A special integral and a Gronwall inequality

Helton, Burrell W.

doi:10.1090/S0002-9947-1976-0407215-8

A special integral and a Gronwall inequality
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by Burrell W. Helton

Trans. Amer. Math. Soc. 217 (1976), 163-181

DOI: https://doi.org/10.1090/S0002-9947-1976-0407215-8

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Abstract:

This paper considers a special integral $(I)\smallint _a^b(fdg + H)$ which is a subdivision-refinement-type limit of the approximating sum \[ \sum \limits _1^n {\{ f({t_i})[g({x_i}) - g({x_{i - 1}})] + H({x_{i - 1}},{x_i})\} ,} \] where ${x_{i - 1}} < {t_i} < {x_i}$. The author shows, with appropriate restrictions, that $(I)\smallint _a^b(fdg + H)$ exists if and only if \[ (R)\smallint _x^y(fdg + H - {A^ - }) = (L)\smallint _x^y(fdg + H + {A^ + })\] for $a \leqslant x < y \leqslant b$, where $A(p,q) = [f(q) - f(p)][g(q) - g(p)],{A^ - }(p,q) = A({q^ - },q)$ and ${A^ + }(p,q) = A(p,{p^ + })$. Furthermore, if either of the equivalent statements is true, then all the integrals are equal. These equivalent statements are used to prove an integration-by-parts theorem and to solve a Gronwall inequality involving this special integral. Product integrals are used in the solution of the Gronwall inequality.

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