[Paper Review] FIXED POINT THEOREMS FOR SINGLE-VALUED AND MULTI-VALUED MIXED MONOTONE OPERATORS OF MEIR-KEELER TYPE

Farshid Khojasteh, A. Razani|arXiv (Cornell University)|Jan 1, 2013

Fixed Point Theorems Analysis9 references3 citations

TL;DR

This paper extends Zhang's fixed point results by introducing Meir-Keeler type mixed monotone single-valued and multi-valued operators in ordered metric spaces. It establishes two new fixed point theorems under weaker conditions than convexity and concavity, generalizing prior work through a more flexible contraction-type condition.

ABSTRACT

In (2008), Zhang proved the existence of xed points of mixed mono- tone operators along with certain convexity and concavity conditions. In this paper, mixed monotone single-valued and multi-valued operators of Meir-Keeler type are dened and two xed point theorems are proved. Our results, extend the results of Zhang.

Motivation & Objective

• To generalize Zhang's fixed point results for mixed monotone operators by relaxing convexity and concavity assumptions.
• To introduce and study Meir-Keeler type contraction conditions in the context of mixed monotone single-valued and multi-valued operators.
• To establish the existence of fixed points under weaker, more flexible conditions than previously required.
• To extend the applicability of fixed point theory to broader classes of operators in ordered metric spaces.

Proposed method

• Define mixed monotone single-valued and multi-valued operators of Meir-Keeler type in ordered metric spaces.
• Utilize a Meir-Keeler type contraction condition that ensures strict control over the distance between images of comparable elements.
• Apply order-theoretic arguments combined with sequential compactness and continuity assumptions to ensure convergence.
• Establish the existence of a fixed point by constructing monotone sequences that converge to a solution.
• Use the properties of the Meir-Keeler condition to guarantee the Cauchy nature of the sequence and its convergence to a fixed point.
• Generalize the framework to multi-valued operators by extending the contraction condition to set-valued mappings.

Experimental results

Research questions

• RQ1 Can the fixed point results for mixed monotone operators be extended beyond convexity and concavity assumptions?
• RQ2 What conditions ensure the existence of fixed points for mixed monotone multi-valued operators?
• RQ3 How does the Meir-Keeler type contraction condition improve upon existing contraction-based fixed point theorems?
• RQ4 To what extent can the order structure and metric properties be combined to prove fixed point existence?
• RQ5 Can the results be generalized to multi-valued mappings while preserving the convergence and fixed point properties?

Key findings

• The paper establishes a new fixed point theorem for mixed monotone single-valued operators under a Meir-Keeler type condition, which is weaker than the convexity and concavity assumptions used by Zhang.
• A corresponding fixed point result is proven for mixed monotone multi-valued operators, extending the theory to set-valued mappings.
• The Meir-Keeler type condition ensures that the operator contracts distances in a uniform way, enabling convergence of iterative sequences to a fixed point.
• The results generalize Zhang’s earlier findings by replacing restrictive structural assumptions with a more flexible contraction condition.
• The existence of a fixed point is guaranteed in ordered complete metric spaces under the new contraction framework.
• The approach provides a unified method for analyzing both single-valued and multi-valued mixed monotone operators using order and metric structures.