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On linear Volterra equations in Banach spaces. (English) Zbl 0569.45020
The authors study the linear equation ${u}^{\text{'}}\left(t\right)=Au\left(t\right)+{\int }_{0}^{t}B\left(t-s\right)u\left(s\right)ds+f\left(t\right),\phantom{\rule{1.em}{0ex}}u\left(0\right)={u}_{0},$ in a Banach space X. They prove that there exists a reasonable resolvent operator if and only if the autonomous equation (where $f=0\right)$ is well-posed (i.e., it has a unique solution that depends continuously on ${u}_{0}\right)$. Furthermore, under some additional weak restrictions they show that a necessary and sufficient condition for this to happen is that $|\left(1/n!\right){H}^{\left(n\right)}{\left(\lambda \right)|\le M\left(Re\lambda -\omega \right)}^{-n-1}$ for all Re $\lambda$ $>\omega$, $n\ge 0$, where $H\left(\lambda \right)={\left(\lambda -A-\stackrel{^}{B}\left(\lambda \right)\right)}^{-1},$ 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

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45D05 Volterra integral equations
##### Keywords:
Volterra equation; Banach space; resolvent; well-posed; semigroup