Riemann-Stieltjes integral


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Riemann-Stieltjes integral

[¦rē‚män ′stēl·tyəs ‚int·i·grəl]
(mathematics)
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where [D.sub.t.sup.[alpha]], [D.sub.t.sup.[beta]], [D.sub.t.sup.[gamma]] are the Riemann-Liouville fractional derivatives of order [alpha], [beta], [gamma] with 0 < [gamma] [less than or equal to] 1 < [alpha] [less than or equal to] 2 < [beta] < 3, [alpha] - [gamma] > 1, [[integral].sup.1.sub.0] [D.sub.t.sup.[gamma]] z(s)dA(s) denotes a Riemann-Stieltjes integral, A is a function of bounded variation.
Sometimes (as in [7]) orthogonal polynomials is used instead of formal orthogonal polynomials, i.e., the meaning of the simpler term is extended beyond the classical setting with a positive definite linear functional and a Riemann-Stieltjes integral with a non-decreasing distribution function; see, e.g., [55], [22], and [41, Section 3.3].
Bai, "Positive solutions of fractional differential equations involving the Riemann-Stieltjes integral boundary condition," Advances in Difference Equations, vol.
By the same method, Zhang [18] discussed the following fractional differential equation with Riemann-Stieltjes integral boundary conditions:
Dragomir, "New bounds for the three-point rule involving the Riemann-Stieltjes integral," in Advances in Statistics, Combinatorics and Related Areas, C.
Clearly, the Riemann-Stieltjes integral boundary conditions includes multi-point boundary conditions as special cases.
Indeed, if there is a stochastic process u (t), whose trajectories are [lambda]-Holder continuous with [lambda] > 1 - H, then the Riemann-Stieltjes integral [[integral].sup.T.sub.0] u(s)d [B.sup.H](s) exists for each trajectory [28].
If f is bounded on [a, b], the for any t [member of] [a, b], the Riemann-Stieltjes Integral [[integral].sub.a.sup.x]df(s) = f(x) - f(a), [[integral].sub.x.sup.b]df(s) = f(b) - f(x).
The integral in (1.1) is understood as the improper vector Riemann-Stieltjes integral. For the details see for instance [8, p.
Fedotov, An inequality of Gruss type for Riemann-Stieltjes integral and applications for special means, Tamkang J.
In this paper, we investigate the eigenvalue problem for Caputo fractional boundary value problem with Riemann-Stieltjes integral boundary conditions
A Stieltjes or Cauchy-Stieltjes function is a function f, with f : C \ (-[infinity],0] [right arrow] C, that can be written as a Riemann-Stieltjes integral as follows: