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

Abstract EN

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.