A Special Integral and a Gronwall Inequality

Abstract

This paper considers a special integral (l)f^(fdg + H) which is a subdivision-refinement-type limit of the approximating sum 2{/(',)[*(*,) -8ix_x)] + H(X_X, x)}, where X_x < t < x.The author shows, with appropriate restrictions, that (I)fba(fdg + H) exists if and only if (R)SX (fdg + H-A~) = (L)fx(fdg + H + A+) for a < x < y < 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.Introduction.This paper considers a special integral (I)fba(fdg + 77) which is a subdivision-refinement-type Umit of the approximating sum {/('IX*,)"***,.,)]+C*f-i.xj)}, iwhere x_x <t<xtfot i=l,2..AU functions are from real numbers to real numbers.Since the function 77 might be defined as 77(jc, y) -u{x)[r(y)-r(x)] + v(y)[s(y)-s(x)], then the Cauchy left and right integrals, the Smith mean integral [9], and the weighted integral of Wright and Baker [13] are special cases of this integral.We define A, A~ and A+ to be the functions A(x, y) = [f(y) -f(x)] \g(y) -g(x)], A~(x, y) = A(y~, y) and A+(x, y) = A(x, x+); then we show, with suitable restrictions, that o) o)Jaifdg+m