Existence and Roughness of Nonuniform Exponential Dichotomies on Time Scales

In this manuscript, we discuss the roughness of nonuniform exponential dichotomies on a time scale in a Banach space and give the existence results for nonuniform and uniform exponential dichotomy for the time scale. We extend and unify the previous results by considering the equation on a time scale. For a given linear equation on a time scale, the existence of exponential dichotomy persists under an adequately small variable linear perturbation. To establish the results, we acquire related roughness results for the case of uniform exponential contractions. In the end, a suitable example is given for illustration.