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

TL;DR

This paper extends Pazy's fixed point theorems to monotone α-nonexpansive mappings in uniformly convex Banach spaces with partial orders, establishing conditions for existence and convergence of fixed points.

Contribution

It generalizes fixed point results for monotone nonexpansive mappings to uniformly convex ordered Banach spaces with partial orders, including convergence theorems.

Findings

Fixed point set is nonempty under bounded Picard iteration.

Existence of fixed points is equivalent to boundedness of Picard iteration starting at zero.

Weak and strong convergence of Picard iteration are established.

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.