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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A special integral and a Gronwall inequality

Author: Burrell W. Helton
Journal: Trans. Amer. Math. Soc. 217 (1976), 163-181
MSC: Primary 26A42; Secondary 26A86
MathSciNet review: 0407215
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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

$\displaystyle \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

$\displaystyle (R)\smallint _x^y(fdg + H - {A^ - }) = (L)\smallint _x^y(fdg + H + {A^ + })$

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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Keywords: Integrals, Gronwall's inequality, product integrals, integration-by-parts, bounded variation, Smith mean integral, Cauchy integrals, Dushkin interior integral, subdivision-refinement-limit
Article copyright: © Copyright 1976 American Mathematical Society

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