# New generalized Trends of Coupled Fixed Point Theorems for Continuous Mappings Satisfying a Property of Mixed Monotone

## TL;DR

This paper presents new generalized results on coupled fixed point theorems for continuous mappings with mixed monotone properties in partially ordered metric spaces.

## Contribution

The paper introduces novel and enhanced coupled fixed point theorems under extended contraction conditions in D∗-complete metric spaces.

## Key findings

- New coupled fixed point theorems are established for continuous mappings with mixed monotone properties.
- The results extend and improve existing theorems in the literature.
- An example is provided to support the validity of the new theorems.

## Abstract

One of the most important results of mathematical analysis is the Banach
fixed point theory, he explained in this theory that a mapping T: X →
X always has a unique fixed point in X. After witnessing the implementations
of this theory in giving the existence and uniqueness solutions for many
integral and differential equations, additionally discovery of solutions for
linear and nonlinear systems, various extensions of this theory were carried
out. Our main results represent one of the most important of these
generalizations in the literature.

Various vital concepts are needed in the sequels which are playing a major
role in verifying our major outcomes have been presented. Throughout this
manuscript,
(X,≼)
indicates to a partially ordered set with the partially
ordered
≼
. In this study, our main objective is to investigate and
verify various new enhanced results of coupled fixed point theorems for
continuous maps having the property of mixed monotone under the influence of
extended contraction circumstances in the context of partially ordered
D∗
-complete metric spaces. Numerous characterizations of
these types of coupled fixed point theorems have been verified. Our major
outcomes are extending and enhancing the various outcomes of coupled fixed
point theorems existing in the literature. Additionally, an appropriate
example that supports the major outcomes was prepared.

Our main results in this manuscript have been explored novel various outcomes
related to the uniqueness of various coupled fixed point theorems for
continuous maps having the property of mixed monotone under the influence of
extended contraction circumstances in the context of partially ordered
D∗
-complete metric spaces. We predict that the discoveries in
this study will aid scientists in enhancing the research on popularized
partially ordered metric spaces to elevate a universal framework for their
practical implementations.