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Abstract

We consider the Darboux problem for a functional differential equation: $$\displaylines{ \frac{\partial^2u}{\partial x \partial y}(x,y)=f\bigg(x,y,u_{(x,y)},u(x,y),\frac{\partial u}{\partial x}(x,y),\frac{\partial u}{\partial y}(x,y)\bigg) \hbox{ a.e. in } [0,a]\times[0,b],\cr u(x,y)=\psi(x,y) \quad\ \hbox{on } [-a_{0},a]\times[-b_{0},b]\setminus(0,a]\times(0,b],\cr} $$ where the function $u_{(x,y)}:[-a_{0},0]\times[-b_{0},0]\to \mathbb R^{k}$ is defined by $u_{(x,y)}(s,t)=u({s+x},{t+y})$ for $ (s,t)\in [-a_{0},0]\times[-b_{0},0]$. We give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.