Mean value theorem and semigroups of operators for interval-valued functions on time scales

Year 2021, , 79 - 91, 04.02.2021

Yonghong Shen

https://doi.org/10.15672/hujms.644665

Abstract

In this paper, a new version of mean value theorem for interval-valued functions on time scales is established. Meantime, some basic concepts and results associated with semigroups of operators for interval-valued functions on time scales are presented. As an application of semigroups of operators, under certain conditions, we consider the initial value problem for interval-valued differential equations on time scales. Finally, two issues worthy of further discussion are presented.

Keywords

interval-valued functions, mean value theorem, semigroup of operators, time scales

Supporting Institution

National Natural Science Foundation of China

Project Number

11701425

References

• [1] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser Boston Inc., Boston, MA, 2001.
• [2] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
• [3] Y. Chalco-Cano, A. Rufián-Lizana, H. Román-Flores and M.D. Jiménez-Gamero, Calculus for interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Sets Syst. 219, 49–67, 2013.
• [4] C.G. Gal and S.G. Gal, Semigroups of operators on spaces of fuzzy-number-valued functions with applications to fuzzy differential equations, J. Fuzzy Math. 13, 647– 682, 2005.
• [5] A.E. Hamza and K.M. Oraby, Semigroups of operators and abstract dynamic equations on time scales, Appl. Math. Comput. 270, 334–348, 2013.
• [6] S. Hilger, Ein Makettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universität Würzburg, 1988.
• [7] V. Lupulescu, Hukuhara differentiability of interval-valued functions and interval differential equations on time scales, Inform. Sciences, 248, 50–67, 2013.
• [8] S. Markov, Calculus for interval functions of a real variable, Computing, 22, 325–337, 1979.
• [9] S. Markov, On the algebraic properties of convex bodies and some applications, J. Convex Anal. 7, 129–166, 2000.
• [10] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst. 161, 1564–1584, 2010.
• [11] L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. 71, 1311–1328, 2009.
• [12] J. Tao and Z. Zhang, Properties of interval-valued function space under the gH-difference and their application to semi-linear interval differential equations, Adv. Differ. Equ. 2016, Article number: 45, 2016.