Paper
Integral equations and semigroups
Published Mar 1, 1963 · J. S. M. Nerney
Illinois Journal of Mathematics
59
Citations
1
Influential Citations
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)where 1 is the n n unit matrix, and provides G and H in the form G(y) G(c) W(c,y) and H(x) W(x,c)H(c).The relationship (i) has been extended by H. S. Wall [9], [10], with the condition of absolute continuity on replaced by that of continuity and bounded variation, the intrinsic nature of the harmonic matrices W so ob- rained being determined explicitly; the reciprocal formulas (involving sum- and product-integrals) (ii) q(y) (x) W( c) .dW(c,), W(x, y) I-IY[1 +were discovered, respectively, by Wall [10] and this author [4].The con- tinuity condition on has been relaxed, in two different directions, by T. H. Hildebrandt [2] and by the present author [5], [6].This paper is concerned with connections between additive and multiplica- rive integration processes, where the integration is directed along intervals in some linearly ordered system and the functions involved satisfy various conditions of boundedness, having their values in a normed algebraic ring which is complete as a metric space.
This paper explores connections between additive and multiplicative integration processes in linearly ordered systems, with functions satisfying various boundedness conditions in a normed algebraic ring.
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