#### Abstract

In this manuscript, we coined pentagonal controlled fuzzy metric spaces and fuzzy controlled hexagonal metric space as generalizations of fuzzy triple controlled metric spaces and fuzzy extended hexagonal b-metric spaces. We use a control function in fuzzy controlled hexagonal metric space and introduce five noncomparable control functions in pentagonal controlled fuzzy metric spaces. In the scenario of pentagonal controlled fuzzy metric spaces, we prove the Banach fixed point theorem, which generalizes the Banach fixed point theorem for the aforementioned spaces. An example is offered to support our main point. We also presented an application to dynamic market equilibrium.

#### 1. Introduction

Fuzzy notions are used to describe the degrees of possession of a certain property. The ability of fuzzy set (FS) theory to address circumstances that fixed point theory finds problematic originates from its attractiveness in tackling control problems. FSs are used to govern ill-defined, convoluted, and nonlinear systems The fabulous idea of FSs was presented by Zadeh [1] in his research paper. A FS extends the concept of a crisp set by associating all elements with membership values in the range of [0,1]. The FS theory has been widely employed in mathematics since then. Schweizer and Sklar [2] presented continuous t-norms (CTNs). Itoh [3] proved fixed point theorems with an application to random differential equations in Banach spaces. Kramosil and Michálek [4] presented the fuzzy metric space (FMS) approach. George and Veeramani [5] modify the notion of FMS and presented the Hausdorff topology in FMS. Grabiec [6] proved the Banach contraction theorem in fuzzy version, and also, he proved the Edelstein theorem in FMS. Han [7] proved the Banach fixed point theorem from the view point of digital topology. Uddin et al. [8] gave a solution of the Fredholm integral inclusions via Suzuki-type fuzzy contractions. Kamran et al. [9] presented the approach of extended metric space and proved several fixed point results for contraction mappings. Mehmood et al. [10] presented fuzzy rectangular b-metric spaces and proved fixed point theorems. Saleem et al. [11] coined the notion of fuzzy double controlled metric spaces and proved several fixed point results. Badshah-e-Rome and Sarwar [12] presented the approach of extended fuzzy rectangular b-metric spaces and proved fixed point results for contraction mappings via α-admissibility. Furqan et al. [13] presented the notion of fuzzy triple controlled metric spaces (FTCMSs) as a generalization of various spaces. Zubair et al. [14] presented fuzzy extended hexagonal b-metric spaces (FEHBMSs) and proved several fixed point results.

In this manuscript, we generalized the ideas of FTCMSs and FEHBMSs and present the approaches of pentagonal controlled fuzzy metric spaces (PCFMSs) and fuzzy controlled hexagonal metric spaces (FCHMSs). We extend the Banach contraction principle in the setting of FCHMSs. At the end, an application to dynamic market equilibrium is given to validate the main result.

#### 2. Preliminaries

This section contains some important definitions that aid comprehension of the main section.

Definition 1 (see [2]). A binary operation is a CTN if (1)(2) is continuous(3)(4)(5)If and with then

Definition 2 (see [13]). Let be a nonempty set. A 3-tuple is named a FTCMS if is a CTN, is a FS on , and are noncomparable and fulfill the following assertions for all , and ; the following circumstances are fulfilled: (S1)(S2) implies (S3)(S4)(S5) is left continuous and

Definition 3 (see [14]). Let be a nonempty set. A 4-tuple is a FEHBMS, if is a CTN, is a FS on , and fulfills the following assertions for all , and ; the following circumstances are fulfilled: (F1)(F2) implies (F3)(F4)(F5) is left continuous

#### 3. Main Results

The definitions of FCHMS and PCFMS are presented in this section, as well as proofs of fixed point findings.

Definition 4. Let be a nonempty set. A 4-tuple is a FCHMS if is a CTN, is a FS on , and fulfills the following assertions for all , and ; the following circumstances are fulfilled: (T1)(T2) implies (T3)(T4)(T5) is left continuous and

Example 5. Let Define as with the CTN such that Then, is FCHMS with control functions

Definition 6. Let be a nonempty set. A triplet is a PCFMS if is a CTN, is a fuzzy set on , and are five noncomparable functions that fulfill the following assertions for all , and ; the following circumstances are fulfilled: (A1)(A2) implies (A3)(A4)(A5) is left continuous and

Example 7. Let Define as with the CTN such that Then, is a PCFMS with noncomparable control functions

Remark 8. From the definition of PCFMS, (i)If we take then it will becomes the definition of FCHMS(ii)If we take then it will becomes the definition of FEHBMS in [14](iii)If and in (A4), then it will becomes FTCMS in [13]

Definition 9. Let be a PCFMS and be a sequence in then is named to be (i)a convergent, if there exists such that (ii)a Cauchy, if and only if for each there exists such that If every Cauchy sequence is convergent in then is a complete PCFMS.

Definition 10. Let be a PCFMS, then we define an open ball with centre radius , and as follows: and the topology that corresponds to it is defined as

Theorem 11. Let be a complete PCFMS and such that Let be a mapping satisfying where Furthermore, if, for it holds where then has a unique fixed point.

Proof. Assume and construct a sequence by Without restricting generality, suppose that With the help of (9), we deduce Continuing in this way, we obtain It implies, if Expressing and by using (11) and (A1), we obtain In similar manner, we can deduce We obtain for each Now, using (15), we deduce that Furthermore, from (11) and (12), we can obtain In similar manner, we can deduce We obtain for each