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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.

The present work studies the stability analysis of equilibrium of ordinary differential equations with the discontinuous right side, also called discontinuous differential equations, using the notion of Carathéodory solution for differential equations. This way, it is studied the stability of equilibrium in the Lyapunov sense for discontinuous systems through nonsmooth Lyapunov functions. Then two existing Lyapunov theorems are obtained. The results established refer to systems determined by nonautonomous differential equations.

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Resumo: Nesse artigo consideramos duas classes de equações integrais de Volterra em escalas temporais. Obtemos para essas classes de equações integrais, propriedades de continuidade e convergência de soluções.