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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 $$v'= Av+ K(t) f+ F_ K(t)$$, $$v(0)= 0$$, $$(F_ K= K* F)$$, $$K$$ 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 $${\mathcal S}_ K(t)$$ giving the $$K$$-mild (local) solutions corresponding to the mentioned regularized problems. For the “$$K$$-convoluted semigroups” $$[{\mathcal S}_ K(t)$$ for $$D({\mathcal S}_ K(t))= X]$$ 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 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)