The Method of Monotone Iterations for Mixed Monotone Operators in Partially Ordered Sets and Order-Attractive Fixed Points

TL;DR

This paper develops a method using monotone iterations to find fixed points of mixed monotone operators in partially ordered sets, providing new criteria for existence, uniqueness, and order-attractiveness without extra assumptions.

Contribution

It introduces a novel approach for fixed point analysis of mixed monotone operators in general partially ordered sets without additional order or convergence assumptions.

Findings

Established criteria for fixed point existence and uniqueness.

Defined the concept of order-attractive fixed points.

Provided an application to ordered linear spaces.

Abstract

We use the method of monotone iterations to obtain fixed point and coupled fixed point results for mixed monotone operators in the setting of partially ordered sets, with no additional assumptions on the partial order and with no convergence structure. We define the concept of attractive fixed point with respect to the partial order and obtain several criteria for the existence, uniqueness and order-attractiveness of the fixed points, both in the presence and in the absence of a coupled lower-upper fixed point. As an application, we present a fixed point result for a class of mixed monotone operators in the setting of ordered linear spaces.