International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 7, Pages 23674, 9 p.
doi:10.1155/IJMMS/2006/23674

A Newton-type method and its application

Abstract

We prove an existence and uniqueness theorem for solving the operator equation F(x)+G(x)=0, where F is a continuous and Gâteaux differentiable operator and the operator G satisfies Lipschitz condition on an open convex subset of a Banach space. As corollaries, a recent theorem of Argyros (2003) and the classical convergence theorem for modified Newton iterates are deduced. We further obtain an existence theorem for a class of nonlinear functional integral equations involving the Urysohn operator.