Some computational aspects of metric fixed-point theory
Let \((X, \| \cdot \| )\) be a normed space, \(C \subset X\) a nonempty convex subset and \((\lambda_n)_{n \in \mathbb N}\) a sequence in \([0,1]\). An operator \(f:C \to C\) is said to be asymptotically nonexpansive with sequence \((k_n)_{n \in \mathbb N}\) if \( \lim_{n \to \infty} k_n=0\) and \[ \| f^n(x)-f^n(y)\| \leq (1+k_n) \| x-y \| ,\;\forall \;n \in \mathbb N, \;\forall \;x,y \in C. \] For such an operator, the Krasnoselskij--Mann iteration is defined as follows \(x_0:=x \in C, \;x_{n+1}:= (1-\lambda_n)x_n+ \lambda_n f^n (x_n) \). In this paper, the author shows how bounds on asymptotic regularity (\(\sum_{n=0}^{\infty}k_n \leq K)\) can effectively be converted into certain bounds on the convergence towards a fixed point.
scientific article; zbMATH DE number 2165059
scientific article