Abstract
We consider the Darboux problem for the hyperbolic partial functional differential
equations
(1)
Dxyz(x,y)=f(x,y,z(x,y)),(x,y)∈[0,a]×[0,b],(2)
z(x,y)=ϕ(x,y),(x,y)∈[−a0,a]×[−b0,b]\(0,a]×(0,b],
where z(x,y):[−a0,0]×[−b0,0]→X is a function defined by z(x,y)(t,s)=z(x+t,y+s),(t,s)∈[−a0,0]×[−b0,0]. If X=ℝ then using the method of functional differential inequalities we prove, under suitable conditions, a theorem on the convergence of the Chaplyghin sequences to the solution of problem (1), (2). In case X is any Banach space we give analogous theorem on the convergence of the Newton method.