Relationship Between Bounded Solutions of Differential Equations and Dynamic Equations on Time Scales

A system of differential equations of the form \(\frac{dx}{dt}=X\left(t,x\right)\) $\frac{\mathit{dx}}{\mathit{dt}}=X\left(t,,,x\right)$ and the corresponding system of dynamic equations with delta-derivative are considered. We establish conditions for the existence of bounded solutions to the system of dynamic equations on the time scale \({\mathbb{T}}_{\lambda }\) ${\mathbb{T}}_{\lambda }$ under the condition of existence of bounded solutions of the original system of differential equations. We obtain conditions for the graininess function under which the existence of bounded solutions of differential equations implies the existence of solutions of the corresponding system of dynamic equations.