Admissible Operators and Solutions of Perturbed Operator Equations

The purpose of this note is to prove an abstract operator version of certain existence theorems for differential and Volterra integral equations which deal with stability properties of solutions. We will do this in such a way that the results unify, as well as generalize, many of the basic theorems concerning stability of such equations. Thus, our main result(Theorem2)is an abstract version of well-known results for differential systems and was obviously motivated by the work begun by Massera and Schaffer and others relating to admissibility [6]. (More precisely, Theorem 2 is appropriately applied to the equivalent Volterra integral system obtained by integrating a differential system.) In the same way, Theorem 2 can by applied to integrodifferential systems to obtain known stability results ($¥mathrm{e}.¥mathrm{g}.$ , Theorems 1 and 2 in [4], Theorem 1 in [5], and Theorem 10 in [10] $)$ . Also as a corollary we can derive a functional analytic theorem of Miller [7] which in turn has many stability theorems of Volterra integral equations as corollaries. These applications are made more explicit in the remarks below. Let $F$ denote a real Frechet space and let $B_{1}$ , $B_{2}$ denote two normed linear subspaces of $F$ whose norms $|¥cdot|_{1}$ , $|¥cdot|_{2}$ respectively yield topologies stronger than that induced by the metric on $F$ . We consider here the problem of describing (locally) the set of those elements $f¥in B_{2}$ for which the operator equation