Classical solutions of initial problems for quasilinear partial functional differential equations of the first order

We consider the initial problem for a quasilinear partial functional differential equation of the first order [formula], z(t, x) = varphi(t, x) ((t, x) ∈ [-h0, 0] x Rn) where z(t, x) : [-h0, 0] x [-h, h] → R is a function defined by z(t, x) (τ, ξ) = z(t + τ, + ξ) for (τ, ξ) ∈ [-h0, 0] x [-h, h]. Using the method of bicharacteristics and the fixed-point theorem we prove, under suitable assumptions, a theorem on the local existence and uniqueness of classical solutions of the problem and its continuous dependence on the initial condition.