Functional equations involving a parameter

Abstract

1. The present note concerns the examination of nonlinear functional equations depending on a parameter. We investigate here the iterative method described in paper [1] and [2], which is a generalization of Newton's classical method. Another abstract formalism for Newton's method has been given first by L. V. Kantorovich (for references see [4]) and applied by him to the examination of operator equations in Banach spaces. The main point here is the application of the majorant method, which was used by Kantorovich [4] and also in paper [3]. The results stated here make it possible to find an error estimation of the exact solution in the case when the solution of a suitable approximate equation is given. An application to approximate solutions of operator equations in Hilbert space will be given in another note. Let X and M be two Banach spaces, and let F(x, ,u) be a nonlinear continuous functional defined on the space X+M, where x and ,u are in some closed spheres in X, M with centres x0, ,uo, respectively. Consider the nonlinear functional equation