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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

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