In 2000, Dragomir (2) answered to the problem of approximating the Stieltjes integral
[[integral].sub.a.sup.b]f(x)du(x) by the quantity [u(b) - u(a)]f (x), which is a natural generalization of the Ostrowski problem (3) analysed in 1937.
The text covers the Stieltjes integral
, fundamental formulas, the moment problem, absolutely and completely monotonic functions, Tauberian theorems, the bilateral Laplace transform, inversion and representation problems for the Laplace transform, and the Stieltjes transform.
and integrating by parts in the Stieltjes integral
[[integral].sub.a.sup.b] [PSI] (t) df (t), which exists, since f is of bounded variation and [PSI] is differentiable on (a, b).
If u and [phi] are two real-valued functions defined on the interval [a, b], then under some appropriate conditions (see ) we can define the Stieltjes integral
(in the Riemann-Stieltjes sense)
The following theorem gives conditions under which we can take the limit of a sequence [([h.sub.n).sub.n[member of]N] of functions inside a Stieltjes integral
Jankowski, Monotone iterative method to second order differential equations with deviating arguments involving Stieltjes integral
boundary conditions, Dynam.
Thus, lower and upper bounds for the quantity [[rho].sub.[mu]] defined by (2.3) can be determined by replacing the Stieltjes integral
in (2.13) by the Gauss and Gauss-Radau quadrature rules (3.14) and (3.15), respectively.
Li, "Positive solutions of second-order problem with dependence on derivative in nonlinearity under Stieltjes integral
boundary condition," Electronic Journal of Qualitative Theory of Differential Equations, no.
Jankowski, Positive solutions for second order impulsive differential equations involving Stieltjes integral
conditions, Nonlinear Anal., 74:3775-3785, 2011.
where [D.sup.[alpha].sub.0+] is the Riemann-Liouville fractional derivative, n - 1 < [alpha] [less than or equal to] n, n [greater than or equal to] 2,[lambda]X[z] = [[integral].sup.1.sub.0] z(t) dA (t) is a linear functional on C[0, 1] given by a Stieltjes integral
with A representing a suitable function of bounded variation, and dA can be a signed measure.
The Integration by parts Corollary 27 yields a new representation theorem without using Stieltjes integral
, which we shall establish next.
Wu, "Positive Solutions for Second-Order Singular Semipositone Differential Equations Involving Stieltjes Integral
Conditions," Abstract and Applied Analysis, vol.