Convergence results for fixed point iterations in \(\mathbb{R}\)
Let \(g:[a,b] \to \mathbb{R}\) satisfy the Lipschitz condition \(|g(x) - g(y)|< L |x - y|\), \(x,y \in [a,b]\). The authors consider the following iteration schemes for computing the fixed point of the mapping \(g\): \[ x_{n+1} = (1 - \lambda) x_n + \lambda g(x_n), \quad \lambda \in \mathbb{R} \tag{a} \] \[ x_{n + 1} = (1 - \alpha_n) x_n + \alpha_n g(x_n) \tag{b} \] for some real parameters \(\alpha_n\). New convergence conditions for (a) and (b) are derived.
scientific article; zbMATH DE number 878517