Uniqueness and existence results for initial value problems on time scales through a reciprocal problem and applications

This paper is devoted to deriving the existence, uniqueness and approximation of solutions to a first-order initial value problem on an interval of an arbitrary time scale, taking values in a different time scale. We obtain the results considering dynamic equations whose nonlinear part is nonnegative and satisfies a type of ``inverseʹʹ Carathéodoryʹs conditions and discontinuity. Several examples are given to illustrate the applicability of these results.