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Zbl 0569.45020
Grimmer, R.; Prüss, J.
On linear Volterra equations in Banach spaces.
(English)
[J] Comput. Math. Appl. 11, 189-205 (1985). ISSN 0898-1221

The authors study the linear equation $u'(t)=Au(t)+\int\sp{t}\sb{0}B(t- s)u(s)ds+f(t),\quad u(0)=u\sb 0,$ 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 $u\sb 0)$. Furthermore, under some additional weak restrictions they show that a necessary and sufficient condition for this to happen is that $\vert (1/n!)H\sp{(n)}(\lambda)\vert \le M(Re \lambda -\omega)\sp{-n-1}$ for all Re $\lambda$ $>\omega$, $n\ge 0$, where $H(\lambda)=(\lambda -A-\hat B(\lambda))\sp{-1},$ 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.
[G.Gripenberg]
MSC 2000:
*45N05 Integral equations in abstract spaces
45D05 Volterra integral equations

Keywords: Volterra equation; Banach space; resolvent; well-posed; semigroup

Cited in: Zbl 0688.45019

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