Integral equations and semigroups

Abstract

Consider an n n mtrix of absolutely continuous functions on n intervM S of rel numbers. If ech of G nd H is lso such mtrix, then, s is well known (cf. [2, p. 352] for comments nd references), the differentiM require- ment that G' (s) G (s)' (s) 0\ almost everywhere S, on H' (s) -k ' (s)H (s) oj where 0 is the n n zero mtrix, is equivalent to the (Stieltjes) integral re- quirement that if c is in S then for 11 x and y in S a() a(c) + a.d4, and H(z) H(c) + d4.H; moreover, there is a fundamental matrix W of eonginuous functions on S N S which saisfieswihout exception (i) W(x,y)