A NOTE ON IMPLICIT ITERATION PROCESSES

Source abstract

Abstract Let C be a closed, bounded, convex subset of a uniformly convex Banach space, and let $\{T_s\}$ be an asymptotic nonexpansive semigroup of nonlinear mappings acting within C . Consider the implicit iteration process defined by the sequence of equations: $$\begin{align*} x_{k+1} = c_k T_{s_{k+1}}(x_{k+1}) + (1 - c_k) x_k,\end{align*}$$ where each $c_k \in (0,1)$ and the initial point $x_0 \in C$ is arbitrarily chosen. In this context, we investigate the conditions under which the sequence $\{x_k\}$ converges, either weakly or strongly, to a common fixed point of the semigroup $\{T_s\}$ . We also touch upon the question of the stability of such processes.