Iterative scheme based on boundary point method for common fixed‎ ‎point of strongly nonexpansive sequences

Document Type : Research Paper

Authors

College of Management and Economics‎, ‎Tianjin University‎, ‎Tianjin 300072‎, ‎China.

Abstract

Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$. Let ${S_n}$ and ${T_n}$ be sequences of nonexpansive self-mappings of $C$, where one of them is a strongly nonexpansive sequence. K. Aoyama and Y. Kimura introduced the iteration process $x_{n+1}=\beta_nx_n+(1-\beta_n)S_n(\alpha_nu+(1-\alpha_n)T_nx_n)$ for finding the common fixed point of ${S_n}$ and ${T_n}$, where $uin C$ is an arbitrarily (but fixed) element in $C$, $x_0\in C$ arbitrarily, ${\alpha_n}$ and ${\beta_n}$ are sequences in $[0,1]$. But in the case where $u\notin C$, the iterative scheme above becomes invalid because $x_n$ may not belong to $C$. To overcome this weakness, a new iterative scheme based on the thought of boundary point method is proposed and the strong convergence theorem is proved. As a special case, we can find the minimum-norm common fixed point of ${S_n}$ and ${T_n}$ whether $0\in C$ or $0\notin C$.

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