Generalized Exponential Dichotomy Theory Roughness Parameters Dependent On The Time Scale

Dynamic equations on time scales is a new research feld. A dynamic equationon a time scale is not only related to diferential equations or diference equations butthose pertaining to equations on more general time scales. The Theory of exponen-tial dichotomy describes characteristics of linear nonautonomous equations and playan important role in the study of qualitative and stability theory for nonautonomousequations.The paper focuses on a new notion called the general exponential dichotomyon time scales, which is more general and contains as special cases most versionsof unform and nonuniform dichotomies on the diferential equations and diferenceequations. We establish the existence of parameter dependence of roughness forthe general exponential dichotomy on time scales under the sufciently small linearperturbation. Moreover, we also show that the stable and unstable subspaces ofgeneral exponential dichotomies for the perturbed equations are Lipschitz continuousfor the parameters.