Chao Wang, Zhien Li, Ravi P Agarwal, and Donal O’Regan. Coupled-Jumping Timescale Theory and Applications to Time-Hybrid Dynamic Equations, Convolution and Laplace Transforms Dynamic Systems and Applications 30 (2021) No.3, 461-508
https://doi.org/10.46719/dsa202130311
ABSTRACT:
In this paper, we initiate the concept of a coupled-jumping timescale space (short for CJTS) T1 -T2 and introduce the theory of calculus and fundamental functions. Based on this, we introduce an initial value problem of time-hybrid dynamic equations whose initial value is given in T2 and the unique solution is located in T1. Moreover, some important integral transforms including the convolution and the Laplace transform on CJTS are also introduced and studied. In particular, we obtain Hilger’s time scale theory by removing the coupled-jumping state (i.e., Hilger’s theory can be obtained by letting T1 = T2 and we find that the Hilger theory is essentially based on a single timescale space). Dynamic equations on CJTS (or called time-hybrid dynamic equations or short for THDEs) has a distinct and independent dynamical behavior which single timescale dynamic equations including all the discrete or continuous dynamic equations do not possess when T1 ≠ T2. The coupled-jumping timescale theory largely deepens and includes the Hilger theory and brings a completely new significance of dynamic equations on time scales.
KEYWORDS. Coupled-jumping timescale space; Time-hybrid dynamic equations; Convolution; Laplace transform.