On the mixed problem for hyperbolic partial differential-functional equations of the first order

Abstract

We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order $D_x z(x,y) = f(x,y,z_{(x,y)} ,D_y z(x,y)),$ where $z_{(x,y)} :[ - \tau ,0] \times [0,h] \to \mathbb{R}$ is a function defined by z _(x,y)(t, s) = z(x + t, y + s), (t, s) ∈ [−τ, 0] × [0, h]. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.