This is an announcement for the paper “Pazy's fixed point theorem with respect to the partial order in uniformly convex Banach spaces” by Yisheng Songhttp://arxiv.org/find/math/1/au:+Song_Y/0/1/0/all/0/1, Rudong Chenhttp://arxiv.org/find/math/1/au:+Chen_R/0/1/0/all/0/1.

Abstract: 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 ${T^n x_0}$ is bounded for some initial point $x_0$ with $x_0\leq Tx_0$ or $Tx_0\leq x_0$. 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 ${T^n x_0}$ are showed for finding a fixed point of $T$ with respect to the partial order "$\leq$" in uniformly convex ordered Banach space.

The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.08216