Abstract
This chapter focuses on differential equations on time scales, including ordinary differential equations and functional differential equations. Section 7.1 briefly introduces the theory of time scales and then fundamental theory of ordinary differential equations on time scales is presented in Section 7.2. Functional differential equations on time scales are discussed in detail in Section 7.3. First, the equation model is formulated in Subsection 7.3.1, and then basic theory about existence and uniqueness of solutions are presented in Subsection 7.3.2. In Subsections 7.3.3 and 7.3.4, various stability results are discussed, such as (uniform) stability, uniform asymptotic stability, and exponential stability. To demonstrate these results, several examples are provided in Subsection 7.3.5 along with numerical simulations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications (Birkhäuser, Boston, 2001)
M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales (Birkhäuser, Boston, 2003)
S. Elaydi, S. Zhang, Stability and periodicity of difference equations with finite delay. Funkcialaj Ekvacioj 37, 401–413 (1994)
J. Hale, Theory of Functional Differential Equations (Springer, New York, 1977)
V. Lakshmikantham, X. Liu, Stability Analysis in Terms of Two Measures (World Scientific, River Edge, 1993)
V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems on Measure Chains (Kluwer Academic Publishers, Boston, 1996)
B. Liu, H.J. Marquez, Razumikhin-type stability theorems for discrete delay systems. Automatica 43(7), 1219–1225 (2007)
Q. Wang, X. Liu, Exponential stability for impulsive delay differential equations by Razumikhin method. J. Math. Anal. Appl. 309(2), 462–473 (2005)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Liu, X., Zhang, K. (2019). Differential Equations on Time Scales. In: Impulsive Systems on Hybrid Time Domains. IFSR International Series in Systems Science and Systems Engineering, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-030-06212-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-06212-5_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-06211-8
Online ISBN: 978-3-030-06212-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)