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Schäffer spaces and uniform exponential stability of linear skew-product semiflows. (English) Zbl 1081.34056

Let Θ be a locally compact metric space and X a Banach space. A mapping σ:Θ× + Θ is called a semiflow if it is continuous and such that

σ(θ,0)=θforθΘ,(1)
σ(θ,t+s)=σ(σ(θ,s),t)forallt,s0andθΘ·(2)

Let σ be a semiflow on Θ and let φ:Θ× + B(X) be a continuous mapping satisfying some extra conditions (for example, φ(θ,0)=I – the identity operator on X; φ(θ,t+s)=φ(σ(θ,t),s)φ(θ,t) for t,s0 and θΘ). The pair π=(φ,σ) is called linear strongly continuous skew-product semiflow on X×Θ. A strongly continuous skew-product semiflow π=(φ,σ) is called uniformly exponentially stable if there exist N>0 and ν>0 such that φ(θ,t)Nexp(-νt) for t0 and θΘ.

Moreover, the authors used the concept that a pair (E,F) of Schäffer spaces is admissible to π. The main result of the paper asserts that π is uniformly exponentially stable if and only if there exists a pair (E,F) of Schäffer spaces admissible to π and satisfying some additional condition. This nice paper contains also some other results and interesting examples explaining the basic concepts and illustrating the obtained results.

MSC:
34G10Linear ODE in abstract spaces
34D20Stability of ODE