Zhu, W., Ling, S. (2016). Iterative scheme based on boundary point method for common fixed point of strongly nonexpansive sequences. Bulletin of the Iranian Mathematical Society, 42(3), 719-730.

W. Zhu; S. Ling. "Iterative scheme based on boundary point method for common fixed point of strongly nonexpansive sequences". Bulletin of the Iranian Mathematical Society, 42, 3, 2016, 719-730.

Zhu, W., Ling, S. (2016). 'Iterative scheme based on boundary point method for common fixed point of strongly nonexpansive sequences', Bulletin of the Iranian Mathematical Society, 42(3), pp. 719-730.

Zhu, W., Ling, S. Iterative scheme based on boundary point method for common fixed point of strongly nonexpansive sequences. Bulletin of the Iranian Mathematical Society, 2016; 42(3): 719-730.

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

^{}College of Management and Economics, Tianjin University, Tianjin 300072, China.

Receive Date: 12 December 2014,
Revise Date: 15 April 2015,
Accept Date: 17 April 2015

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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