Bounded and exponentially decaying solutions of almost linear dynamic systems on time scales

We consider small nonlinear perturbations of linear systems on a time scale with the phase space being finite or infinite-dimensional. For $\Delta$-differential operators, corresponding to linear dynamic systems we consider their solvability in various functional spaces. Based on these techniques, we prove several results on the existence of a bounded solution for some systems with small nonlinearities. Thus we introduce some generalizations of the classic hyperbolicity property (exponential dichotomy) for systems of ordinary differential equations. Besides, we introduce the Lyapunov regularity condition that coincides with the classic one for ordinary differential equations. For regular systems, we prove some criteria for the existence of bounded solutions of perturbed systems.