Fixed point results in intuitionistic fuzzy pentagonal controlled metric spaces with applications to dynamic market equilibrium and satellite web coupling

This manuscript contains several new spaces as the generalizations of fuzzy triple controlled metric space, fuzzy controlled hexagonal metric space, fuzzy pentagonal controlled metric space and intuitionistic fuzzy double controlled metric space. We prove the Banach fixed point theorem in the context of intuitionistic fuzzy pentagonal controlled metric space, which generalizes the previous ones in the existing literature. Further, we provide several non-trivial examples to support the main results. The capacity of intuitionistic fuzzy pentagonal controlled metric spaces to model hesitation, capture dual information, handle imperfect information, and provide a more nuanced representation of uncertainty makes them important in dynamic market equilibrium. In the context of changing market dynamics, these aspects contribute to a more realistic and flexible modelling approach. We present an application to dynamic market equilibrium and solve a boundary value problem for a satellite web coupling.


Introduction
Fuzzy sets (FSs) are very beneficial when dealing with data or information that contains uncertainty or ambiguity.They give a framework for dealing with circumstances in which the borders between categories are hazy or ambiguous.Fuzzy sets are therefore useful in fields such as artificial intelligence, expert systems, decision-making, and control systems.Zadeh [1] presented FSs as an extension of classical set theory in 1965.Unlike classical sets, which are binary and feature elements that either belong or do not belong to the set, FSs allow for degrees of membership.In other words, an element can have partial membership in a FS, which is represented by a value between 0 and 1, indicating the degree to which it belongs to the set.
In 1979, Itoh [2] proved fixed point theorems with an application to random differential equations in Banach spaces.Schweizer and Saklar [3] itroduced the notion of continuous tnorms (CTNs).Kramosil and Micha ´lek [4] introduced the concept of fuzzy metric space (FMS) by utilizing CTNs.George and Veeramani [5] modify the notion of FMS and presented Hausdorff topology in FMS.Grabiec [6] proved the Banach contraction theorem and Edelstein theorem in FMS.Han [7] demonstrated Banach fixed point theorem from the point of view of digital topology.Kamran et al. [8] developed the extended b-metric space and demonstrated numerous fixed point findings for contraction mappings.Mehmood et al. [9] proposed and demonstrated fixed point theorems for fuzzy rectangular b-metric spaces.Badshah-e-Rome et al. [10] defined extended fuzzy rectangular b-metric spaces and demonstrated numerous fixed point findings using α-admissibility.Furqan et al. [11] defined fuzzy triple controlled metric spaces (FTCMSs) as a generalization of various spaces.Zubair et al. [12] introduced and proved the Banach fixed point result for fuzzy extended hexagonal b-metric spaces (FEHbMSs).Hussain et al. [13] defined pentagonal controlled fuzzy metric spaces (PCFMSs) and fuzzy controlled hexagonal metric spaces (FCHMSs) and extended the Banach contraction concept to PCFMSs.
In 2004, Park [14] introduced the concept of intuitionistic fuzzy metric spaces (IFMSs) and discussed the topological structure.Konwar [15] proposed the notion of intuitionistic fuzzy bmetric spaces (IFbMSs) as a generalization of IFMSs.Shatanawi et al. [16] used an E.A property and the common E.A property for coupled maps to obtain new results on generalized IFMSs.Gupta et al. [17] obtained some coupled fixed-point results on modified IFMSs and applied them to the integral-type contractions.Farheen et al. [18] introduced the concept of intuitionistic fuzzy double-controlled metric spaces (IFDCMSs) and proved some fixed-point results.Ishtiaq et al. [19] coined the concept of intuitionistic fuzzy double-controlled metriclike spaces and provided several non-trivial examples with their graphical views, for more related knowledge, see [20].Younis and Abdou [21] presented novel fuzzy contractions and applications to engineering science.Ahmad et al. [22] presented the concept of bipolar b-metric spaces in the graph setting and related fixed point results.
We divide the paper into the six parts.In the first part, we present the introduction section.In the second part, we provide some basic and related definitions from the existing literature including CTN, CTCN, FTCMS, FEHBMS, CHFMS, and PCFMS.In the third part, we generalize the concepts of PCFMSs, FCHMSs and IFDCMSs and present the concepts of intuitionistic fuzzy pentagonal controlled metric spaces (IFPCMSs) and intuitionistic fuzzy controlled hexagonal metric spaces (IFCHMSs).We extend the Banach contraction principle in the setting of IFPCMSs.In the fourth part, we present an application to dynamic market equilibrium.In the fifth part, we provide an application to satellite web coupling.In the sixth part, we present the discussion and conclusion.

Preliminaries
This section contains some definitions from the existing literature that are useful for main section.
(b) a Cauchy, if and only if for each ω > 0, α > 0, there exists n 0 2 N such that Definition 3.6 Let (X, M, N, �, Δ) is an IFPCMS, then we define an open ball B ϰ; r; a ð Þ with centre ϰ; radius r, 0 < r < 1 and α > 0 as follows: and the topology that corresponds to it is defined as Let F: X !X be a mapping satisfying and where 0 < p <1. Furthermore, if, for ϰ 0 2 X and n; q 2 1; 2; 3; Proof: Let ϰ 0 2 X and define a sequence ϰ n f g by Without loss of generality, assume that ϰ n 6 ¼ ϰ nþ1 for all n 2 0; 1; 2; 3; � � � f g: With the help of (2), and (3), we deduce Continuing on the same lines, we obtain and Continuing on the same lines, we obtain and by using ( 4), ( 12), (A2) and (A7), we obtain and In similar manner, we can deduce Now, using (8), and ( 16), we deduce that and Furthermore, from (4), ( 5), ( 12) and ( 13), we can obtain and In similar manner, we can deduce Now, using ( 9) and ( 17), we deduce that and Accordingly, we get and Furthermore, for every q and from the inequalities ( 20)-( 27), we have p for all n; q 2 N and p 2 0; 1 ð Þ i:e:; ϰ n is a Cauchy sequence in X. Since, (X, M, N, �, Δ) is complete, there exists ϰ 2 X such that ϰ n !ϰ as n ! 1.Now, we investigate that ϰ is the fixed point of F. By applying Eqs (28) and (29) and conditions (IFP4),(IFP8), we have and Letting n ! 1 in the above inequalities, we deduce Fϰ ¼ ϰ; i:e:; ϰ is a fixed point of F. By applying the inequalities (2) and (3), it is easy to show that ϰ is a unique fixed point of F.

Application to dynamic market equilibrium
In this section, we demonstrate how our previously proven result can be used to identify the unique solution to an integral equation in dynamic market equilibrium the field of Economics.Supply Q β and demand Q d , in many markets, current prices and pricing trends (whether prices are rising or dropping and whether they are rising or falling at an increasing or decreasing rate) have an impact.The economist, therefore, wants to know what the current price is P(α), the first derivative dPðaÞ da , and the second derivative d 2 PðaÞ da 2 .Assume where g 1 , g 2 , γ 1 , γ 2 , e 1 and e 2 are constants.If pricing clears the market at each point in time, we can conclude that the market is dynamically stable.In equilibrium, Since Letting ϰ ¼ ϰ 1 À ϰ 2 ; e ¼ e 1 À e 2 ; g ¼ g 1 À g 2 and g ¼ g 1 À g 2 in above, we have Dividing through by ϰ; P a ð Þ is governed by the following initial value problem where e 2 ϰ ¼ 4g ϰ and g e ¼ m is a continuous function.It is straightforward to demonstrate that the problem (34) is identical to the integral equation: where ψ(α, r) is Green's function given by ( The integral equation has a solution, as we shall demonstrate: Let X = C ([0, T]) set of real continuous functions defined on [0, T] for α > 0, we define It is easy to prove that (X, M, N, �, Δ) is a complete IFPCMS and F: X !X defined by Then, the integral Eq (35) has a unique solution.Proof: For h; ϰ 2 X; by using of assumptions (i) to (iii), we have and all h; ϰ 2 X; and all conditions of Theorem 3.1 are satisfied.Therefore, Eq (34) has a unique fixed point.

Application to a satellite web coupling problem
We use Theorem 3.1 to solve a satellite web coupling boundary value problem [20] since fixed point techniques have been applied to a variety of real-world challenges.A thin sheet linking two cylinder-shaped satellites is an ideal representation of a satellite web coupling.The following non-linear boundary value problem is caused by the radiation from the web coupling issue between two satellites: where w(t) shows the temperature of radiation at any point t 2 0; 1 ½ �; m ¼ 2al 2 K 3 ch > 0 is a nondimensional positive constant, K is the constant absolute temperature of both satellites, while heat is radiated from the surface of the web into space at 0 absolute temperature, l is the distance between two satellites, a is a positive constant describing the radiation properties of the surface of the web, factor 2 is required because there is radiation from both the top and bottom surfaces, ψ is thermal conductivity, and h is the thickness.The Green function for all h; ϰ 2 X with the CTN 0 � 0 such that α 1 � α 2 = α 1 α 2 , and Δ is a CTCN such that α 1 Δ α 2 = max{α 1 , α 2 }.Define Q, W, E, R, T: X × X ![1, 1) by It is easy to prove that (X, M, N, �, Δ) is a complete IFPCMS.Theorem 5.1: Let f: X !X be a self-mapping in a complete IFPCMS, satisfying Then, the satellite web coupling boundary value problem (36) has a unique solution.
Proof: Define a self-mapping f: X !X by Clearly, a solution to the satellite web coupling problem (35) is a fixed point of a self-mapping f.However, ϰ t ð Þ À e t ð Þ > 0; so  Therefore, all the conditions of Theorem 3.1 are satisfied.Hence, f has a unique fixed point, and a satellite web coupling problem (36) has a unique solution.

Discussion and conclusions
In this paper, we introduced the notions of IFTCMS, IFHEbMS, IFHCMS, and IFPCMS as a generalization of several notions existing in the previous literature [1,2,9,14,15,18], in which we extended the triangular inequality and used membership and non-membership functions.In the definition of an IFPCMS: then it will become the definition of IFCHMS.

• If we take
then it will become the definition of IFEHbMS.
• Every PCFMS is an IFPCMS of the form (X, M, 1 -M, �, Δ), if we take Further, we proved a Banach fixed point theorem in the framework of IFPCMS that is a most generalized notion in IFMSs theory.Furthermore, we provided several examples for introduced notions and graphical representation to show the existence of fixed point of our main result.At the end, we presented applications to dynamic market equilibrium and satellite web coupling problem.This work is extendable in the context of intuitionistic fuzzy pentagonal controlled cone metric spaces, intuitionistic fuzzy pentagonal controlled partial metric spaces, pentagonal neutrosophic metric spaces and many other structures.
by plotting the below Fig 2. Hence, by Theorem 3.1, F has unique fixed point, which is 0 as shown in the below Fig 3.

Theorem 4 . 1
Assume an Eq (35) and let that(i) G: [0, T] × [0, T] !R + is continuous function;(ii) there exists a continuous function c : 0 and lim and lim It is immediate if we take g = ϖ = d and γ + δ+ w = r 0 in Theorem 3.1.
1] be a set of Riemann integrable functions defined on [0, 1].we define