Discontinuous systems and the Henstock-Kurzweil integral (Q1276330)

The authors study the initial value problem for systems (1) \(x'(t)= f(t,x(t))\), (2) \(x(\tau)= \xi\), with discontinuous right-hand sides in the framework of the Henstock-Kurzweil integral, generalizing similar approaches by other researchers. In the Carathéodory extension of the classical theory \(f: G\to \mathbb{R}^n\) (\(G\subset\mathbb{R}^{n+1}\) open) (i) is continuous in \(x\) for almost all \(t\) and measurable in \(t\) for fixed \(x\), and (ii) it satisfies the estimate \(\|f(t,x)\|\leq m(t)\in L^1\) in compact subsets \(G_0\subset G\). In the present theory (ii) is replaced by \(g(t)\leq f(t,x)\leq h(t)\) in \(G_0\) (componentwise), where \(g\) and \(h\) (depending on \(G_0\)) are HK-integrable. Under these conditions problem (1), (2) has a generalized Carathéodory solution \(x(t)\) in the interval \(I= [\tau-\beta, \tau+\beta]\) that satisfies (1) a.e. in \(I\) and (2) and is ``generalized absolutely continuous'' (has a lengthy definition). The proof by reduction to the corresponding existence theorem in Carathéodory theory is remarkably simple; it uses the fact that an HK-integrable function that is bounded from one side is \(L\)-integrable. A uniqueness theorem to the right for problem (1), (2) uses the estimate \(\|f(t,x)- f(t,y)\|\leq \omega(\|x-y\|)(h(v)- h(u))/(v- u)\) for \((u\leq t\leq v)\) where \(\omega\) is an Osgood function, \(h(s)\) is increasing and continuous from the left.