Georgian Mathematical Journal 13 (2006), No. 3, 529--537
Copyright Heldermann Verlag 2006
Approximating Common Fixed Points of Nonexpansive Mappings in Banach Spaces
Naseer Shahzad
Dept. of Mathematics, King Abdul Aziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
nshahzad@kau.edu.sa
Reem Al-Dubiban
Dept. of Mathematics, King Abdul Aziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
[Abstract-pdf]
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: 47H09, 47J25
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