Author(s)

Le Huy Tien, Le Duc Nhien

Department of Mathematics, Mechanics and Informatics, Vietnam National University at Hanoi, Hanoi, Vietnam.

For time-varying non-regressive linear dynamic equations on a time scale with bounded graininess, we introduce the concept of the associative operator with linear systems on time scales. The purpose of this research is the characterizations of the exponential dichotomy obtained in terms of Fredholm property of that associative operator. Particularly, we use Perron’s method, which was generalized on time scales by J. Zhang, M. Fan, H. Zhu in [1], to show that if the associative operator is semi-Fredholm then the corresponding linear nonautonomous equation has an exponential dichotomy on both T ⁺ and T^-.  Moreover, we also give the converse result that the linear systems have an exponential dichotomy on both T ⁺ andT^-  then the associative operator is Fredholm on T.

Exponential Dichotomy, Fredholm Operator, Time Scales, Linear Dynamic Equation

Tien, L. and Nhien, L. (2019) Exponential Dichotomies and Fredholm Operators of Dynamic Equations on Time Scales. Applied Mathematics, 10, 39-50. doi: 10.4236/am.2019.101004.