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Integrated solutions of Volterra integrodifferential equations and applications. (English) Zbl 0675.45017
Volterra integrodifferential equations in Banach spaces and applications, Proc. Conf., Trento/Italy 1987, Pitman Res. Notes Math. Ser. 190, 21-51 (1989).
[For the entire collection see Zbl 0664.00018.]
The authors extend the concept of an integrated semigroup to the Volterra integrodifferential equation P(A,$$\eta)$$ $$u'(t)=\int^{t}_{0}Au(t- s)d\eta (s),$$ $$u(0)=x$$. The crucial condition for a strongly continuous family S(t) of bounded linear operators to be an n-times integrated solution family of P(a,$$\eta)$$ is that $(\lambda -{\hat \eta}(\lambda)A)^{-1}/\lambda^ n=\int e^{-\lambda t}S(t)dt,\quad \lambda >w.$ A number of properties of these solution families, in particular with respect to solutions of the nonhomogeneous equation, are established.
Some necessary and sufficient conditions for the problem P(A,$$\eta)$$ to be governed by an integrated solution family are given. The relationship to distribution semigroups and to integrated cosine functions is discussed. Finally a number of applications are given.
Reviewer: G.Gripenberg

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations