Existence, uniqueness and continuous dependence for a hereditary nonlinear functional partial differential equation of the first order

We consider the equation (including various functional-differential equations of retarded type) z_x(x, y) = f(x, y, T^(1)(x, y, z), · · · , T^(r)(x, y, z), z_y(x, y)) on the strip I = [0, a]xR, where T^(j) (j = 1, 2, · · · , r) are Volterra type operators. Existence, uniqueness and continuous dependence of a smooth solution on I for the Cauchy problem of the above equation with initial value z = g on [-p_0, 0]xR are proved. A quantitative estimate on the domain strip of the solution is given.