banach algebra B(X) of all bounded linear operators from X into itself will be denoted by ‖ · ‖. We recall that if ∆ = {(t, t0) ∈ R+ : t ≥ t0} then an application Φ : ∆ → B(X) is called evolution operator on X if it satisfies the following conditions: e1) Φ(t, t) = I (the identity operator on X) for every t ≥ 0; e2) Φ(t, s)Φ(s, t0) = Φ(t, t0), for all (t, s), (s, t0) ∈ ∆; e3) for every t, t0 ∈ R+ and every x ∈ X the mappings s 7→ Φ(t, s)x and s 7→ Φ(s, t0)x are continuous on [0, t] and on [t0,∞), respectively. If Φ : ∆ → B(X) is an evolution operator on X , then we introduce the mapping 2000 Mathematics Subject Classification. 34D05, 34D20.