### Resolvent Operators for Integral Equations in a Banach Space

R. C. Grimmer
1982 Transactions of the American Mathematical Society
Conditions are given which ensure the existence of a resolvent operator for an integrodifferential equation in a Banach space. The resolvent operator is similar to an evolution operator for nonautonomous differential equations in a Banach space. As in the finite dimensional case, this operator is used to obtain a variation of parameters formula which can be used to obtain results concerning the asymptotic behaviour of solutions and weak solutions. 1. Introduction. In this paper we shall be
more » ... er we shall be concerned with the integrodifferential equation in a Banach space X. A(t) and B(t, s) are closed linear operators on X with fixed domain which we will denote D(A) while the function/: R+ -> X is continuous. In the convolution case where A(t) = A and B(t, s) = B(t -s) this equation has been studied by numerous authors under various hypotheses concerning A and B. Of interest to us here are the papers by Chen and Grimmer , Hannsgen [10,11], Miller [14,15], and Miller and Wheeler [17,18]. For the nonconvolution case, the papers by Chen and Grimmer  and Friedman and Shinbrot  are of particular importance to the problem studied here. If X is finite dimensional the work of Grossman and Miller  is of significance to us as they develop perturbation theory for (VE) using the resolvent operator for (VE). It is this theory which shall be developed for (VE) when X is not finite dimensional. The resolvent operator will satisfy an equation like equation (A) of Grossman and Miller pointwise on D(A). It will also resemble an evolution operator for a nonautonomous linear differential equation in a Banach space (cf. e.g. Tanabe  ). It will not, however, be an evolution operator because it will not satisfy an evolution or semigroup property. Because a number of results follow directly from the definition of the resolvent operator we shall first hypothesize the existence of a resolvent and obtain the variation of parameters formula along with proving