The method of quasilinearization for system of hyperbolic functional differential equations

We deal with monotone iterative method for the Darboux problem for the system of hyperbolic partial functional-differential equations $$\{\begin{array}{l}\frac{{\partial }^{2}u}{\partial x\partial y}(x,y)=f(x,y,u_{(x,y)},u_{(x,y)},\text{a.e. in}[0,1]\times [0,b]\\ u(x,y)=\psi (x,y),\text{on}[-a_0,a]\times [-b_0,b]\setminus (0,a]\times (0,b],\end{array}$$ 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 $(s,t)\in [-a_0,0]\times [-b_0,0]$.