On the local Cauchy problem for nonlinear hyperbolic functional differential equations

We consider the local initial value problem for the hyperbolic partial functional differential equation of the first order (1) ${\displaystyle Dₓz(x,y)=f(x,y,z(x,y),\left(Wz\right)(x,y),D_{y}z(x,y))}$ on E, (2) z(x,y) = ϕ(x,y) on [-τ₀,0]×[-b,b], where E is the Haar pyramid and τ₀ ∈ ℝ₊, b = (b₁,...,bₙ) ∈ ℝⁿ₊. Using the method of bicharacteristics and the method of successive approximations for a certain functional integral system we prove, under suitable assumptions, a theorem on the local existence of weak solutions of the problem (1),(2).