On the Darboux problem for partial differential-functional equations with infinite delay at derivatives (Q5936423)

The second-order Darboux problem NEWLINE\[NEWLINE\begin{aligned} D_{xy} z(x,y) & =(f(x,y,z_{(x,y)}, (D_xz)_{(x,y)}, (D_yz)_{(x,y)}),\quad (x,y) \in\mathbb{R},\\ z(x,y) & =\varphi(x,y),\quad (x,y)\in\bigl\{(-\infty,a]\times(-\infty,b]\bigr\} \setminus \bigl\{(0,a] \times (0,b]\bigr\} \end{aligned}NEWLINE\]NEWLINE is considered, and by using the Banach fixed-point theorem, a unique solution is shown to exist if \(f\) satisfies a Lipschitz-type condition in the last three variables and certain consistency conditions hold. If \(f\) is bounded and satisfies a Lipschitz condition in the last two variables, a similar result is proved by applying the Schauder fixed-point theorem.