Ważewski theorem on time scales with a set of egress points that is not the whole boundary

In the paper the celebrated Ważewski retract method is developed for planar dynamic equations on an arbitrary time scale without a restrictive assumption that the whole boundary of a set of constraints, where we look for solutions, is a set of egress points. As a consequence, some recently published results are essentially generalized. One transparent example illustrating the main theorem is presented. The results are new and more general even for ordinary difference equations.