First-order partial differential equations and Henstock-Kurzweil integrals. (English) Zbl 0890.35025

Chew, T. S.; Van-Brunt, B.; Wake, G. C.

Differ. Integral Equ. 10, No. 5, 947-960 (1997).

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.

Reviewer: Š.Schwabik (Praha)

MSC:

35F25	Initial value problems for nonlinear first-order PDEs
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

Carathéodory conditions