The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space

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.