First-order partial differential equations and Henstock-Kurzweil integrals (Q1386301)

The Cauchy problem given by \[ \frac {\partial z}{\partial x} + a(x,y,z)\frac {\partial z}{\partial y}= b(x,y,z),\qquad z(x_0,y)=w(y) \] is considered, where the functions \(a\) and \(b\) satisfy Carathéodory conditions together with estimates of the form \(g_1(x) \leq a(x,y,z) \leq h_1(x) \), \(g_2(x) \leq b(x,y,z) \leq h_2(x) \) on a certain Cartesian product of intervals \(I=I_1\times I_2 \times I_3\) with Henstock-Kurzweil integrable \(h_i, g_i\), \(a,b\) Lipschitzian in the second two variables. Conditions for the existence and uniqueness of solutions are presented together with other fundamental results concerning the given equation.