1. 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 t-norms (CTNs). Kramosil and Michálek [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 b-metric 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 metric-like 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.

2. Preliminaries

This section contains some definitions from the existing literature that are useful for main section.

Definition 2.1 [3] A binary operation *: [0, 1] × [0, 1] → [0, 1] is a CTN if it verifies the below axioms:

1. τ * θ = θ * τ, (∀) τ, θ ∈ [0, 1];
2. * is continuous;
3. τ * 1 = τ, (∀) τ ∈ [0, 1];
4. (τ * θ) * ρ = τ * (θ * ρ), (∀) τ, θ, ρ ∈ [0, 1];
5. if τ ≤ ρ and θ ≤ σ, with τ, θ, ρ, σ ∈ [0, 1], then τ * θ ≤ ρ * σ.

Definition 2.2 [3] A binary operation Δ: [0, 1] × [0, 1] → [0, 1] is a CTCN if it verifies the below axioms:

1. τΔθ = θ Δ τ, (∀) τ, θ ∈ [0, 1];
2. Δ is continuous;
3. τΔ0 = τ, (∀) τ ∈ [0, 1];
4. (τΔθ) Δ ρ = τΔ(θΔρ), (∀) τ, θ, ρ ∈ [0, 1];
5. if τ ≤ ρ and θ ≤ σ, with τ, θ, ρ, σ ∈ [0, 1], then τΔθ ≤ ρΔσ.

Definition 2.3 [11] Let X be a non-empty set. A 3-tuple (X, N_t,*,) is called a FTCMS if * is a CTN, N_t is a FS on X × X ×[0,∞) and Q, W, E: X × X→[1,∞) are non-comparable functions, satisfies the below conditions for all and α, β, γ > 0:

1. S1.
2. S2. iff
3. S3.
4. S4.
5. S5. N_t(ϰ, d,.): (0, ∞) → [0,1] is left continuous and

Definition 2.4 [12] Let X be a non-empty set. A 3-tuple (X, N_l, *) is called a FEHBMS, if * is a CTN, N_l is a FS on X × X ×[0,∞) and Q: X × X → [1, ∞) is a function, satisfies the below conditions for all and α, β, γ, δ, w > 0:

1. F1.
2. F2. iff
3. F3.
4. F4.
5. F5. N_l(ϰ, d,.): (0, ∞) → [0,1] is left continuous.

Definition 2.5 [13] Let X be a non-empty set. A 3-tuple (X, N_h, *) is called a CHFMS if * is a CTN, N_h, is a FS on X × X ×[0, ∞) and Q: X × X → [1, ∞) be a function, satisfies the below conditions for all and α, β, γ, δ, w > 0:

1. T1.
2. T2. iff
3. T3.
4. T4.
5. T5. N_h(ϰ, d,.): (0, ∞) → [0,1] is left continuous and

Definition 2.6 [13] Let X be a non-empty set. A 3-tuple (X, N_p, *) is called a PCFMS if * is a CTN, N_p is a FS on X × X ×[0, ∞), and Q, W, E, R, T: X × X →[1, ∞) are five non-comparable functions, satisfies the below conditions for all and α, β, γ, δ, w > 0:

1. A1.
2. A2. iff
3. A3.
4. A4.
5. A5. N_p(ϰ, d,.): (0, ∞) → [0,1] is left continuous and

3. Main results

In this section, we introduce the definitions of IFTCMS, IFCHMS and IFPCMS and provide fixed point theorems.

Definition 3.1 Let X be a non-empty set. A 5-tuple (X, N_t, M_t, *, Δ) is called an IFTCMS if * is a CTN, Δ is a CTCN, N_t, M_t are FSs on X × X × [0, ∞) and Q, W, E: X × X →[1, ∞) are non-comparable functions, satisfies the below conditions for all and α, β, γ > 0:

1. S1.
2. S2.
3. S3. iff
4. S4.
5. S5.
6. S6. N_t(ϰ, d,.): (0, ∞) → [0,1] is left continuous and
7. S7.
8. S8. iff
9. S9.
10. S10.
11. S11. M_t(ϰ, d,.): (0, ∞) → [0,1] is left continuous and

Example 3.1 Let X = [0,1]. Define N_t, M_t: X × X × [0,∞) →[0,1] as with the CTN * such that α₁ * α₂ = α₁α₂, and CTCN Δ such that α₁ Δ α₂ = max{α₁, α₂}. Then, (X, N_t, M_t,*, Δ) is an IFPCMS with non-comparable control functions

Definition 3.2: Let X be a non-empty set. A 5-tuple (X, H, N,*,Δ) is an IFEHbMS if * is a CTN, Δ is a CTCN, H, N are FSs on X × X × [0, ∞) and Q: X × X → [1, ∞), satisfies the below conditions for all and α, β, γ, δ, w > 0:

1. C1.
2. C2.
3. C3. iff
4. C4.
5. C5.
6. C6. H(ϰ, d,.): (0, ∞) → [0,1] is left continuous and
7. C7.
8. C8. iff
9. C9.
10. C10.
11. C11. N(ϰ, d,.): (0, ∞) → [0,1] is left continuous and

Definition 3.3: Let X be a non-empty set. A 5-tuple (X, H, N, *, Δ) is an IFCHMS if * is a CTN, Δ is a CTCN, H, N are FSs on X × X × [0, ∞) and Q: X × X → [1, ∞) satisfies the below conditions for all and α, β, γ, δ, w > 0:

1. C12.
2. C13.
3. C14. iff
4. C15.
5. C16.
6. C17. H(ϰ, d,.):(0, ∞) → [0,1] is left continuous and
7. C18.
8. C19. iff
9. C20.
10. C21.
11. C22. N(ϰ, d,.): (0, ∞) → [0,1] is left continuous and

Example 3.2: Let X = {1,2,3,4,5,6}. Define H, N: X × X × [0,∞) → [0,1] as with the CTN * such that α₁ * α₂ = α₁α₂, and CTCN Δ such that α₁ Δ α₂ = max{α_1, α₂}. Then, (X, H, N, *, Δ) is an IFCHMS with a control function

Definition 3.4: Let X be a non-empty set. A 5-tuple (X, M, N,*, Δ) is an IFPCMS if * is a CTN, Δ is a CTCN, M and N are FSs on X × X × [0, ∞) and Q, W, E, R, T: X × X →[1, ∞) are five non-comparable functions, satisfies the below conditions for all and α, β, γ, δ, w > 0:

1. IFP1.
2. IFP2.
3. IFP3. iff
4. IFP4.
5. IFP5.
6. IFP6. M(ϰ, d,.): (0, ∞) → [0,1] is left continuous and
7. IFP7.
8. IFP8. iff
9. IFP9.
10. IFP10.
11. IFP11. N(ϰ, d,.): (0, ∞) → [0,1] is left continuous,

Example 3.3 Let X = [0, 1]. Define M, N: X × X × [0, ∞) → [0, 1] as with the CTN * such that α₁ * α₂ = α₁α₂, and CTCN Δ such that α₁ Δ α₂ = max{α_1, α₂}. Then, (X, H, N, *, Δ) is an IFPCMS with non-comparable control functions

Remark 3.1 From the definition of IFPCMS,

1. If we take then it will become the definition of IFCHMS.
2. If we take then it will become the definition of IFEHbMS.
3. If and γ + δ +w = r′, then it will become IFTCMS.
4. If and β + γ + δ + w = t′, then it will become IFDCMS in [18].
5. If β + γ + δ + w = t′, and W(e, κ) = E(κ, g) = b ≥ 1 then it will become IFbMS in [15].
6. If β + γ + δ + w = t′, and W(e, κ) = E(κ, g) = 1 then it will become IFMS in [14].
7. Every PCFMS is an IFPCMS of the form (X, M, 1 − M, *, Δ), if we take

Definition 3.5 Let (X, M, N, *, Δ) is a IPCFMS and be a sequence in X, then is called:

1. (a) a convergent, if there exists such that
2. (b) a Cauchy, if and only if for each ω > 0, α > 0, there exists such that
3. (c) if every Cauchy sequence is convergent in X, then (X, M, N, *, Δ) is a complete IFPCMS.

Definition 3.6 Let (X, M, N, *, Δ) is an IFPCMS, then we define an open ball with centre radius r, 0 < r < 1 and α > 0 as follows: and the topology that corresponds to it is defined as

Theorem 3.1 Let (X, M, N,*,Δ) be a complete IFPCMS and Q, W, E, R, T:X × X → [1,∞) such that (1)

Let F: X → X be a mapping satisfying (2) and (3) where 0 < p <1. Furthermore, if, for It holds where then F has a unique fixed point.

Proof: Let and define a sequence by

Without loss of generality, assume that With the help of (2), and (3), we deduce (4)

Continuing on the same lines, we obtain (5) (6) (7)

It implies, if m = 1,2,3,⋯, (8) (9) (10) (11) and (12)

Continuing on the same lines, we obtain (13) (14) (15)

It implies, if m = 1,2,3,⋯, (16) (17) (18) (19)

Expressing and by using (4), (12), (A2) and (A7), we obtain and

In similar manner, we can deduce and

We obtain for each m = 1,2,3,⋯, and

Now, using (8), and (16), we deduce that (20) and (21)

Furthermore, from (4), (5), (12) and (13), we can obtain and

In similar manner, we can deduce and

We obtain for each m = 1,2,3,⋯, and

Now, using (9) and (17), we deduce that (22) and (23)

Accordingly, we get (24) (25) and (26) (27)

Furthermore, for every q and from the inequalities (20)–(27), we have (28) (29) as is a Cauchy sequence in X. Since, (X, M, N, *, Δ) is complete, there exists as n → ∞. Now, we investigate that is the fixed point of F. By applying Eqs (28) and (29) and conditions (IFP4),(IFP8), we have and

Letting n → ∞ in the above inequalities, we deduce is a fixed point of F. By applying the inequalities (2) and (3), it is easy to show that is a unique fixed point of F.

Corollary 3.1 Let (X, M, N, *, Δ) be a complete IFCHMS and Q: X × X → [1, ∞) such that

Let F: X → X be a mapping satisfying (30) (31) where 0 < p < 1 Furthermore, if, for and n, q ∈ {1,2,3,⋯}, It holds where then F has a unique fixed point.

Proof: It is immediate if we take in Theorem 3.1.

Corollary 3.2 Let (X, M, N, *, Δ) be a complete IFEHbMS and Q: X × X → [1, ∞) such that

Let F: X → X be a mapping satisfying (32) where 0 < p < 1. Furthermore, if, for and n, q, ∈ {1,2,3,⋯}, It holds where then F has a unique fixed point.

Proof: It is immediate if we take in Theorem 3.1.

Corollary 3.3: Let are three non comparable functions where k ∈ (0,1) and (X, M, N, *, Δ) is a complete IFTCMS, such that

Let F: X → X be a mapping satisfying then F has a unique fixed point.

Proof: It is immediate if we take g = ϖ = d and γ + δ+ w = r′ in Theorem 3.1.

Example 3.4 Let X = [0,1]. Define M: X × X × [0, ∞) → [0,1] as with the CTN * such that α₁ * α₂ = α₁α₂, and CTCN Δ such that α₁ Δ α₂ = max{α₁, α₂}, Then (X, M, N, *, Δ) is a complete IFPCMS with non-comparable control functions .

Define F: X → X by Consider

Now, we show that by plotting the below Fig 1.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Shows the graphical behavior of an inequality for ϖ = 1, α = 1, and q = 0.01.

https://doi.org/10.1371/journal.pone.0303141.g001

Now, we show that by plotting the below Fig 2.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Shows the graphical behavior of an inequality for ϖ = 1, α = 1, and q = 0.01.

https://doi.org/10.1371/journal.pone.0303141.g002

Hence, by Theorem 3.1, F has unique fixed point, which is 0 as shown in the below Fig 3.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Shows that 0 is a unique fixed point of F.

https://doi.org/10.1371/journal.pone.0303141.g003

4. 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 , and the second derivative . Assume where g₁, g₂, γ₁, γ₂, e₁ and e₂ are constants. If pricing clears the market at each point in time, we can conclude that the market is dynamically stable. In equilibrium, Q_β = Q_d. So

Since

Letting in above, we have

Dividing through by is governed by the following initial value problem (33) where 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: (34)

Let X = C ([0, T]) set of real continuous functions defined on [0, T] for α > 0, we define

For all with α₁ * α₂ = α₁α₂, and α₁ Δ α₂ = max{α₁, α₂}. Define Q, W, E, R, T: X × X → [1, ∞). As

It is easy to prove that (X, M, N, *, Δ) is a complete IFPCMS and F: X → X defined by

Theorem 4.1 Assume an Eq (35) and let that

1. (i) G: [0, T] × [0, T] → ℝ⁺ is continuous function;
2. (ii) there exists a continuous function such that
3. (iii)

Then, the integral Eq (35) has a unique solution.

Proof: For by using of assumptions (i) to (iii), we have and

Thus and all conditions of Theorem 3.1 are satisfied. Therefore, Eq (34) has a unique fixed point.

5. 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: (35) where w(t) shows the temperature of radiation at any point is a non-dimensional 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

Eq (35) is equivalent to

Let X = ℛ[0,1] be a set of Riemann integrable functions defined on [0, 1]. we define for all with the CTN ′*′ such that α₁ * α₂ = α₁α₂, and Δ is a CTCN such that α₁ Δ α₂ = max{α₁, α₂}. Define Q, W, E, R, T: X × X → [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 (36)

Then, the satellite web coupling boundary value problem (36) has a unique solution.

Proof: Define a self-mapping f: X → X by (37)

Clearly, a solution to the satellite web coupling problem (35) is a fixed point of a self-mapping f. However, 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.

6. 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:

• If we take then it will become the definition of IFCHMS.
• If we take then it will become the definition of IFEHbMS.
• If g = ϖ = d and γ + δ + w = r′, then it will become IFTCMS.
• If κ = g = ϖ = d and β + γ + δ + w = t′, then it will become IFDCMS in [18].
• If κ = g = ϖ = d, β + γ + δ + w = t′, and W(e, κ) = E(κ, g) = b ≥ 1 then it will become IFbMS in [15].
• If κ = g = ϖ = d, β + γ + δ + w = t′, and W(e, κ) = E(κ, g) = 1 then it will become IFMS in [14].
• 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.