Integration of functions \(rd\)-continuous on time scales (Q656254)

This paper describes the possibility of extending time scale functions into functions defined on the whole real line. Using this extension, dynamic equations on time scales can be transformed to integral equations with the Lebesgue-Stieltjes integral. Although this basic idea is correct, I found the paper rather obscure and confusing. The authors start with the intention to investigate the equation \[ x^\Delta(t)=A(t)x(\sigma(t))+f(t),\eqno(1) \] and then proceed to recall the basic time scale concepts. I strongly disagree with their statement that the definition of the \(\Delta\)-integral ``\(\ldots\) raises doubts in the correctness of results based on such an integration method. The very peculiar methods of analysis related to \(\Delta\)-differentiation and \(\Delta\)-integration lead to a~cumbersome difficult-to-comprehend apparatus, which makes it possible to investigate only asymptotic (as \(t\to\infty\)) properties of solutions.'' Given a time scale function \(x:\mathbb T\to\mathbb R\), the authors define its \(C\)-extension \(\hat x:\mathbb R\to\mathbb R\) to be a function which coincides with \(x\) on \(\mathbb T\) and is constant on every open interval contained in the complement of \(\mathbb T\). However, these requirements do not determine the extension uniquely. At a later stage, the authors switch to the equation \[ x'=A'(t)x+f(t),\eqno(2) \] which they claim to be ``of the same form as (1)'' and where ``\(A'(t)\) is the generalized derivative of the function \(A(t)\).'' Finally, they assert that ``the correct form'' of (2) is \[ x(t)-x(a)=\int_a^t x(s){\text d}A(s)+F(t), \] and present an existence-uniqueness theorem for this type of equation. They assume that \(A\) belongs to a~certain class \(C_{\text{rd}}^0\) whose definition does not make sense as it refers to \(C_{\text{rd}}^0\) again.