On qualitative and quantitative results for solutions to first-order dynamic equations on time scales

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(Delta) = f (t, x, x(sigma)). 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 Schafer's fixed-point theorem, we obtain the existence of at least one solution to initial value problems.