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

Author: Burrell W. Helton
Journal: Trans. Amer. Math. Soc. 217 (1976), 163-181
MSC: Primary 26A42; Secondary 26A86
DOI: https://doi.org/10.1090/S0002-9947-1976-0407215-8
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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