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

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 ${T^nx_0}$ 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 ${T^nx_0}$ are showed for finding a fixed point of $T$ with respect to the partial order "$\leq$" in uniformly convex ordered Banach space.