A Newton-type method and its application

We prove an existence and uniqueness theorem for solving the operator equationF(x)+G(x)=0, whereFis a continuous and Gâteaux differentiable operator and the operatorGsatisfies 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.