Introduction to the Stieltjes extension of Dunford and Pettis-type integrals on time scales

Hemen Bharali; Vikuozonuo  Sekhose; Hemanta  Kalita

doi:10.5269/bspm.76701

Hemen Bharali Assam Don Bosco University https://orcid.org/0000-0002-4800-9536
Vikuozonuo Sekhose https://orcid.org/0009-0007-1465-1604
Hemanta Kalita https://orcid.org/0000-0002-9798-6608

Résumé

Riemann-Stieltjes type integrals on time scales are explored in this paper. Theoretical definitions of the Riemann-Stieltjes-Dunford, Riemann-Stieltjes-Pettis and Riemann-Stieltjes-Gelfand integrals are presented. Scalarly Riemann-Stieltjes integral, weak Riemann-Stieltjes integral and weak∗ scalarly RiemannStieltjes integral definitions are put forward; and a few relations between these integrals are established. Uniform convergence of a sequence of functions which are Riemann-Stieltjes-Dunford, Riemann-Stieltjes-Pettis and Riemann-Stieltjes-Gelfand integrable are also formulated.

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Références

C. D. Ahlbrandt, M. Bohner, Hamiltonian Systems on Time Scales. Journal of Mathematical Analysis and Applications 250, 561-578, (2000).

F. M. Atici, G. Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics 141, 75-99, (2002).

B. Aulbach, L. Neidhart, Integration on Measure Chains, in: Proc. Sixth Int. Conf. Difference Equations, CRC, Boca Raton, FL, 239-252, (2004).

M. Bohner, S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer International Publishing Switzerland, 2016.

M. Cichon, On integrals of vector-valued functions on time scales. Communications in Mathematical Analysis 11(1), 94-110, (2011).

G. Sh. Guseinov, B. Kaymak¸calan, Basics of Riemann Delta and Nabla Integration on Time Scales. Journal of Difference Equations and Applications 8(11), 1001-1017, (2002).

G. Sh. Guseinov, Integration on time scales. Journal of Mathematical Analysis and Applications 285, 107-127, (2003).

S. Hilger, Ein Maßkettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten, PhD thesis, University of Wurzburg, Germany, 1988.

S. Hilger, Analysis on Measure Chains- A unified approach to continuous and discrete calculus. Results in Mathematics 18, 18-56, (1990).

S. Hilger, Differential and Difference Calculus- Unified!. Nonlinear Analysis, Theory, Methods & Appications 30(5), 2683-2694, (1997).

D. Mozyrska, E. Paw luszewicz, D. F. M. Torres, The Riemann-Stieltjes Integral on Time Scales. Australian Journal of Mathematical Analysis and Applications 7(1), 1-14, (2010).

D. Mozyrska, E. Paw luszewicz, D. F. M. Torres, Inequalities and majorisations for the Riemann Stieltjes integral on time scales. Mathematical Inequalities and Applications 14(2), 281–293, (2011).

V. Sekhose, H. Bharali, H. Kalita, Extensibility of Banach-valued Riemann type integrals on time scales. Journal of Nonlinear and Convex Analysis 24(9), 2037-2057, (2023).

V. Sekhose, H. Kalita, H. Bharali, Riemann-Stieltjes integrability of finite discontinuous functions on time scales. Facta Universitatis (Nis) Series Mathematics and Informatics 38(3), 559-575, (2023).