The existence of Carathéodory solutions of hyperbolic functional differential equations

We consider the following Darboux problem for the functional differential equation ${\displaystyle \partial ²\frac{u}{\partial }x\partial y(x,y)=f(x,y,u_{(x,y)},\partial \frac{u}{\partial }x(x,y),\partial \frac{u}{\partial }y(x,y))}$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]\(0,a]×(0,b], where the function $u_{(x,y)}:[-a₀,0]×[-b₀,0] → ℝ^{k}$ is defined by $u_{(x,y)}(s,t) = u(s+x,t+y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.