Application To Lipschitzian and Integral Systems Via Quadruple Coincidence Point in Fuzzy Metric Spaces

Without a partially ordered set, in this manuscript, we investigate quadruple coincidence point (QCP) results for commuting mapping in the setting of fuzzy metric spaces (FMSs). Furthermore, some relevant ﬁndings are presented to generalize some of the previous results in this direction. Ultimately, non-trivial examples and applications to ﬁnd a unique solution for Lipschitzian and integral quadruple systems are provided to support and strengthen our theoretical results.


Introduction
There is no doubt that the study of fuzzy sets is extremely important for its multiple applications, such as control ill-defined, complex and non-linear systems. It is more common to find solutions to control problems that are difficult to solve with the classical control theory. Fuzzy set theory is becoming an increasingly important tool, especially in the rapidly evolving disciplines of artificial intelligence: expert systems and neural networks. It creates completely new opportunities for the application of fuzzy sets in chemical engineering [1,2,3,4].
The concept of fuzzy sets initiated by Zadeh [5] in 1965. Many mathematicians considered these sets to introduce interesting concepts into the field of mathematics like fuzzy logic, fuzzy differential equations, and fuzzy metric spaces. It is known that the FMS is an important generalization of the ordinary metric space where the extended topological definitions and possible applications in several areas. Many mathematicians have considered this problem in many ways. For example, the authors in [6] modified the concept of a FMS that initiated by Kramosil and Michalek [7] and defined the Hausdorff topology of a FMS. For more details about this idea we advise the reader to view [8,9,10,11,12].
In the setting of FMSs, coupled FP results are presented and some important theorems are given by Zhu and Xiao [21] and Hu [22]. Elagan et al. [23] studied the existence of a FP in locally convex topology generated by fuzzy n normed spaces.
Motivated by the results of coupled and tripled FP notions in partially ordered metric spaces, Karapınar [24] suggested the concept of quadruple FP and proved some related FP consequences in the same spaces. Also, see [25,26,27].
Based on the last two paragraphs above, in this manuscript, a QCP is considered and some new related FP results are presented in FMSs. The strength of our paper depends on two directions. Firstly, we can customize it to complete metric spaces (CMSs) such as obtaining the results of Karapınar [24] (in non fuzzy sets). So our paper covered and unify a lot of results in the same direction, also covers coupled FP results. Secondly, we can use the theoretical results to find a unique solution for Lipschitzian and integral quadruple systems. Ultimately, non-trivial examples are stated and discussed.

Main results
We begin this section with the simple definition below Definition 3.1. Assume that Ω : ⇣ 4 ! ⇣ and k : ⇣ ! ⇣ are two mappings.
The proof of the corollary below follows immediately by Theorem 4.1.

Supportive applications
This section is specially prepared to highlight the importance of theoretical results and how to use them in obtaining the existence of the solution to a Lipschitzian and an integral quadruple system.
According to the above results we can state the corollary below.