Discontinuous dynamic equations on time scales (Q903062)

The paper is devoted to initial-value problems of the form \[ \begin{aligned} x^\Delta(t)&=f(t,x(t)),\quad t\in[a,b)_{\mathbb{T}},\\ x(a)&=x_0, \end{aligned}\eqno(1) \] with \(\mathbb T\subseteq\mathbb R\) being a time scale, \(x:[a,b]_{\mathbb T}\to\mathbb R^n\), and \(f:\mathbb T\times\mathbb R^n\to\mathbb R^n\). The right-hand side \(f\) is not assumed to be continuous. In the classical case when \(\mathbb T=\mathbb R\), there exists various concepts of generalized solutions to the problem (1), such as solutions in the sense of Carathéodory, Hermes, Filippov, Krasovskii, etc. The author extends these notions to the context of dynamic equations on time scales. The main results obtained in the paper are as follows: \begin{itemize} \item[{\(\bullet\)}] Existence of local solutions in Carathéodory's sense. \item[{\(\bullet\)}] Existence of global solutions in the sense of Hermes, provided that \(f\) satisfies the linear growth condition \[ \|f(t,x)\|\leq \gamma\|x\|+c, \quad(t,x)\in\mathbb T\times\mathbb R^n.\eqno(2) \] (The existence of global Krasovskii and Filippov solutions under the same assumption (2) was already proved in the paper [the author and \textit{G. N. Silva}, Math. Ann. 356, No. 1, 373--399 (2013; Zbl 1277.34126)]). \item[{\(\bullet\)}] A closure theorem for global Krasovskii solutions (a uniform limit of solutions is a solution again). \end{itemize} Finally, the author explains the relation between various types of generalized solutions and provides some examples showing that these concepts do not coincide.