#### Vol. 64, No. 2, 1976

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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 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).

Primary: 45D05
##### Milestones 