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X and Y denote Banach spaces; we consider systems of the form (K) yit)-y(t 0) + ƒ ^ d a K(t, a) • y(&) = fit)-ƒ (f 0), (F) %]=c, where j, ƒ G G([fl, 6], ^) (the space of regulated functions g: [a, b] — • X, i.e., g has only discontinuities of the first kind); K G G uo (see §2) and F G L[G([a, b], X), Y] (linear constraint). (K) includes linear Volterra integral equations, linear delay differential equations, differential equations./ 4-A 1 y —f 1 , with the meaning that we have (L) y(t)-y(s) + ƒ I dA(a) • y(o) = ƒ (r)-ƒ (0 for all s, r G [a, &]. In §2 we give the existence of the resolvent for (K) and in §3 for (L); in §4 we find the Green function for the system (K), (F). The results of §1 are used in the proofs. All results of this announcement may be extended to open intervals and Y a separated sequentially complete locally convex TVS. The proofs will appear in [H.3]. 1. A division of [a, b] is a finite sequence d: t 0 = a <t x <* • • <t n =b. We write \d\ = n and Ad = sup 1</<w \t t-t i ^. l |. The set D of all divisions of [a, b] is ordered by refinement and lim^^x^ denotes the limit according to the associated net. For a: [a, b] —»L(X, Y) and ƒ: [a, b]->Iwe define the usual Riemann-Stieltjes operator integral f b da(t)-f(t) = lim EM'/)-*('/-!)] • /«/) where % t G [^__ x , /J (see [G], [H.l], [D]), and the interior integral

Volterra-Stieltjes integral equations with linear constraints and discontinuous solutions