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 [2], Hannsgen [10,11], Miller [14,15], and Miller and Wheeler [17,18]. For the nonconvolution case, the papers by Chen and Grimmer [3] and Friedman and Shinbrot [4] are of particular importance to the problem studied here. If X is finite dimensional the work of Grossman and Miller [8] 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 [21] ). 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
doi:10.2307/1999209 fatcat:y7r3hjdm6zcthnb5vizqul34we