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Evolution problems regularized via a general kernel $K(t)$. Duhamel formula, extensions, generation results. (Problèmes d’évolution régularisés par un noyau général $K(t)$. Formule de Duhamel, prolongements, théorèmes de génération.) (French. Abridged English version) Zbl 0821.47032
Summary: For problems of the type $u'= Au+ F(t)$, $u(0)= f$, in a Banach space $X$, we consider the regularized problems $v'= Av+ K(t) f+ F\sb K(t)$, $v(0)= 0$, $(F\sb K= K* F)$, $K$ being a kernel operator-valued in general. (This contains the well-known “integrated solutions”, “integrated semigroups”, and “$C$-semigroups”; other situations are described below.) We study the evolution operators ${\cal S}\sb K(t)$ giving the $K$-mild (local) solutions corresponding to the mentioned regularized problems. For the “$K$-convoluted semigroups” $[{\cal S}\sb K(t)$ for $D({\cal S}\sb K(t))= X]$ we obtain a generation theorem in terms of the resolvent $R(z, A)$ of a generator (Hille-Yosida type results of very general kind). We give several applications, improving among others well-known results of J. Chazarain and R. Beals.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 47E05 Ordinary differential operators