Strong convergence for hybrid implicit \(S\)-iteration scheme of nonexpansive and strongly pseudocontractive mappings
Summary: Let \(K\) be a nonempty closed convex subset of a real Banach space \(E\), let \(S : K \rightarrow K\) be nonexpansive, and let \(T : K \rightarrow K\) be Lipschitz strongly pseudocontractive mappings such that \(p \in F \left(S\right) \cap F \left(T\right) = \left\{x \in K : S x = T x = x\right\}\) and \(\| x-Sy\| \leq \| x-Sy\|\) and \(\|x-Ty\| \leq \|Tx-Ty\|\) for all \(x, y \in K\). Let \(\left\{\beta_n\right\}\) be a sequence in \(\left[0, 1\right]\) satisfying (i) \(\sum_{n = 1}^\infty \beta_n = \infty\); (ii) \(\lim_{n \rightarrow \infty} \beta_n = 0 \). For arbitrary \(x_0 \in K\), let \(\left\{x_n\right\}\) be a sequence iteratively defined by \(x_n = S y_n, y_n = \left(1 - \beta_n\right) x_{n - 1} + \beta_n T x_n, n \geq 1 \). Then the sequence \(\left\{x_n\right\}\) converges strongly to a common fixed point \(p\) of \(S\) and \(T\).
Lipschitz strongly pseudocontractive mappings
scientific article; zbMATH DE number 7022976
scientific article