Vol. 64, No. 2, 1976

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ISSN: 0030-8730
The solution of a Stieltjes-Volterra integral equation for rings

Burrell Washington Helton

Vol. 64 (1976), No. 2, 419–436
Abstract

For a triple (h,K,g) of functions and an interval [a,x], the author defines a subdivision-refinement-type limit V (a,x;h,K,dg) of the set {V (D,h,K,Δg)} of determinants, where each subdivision D = {xi}0n of [a,x] defines an n×n determinant of the set and each determinant has the form

|                                      |
||h1 + h0K10Δg1    − 1       0       0   ||
||h2 + h0K20Δg1  K21Δg2     − 1      0   ||
||h3 + h0K30Δg1  K31Δg2   K32Δg3    − 1  ||
|h4 + h0K40Δg1  K41Δg2   K42Δg3  K43Δg4 |

The following theorem is proved. If f, g, h and K are functions to a ring and g has bounded variation on [a,b], then (f,K,g) OA and f(x) = h(x) + (L) axf(t)K(x,t)dg(t) on [a,b] iff (h,K,g) OM and f(x) = V (a,x;h,K,dg) on [a,b]. The OA and OM sets are defined and sufficient conditions are proved for (f,K,g) OA and (h,K,g) OM, and for the existence of the limit V (a,x;h,K,dg), and for V (a,x;h,K,dg) = h(x) (L) axh(t)dV (t,x;1,K,dg).

Mathematical Subject Classification 2000
Primary: 45D05
Milestones
Received: 15 October 1974
Revised: 11 February 1976
Published: 1 June 1976
Authors
Burrell Washington Helton