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Pazy's fixed point theorem with respect to the partial order in uniformly convex Banach spaces

Abstracts

In this paper, the Pazy's Fixed Point Theorems of monotone $\alpha-$nonexpansive mapping $T$ are proved in a uniformly convex Banach space $E$ with the partial order ``$\leq$". That is, we obtain that the fixed point set of $T$ with respect to the partial order ``$\leq$" is nonempty whenever the Picard iteration $\{T^nx_0\}$ is bounded for some comparable initial point $x_0$ and its image $Tx_0$. When restricting the demain of $T$ to the cone $P$, a monotone $\alpha-$nonexpansive mapping $T$ has at least a fixed point if and only if the Picard iteration $\{T^n0\}$ is bounbed. Furthermore, with the help of the properties of the normal cone $P$, the weakly and strongly convergent theorems of the Picard iteration $\{T^nx_0\}$ are showed for finding a fixed point of $T$ with respect to the partial order ``$\leq$" in uniformly convex ordered Banach space.
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From: 宋义生
DOI:10.12074/201606.00325
Recommended references: Yisheng Song,Rudong Chen.(2016).Pazy's fixed point theorem with respect to the partial order in uniformly convex Banach spaces.[ChinaXiv:201606.00325] (Click&Copy)
Version History
[V2] 2016-07-05 14:43:42 chinaXiv:201606.00325V2 Download
[V1] 2016-06-27 07:21:11 chinaXiv:201606.00325v1(View This Version) Download
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