Exponential Dichotomy And Permissibility On Time Scales

Dynamic equations on time scales is not only related to differential equation-s or difference equations but those pertaining to equations on more general time scales. The Theory of exponential dichotomy describes characteristics of linear nonautonomous dynamic equations on time scales and play an important role in the study of qualitative and stability theory for nonautonomous dynamic equa-tions on time scales. In this paper, we study a relation between exponential di-chotomy on time scales and admissibility of the pair (Crdb(T+,X),Lp(T+,X)) for an evolution family on time scales. We establish a sufficient criterion for the ex-istence of exponential dichotomy on time scales in terms of the admissibility of the pair (Crdb(T+,X),Lp(T+,X)) for the evolution family. Conversely, with the help of exponential dichotomy on time scales, we give the admissibility of the pair (Crdb(T+,X),Lp(T+,X)) for an input-output equation on time scales.