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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.

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)