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.