On interval Riemann double integration on time scales
DOI:
https://doi.org/10.31489/2025m3/200-213Keywords:
interval valued functions, Hausdorff-Pompeiu distance, Riemann ∆∆-integral, Riemann ∇∇-integral, generalized Hukuhara difference, interval Riemann ∆∆-integral, interval Riemann ∇∇-integral, time scalesAbstract
This article explores the theory of Riemann double integration for functions whose values are intervals in the framework of time scale calculus. We define the Riemann double ∆-integral and Riemann double ∇-integral for interval valued functions, namely interval Riemann ∆∆-integral and interval Riemann ∇∇integral. Some key theorems in the article discuss the uniqueness of the integral, the equality of the interval Riemann double integral to the Riemann double integral when function is degenerate, necessary and sufficient conditions for integrability, proving integrability of a function without knowing the actual value of the integral. Additionally the relationship between the interval Riemann double integral and Riemann double integral for two interval-valued functions is estableshed via Hausdorff-Pompeiu distance. Elementary properties of the integral such as linearity property, subset property and others are established. Using the concept of generalized Hukuhara difference, alternate definitions of the interval Riemann ∆∆-integral and interval Riemann ∇∇-integral are formulated and theorems proving the equivalence of the integrals defined in both approaches are established. Theorems proving the equivalence of interval Riemann ∆- and ∇integrals previously defined in both approaches are also shown.
References
Hilger, S. (1988). Ein Maßkettenkalku¨l mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD thesis. University of Wu¨rzburg, Germany.
Hilger, S. (1990). Analysis on Measure Chains- A unified approach to continuous and discrete calculus. Results in Mathematics, 18, 18–56. https://doi.org/10.1007/BF03323153
Agarwal, R.P., Hazarika, B., & Tikare, S. (2025). Dynamic equations on time scales and applications. CRC Press, Taylor & Francis Group. https://doi.org/10.1201/9781003467908
Hilger, S. (1997). Differential and difference calculus — Unified! Nonlinear Analysis: Theory, Methods & Applications, 30(5), 2683–2694. https://doi.org/10.1016/S0362-546X(96)00204-0
Ahlbrandt, C.D., & Bohner, M. (2000). Hamiltonian Systems on Time Scales. Journal of Mathematical Analysis and Applications, 250(2), 561–578. https://doi.org/10.1006/jmaa.2000.6992
Atici, F.M., & Guseinov, G.Sh. (2002). On Green’s functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics, 141(1-2), 75–99. https://doi.org/10.1016/S0377-0427(01)00437-X
Sailer, S. (1992). Riemann-Stieltjes Integrale auf Zeitmengen (PhD thesis). University of Augsburg, Germany.
Guseinov, G.S., & Kaymak¸calan, B. (2002). Basics of Riemann Delta and Nabla Integration on Time Scales. Journal of Difference Equations and Applications, 8(11), 1001–1017. https://doi.org/10.1080/10236190290015272
Guseinov, G.S. (2003). Integration on time scales. Journal of Mathematical Analysis and Applications, 285, 107–127. https://doi.org/10.1016/S0022-247X(03)00361-5
Bohner, M., & Guseinov, G.Sh. (2007). Double integral calculus of variations on time scales. Computers and Mathematics with Applications, 54, 45–57. https://doi.org/10.1016/j.camwa.2006.10.032
Bohner, M., & Georgiev, S.G. (2016). Multivariable Dynamic Calculus on Time Scales. Springer International Publishing Switzerland. https://doi.org/10.1007/978-3-319-47620-9
Moore, R.E. (1966). Interval Analysis. Prentice Hall International, Englewood Cliffs.
Moore, R.E., Kearfott, R.B., & Cloud, M.J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics Philadelphia. https://doi.org/10.1137/1.9780898717716
Moore, R.E. (1962). Interval arithmetic and automatic error analysis in digital computing. PhD thesis. Stanford University, California.
Caprani, O., Madsen, K., & Rall, L.B. (1981). Integration of Interval Functions. SIAM Journal on Mathematical Analysis, 12(3), 321–341. https://doi.org/10.1137/0512030
Stefanini, L., & Bede, B. (2008). Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Analysis, 71(3-4). https://doi.org/10.1016/j.na.2008.12.005
Stefanini, L. (2010). A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets and Systems, 161(11), 1564–1584. https://doi.org/10.1016/j.fss.2009.06.009
Hukuhara, M. (1967). Intu´gration des applications mesurables dont la valeur est un compact convex. Funkcial. Ekvac., 10, 205–229.
Lupulescu, V. (2013). Hukuhara differentiability of interval-valued functions and interval differential equations on time scales. Information Sciences, 248, 50–67. https://doi.org/10.1016/j.ins.2013.06.004
Bohner, M., Nguyen, L., Schneider, B., & Truong, T. (2023). Inequalities for interval-valued Riemann diamond-alpha integrals. Journal of Inequalities and Applications, 86. https://doi.org/10.1186/s13660-023-02993-3
Wu, C., & Gong, Z. (2000). On Henstock integrals of interval-valued functions and fuzzyvalued functions. Fuzzy Sets and Systems, 115(3), 377–391. https://doi.org/10.1016/S0165-0114(98)00277-2
Hamid, M.E., & Elmuiz, A.H. (2016). On Henstock-Stieltjes integrals of interval-valued functions and fuzzy-number-valued functions. Journal of Applied Mathematics and Physics, 4(4), 779–786. http://dx.doi.org/10.4236/jamp.2016.44088
Hamid, M.E., Elmuiz, A.H., & Sheima, M.E. (2016). On AP-Henstock integrals of intervalvalued functions and fuzzy-number-valued functions. Applied Mathematics, 7(18), 2285–2295. http://dx.doi.org/10.4236/am.2016.718180
Eun, G.S., Yoon, J.H., Park, J.M., & Lee, D.H. (2012). On AP-Henstock-Stieltjes integral of interval-valued functions. Journal of the Chungcheong Mathematical Society, 25(2), 291–298. https://doi.org/10.14403/jcms.2012.25.2.291
Park, C.K. (2004). On McShane-Stieltjes integrals of interval valued functions and fuzzy-numbervalued functions. Bull. Korean Math. Soc., 41(2), 221–233. https://doi.org/10.4134/BKMS.2004.41.2.221
Zhao, D., Ye, G., Liu, W., & Torres, D.F.M., (2019). Some inequalities for interval-valued functions on time scales. Soft Computing, 23, 6005–6015. https://doi.org/10.1007/s00500-018-3538-6
Sekhose, V., & Bharali, H., (2024). Interval Riemann-Stieltjes integration on time scales. Conference Proceeding: 18th International Conference MSAST 2024, 139–151, Kolkata, India.
Oh, W.T., & Yoon, J.H., (2014). On Henstrock integrals of interval-valued functions on time scales. Journal of the Chungcheong Mathematical Society, 27(4), 745–751. http://dx.doi.org/10.14403/jcms.2014.27.4.745
Yoon, J.H. (2016). On Henstock-Stieltjes integrals of interval-valued functions on time scales. Journal of the Chungcheong Mathematical Society, 29(1), 109–115. http://dx.doi.org/10.14403/jcms.2016.29.1.109
Hamid, M.E., Xu, L., & Elmuiz, A.H. (2017). On McShane integrals of interval-valued functions and fuzzy-number-valued functions on Time Scales. Journal of Progressive Research in Mathematics, 12(1), 1780–1788.
Hamid, M.E. (2018). On McShane-Stieltjes integrals of interval-valued functions and fuzzynumber-valued functions on time scales. European Journal of Pure and Applied Mathematics, 11(2), 493–504. https://doi.org/10.29020/nybg.ejpam.v11i2.3200
Afariogun, D.A., Alao, M.A., & Olaoluwa, H.O. (2021). On Henstock-Kurzweil-Stieltjes-♦Double Integrals of interval-valued functions on time scales. Annals of Mathematics and Computer Science, 2, 28–40.
Afariogun, D.A., & Alao, M.A. (2021). An Existence Result for Henstock-Kurzweil-Stieltjes-♦Double Integral of Interval-Valued Functions on Time Scales. Journal of Nepal Mathematical Society, 4(2), 1–7. https://doi.org/10.3126/jnms.v4i2.41458
