# Riemann-Stieltjes integral

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## Riemann-Stieltjes integral

[¦rē‚män ′stēl·tyəs ‚int·i·grəl]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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where [mathematical expression not reproducible] is the Riemann-Liouville differential operator of 2 < [alpha] < 3, A(t) is right continuous on [0,1), left continuous at t = 1, and nondecreasing on [0,1] with A(0) = 0, and [[integral].sup.1.sub.0] u(t)dA(t) denotes the Riemann-Stieltjes integrals of u with respect to A.
Wu, "The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition," Applied Mathematics and Computation, vol.
involving Riemann-Stieltjes integrals defined via positive Stieltjes measures of A, B.
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are continuous functions, [a.sub.i], [b.sub.i], [c.sub.i], [d.sub.i] are nonnegative constants satisfying [b.sub.i] > 2-[[alpha].sub.i]/[[alpha].sub.i]-1 [a.sub.i] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for i = 1,2 are the Caputo fractional derivatives, A, B : [0, 1] [right arrow] [R.sup.+] are nondecreasing functions of bounded variation and the integrals are the Riemann-Stieltjes integrals.
Xiao starts by describing sets, relations, functions, cardinals, ordinals, reals, basic theorems and sequence limits, proceeding to Riemann integrals, Riemann-Stieltjes integrals, Lebesque-Radon-Stieltjes integrals, metric spaces, continuous maps, normed linear spaces, Banach spaces via operators and functionals, and Hilbert spaces and their operators.
Jensen-Steffensen Inequality for Riemann-Stieltjes Integrals
where 2 < p < 3, [D.sup.p.sub.0+] is the standard Riemann-Liouville differentiation, f : [0,1] x [R.sup.3] [right arrow] R, and f satisfies Caratheodory conditions; A(t) is right continuous on [0, 1) and left continuous at t = 1; [[integral].sup.1.sub.0] x(t) dA(t) denotes the Riemann-Stieltjes integrals of x with respect to A.

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