Abstract
Abstract The aim of this paper is to give several characterizations for weak exponential expansiveness properties of skew-evolution semiflows in Banach spaces. Variants for weak exponential expansiveness of some well-known results in uniform exponential stability theory (Datko (1973)) and exponential instability theory (Lupa (2010), Megan et al. (2008)) are obtained. MSC:93D20, 34D20.
Highlights
It is well known that in recent years, the exponential stability theory of one parameter semigroups of operators and evolution operators has witnessed significant development.A number of long-standing open problems have been solved and the theory seems to have obtained a certain degree of maturity
One of the most important results of the stability theory is due to Datko, who proved in in [ ] that a strongly continuous semigroup of operators {T(t)}t≥ is uniformly exponentially stable if and only if for each vector x from the Banach space X, the function t → T(t)x lies in L (R+)
The weak exponential stability of evolution operators in Banach spaces has been investigated and several important results have been obtained by Lupa, Megan and Popa in [ ]
Summary
It is well known that in recent years, the exponential stability theory of one parameter semigroups of operators and evolution operators has witnessed significant development.A number of long-standing open problems have been solved and the theory seems to have obtained a certain degree of maturity. A skew-evolution semiflow C = (φ, ) is said to be uniformly exponentially expansive if there are N, α > such that (t, t , x )v ≥ N eα(t–r) (r, t , x )v , for all (t, r), (r, t ) ∈ T and all (x , v ) ∈ Y .
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