The Numerical Evaluation of Fixed Points for Completely Continuous Operators

Abstract

Given a completely continuous nonlinear operator $\mathcal{K}$ on a domain D contained in a Banach space $\mathcal{X}$, one common way of approximating $\mathcal{K}$ leads to a sequence $\{ \mathcal{K}_n \} $ which is collectively compact and pointwise convergent to $\mathcal{K}$ on D. If $\mathcal{K}$ is an integral operator, then $\mathcal{K}_n $ would usually be the result of replacing the integral with a numerical integral which depended on n. The results given here concern local existence and uniqueness of fixed points of $\mathcal{K}_n $ and convergence of them to fixed points of $\mathcal{K}$. The approach used is the invariance under homotopy of the rotation of a completely continuous vector field.