Gronwall's inequality and the Henstock integral (Q1106351)

The Gronwall inequality, namely \(u(t)\leq ce^{kt}\) for \(t\in [0,b]\) whenever u: [0,b]\(\to {\mathbb{R}}\) is continuous and satisfies \[ u(t)\quad \leq \quad c\quad +\quad k\int^{t}_{0}u(s) ds,\quad t\in [0,b[, \] is generalized to the case where u: [0,b]\(\to {\mathbb{R}}\) is integrable in the sense of Kurzweil-Henstock on [0,b]. An ``almost-everywhere'' version also holds in this situation. The proof makes use of an integration by parts formula for the product of a Kurzweil-Henstock integrable function with a function of bounded variation.