Vikuozonuo Sekhose, Hemanta Kalita, Hemen Bharali

DOI Number
First page
Last page


We establish the equivalence of the Riemann-Stieltjes $\Delta$-integral as defined in \cite{Delfim1, Delfim2} in terms of the Darboux sum definition and the Riemann sum definition, and provide the definition of the Riemann-Stieltjes $\nabla$-integral in terms of the Riemann sum definition and prove its equivalence with the Riemann-Stieltjes $\nabla$-integral as defined in \cite{Delfim1} in terms of the Darboux sum definition. We establish a few results concerning finite discontinuity.


Riemann-Stieltjes Delta Integral, Riemann-Stieltjes Nabla Integral, Time Scale.

Full Text:



bibitem{Bohner1} {sc R. P. Agarwal {rm and} M. Bohner}: textit{Basic Calculus on Time Scales and some of its Applications}. Results in Mathematics, {bf 35} (1999), 3--22.

bibitem{calvin} {sc C. D. Ahlbrandt {rm and} M. Bohner}: textit{Hamiltonian Systems on Time Scales}. Journal of Mathematical Analysis and Applications, {bf 250} (2000), 561--578.

bibitem{Guseinov1} {sc F. M. Atici {rm and} G. Sh. Guseinov}: textit{On Green s functions and positive solutions for boundary value problems on time scales}. Journal of Computational and Applied Mathematics, {bf 141} (2002), 75--99.

bibitem{luciano} {sc L. Barbanti and B. C. Damasceno and F. R. Chavarette and J. M. Balthazar}: textit{A generalized Riemann-Stieltjes Integral on Time Scales and discontinuous dynamical equations}. International Journal of Pure and Applied Mathematics, {bf 68}(3) (2011), 253--263.

bibitem{Bohner2} {sc M. Bohner and S. G. Georgiev}: textit{Multivariable Dynamic Calculus on Time Scales}. Springer, Switzerland, 2016.

bibitem{Bohner4} {sc M. Bohner and A. Peterson}: textit{Advances in Dynamic Equations on Time Scales}. Birkh$ddot{a}$user, Boston, 2003.

%bibitem{gil}M.l. Gil and P.E. Kloeden, textit{Stability and boundedness of solutions of Stieltjes Differential Equations*}, Results in Mathematics, 43, (2003), pp. 101-113

bibitem{Guseinov2} {sc G. Sh. Guseinov {rm and} B. Kaymakc{c}alan}: textit{Basics of Riemann Delta and Nabla Integration on Time Scales}. Journal of Difference Equations and Applications, {bf 8}(11) (2002), 1001--1017.

bibitem{Guseinov3} {sc G. Sh. Guseinov}: textit{Integration on time scales}. Journal of Mathematical Analysis and Applications, {bf 285} (2003), 107--127.

bibitem{Hilger1} {sc S. Hilger}: textit{Analysis on Measure Chains- A unified approach to continuous and discrete calculus}. Results in Mathematics, {bf 18}(1-2) (1990).

bibitem{Hilger2} {sc S. Hilger}: textit{Differential and Difference Calculus- Unified!}. Nonlinear Analysis: Theory, Methods & Applications, {bf 30}(5) (1997), 2683--2694.

bibitem{Adil} {sc A. Huseynov}: textit{The Riesz representation theorem on time scales}. Mathematical and Computer Modelling, {bf 55} (2012), 1570--1579.

bibitem{firstmonograph} {sc V. Lakshmikantham and S. Sivasundaram and B. Kaymakcalan}: textit{Dynamic Systems on Measure Chains}; Springer, B.Y., 1996.

bibitem{Delfim1} {sc D. Mozyrska {rm and} E. Paw{l}uszewicz {rm and} D. F. M. Torres}: textit{The Riemann-Stieltjes Integral on Time Scales}. The Australian Journal of Mathematical Analysis and Applications, {bf 7}(1) (2010), 1--14.

bibitem{Delfim2} {sc D. Mozyrska {rm and} E. Paw{l}uszewicz {rm and} D. F. M. Torres}: textit{Inequalities and majorisations for the Riemann Stieltjes integral on time scales*}. Mathematical Inequalities and Applications, (2010).

DOI: https://doi.org/10.22190/FUMI230125036S


  • There are currently no refbacks.

© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)