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In the present work, we study qualitative and quantitative results proposed in the paper Tisdell and Zaidi (Nonlinear Anal 68(11):3504-3524, 2008) of first-order dynamic equations on time scales. Thus, we examine initial value problems described by dynamic equations on time scales of the form x = f (t, x, x σ ). We obtain a result on the dependency of solutions to initial value problems with respect to initial values. Using Banach's fixed-point theorem, we prove the existence and uniqueness of solutions to initial value problems. On the other hand, under weaker hypothesis on f , using Schäfer's fixed-point theorem, we obtain the existence of at least one solution to initial value problems.

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We introduce and prove the existence of Hermes, Filippov, and Krasovskii generalized solutions to discontinuous dynamic equations on time scales. We also consider comparisons between the Carathéodory, Euler, Filippov, Hermes, and Krasovskii generalized solutions to discontinuous dynamic equations on time scales.