Fixed Point Iterations Using Infinite Matrices

Abstract

Let E be a closed, bounded, convex subset of a Banach space <!-- MATH $X,f:E \to E$ --> . Consider the iteration scheme defined by <!-- MATH ${\bar x_0} = {x_0} \in E,{\bar x_{n + 1}} = f({x_n}),{x_n} = \Sigma _{k = 0}^n{a_{nk}}{\bar x_k},\;n \geq 1$ --> , where A is a regular weighted mean matrix. For particular spaces X and functions f we show that this iterative scheme converges to a fixed point of f.