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On linear Volterra equations in Banach spaces. (English) Zbl 0569.45020
The authors study the linear equation u ' (t)=Au(t)+ 0 t B(t-s)u(s)ds+f(t),u(0)=u 0 , in a Banach space X. They prove that there exists a reasonable resolvent operator if and only if the autonomous equation (where f=0) is well-posed (i.e., it has a unique solution that depends continuously on u 0 ). Furthermore, under some additional weak restrictions they show that a necessary and sufficient condition for this to happen is that |(1/n!)H (n) (λ)|M(Reλ-ω) -n-1 for all Re λ >ω, n0, where H(λ)=(λ-A-B ^(λ)) -1 , that is a result of Hille-Yosida type. The authors also give an example showing that this condition can be satisfied although A does not generate a semigroup.
Reviewer: G.Gripenberg

45N05Abstract integral equations, integral equations in abstract spaces
45D05Volterra integral equations