A new unification of continuous, discrete and impulsive calculus through Stieltjes derivatives (Q1704439)

The main objective of this paper is to establish a couple of fundamental theorems of calculus for the Lebesgue-Stieltjes and the Kurzweil-Stieltjes integral, with respect to a function derivative. Let \(F:[a,b]\to \mathbb{R}\) be a function; and \(g:\mathbb{R}\to \mathbb{R}\) be a nondecreasing function which is continuous from the left everywhere. Theorem 1. The function \(F\) is \(g\)-absolutely continuous on \([a,b]\) iff the following conditions are fulfilled {\parindent=8mm \begin{itemize}\item[(i)] \(F_g'(x)\) exists for \(g\)-almost all \(x\in [a,b]\) \item[(ii)] \(F_g'\in {\mathcal L}_g^1([a,b])\) \item[(iii)] \(F(x)=F(a)+\int_{[a,x)}F_g'(x)d\mu_g\),\ \(\forall x\in [a,b]\). \end{itemize}} Theorem 2. Suppose that {\parindent=8mm \begin{itemize}\item[(I)] \(F\) is \(g\)-differentiable everywhere in \([a,b]\setminus C_g\) \item[(II)] \(F\) is left continuous at the points of \((a,b]\cap D_g\) \item[(III)] \(F\) is constant on every subinterval of \([a,b]\) where \(g\) is. \end{itemize}} If \(h:[a,b]\to \mathbb{R}\) coincides with \(F_g'\) in \([a,b]\setminus C_g\), then {\parindent=8mm \begin{itemize}\item[(j)] \(h\in K_g([a,b])\) \item[(jj)] \(F(x)=F(a)+\int_a^x h(t)dg(t)\),\ \(\forall x\in [a,b]\). \end{itemize}} Further applications of these results to differential equations constructed by means of such function derivatives are also considered.