Journal of Inequalities and Applications
Volume 4 (1999), Issue 2, Pages 141-161
doi:10.1155/S1025583499000338

Iterative methods for the darboux problem for partial functional differential equations

Tomasz Człapiński

Institute of Mathematics, University of Gdańsk, Wit Stwosz St. 57, Gdańsk 80-952, Poland

Received 24 September 1998; Revised 22 October 1998

Copyright © 1999 Tomasz Człapiński. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.