Copyright © 2010 S. J. Aneke. 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

The equation Lu=f, where L=A+B, with A being a K-positive definite operator and B being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn (1962), as well as Lax and Milgram (1954). Furthermore, an application of the inverse function theorem provides simultaneously a general solution to this equation in some neighborhood of a point xo, where L is Fréchet differentiable and an iterative scheme which converges strongly to the unique solution of this equation.