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 xΔ=f(t,x,xσ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{\Delta } = f(t,x,x^{\sigma })$$\end{document}. 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.