Fixed point iterations using infinite matrices
Billy E. Rhoades · Transactions of the American Mathematical Society · 1974
Let E be a closed, bounded, convex subset of a Banach space X , f : E → E X,f:E \to E . Consider the iteration scheme defined by x ¯ 0 = x 0 ∈ E , x ¯ n + 1 = f ( x n ) , x n = Σ k = 0 n a n k x ¯ k , n ≥ 1 {\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 .