Naseer Shahzad, Reem Al-Dubiban

Approximating Common Fixed Points of Nonexpansive Mappings in Banach Spaces

Abstract:
Let $K$ be a nonempty closed convex subset of a real uniformly convex Banach space $E$ and $S, T:K\rightarrow K$ two nonexpansive mappings such that $F(S)\cap F(T):=\{x\in K: Sx=Tx=x\}\neq \varnothing$. Suppose $\{x_n\}$ is generated iteratively by
$$ x_1\in K,\;\; x_{n+1}=(1-\alpha_n) x_n+\alpha_n S[(1-\beta_{n})x_n+\beta_{n}Tx_n], $$
$n\geq 1,$ where $\{{\alpha_n}\}$, $\{{\beta_n}\}$ are real sequences in $[0,1]$.
In this paper, we discuss the weak and strong convergence of $\{x_n\}$ to some $x^*\in F(S)\cap F(T)$.

Keywords:
Common fixed point, nonexpansive mapping, Banach space.

MSC 2000: 47H09, 47J25