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Journal articles on the topic 'Riemann-Stieltjes integral'

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Published: 4 June 2021

Last updated: 27 April 2022

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1

Rumlawang, Francis Y., and Harimanus Batkunde. "SIFAT-SIFAT INTEGRAL RIEMANN-STIELTJES." BAREKENG: Jurnal Ilmu Matematika dan Terapan 1, no. 2 (December 1, 2007): 25–30. http://dx.doi.org/10.30598/barekengvol1iss2pp25-30.

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If is limited and []ℜ→baf,:[]ℜ→ba,:α Monotone increase in [, is Riemann-Stieltjes integral able to α on ] ba,[]ba, simply written by[]αRSf∈ if . With JI=()()xdxfIbaα∫= is called Riemann Stieltjes lower integral f to α and ()()xdxfJbaα∫= is called Riemann Stieltjes upper integral f to α. Then is called Riemann Stieltjes upper integral f to ()()∫==baxdxfJIαα on [. if f ang g is Riemann Stieltjes integralable, and, k oe √ then f + g, kf, and fg is also Riemann Stieltjes integralable. But if f and ] ba,α have united discontinue point then f is not Riemann Stieltjes integralable on α

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2

Narita, Keiko, Kazuhisa Nakasho, and Yasunari Shidama. "Riemann-Stieltjes Integral." Formalized Mathematics 24, no. 3 (September 1, 2016): 199–204. http://dx.doi.org/10.1515/forma-2016-0016.

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Abstract In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties. In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].

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3

Pirade, Septian, Tohap Manurung, and Jullia Titaley. "Integral Riemann-Stieltjes Pada Fungsi Bernilai Real." d'CARTESIAN 6, no. 1 (February 1, 2017): 1. http://dx.doi.org/10.35799/dc.6.1.2017.14987.

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Integral Riemann-Stieltjes merupakan generalisasi dari Integral Riemann, kedua Integral ini memiliki hubungan, juga beberapa sifat dasar pada Integral Riemann dapat diberlakukan pada Integral Riemann-Stieltjes. Misalkan f dan a adalah fungsi bernilai real yang terbatas pada interval [a,b]. Jika f ϵ R[a,b] dan f ϵ Ra[a,b], maka sifat terbatas, monoton naik, linear penjumlahan dan linear perkalian terhadap konstanta yang berlaku pada fungsi f yang terintegral Riemann, berlaku juga pada fungsi f yang terintegral Riemann-Stieltjes. Jika a(x) = x, maka integral Riemann-Stieltjes ekuivalen dengan integral Riemann, dan dapat direduksi menjadi integral Riemann ketika a mempunyai turunan dan terbatas pada interval terbuka (a,b). Kata kunci : Fungsi Bernilai Real, Integral Riemann, Integral Riemann-Stieltjes.

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4

Bradley, R. E. "The Riemann-Stieltjes Integral." Missouri Journal of Mathematical Sciences 6, no. 1 (February 1994): 20–28. http://dx.doi.org/10.35834/1994/0601020.

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5

Štajner-Papuga, Ivana, Tatjana Grbić, and Martina Daňková. "Pseudo-Riemann–Stieltjes integral." Information Sciences 179, no. 17 (August 2009): 2923–33. http://dx.doi.org/10.1016/j.ins.2008.09.009.

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6

Wu, Hsien-Chung. "The Fuzzy Riemann-Stieltjes Integral." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 06, no. 01 (February 1998): 51–67. http://dx.doi.org/10.1142/s0218488598000045.

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In the classcial sense, the expectation is sometimes defined as Riemann-Stieltjes integral. In this paper, we propose the concept of fuzzy Riemann-Stieltjes integral to make the expectation of fuzzy random variable applicable to statistical analysis for imprecise data.

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7

Muchtar, Kalfin, Jullia Titaley, and Mans Mananohas. "Integral Baire-1 Stieltjes, Henstock-Stieltjes dan Riemann-Stieltjes." d'CARTESIAN 5, no. 1 (April 29, 2016): 7. http://dx.doi.org/10.35799/dc.5.1.2016.11937.

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Beberapa sifat dasar termasuk Kriteria Cauchy dan Teorema Aditif dapat diberlakukan pada konsep integral Baire-1 Stieltjes. Misalkan 𝑓 dan 𝑔 merupakan fungsi-fungsi bernilai real yang didefinisikan pada [𝑎,𝑏]⊂ℝ. Jika 𝑓 terintegral Baire-1 Stieltjes terhadap 𝑔 pada [𝑎,𝑏], maka 𝑓 terintegral Henstock-Stieltjes terhadap 𝑔 pada [𝑎,𝑏] dengan nilai integralnya sama. Syarat cukup agar fungsi 𝑓 yang terintegral Henstock-Stieltjes terhadap 𝑔 pada [𝑎,𝑏] terintegral Baire-1 Stieltjes terhadap 𝑔 pada [𝑎,𝑏] yaitu 𝑓 fungsi kelas Baire-1 dan 𝑔 fungsi bervariasi terbatas pada [𝑎,𝑏]. Jika 𝑓 terintegral Riemann-Stieltjes terhadap fungsi 𝑔 pada [𝑎,𝑏], maka 𝑓 terintegral Baire-1 Stieltjes terhadap 𝑔 pada [𝑎,𝑏] dengan nilai integralnya sama. Kata Kunci: Integral Baire-1 Stieltjes, Integral Henstock-Stieltjes, Integral Riemann-Stieltjes

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8

Zhao, Weijing, and Zhaoning Zhang. "Derivative-Based Trapezoid Rule for the Riemann-Stieltjes Integral." Mathematical Problems in Engineering 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/874651.

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The derivative-based trapezoid rule for the Riemann-Stieltjes integral is presented which uses 2 derivative values at the endpoints. This kind of quadrature rule obtains an increase of two orders of precision over the trapezoid rule for the Riemann-Stieltjes integral and the error term is investigated. At last, the rationality of the generalization of derivative-based trapezoid rule for Riemann-Stieltjes integral is demonstrated.

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9

Rodionov, V. I. "The adjoint Riemann-Stieltjes integral." Russian Mathematics 51, no. 2 (February 2007): 75–79. http://dx.doi.org/10.3103/s1066369x07020107.

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10

Pfeffer. "THE GENERALIZED RIEMANN-STIELTJES INTEGRAL." Real Analysis Exchange 21, no. 2 (1995): 521. http://dx.doi.org/10.2307/44152664.

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11

Mozer, Víctor Marques Fernandes, and Marisa de Souza Costa. "A Integral de Riemann-Stieltjes." Brazilian Journal of Development 6, no. 6 (2020): 37747–55. http://dx.doi.org/10.34117/bjdv6n6-346.

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12

Ren, Xuekun, and Chong Wu. "The Fuzzy Riemann-Stieltjes Integral." International Journal of Theoretical Physics 52, no. 6 (February 14, 2013): 2134–51. http://dx.doi.org/10.1007/s10773-013-1511-9.

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13

Alomari, Mohammad. "Inequalities for Riemann - Stieltjes integral." International Journal of Emerging Multidisciplinaries: Mathematics 1, no. 1 (January 14, 2022): 12–16. http://dx.doi.org/10.54938/ijemdm.2022.01.1.14.

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14

Alomari, Mohammad W. "A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 15, no. 1 (December 1, 2016): 69–78. http://dx.doi.org/10.1515/aupcsm-2016-0006.

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AbstractA sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral $\int_a^b {f(t)\;du(t)} $, where f is assumed to be of r-H-Hölder type on [a, b] and u is of bounded variation on [a, b], is proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.

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15

Nakasho, Kazuhisa, Keiko Narita, and Yasunari Shidama. "The Basic Existence Theorem of Riemann-Stieltjes Integral." Formalized Mathematics 24, no. 4 (December 1, 2016): 253–59. http://dx.doi.org/10.1515/forma-2016-0021.

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Summary In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11].

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16

Gal, Sorin G. "On a Choquet-Stieltjes type integral on intervals." Mathematica Slovaca 69, no. 4 (August 27, 2019): 801–14. http://dx.doi.org/10.1515/ms-2017-0269.

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Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

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17

Fedorov, D. L. "On the Riemann-Stieltjes double integral." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 26, no. 3 (September 2016): 366–78. http://dx.doi.org/10.20537/vm160306.

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18

Grobler, Trienko. "Visualization of the Riemann–Stieltjes Integral." College Mathematics Journal 50, no. 3 (May 27, 2019): 198–209. http://dx.doi.org/10.1080/07468342.2019.1580109.

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19

Valdivia, Manuel. "On the integral of Riemann–Stieltjes." Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 108, no. 2 (May 17, 2013): 567–75. http://dx.doi.org/10.1007/s13398-013-0128-4.

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20

Ma, Wenjie, Shuman Meng, and Yujun Cui. "Resonant Integral Boundary Value Problems for Caputo Fractional Differential Equations." Mathematical Problems in Engineering 2018 (August 7, 2018): 1–8. http://dx.doi.org/10.1155/2018/5438592.

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This paper deals with the following Caputo fractional differential equations with Riemann-Stieltjes integral boundary conditions Dc0+αut=ft,ut,u′t,u′′t, t∈0,1, u0=u′′0=0, u1=∫01‍utdAt, where Dc0+α denotes the standard Caputo derivative, α∈(2,3]; ∫01x(t)dA(t) denotes the Riemann-Stieltjes integrals of x with respect to A. By mean of coincidence degree theory, we obtain the existence of solutions for the above fractional BVP at resonance. In the end, according to the main results, we give a typical example.

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21

Hakim, Denny Ivanal, and Hendra Gunawan. "KETAKSAMAAN HERMITE-HADAMARD TERHADAP INTEGRAL RIEMANN-STIELTJES." Jurnal Ilmiah Matematika dan Pendidikan Matematika 4, no. 1 (June 29, 2012): 59. http://dx.doi.org/10.20884/1.jmp.2012.4.1.2942.

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The Hermite-Hadamard inequality is an inequality for convex functions that gives an estimate for the integral mean value of a convex function on a closed interval by its value at the middle of interval and the average of its values at the endpoints. The Hermite-Hadamard inequality can be generalized by using the Riemann-Stieltjes integral mean value. An application of the Hermite-Hadamard inequality with respect to Riemann-Stieltjes integral for estimating the power mean of positive real numbers by the aritmethic mean is given at the end of discussion.

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22

Alomari, Wajeeh. "On approximation of the Riemann-Stieltjes integral and applications." Publications de l'Institut Math?matique (Belgrade) 92, no. 106 (2012): 145–56. http://dx.doi.org/10.2298/pim1206145a.

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Several inequalities of Gr?ss type for the Stieltjes integral with various type of integrand and integrator are introduced. Some improvements inequalities are proved. Applications to the approximation problem of the Riemann-Stieltjes integral are also pointed out.

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23

Alomari, Mohammad. "Difference between two Riemann-Stieltjes integral means." Kragujevac Journal of Mathematics 38, no. 1 (2014): 35–49. http://dx.doi.org/10.5937/kgjmath1401035a.

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24

Zhao, Weijing, and Zhaoning Zhang. "Simpson’s rule for the Riemann-Stieltjes integral." Journal of Interdisciplinary Mathematics 24, no. 5 (March 18, 2021): 1305–14. http://dx.doi.org/10.1080/09720502.2020.1848317.

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25

LUKKARINEN, JANI, and MIKKO S. PAKKANEN. "ON THE POSITIVITY OF RIEMANN–STIELTJES INTEGRALS." Bulletin of the Australian Mathematical Society 87, no. 3 (August 2, 2012): 400–405. http://dx.doi.org/10.1017/s0004972712000639.

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26

Bullock, Gregory L. "A Geometric Interpretation of the Riemann-Stieltjes Integral." American Mathematical Monthly 95, no. 5 (May 1988): 448. http://dx.doi.org/10.2307/2322483.

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27

Bullock, Gregory L. "A Geometric Interpretation of the Riemann-Stieltjes Integral." American Mathematical Monthly 95, no. 5 (May 1988): 448–55. http://dx.doi.org/10.1080/00029890.1988.11972030.

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28

Valdivia, Manuel, and Manuel Fúnez. "A note on the integral of Riemann–Stieltjes." Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 108, no. 2 (September 24, 2013): 827–32. http://dx.doi.org/10.1007/s13398-013-0145-3.

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29

Munteanu*, Marilena. "Quadrature formulas for the generalized Riemann-Stieltjes integral." Bulletin of the Brazilian Mathematical Society, New Series 38, no. 1 (March 2007): 39–50. http://dx.doi.org/10.1007/s00574-007-0034-5.

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30

Alomari, Mohammad Wajeeh. "Bounds for the Riemann–Stieltjes integral via s-convex integrand or integrator." Acta et Commentationes Universitatis Tartuensis de Mathematica 16, no. 2 (September 25, 2012): 181–89. http://dx.doi.org/10.12697/acutm.2012.16.10.

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31

DRAGOMIR, S. S. "APPROXIMATING THE STIELTJES INTEGRAL FOR (φ,Φ)-LIPSCHITZIAN INTEGRATORS." Bulletin of the Australian Mathematical Society 77, no. 1 (February 2008): 73–90. http://dx.doi.org/10.1017/s0004972708000063.

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AbstractApproximations for the Stieltjes integral with (φ,Φ)-Lipschitzian integrators are given. Applications for the Riemann integral of a product and for the generalized trapezoid and Ostrowski inequalities are also provided.

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32

Ashirbayev, Nurgali K., Józef Banaś, and Raina Bekmoldayeva. "A Unified Approach to Some Classes of Nonlinear Integral Equations." Journal of Function Spaces 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/306231.

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We are going to discuss some important classes of nonlinear integral equations such as integral equations of Volterra-Chandrasekhar type, quadratic integral equations of fractional orders, nonlinear integral equations of Volterra-Wiener-Hopf type, and nonlinear integral equations of Erdélyi-Kober type. Those integral equations play very significant role in applications to the description of numerous real world events. Our aim is to show that the mentioned integral equations can be treated from the view point of nonlinear Volterra-Stieltjes integral equations. The Riemann-Stieltjes integral appearing in those integral equations is generated by a function of two variables. The choice of a suitable generating function enables us to obtain various kinds of integral equations. Some results concerning nonlinear Volterra-Stieltjes integral equations in several variables will be also discussed.

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33

Zając, Tomasz. "On Monotonic and Nonnegative Solutions of a Nonlinear Volterra-Stieltjes Integral Equation." Journal of Function Spaces 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/601824.

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We study the existence of monotonic and nonnegative solutions of a nonlinear quadratic Volterra-Stieltjes integral equation in the space of real functions being continuous on a bounded interval. The main tools used in our considerations are the technique of measures of noncompactness in connection with the theory of functions of bounded variation and the theory of Riemann-Stieltjes integral. The obtained results can be easily applied to the class of fractional integral equations and Volterra-Chandrasekhar integral equations, among others.

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34

Zhao, Daliang, and Juan Mao. "Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions." Symmetry 13, no. 1 (January 8, 2021): 107. http://dx.doi.org/10.3390/sym13010107.

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In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

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35

Zhao, Daliang, and Juan Mao. "Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions." Symmetry 13, no. 1 (January 8, 2021): 107. http://dx.doi.org/10.3390/sym13010107.

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In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

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36

Dragomir, Silvestru. "INEQUALITIES FOR THE RIEMANN-STIELTJES INTEGRAL OF S-DOMINATED INTEGRATORS WITH APPLICATIONS. I." Issues of Analysis 22, no. 1 (October 2015): 11–37. http://dx.doi.org/10.15393/j3.art.2015.2809.

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37

Tudorache, Alexandru, and Rodica Luca. "Positive Solutions of a Fractional Boundary Value Problem with Sequential Derivatives." Symmetry 13, no. 8 (August 13, 2021): 1489. http://dx.doi.org/10.3390/sym13081489.

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We investigate the existence of positive solutions of a Riemann-Liouville fractional differential equation with sequential derivatives, a positive parameter and a nonnegative singular nonlinearity, supplemented with integral-multipoint boundary conditions which contain fractional derivatives of various orders and Riemann-Stieltjes integrals. Our general boundary conditions cover some symmetry cases for the unknown function. In the proof of our main existence result, we use an application of the Krein-Rutman theorem and two theorems from the fixed point index theory.

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38

DRAGOMIR, S. S., and I. FEDOTOV. "THE QUASILINEARITY OF SOME FUNCTIONALS ASSOCIATED WITH THE RIEMANN–STIELTJES INTEGRAL." Bulletin of the Australian Mathematical Society 84, no. 1 (April 5, 2011): 53–66. http://dx.doi.org/10.1017/s0004972710002042.

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AbstractThe superadditivity and subadditivity of some functionals associated with the Riemann–Stieltjes integral are established. Applications in connection to Ostrowski’s and the generalized trapezoidal inequalities and for special means are provided.

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39

Makogin, Vitalii, and Yuliya Mishura. "Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions." Nonlinear Analysis: Modelling and Control 25, no. 6 (November 1, 2020): 1079–106. http://dx.doi.org/10.15388/namc.2020.25.20566.

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In this paper, we find fractional Riemann–Liouville derivatives for the Takagi–Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi–Landsberg functions, which have arbitrary bounded coefficients in the expansion under Schauder basis. The class of weighted Takagi–Landsberg functions of order H > 0 on [0; 1] coincides with the class of H-Hölder continuous functions on [0; 1]. Based on computed fractional integrals and derivatives of the Haar and Schauder functions, we get a new series representation of the fractional derivatives of a Hölder continuous function. This result allows us to get a new formula of a Riemann–Stieltjes integral. The application of such series representation is a new method of numerical solution of the Volterra and linear integral equations driven by a Hölder continuous function.

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40

MEMON, K., M. M. SHAIKH, M. S. CHANDIO, and A. W. SHAIKH. "A Modified Derivative-BasedScheme for the Riemann-Stieltjes Integral." SINDH UNIVERSITY RESEARCH JOURNAL -SCIENCE SERIES 52, no. 1 (March 21, 2020): 37–40. http://dx.doi.org/10.26692/surj/2020.03.06.

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41

Zhao, Dafang, Xuexiao You, and Jian Cheng. "THE RIEMANN-STIELTJES DIAMOND-ALPHA INTEGRAL ON TIME SCALES." Journal of the Chungcheong Mathematical Society 28, no. 1 (February 15, 2015): 53–63. http://dx.doi.org/10.14403/jcms.2015.28.1.53.

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42

Dragomir, Sever Silvestru. "A generalized Čebyšev functional for the Riemann-Stieltjes integral." Mathematical Inequalities & Applications, no. 3 (2015): 959–73. http://dx.doi.org/10.7153/mia-18-72.

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43

Ahmad, Bashir, Ahmed Alsaedi, and Ymnah Alruwaily. "On Riemann-Stieltjes integral boundary value problems of Caputo-Riemann-Liouville type fractional integro-differential equations." Filomat 34, no. 8 (2020): 2723–38. http://dx.doi.org/10.2298/fil2008723a.

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Under different criteria, we prove the existence and uniqueness of solutions for a Riemann-Stieltjes integro-multipoint boundary value problem of Caputo-Riemann-Liouville type fractional integrodifferential equations. Our results rely on the modern methods of functional analysis and are well-illustrated with the help of examples. Some interesting observations are also presented.

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44

Shammakh, Wafa, Hadeel Z. Alzumi, and Bushra A. AlQahtani. "A New Class of ψ -Caputo Fractional Differential Equations and Inclusion." Journal of Mathematics 2021 (January 16, 2021): 1–18. http://dx.doi.org/10.1155/2021/6677959.

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In the present research work, we investigate the existence of a solution for new boundary value problems involving fractional differential equations with ψ -Caputo fractional derivative supplemented with nonlocal multipoint, Riemann–Stieltjes integral and ψ -Riemann–Liouville fractional integral operator of order γ boundary conditions. Also, we study the existence result for the inclusion case. Our results are based on the modern tools of the fixed-point theory. To illustrate our results, we provide examples.

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45

Coghetto, Roland. "Gauge Integral." Formalized Mathematics 25, no. 3 (October 1, 2017): 217–25. http://dx.doi.org/10.1515/forma-2017-0021.

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Summary Some authors have formalized the integral in the Mizar Mathematical Library (MML). The first article in a series on the Darboux/Riemann integral was written by Noboru Endou and Artur Korniłowicz: [6]. The Lebesgue integral was formalized a little later [13] and recently the integral of Riemann-Stieltjes was introduced in the MML by Keiko Narita, Kazuhisa Nakasho and Yasunari Shidama [12]. A presentation of definitions of integrals in other proof assistants or proof checkers (ACL2, COQ, Isabelle/HOL, HOL4, HOL Light, PVS, ProofPower) may be found in [10] and [4]. Using the Mizar system [1], we define the Gauge integral (Henstock-Kurzweil) of a real-valued function on a real interval [a, b] (see [2], [3], [15], [14], [11]). In the next section we formalize that the Henstock-Kurzweil integral is linear. In the last section, we verified that a real-valued bounded integrable (in sense Darboux/Riemann [6, 7, 8]) function over a interval a, b is Gauge integrable. Note that, in accordance with the possibilities of the MML [9], we reuse a large part of demonstrations already present in another article. Instead of rewriting the proof already contained in [7] (MML Version: 5.42.1290), we slightly modified this article in order to use directly the expected results.

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46

DRAGOMIR, S. S., and I. FEDOTOV. "AN INEQUALITY OF GRUSS' TYPE FOR RIEMANN-STIELTJES INTEGRAL AND APPLICATIONS FOR SPECIAL MEANS." Tamkang Journal of Mathematics 29, no. 4 (December 1, 1998): 287–92. http://dx.doi.org/10.5556/j.tkjm.29.1998.4257.

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In this paper we derive a new inequality ofGruss' type for Riemann-Stieltjes integral and apply it for special means (logarithmic mean, identric mean, etc·. ·).

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47

BESALÚ, MIREIA, and CARLES ROVIRA. "STOCHASTIC VOLTERRA EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION WITH HURST PARAMETER H > 1/2." Stochastics and Dynamics 12, no. 04 (October 10, 2012): 1250004. http://dx.doi.org/10.1142/s0219493712500049.

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In this note we prove an existence and uniqueness result of solution for stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2, showing also that the solution has finite moments. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.

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48

Yang, Xiao-Jun. "The vector power-law calculus with applications in power-law fluid flow." Thermal Science 24, no. 6 Part B (2020): 4289–302. http://dx.doi.org/10.2298/tsci2006289y.

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In this article, based on the Leibniz derivative and Stieltjes-Riemann integral, we suggest the vector power-law calculus to consider the conservations of the mass and angular momentums for the power-law fluid.

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49

Wang, Wei, and Li Huang. "Existence of Positive Solution for Semipositone Fractional Differential Equations Involving Riemann-Stieltjes Integral Conditions." Abstract and Applied Analysis 2012 (2012): 1–17. http://dx.doi.org/10.1155/2012/723507.

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The existence of at least one positive solution is established for a class of semipositone fractional differential equations with Riemann-Stieltjes integral boundary condition. The technical approach is mainly based on the fixed-point theory in a cone.

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50

Yang, Ge-Feng. "Nontrivial Solution of Fractional Differential System Involving Riemann-Stieltjes Integral Condition." Abstract and Applied Analysis 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/719192.

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We study the existence and uniqueness of nontrivial solutions for a class of fractional differential system involving the Riemann-Stieltjes integral condition, by using the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle, some sufficient conditions of the existence and uniqueness of a nontrivial solution of a system are obtained.

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