Existence on solutions of a class of casual differential equations on a time scale (Q2054661)

The theory of a class of casual differential equations of the form \[ \left[\frac{u(t)-k(t,u(t))}{f(t,u(t))}\right]^\Delta=(Qu)(t);\qquad u(t_0)=u_0 \tag{1} \] is developed for \(t\in J=[t_0,t_0+a]_{\mathbb T}:=[t_0,t_0+a]\cup{\mathbb T}\), where \(\mathbb T\) is a time scale, \(Q:\mathrm{C}_{\mathrm{rd}}(J\times{\mathbb R})\to\mathrm{C}_{\mathrm{rd}}(J\times{\mathbb R})\) is a continuous operator, \(f\in\mathrm{C}_{\mathrm{rd}}(J\times{\mathbb R},{\mathbb R}\setminus\{0\})\), and \(k,g\in\mathrm{C}_{\mathrm{rd}}(J\times{\mathbb R},{\mathbb R})\). An existence theorem is given for Eq. (1) under mixed Lipschitz (on the functions \(f\) and \(k\)) and compactness conditions by the fixed point theorem in Banach algebra due to \textit{B. C. Dhage} [Kyungpook Math. J. 55, No. 4, 1069--1088 (2015; Zbl 1344.39002)]. Some basic differential inequalities on a time scale are presented which are utilized to investigate the existence of extremal solutions. The comparison principle on the equations of type (1) is established. The results in this paper extend and improve some well-known results by Dhage, and those reduces to results given in the monographs by \textit{V. Lakshmikantham} and \textit{S. Leela} [Differential and integral inequalities. Theory and applications. Vol. I: Ordinary differential equations. New York-London: Academic Press (1969; Zbl 0177.12403)], and \textit{W. G. Kelley} and \textit{A. C. Peterson} [Difference equations. An introduction with applications. 2nd ed. San Diego, CA: Harcourt/Academic Press (2001; Zbl 0970.39001)] in the real case.