On the existence of the solution of nonlinear functional partial differential equations of the first order

We prove a theorem of existence and uniqueness of the solution of a Cauchy problem (j) for nonlinear partial differential equation of the first order with a hereditary structure. We associate to (j) problem a sequence of Cauchy problems without functional argument. The solutions z_n of these problems are established by Baiada's approximation method. Finally the solution of problem (j) is given as a limit surface of the sequence (z_n)_n. The present result extends the existence theorem given by Z. Kamont in the linear case [Ann. Polon. Math. 35 (1977/78), no. 1, 27–48].