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2326-9865

Vol. 71 No. 3s2 (2022) 756 http://philstat.org.ph

Some Compatible and Weakly-Compatible Four Self-Mapping Results Approach to Nonlinear Integral Equations in Intuitionistic

Fuzzy Cone Metric Spaces

𝐃𝐫. 𝐀. 𝐌𝐨𝐡𝐚𝐧

Assistant Professor, PG & Research Department of Mathematics, Urumu Dhanalakshmi College, Bharathidasan University, Trichy, India.

Email id: [email protected].,

Article Info

Page Number: 756 - 777 Publication Issue:

Vol 71 No. 3s2 (2022)

Article History

Article Received: 28 April 2022 Revised: 15 May 2022

Accepted: 20 June 2022 Publication: 21 July 2022

Abstract

This study uses the compatible and weakly compatible four self-mappings in intuitionistic fuzzy cone metric (IFCM) space to prove several unique common fixed point theorems. With the use of one self-map, we show that the findings in IFCM spaces under generalised rational contraction conditions are continuous.

Furthermore, given the weaker requirement of self-mapping continuity, we show several logical contraction findings. Finally, our theoretical work was used to demonstrate that the two nonlinear integral equations had a solution.

This is an example of how IFCM spaces may be used to different integral type operators.

2020 Mathematical Sciences Classification : 46N20, 46S40, 47H10, 37C25, 54H25, 55M20, 58C30.

Keywords- Intuitionistic fuzzy metric, Intuitionistic fuzzy cone, compatiple, fixed point

1. Introduction

In the year 1965, Zadeh [16] introduced the concept of fuzzy sets which permit the gradual assessment of the membership of the elements in a set. To use this concept in topology, Kramosil and Michálek in [8] introduced the class of fuzzy metric spaces. After that, George and Veeramani [5] modified the concept of fuzzy metric spaces and defined a Hausdorff topology on this fuzzy space.

The concept of cone metric space is introduced by Huang and Zhang [6] also they proved fixed point results. In 1986, the concept of an intuitionistic fuzzy set (IFS) was put forward by Atanassov [3], which can be viewed as an extension of fuzzy set. Park [10-11]

proved various types of fixed point results in Intuitionistic fuzzy metric space. After that In 2015, the notion of fuzzy cone metric space (FCM space) was introduced by Oner [9] et al.

Some triangular property and integral type application results in the theory of fixed point can be found in [7,8,12-14]. Some Basic property’s fount in [1,2,15].

2. Preliminaries

We'll go over some fundamental definitions and lemmas in this section.

Definition 1[15]

An operation ∗,◊∶ [0, 1]² ⟶ [0, 1] is called a continuous t-norm and t-conorm if (i) ∗ is associative, commutative, and continuous

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2326-9865

(ii) 1 ∗ 𝑞₁ = 𝑞₁ and 𝑞₁∗ 𝑞₂ ≤ 𝑞₃ ∗ 𝑞₄, whenever 𝑞₁ ≤ 𝑞₃ and 𝑞₂ ≤ 𝑞₄, for all 𝑞₁, 𝑞₂, 𝑞₃, 𝑞₄ ∈ [0, 1]

(iii) The minimum; 𝑞₁ ∗ 𝑞₂ = 𝑚𝑖𝑛 {𝑞₁, 𝑞₂} (iv) The product; 𝑞₁ ∗ 𝑞₂ = 𝑞₁𝑞₂

(v) The Lukasiewicz; 𝑞₁ ∗ 𝑞₂ = 𝑚𝑎𝑥 {𝑞₁ + 𝑞₂ − 1, 0}

(vi) ◊ is associative, commutative, and continuous

(vii) 0 ◊ 𝑞₁ = 𝑞₁ and 𝑞₁◊ 𝑞₂ ≤ 𝑞₃ ◊ 𝑞₄, whenever 𝑞₁ ≤ 𝑞₃ and 𝑞₂ ≤ 𝑞₄, for all 𝑞₁, 𝑞₂, 𝑞₃, 𝑞₄ ∈ [0, 1]

(viii) The minimum; 𝑞₁ ◊ 𝑞₂ = 𝑚𝑎𝑥 {𝑞₁, 𝑞₂} (ix) The product; 𝑞₁ ◊ 𝑞₂ = 𝑞₁+ 𝑞₂− 𝑞₁𝑞₂

(x) The Lukasiewicz; 𝑞₁ ∗ 𝑞₂ = 𝑚𝑖𝑛 {𝑞₁ + 𝑞₂ ,1}. Schweizer are Sklar [15] define the above basic continuous t-norms and t-conorms.

Definition 2 [1]

A 5-tuple (𝑈, 𝑀₀, 𝑁₀,∗,◊) is called a IFCM space if C is a cone of E, 𝑈 is an arbitrary set, (∗,◊) is a continuous t-norm, t-conorm and 𝑀₀, 𝑁₀ is a intuitionistic fuzzy set on 𝑈 ²× 𝑖𝑛𝑡 (𝑃) satisfying the following conditions

i. 𝑀₀(λ₁,λ₂, t) > 0 and 𝑀₀(λ₁,λ₂, t) = 1 ⇔λ₁ = λ₂ ii. 𝑀₀(λ₁,λ₂, t) = 𝑀₀(λ₂,λ₁, t)

iii. 𝑀₀(λ₁,λ₂, t) ∗ 𝑀₀(λ₂,λ₃, s) ≤ 𝑀₀(λ₁,λ₃, t + s)

iv. 𝑀₀(λ₁,λ₂, . ): 𝑖𝑛𝑡 (𝑝) → [0,1] is continuous ∀ λ₁,λ₂,λ₃ ∈ U and 𝑡, 𝑠 ∈ 𝑖𝑛𝑡(𝑝).

v. 𝑁₀(λ₁,λ₂, t) ≤ 1 and 𝑁₀(λ₁,λ₂, t) = 0 ⇔λ₁ = λ₂ vi. 𝑁₀(λ₁,λ₂, t) = 𝑁₀(λ₂,λ₁, t)

vii. 𝑁₀(λ₁,λ₂, t)◊𝑁₀(λ₂,λ₃, s) ≥ 𝑁₀(λ₁,λ₃, t + s)

viii. 𝑁₀(λ₁,λ₂, . ): 𝑖𝑛𝑡 (𝑝) → [0,1] is continuous ∀ λ₁,λ₂,λ₃ ∈ U and 𝑡, 𝑠 ∈ 𝑖𝑛𝑡(𝑝).

Definition 3[2]

Let (𝑈, 𝑀₀, 𝑁₀,∗,◊) be a IFCM space, ∃ λ₁ ∈ U and {λ_𝑗} be any sequence in U.

(i) {λ_𝑗} converges to λ₁ if for any c ∈ (0, 1), t ≫θ, and ∃ j₁ ∈ N such that M₀(λ_𝑗,λ₁, t) > 1 − c and 𝑁₀(λ_𝑗,λ₁, t) < c,for j ≥ j₁.

This can be written as lim

j⟶∞λ_𝑗 =λ₁ , or λ_𝑗 ⟶λ₁ as j ⟶∞

(ii) (λ_𝑗) is Cauchy if for any c ∈ (0, 1), t ≫θ and ∃j₁ ∈ N such that M₀(λ_𝑗,λ₁, t) >

1 – c and 𝑁₀(λ_𝑗,λ₁, t) < c for j, k ≥ j₁.

(iii) (𝑈, 𝑀₀, 𝑁₀,∗,◊) is complete if every Cauchy sequence is convergent in U (iv) {λ_𝑗} is IFC contractive if ∃a ∈ (0, 1) so that

1

𝑀₀(λ_𝑗,λ_𝑗+1,𝑡)− 1 ≤ 𝑎 ( ¹

𝑀₀(λ_𝑗−1,λ_𝑗,𝑡)− 1) for t ≫ θ, j ≥ 1 and

𝑁₀(λ_𝑗,λ_𝑗+1, 𝑡) ≤ 𝑎(𝑁₀(λ_𝑗−1,λ_𝑗, 𝑡)) for t ≫θ, j ≥ 1 (1)

Lemma 4 [3]

“Let (𝑈, 𝑀₀, 𝑁₀,∗,◊) be a IFCM space and a sequence λ_𝑗 ⟶λ₁ ∈ U iff M₀(λ_𝑗,λ₁, t) ⟶ 1 𝑎𝑛𝑑 𝑁₀(λ_𝑗,λ₁, t) ⟶ 0 as j ⟶∞ for each t ≫θ”.

Let (𝑈, 𝑀₀, 𝑁₀,∗,◊) be a IFCM space. The IFCM 𝑀₀, 𝑁₀ is triangular if

1

𝑀₀(λ₁,λ₃,𝑡)− 1 ≤ ( ¹

𝑀₀(λ₁,λ₂,𝑡)− 1) + ( ¹

𝑀₀(λ₂,λ₃,𝑡)− 1) ∀ λ₁,λ₂,λ₃ ∈ 𝑈 t ≫θ, and

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2326-9865

𝑁₀(λ₁,λ₃, 𝑡) ≤ 𝑁₀(λ₁,λ₂, 𝑡) + 𝑁₀(λ₂,λ₃, 𝑡)∀ λ₁,λ₂,λ₃ ∈ 𝑈 t ≫θ, (2)

Let (𝑈, 𝑀₀, 𝑁₀,∗,◊) be a IFCM space and A ∶ U ⟶ U. Then, A is said to be IFC contractive if there is a ∈ (0, 1) so that

1

𝑀₀(𝐴λ₁,𝐴λ₂,𝑡)− 1 ≤ 𝑎 ( ¹

𝑀₀(λ₁,λ₂,𝑡)− 1) ∀ λ₁,λ₂, ∈ 𝑈 and t ≫θ,

𝑁₀(𝐴λ₁, 𝐴λ₂, 𝑡) ≤ 𝑎(𝑁₀(λ₁,λ₂, 𝑡)) ∀ λ₁,λ₂, ∈ 𝑈 and t ≫θ, (3)

Definition 7[12]

Let U ≠ ∅ set and let B, h ∶ U ⟶ U be the self-mappings on U. If there exists ξ ∈ U such that B ρ = hρ = ξ for some ρ ∈ U. Then, ρ is called a coincidence point of B and h, and ξ is known as a point of coincidence of the mappings B, h. A pair of self-mappings (B, h) is known to be weakly-compatible if the self-mappings commute at their coincidence point, i.e., hB(ρ) = Bh(ρ) for ρ ∈ U.

Proposition 8 [13]

Let B, h be weakly-compatible self-mappings on U. If B and h have a unique point of coincidence, that is, Bρ = hρ = ξ, then, ξ is a unique CFP of the mappings B and ℎ.

A self-mapping pair (h, B) is said to be compatible on a IFCM space (𝑈, 𝑀₀, 𝑁₀,∗,◊) if, lim

𝑗→∞ 𝑀₀(Bhλ_𝑗, hBλ_𝑗, t) = 1 and lim

𝑗→∞ 𝑁₀(Bhλ_𝑗, hBλ_𝑗, t) = 0 for t ≫ θ, whenever {λ_𝑗} is a sequence in U so that lim

𝑗→∞𝐵λ_𝑗 = lim

𝑗→∞ℎλ_𝑗 = 𝜉 for some ξ ∈ U.

3. Main Results

We are now in a position to communicate our key findings.

Theorem 10.

Let A, B, g, h ∶ U ⟶ U be the four self mappings on a complete IFCM space (𝑈, 𝑀₀, 𝑁₀,∗,◊) in which a IFCM, 𝑀₀, 𝑁₀ is triangular and satisfies

1

𝑀₀(𝐴λ,𝐵𝜇,𝑡)− 1 ≤ {

𝑎 ( ¹

𝑀₀(ℎλ,𝑔𝜇,𝑡)− 1) + 𝑏 ( ^𝑀⁰^{(ℎλ,𝑔𝜇,𝑡)}

𝑀₀(ℎλ,𝐵𝜇,2𝑡)∗𝑀₀(𝑔𝜇,𝐴λ,2𝑡)− 1) +𝑐 ( ¹

𝑀_𝑟(ℎλ,𝐴λ,𝑡)− 1 + ¹

𝑀_𝑟(𝑔𝜇,𝐵𝜇,𝑡)− 1) + 𝑑 ( ¹

𝑀_𝑟(𝑔𝜇,𝐴λ,𝑡)− 1 + ¹

𝑀_𝑟(ℎλ,𝐵𝜇,𝑡)− 1)} and 𝑁₀(𝐴λ, 𝐵𝜇, 𝑡) ≤ { 𝑎(𝑁₀(ℎλ, 𝑔𝜇, 𝑡)) + 𝑏 (^𝑁⁰(ℎλ,𝐵𝜇,2𝑡)◊𝑁₀(𝑔𝜇,𝐴λ,2𝑡)

𝑁₀(ℎλ,𝑔𝜇,𝑡) )

+𝑐(𝑁_𝑟(ℎλ, 𝐴λ, 𝑡) + 𝑁_𝑟(𝑔𝜇, 𝐵𝜇, 𝑡)) + 𝑑(𝑁_𝑟(𝑔𝜇, 𝐴λ, 𝑡) + 𝑁_𝑟(ℎλ, 𝐵𝜇, 𝑡))} (4)

∀ λ,μ ∈ U, t ≫ θ, and 0 ≤ a, b, c, d < 1 with (a + b + 2c + 2d) < 1. If A(U) ⊆ g(U), B(U) ⊆ h(U) and consider that

(1) h is a continuous self-mapping, (2) A pair (A, h) is compatible, and

(3) A pair (B, g) is weakly-compatible, Then, the mappings A, B, g, and h have a unique CFP in U.

(4)

2326-9865

Vol. 71 No. 3s2 (2022) 759 http://philstat.org.ph Proof.

Fix λ₀ ∈ U and by the hypothesis A(U) ⊆ g(U), B(U) ⊆ h(U), we define the iterative sequences in U so that

ξ_2j+1 = gλ_2j+1 = Aλ_2j and ξ_2j+2 = hλ_2j+2 = Bλ_2j+1 , j ≥ 0:

(5)

Then by (4)

1

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1 = ¹

𝑀₀(𝐴λ_2𝑗,𝐵λ_2𝑗+1,𝑡)− 1 ≤

{

𝑎 (_𝑀 ¹

0(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡)− 1) + 𝑏 ( ^𝑀⁰^(ℎλ^2𝑗^,𝑔λ^2𝑗+1^,𝑡)

𝑀₀(ℎλ_2𝑗,𝐵λ_2𝑗+1,2𝑡)∗𝑀₀(𝑔λ_2𝑗+1,𝐴λ_2𝑗,2𝑡)− 1)

+𝑐 ( ¹

𝑀₀(ℎλ_2𝑗,𝐴λ_2𝑗,𝑡)− 1 + ¹

𝑀₀(𝑔λ_2𝑗+1,𝐵λ_2𝑗+1,𝑡)− 1)

+𝑑 ( ¹

𝑀₀(𝑔λ_2𝑗+1,𝐴λ_2𝑗,𝑡)− 1 + ¹

𝑀₀(ℎλ_2𝑗,𝐵λ_2𝑗+1,𝑡)− 1)

}

= {

𝑎 (_𝑀 ¹

0(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡)− 1) + 𝑏 (^𝑀⁰^(ℎλ^2𝑗^,𝑔λ^2𝑗+1^,𝑡)

𝑀₀(ℎλ_2𝑗,ℎλ_2𝑗+2,2𝑡))

+𝑐 ( ¹

𝑀₀(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡)− 1 + ¹

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1)

+𝑑 ( ¹

𝑀₀(ℎλ_2𝑗,𝐵λ_2𝑗+2,𝑡)− 1)

}

and

𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡) = 𝑁₀(𝐴λ_2𝑗, 𝐵λ_2𝑗+1, 𝑡)

≤ {

𝑎 (𝑁0(ℎλ_2𝑗, 𝑔λ_2𝑗+1, 𝑡)) + 𝑏 (^𝑁⁰^(ℎλ^2𝑗^,𝐵λ^2𝑗+1^{,2𝑡)◊ 𝑁}⁰^(𝑔λ^2𝑗+1^,𝐴λ^2𝑗^,2𝑡)

𝑁₀(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡) ) +𝑐 (𝑁₀(ℎλ_2𝑗, 𝐴λ_2𝑗, 𝑡) + 𝑁₀(𝑔λ_2𝑗+1, 𝐵λ_2𝑗+1, 𝑡))

+𝑑(𝑁₀(𝑔λ_2𝑗+1, 𝐴λ_2𝑗, 𝑡) + 𝑁₀(ℎλ_2𝑗, 𝐵λ_2𝑗+1, 𝑡)) }

= {

𝑎 (𝑁0(ℎλ_2𝑗, 𝑔λ_2𝑗+1, 𝑡)) + 𝑏 (^𝑁⁰^(ℎλ^2𝑗^,ℎλ^2𝑗+2^,2𝑡)

𝑁₀(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡)) +𝑐 (𝑁₀(ℎλ_2𝑗, 𝑔λ_2𝑗+1, 𝑡) + 𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡))

+𝑑(𝑁₀(ℎλ_2𝑗, 𝐵λ_2𝑗+2, 𝑡)) }

(6)

By Definition 2 (iii & vii), 𝑀₀(ℎλ_2𝑗, ℎλ_2𝑗+2, 2𝑡) ≥ 𝑀₀(ℎλ_2𝑗, 𝑔λ_2𝑗+1, 𝑡) ∗ 𝑀₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡) and 𝑁₀(ℎλ_2𝑗, ℎλ_2𝑗+2, 2𝑡) ≤ 𝑁₀(ℎλ_2𝑗, 𝑔λ_2𝑗+1, 𝑡) ◊ 𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡) for t ≫θ. One writes

(5)

2326-9865

1

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡),− 1 ≤ {

𝑎 (_𝑀 ¹

0(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡)− 1) +𝑏 ( ^𝑀⁰^(ℎλ^2𝑗^,𝑔λ^2𝑗+1^,𝑡)

𝑀₀(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡)∗𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1)

+𝑐 ( ¹

𝑀₀(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡)− 1 + ¹

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1)

+𝑑 ( ¹

𝑀₀(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡)− 1 + ¹

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1) }

and

𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡) ≤ {

𝑎 (𝑁₀(ℎλ_2𝑗, 𝑔λ_2𝑗+1, 𝑡)) +𝑏 (^𝑁⁰^(ℎλ^2𝑗^,𝑔λ^2𝑗+1^{,𝑡) ◊ 𝑁}⁰^(𝑔λ^2𝑗+1^,ℎλ^2𝑗+2^,𝑡)

𝑁₀(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡) ) +𝑐 (𝑁₀(ℎλ_2𝑗, 𝑔λ_2𝑗+1, 𝑡) + 𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡)) +𝑑 (𝑁₀(ℎλ_2𝑗, 𝑔λ_2𝑗+1, 𝑡) + 𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡))}

(7) This implies that

1

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1 ≤ 𝛽 ( ¹

𝑀₀(ℎλ_2𝑗,𝑔λ_2𝑗+1,𝑡)− 1) For t ≫θ and

𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡) ≤ 𝛽 (𝑁₀(ℎλ_2𝑗, 𝑔λ_2𝑗+1, 𝑡)) (8)

Where γ = a + c + d/1 − b − c − d < 1 since (a + b + 2c + 2 d) < 1.

Similarly,

1

𝑀₀(ℎλ_2𝑗+1,𝑔λ_2𝑗+3,𝑡)− 1 = ¹

𝑀₀(𝐴λ_2𝑗+2,𝐵λ_2𝑗+1,𝑡)− 1

≤

{

𝑎 ( 1

𝑀₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡)− 1) + 𝑏 ( 𝑀₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡)

𝑀₀(ℎλ_2𝑗+2, 𝐵λ_2𝑗+1, 2𝑡) ∗ 𝑀₀(𝑔λ_2𝑗+1, 𝐴λ_2𝑗+2, 2𝑡)− 1)

+𝑐 ( 1

𝑀₀(ℎλ_2𝑗+2, 𝐴λ_2𝑗+2, 𝑡)− 1 + 1

𝑀₀(𝑔λ_2𝑗+1, 𝐵λ_2𝑗+1, 𝑡)− 1)

+𝑑 ( 1

𝑀₀(𝑔λ_2𝑗+1, 𝐴λ_2𝑗+2, 𝑡)− 1 + 1

𝑀₀(ℎλ_2𝑗+2, 𝐵λ_2𝑗+1, 𝑡)− 1)

}

= {

𝑎 (_𝑀 ¹

0(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1) + 𝑏 (^𝑀⁰^(𝑔λ^2𝑗+1^,ℎλ^2𝑗+2^,𝑡)

𝑀₀(𝑔λ_2𝑗+1,𝑔λ_2𝑗+3,2𝑡))

+𝑐 ( ¹

𝑀₀(ℎλ_2𝑗+2,𝑔λ_2𝑗+3,𝑡)− 1 + ¹

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1)

+𝑑 ( ¹

𝑀₀(𝑔λ_2𝑗+1,𝑔λ_2𝑗+3,𝑡)− 1)

}

and

1

𝑁₀(ℎλ_2𝑗+1,𝑔λ_2𝑗+3,𝑡)− 1 = ¹

𝑁₀(𝐴λ_2𝑗+2,𝐵λ_2𝑗+1,𝑡)− 1

(6)

2326-9865

≤ {

𝑎 (𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡)) + 𝑏 (𝑁₀(ℎλ_2𝑗+2, 𝐵λ_2𝑗+1, 2𝑡)◊𝑁₀(𝑔λ_2𝑗+1, 𝐴λ_2𝑗+2, 2𝑡) 𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡) ) +𝑐 (𝑁₀(ℎλ_2𝑗+2, 𝐴λ_2𝑗+2, 𝑡) + 𝑁₀(𝑔λ_2𝑗+1, 𝐵λ_2𝑗+1, 𝑡))

+𝑑(𝑁₀(𝑔λ_2𝑗+1, 𝐴λ_2𝑗+2, 𝑡) + 𝑁₀(ℎλ_2𝑗+2, 𝐵λ_2𝑗+1, 𝑡)) }

= {

𝑎 (𝑁0(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡)) + 𝑏 (^𝑁⁰^(𝑔λ^2𝑗+1^,𝑔λ^2𝑗+3^,2𝑡)

𝑁₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)) +𝑐 (𝑁₀(ℎλ_2𝑗+2, 𝑔λ_2𝑗+3, 𝑡) + 𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡))

+𝑑(𝑁₀(𝑔λ_2𝑗+1, 𝑔λ_2𝑗+3, 𝑡)) }

(9)

Again, By Definition 2 (iii & vii), 𝑀₀(𝑔λ_2𝑗+1, 𝑔λ_2𝑗+3, 2𝑡) ≥ 𝑀₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡)∗ 𝑀₀(ℎλ_2𝑗+2, 𝑔λ_2𝑗+3, 𝑡) and 𝑁₀(𝑔λ_2𝑗+1, 𝑔λ_2𝑗+3, 2𝑡) ≤ 𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡) ◊ 𝑁₀(ℎλ_2𝑗+2, 𝑔λ_2𝑗+3, 𝑡) for t ≫θ. we have

1

𝑀₀(ℎλ_2𝑗+1,𝑔λ_2𝑗+3,𝑡)− 1 ≤ {

𝑎 ( ¹

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1) +𝑏 ( ^𝑀⁰^(𝑔λ^2𝑗+1^,ℎλ^2𝑗+2^,𝑡)

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)∗𝑀₀(ℎλ_2𝑗+2,𝑔λ_2𝑗+3,𝑡)− 1)

+𝑐 ( ¹

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1 + ¹

𝑀₀(ℎλ_2𝑗+2,𝑔λ_2𝑗+3,𝑡)− 1)

+𝑑 ( ¹

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1 + ¹

𝑀₀(ℎλ_2𝑗+2,𝑔λ_2𝑗+3,𝑡)− 1) }

and

𝑁₀(ℎλ_2𝑗+1, 𝑔λ_2𝑗+3, 𝑡) ≤ {

𝑎 (𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡)) +𝑏 (^𝑁⁰^(𝑔λ^2𝑗+1^,ℎλ^2𝑗+2^{,𝑡)◊ 𝑁}⁰^(ℎλ^2𝑗+2^,𝑔λ^2𝑗+3^,𝑡)

𝑁₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡) ) +𝑐 (𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡) + 𝑁₀(ℎλ_2𝑗+2, 𝑔λ_2𝑗+3, 𝑡))

+𝑑(𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡) + 𝑁₀(ℎλ_2𝑗+2, 𝑔λ_2𝑗+3, 𝑡)) }

(10) This implies that,

1

𝑀₀(ℎλ_2𝑗+2,𝑔λ_2𝑗+3,𝑡)− 1 ≤ 𝛽 ( ¹

𝑀₀(𝑔λ_2𝑗+1,ℎλ_2𝑗+2,𝑡)− 1) For t ≫θ and 𝑁₀(ℎλ_2𝑗+2, 𝑔λ_2𝑗+3, 𝑡) ≤ 𝛽 (𝑁₀(𝑔λ_2𝑗+1, ℎλ_2𝑗+2, 𝑡))

(11)

where the value of 𝛽 is the same as in (8). Now, from (3), (8), (11), and by induction, we have

1

𝑀₀(ξ_2j+2,𝜉_2𝑗+3,𝑡)− 1 ≤ 𝛽 ( ¹

𝑀₀(𝜉_2𝑗+1,𝜉_2𝑗+2,𝑡)− 1)

≤ 𝛽²( ¹

𝑀₀(𝜉_2𝑗+1,𝜉_2𝑗+2,𝑡)− 1) ≤ ⋯ ≤ 𝛽^2𝑗+2( ¹

𝑀₀(𝜉₀,𝜉₁,𝑡)− 1) → 0. As 𝑗 →∞.

and 𝑁₀(ξ_2j+2, 𝜉_2𝑗+3, 𝑡) ≤ 𝛽 (𝑁₀(𝜉_2𝑗+1, 𝜉_2𝑗+2, 𝑡))

≤ 𝛽²(𝑁₀(𝜉_2𝑗+1, 𝜉_2𝑗+2, 𝑡)) ≤ ⋯ ≤ 𝛽^2𝑗+2(𝑁₀(𝜉₀, 𝜉₁, 𝑡)) → 0. As 𝑗 →∞. (12)

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2326-9865

Its prove that a sequence {ξ_𝑗}_𝑗≥0 is a IFC contractive, and we get that,

𝑗→∞lim𝑀₀(𝜉_𝑗, 𝜉_𝑗+1, 𝑡) = 1 and lim

𝑗→∞𝑁₀(𝜉_𝑗, 𝜉_𝑗+1, 𝑡) = 0 for t ≫θ (13)

Since 𝑀₀, 𝑁₀ is triangular, then ∀ k > j ≥ j₀,

1

𝑀₀(𝜉_𝑗,𝜉_𝑘,𝑡)− 1 ≤ ( ¹

𝑀₀(𝜉_𝑗,𝜉_𝑗+1,𝑡)− 1) + ( ¹

𝑀₀(𝜉_𝑗+1,𝜉_𝑗+2,𝑡)− 1) + ⋯ + ( ¹

𝑀₀(𝜉_𝑘−1,𝜉_𝑘,𝑡)− 1) ≤ (𝛽^𝑗+ 𝛽^𝑗+1+ ⋯ + 𝛽^𝑘−1) ( ¹

𝑀₀(𝜉₀,𝜉₁,𝑡)− 1) → 0. As 𝑗 →

∞ and

𝑁₀(𝜉_𝑗, 𝜉_𝑘, 𝑡) ≤ 𝑁₀(𝜉_𝑗, 𝜉_𝑗+1, 𝑡) + 𝑁₀(𝜉_𝑗+1, 𝜉_𝑗+2, 𝑡) + ⋯ + 𝑁₀(𝜉_𝑘−1, 𝜉_𝑘, 𝑡)

≤ (𝛽^𝑗 + 𝛽^𝑗+1+ ⋯ + 𝛽^𝑘−1)(𝑁₀(𝜉₀, 𝜉₁, 𝑡)) → 0. As 𝑗 →∞ (14)

Hence, proved that {ξ_𝑗} is a Cauchy sequence. Now by the completeness of IFCM space (𝑈, 𝑀₀, 𝑁₀,∗,◊) , ∃ξ ∈ U so that ξ_𝑗 ⟶ξ as j ⟶∞. Now for its subsequences, we have that gλ_2𝑗+1⟶ξ, hλ_2𝑗+2 ⟶ξ, Aλ_2𝑗⟶ξ, and Bλ_2𝑗+1⟶ ξ as j ⟶∞.

(15)

Since, a self-mapping h ∶ U ⟶ U is continuous, therefore

h(gλ_2𝑗+1) ⟶ hξ, h(hλ_2𝑗+2) ⟶ hξ, h(Aλ_2𝑗) ⟶ hξ, and ℎ(Bλ_2𝑗+1) ⟶ hξ as j ⟶∞.

(16)

By hypothesis (2), a (A, h) is compatible, therefore,

𝑗→∞lim𝑀₀(𝐴(ℎλ_2𝑗), ℎ(𝐴λ_2𝑗), 𝑡) = lim

𝑗→∞𝑀₀(𝐴(ℎλ_2𝑗), ℎξ , 𝑡) = 1, ⟹ lim

𝑗→∞𝑀₀(ℎ(𝐴λ_2𝑗), ℎξ , 𝑡) = 1 for t ≫θ and

𝑗→∞lim𝑁₀(𝐴(ℎλ_2𝑗), ℎ(𝐴λ_2𝑗), 𝑡) = lim

𝑗→∞𝑁₀(𝐴(ℎλ_2𝑗), ℎξ , 𝑡) = 0, ⟹ lim

𝑗→∞𝑁₀(ℎ(𝐴λ_2𝑗), ℎξ , 𝑡) = 0 for t ≫θ (17)

Next, we have to prove that hξ = ξ, then, by Definition 2 (iii & vii), 𝑀₀(ℎ𝜉, 𝜉, 2𝑡) ≥ 𝑀_𝑟(ℎ𝜉, 𝐴(ℎλ_2𝑗), 𝑡) ∗ 𝑀_𝑟(𝐴(ℎλ_2𝑗),ξ , 𝑡) for t ≫ θ and

𝑁₀(ℎ𝜉, 𝜉, 2𝑡) ≤ 𝑁_𝑟(ℎ𝜉, 𝐴(ℎλ_2𝑗), 𝑡)◊𝑁_𝑟(𝐴(ℎλ_2𝑗),ξ , 𝑡) for t ≫θ (18)

Since, a pair (A, h) is compatible, by using lim

𝑗→∞, and by the view of (15), (17), and (18), we have

𝑀₀(ℎ𝜉, 𝜉, 2𝑡) ≥ lim

𝑗→∞ 𝑀_𝑟(ℎ𝜉, 𝐴(ℎλ_2𝑗), 𝑡) ∗ lim

𝑗→∞ 𝑀_𝑟(𝐴(ℎλ_2𝑗),ξ , 𝑡) = 1 ∗ 1 = 1 for t ≫θ and 𝑁₀(ℎ𝜉, 𝜉, 2𝑡) ≤ lim

𝑗→∞ 𝑁_𝑟(ℎ𝜉, 𝐴(ℎλ_2𝑗), 𝑡)◊lim

𝑗→∞ 𝑁_𝑟(𝐴(ℎλ_2𝑗),ξ , 𝑡) = 0◊0 = 0 for t ≫θ (19)

Hence, M₀(hξ,ξ, 2t) = 1 ⇒ hξ = ξ, for t ≫θ. Now, we prove that Aξ = ξ, then again by Definition 2 (iii & vii),

𝑀₀(𝐴𝜉, 𝜉, 2𝑡) ≥ 𝑀_𝑟(𝐴𝜉, ℎ(𝐴λ_2𝑗), 𝑡) ∗ 𝑀_𝑟(ℎ(𝐴λ_2𝑗),ξ , 𝑡) for t ≫θ and

𝑁₀(𝐴𝜉, 𝜉, 2𝑡) ≤ 𝑁_𝑟(𝐴𝜉, ℎ(𝐴λ_2𝑗), 𝑡)◊𝑁_𝑟(ℎ(𝐴λ_2𝑗),ξ , 𝑡) for t ≫θ (20)

(8)

2326-9865

Vol. 71 No. 3s2 (2022) 763 http://philstat.org.ph Again by using lim

𝑗→∞, and by the view of (15), (17), and (20), we have 𝑀₀(𝐴𝜉, 𝜉, 2𝑡) ≥ lim

𝑗→∞ 𝑀_𝑟(𝐴𝜉, ℎ(𝐴λ_2𝑗), 𝑡) ∗ lim

𝑗→∞ 𝑀₀(ℎ(𝐴λ_2𝑗),ξ , 𝑡) = 1 ∗ 1 = 1 for t ≫θ and

𝑁₀(𝐴𝜉, 𝜉, 2𝑡) ≤ lim

𝑗→∞ 𝑁_𝑟(𝐴𝜉, ℎ(𝐴λ_2𝑗), 𝑡)◊lim

𝑗→∞ 𝑁₀(ℎ(𝐴λ_2𝑗),ξ , 𝑡) = 0◊0 = 0 for t ≫θ (21)

Hence, 𝑀₀(𝐴𝜉, 𝜉, 2𝑡) = 1 ⇒ Aξ = ξ, and 𝑁₀(𝐴𝜉, 𝜉, 2𝑡) = 0 ⇒ Aξ = ξ for t ≫θ. Thus, we get that Aξ = hξ = ξ. Next, we have to prove that Bξ = gξ. Now by hypothesis (1), i.e., A(U) ⊆ g(U), and there exists ρ ∈ U such that ξ = Aξ = gρ. Then, by view of (4), for t ≫θ,

1

𝑀₀(Bρ,gρ,𝑡)− 1 = ¹

𝑀₀(𝐴𝜉,Bρ,𝑡)− 1 ≤ {

𝑎 ( ¹

𝑀₀(ℎ𝜉,gρ,𝑡)− 1) + 𝑏 ( ^𝑀⁰^{(ℎ𝜉,gρ,𝑡)}

𝑀₀(ℎ𝜉 ,Bρ,2𝑡)∗𝑀₀(gρ,𝐴𝜉,2𝑡)− 1) +𝑐 ( ¹

𝑀₀(ℎ𝜉,𝐴𝜉,𝑡)− 1 + ¹

𝑀₀(gρ,Bρ,𝑡)− 1) + 𝑑 ( ¹

𝑀₀(gρ,𝐴𝜉,𝑡)− 1 + ¹

𝑀₀(ℎ𝜉,𝐵ρ,𝑡)− 1)}

= {

𝑎 ( 1

𝑀₀(ℎ𝜉, 𝜉, 𝑡)− 1) + 𝑏 ( 𝑀₀(ℎ𝜉, gρ, 𝑡)

𝑀₀(ℎ𝜉 , Bρ, 2𝑡) ∗ 𝑀₀(𝜉, 𝐴𝜉, 2𝑡)− 1)

+𝑐 ( 1

𝑀₀(ℎ𝜉, 𝜉, 𝑡)− 1 + 1

𝑀₀(gρ, Bρ, 𝑡)− 1) + 𝑑 ( 1

𝑀₀(𝜉, 𝐴𝜉, 𝑡)− 1 + 1

𝑀₀(ℎ𝜉, 𝐵ρ, 𝑡)− 1) } = {𝑏 (^𝑀⁰^{(ℎ𝜉,gρ,𝑡)}

𝑀₀(ℎ𝜉,Bρ,2𝑡)− 1) + 𝑐 ( ¹

𝑀₀(gρ,𝐵ρ,𝑡)− 1) + 𝑑 ( ¹

𝑀₀(gρ,𝐵ρ,𝑡)− 1)} and 𝑁₀(Bρ, gρ, 𝑡) = 𝑁₀(𝐴𝜉, Bρ, 𝑡)

≤ { 𝑎(𝑁₀(ℎ𝜉, gρ, 𝑡)) + 𝑏 (^𝑁⁰(ℎ𝜉 ,Bρ,2𝑡)∗𝑁₀(gρ,𝐴𝜉,2𝑡) 𝑁₀(ℎ𝜉,gρ,𝑡) )

+𝑐(𝑁₀(ℎ𝜉, 𝐴𝜉, 𝑡) + 𝑁₀(gρ, Bρ, 𝑡)) + 𝑑(𝑁₀(gρ, 𝐴𝜉, 𝑡) + 𝑁₀(ℎ𝜉, 𝐵ρ, 𝑡))} = { 𝑎(𝑁₀(ℎ𝜉, 𝜉, 𝑡)) + 𝑏 (^𝑁⁰(ℎ𝜉 ,Bρ,2𝑡)∗𝑁₀(𝜉,𝐴𝜉,2𝑡)

𝑁₀(ℎ𝜉,gρ,𝑡) )

+𝑐(𝑁₀(ℎ𝜉, 𝜉, 𝑡) + 𝑁₀(gρ, Bρ, 𝑡)) + 𝑑(𝑁₀(𝜉, 𝐴𝜉, 𝑡) + 𝑁₀(ℎ𝜉, 𝐵ρ, 𝑡))} = {𝑏 (^𝑁⁰^{(ℎ𝜉,Bρ,2𝑡)}

𝑁₀(ℎ𝜉,gρ,𝑡)) + 𝑐(𝑁₀(gρ, 𝐵ρ, 𝑡)) + 𝑑(𝑁₀(gρ, 𝐵ρ, 𝑡))}

(22)

Again , by Definition 2 (iii & vii), 𝑀₀(hξ, Bρ, 2t) ≥ 𝑀₀(hξ, gρ, t) ∗ 𝑀₀(gρ, Bρ, t) and 𝑁₀(hξ, Bρ, 2t) ≤ 𝑁₀(hξ, gρ, t) ◊𝑁₀(gρ, Bρ, t) for t ≫θ: It follows that

= ¹

𝑀₀(Bρ,gρ,𝑡)− 1 + 𝑏 ( ^𝑀⁰^{(ℎ𝜉,gρ,𝑡)}

𝑀₀(ℎ𝜉 ,gρ,2𝑡)∗𝑀₀(gρ,Bρ,2𝑡)− 1) + (𝑐 + 𝑑) ( ¹

𝑀₀(gρ,Bρ,𝑡)− 1)

= (𝑏 + 𝑐 + 𝑑) ( ¹

𝑀₀(gρ,Bρ,𝑡)− 1) For t ≫θ and

= 𝑁₀(Bρ, gρ, 𝑡) + 𝑏 (^𝑁⁰(ℎ𝜉 ,gρ,2𝑡)◊ 𝑁₀(gρ,Bρ,2𝑡)

𝑁₀(ℎ𝜉,gρ,𝑡) ) + (𝑐 + 𝑑)(𝑁₀(gρ, Bρ, 𝑡))

= (𝑏 + 𝑐 + 𝑑)(𝑁₀(gρ, Bρ, 𝑡))For t ≫θ (23)

Noticing that (b + c + d) < 1, therefore, 𝑀₀(Bρ, gρ, t) = 1 ⇒ Bρ = gρ and 𝑁₀(Bρ, gρ, t) = 0 ⇒ Bρ = gρ for t ≫θ, hence, Bρ = gρ = ξ.. Now by hypothesis (3), a pair (B, g) is weakly compatible, therefore,

gξ = g(Bρ) = B(gρ) = Bξ:

(24)

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2326-9865

Next, we have to prove that Bξ = ξ, then again by view of (4) and by using Definition 2 (iii

& v), for t ≫θ,

1

𝑀₀(B𝜉,𝜉,𝑡)− 1 = ¹

𝑀₀(B𝜉,A𝜉,𝑡)− 1

≤ {

𝑎 ( 1

𝑀₀(ℎ𝜉, g𝜉, 𝑡)− 1) + 𝑏 ( 𝑀₀(ℎ𝜉, g𝜉, 𝑡)

𝑀₀(ℎ𝜉 , B𝜉, 2𝑡) ∗ 𝑀₀(g𝜉, 𝐴𝜉, 2𝑡)− 1)

+𝑐 ( 1

𝑀_𝑟(ℎ𝜉, 𝐴𝜉, 𝑡)− 1 + 1

𝑀_𝑟(g𝜉, B𝜉, 𝑡)− 1) + 𝑑 ( 1

𝑀_𝑟(g𝜉, 𝐴𝜉, 𝑡)− 1 + 1

𝑀_𝑟(ℎ𝜉, 𝐵𝜉, 𝑡)− 1) } ≤ {

𝑎 ( ¹

𝑀₀(𝜉,B𝜉,𝑡)− 1) + 𝑏 ( ^𝑀⁰^{(ℎ𝜉,g𝜉,𝑡)}

𝑀₀(ℎ𝜉,g𝜉,𝑡)∗𝑀₀(𝑔𝜉 ,B𝜉,𝑡)∗𝑀₀(g𝜉,𝜉,𝑡)∗𝑀₀(𝜉,A𝜉,𝑡)− 1) +𝑐 ( ¹

𝑀_𝑟(ℎ𝜉,𝜉,𝑡)− 1 + ¹

𝑀_𝑟(g𝜉,B𝜉,𝑡)− 1) + 𝑑 ( ¹

𝑀_𝑟(B𝜉,𝜉,𝑡)− 1 + ¹

𝑀_𝑟(𝜉,𝐵𝜉,𝑡)− 1)} and

𝑁₀(B𝜉, 𝜉, 𝑡) = 𝑁₀(B𝜉, A𝜉, 𝑡)

≤ { 𝑎(𝑁₀(ℎ𝜉, g𝜉, 𝑡)) + 𝑏 (𝑁₀(ℎ𝜉 , B𝜉, 2𝑡)◊𝑁₀(g𝜉, 𝐴𝜉, 2𝑡) 𝑁₀(ℎ𝜉, g𝜉, 𝑡) )

+𝑐(𝑁_𝑟(ℎ𝜉, 𝐴𝜉, 𝑡) + 𝑁_𝑟(g𝜉, B𝜉, 𝑡)) + 𝑑(𝑁_𝑟(g𝜉, 𝐴𝜉, 𝑡) + 𝑁_𝑟(ℎ𝜉, 𝐵𝜉, 𝑡)) }

≤ { 𝑎(𝑁₀(𝜉, B𝜉, 𝑡)) + 𝑏 (^𝑁⁰(ℎ𝜉,g𝜉,𝑡)◊𝑁₀(𝑔𝜉 ,B𝜉,𝑡)◊𝑁₀(g𝜉,𝜉,𝑡)◊𝑁₀(𝜉,A𝜉,𝑡)

𝑀₀(ℎ𝜉,g𝜉,𝑡) )

+𝑐(𝑁_𝑟(ℎ𝜉, 𝜉, 𝑡) + 𝑁_𝑟(g𝜉, B𝜉, 𝑡)) + 𝑑(𝑁_𝑟(B𝜉, 𝜉, 𝑡) + 𝑁_𝑟(𝜉, 𝐵𝜉, 𝑡))

} (25)

After simplification, we obtain

1

𝑀_𝑟(B𝜉,𝜉,𝑡)− 1 ≤ (𝑎 + 𝑏 + 2𝑑) ( ¹

𝑀_𝑟(B𝜉,𝜉,𝑡)− 1) ⟹ (1 − 𝑎 − 𝑏 − 2𝑑) ( ¹

𝑀_𝑟(B𝜉,𝜉,𝑡)− 1) ≤ 0.for t ≫θ and

𝑁_𝑟(B𝜉, 𝜉, 𝑡) ≤ (𝑎 + 𝑏 + 2𝑑)(𝑁_𝑟(B𝜉, 𝜉, 𝑡)) ⟹ (1 − 𝑎 − 𝑏 − 2𝑑)(𝑁_𝑟(B𝜉, 𝜉, 𝑡)) ≤ 0.for t ≫

θ, (26) Since (1 − a − b − 2d) ≠ 0, therefore, 𝑀_𝑟(Bξ,ξ, t) = 1 ⇒ Bξ = ξ and 𝑁_𝑟(Bξ,ξ, t) =

0 ⇒ Bξ = ξ for t ≫θ, which further implies that gξ = ξ. Hence, proved that hξ = gξ = Aξ = Bξ = ξ, that is, ξ is the CFP of the mappings 𝐴, 𝐵, 𝑔 and ℎ.

Uniqueness:

let η ∈ U be the other CFP of the mappings A, B, g and h in U such that hη = gη = Aη = Bη = η. Then by view of (4) and by using Definition 2 (iii & v), for

t ≫θ,

1

𝑀_𝑟(𝜉,η,𝑡)− 1 = ¹

𝑀_𝑟(A𝜉,Bη,𝑡)− 1

≤ {

𝑎 ( 1

𝑀₀(ℎ𝜉, gη, 𝑡)− 1) + 𝑏 ( 𝑀₀(ℎ𝜉, gη, 𝑡)

𝑀₀(ℎ𝜉 , Bη, 2𝑡) ∗ 𝑀₀(gη, 𝐴𝜉, 2𝑡)− 1)

+𝑐 ( 1

𝑀_𝑟(ℎ𝜉, 𝐴𝜉, 𝑡)− 1 + 1

𝑀_𝑟(gη, Bη, 𝑡)− 1) + 𝑑 ( 1

𝑀_𝑟(gη, 𝐴𝜉, 𝑡)− 1 + 1

𝑀_𝑟(ℎ𝜉, 𝐵η, 𝑡)− 1) } ≤ {

𝑎 ( ¹

𝑀₀(𝜉,η,𝑡)− 1) + 𝑏 ( ^𝑀⁰^{(𝜉,η,𝑡)}

𝑀₀(𝜉,𝜉,𝑡)∗𝑀₀(𝜉 ,η,𝑡)∗𝑀₀(η,η,𝑡)∗𝑀₀(η,𝜉,𝑡)− 1) +2𝑑 ( ¹

𝑀_𝑟(η,𝜉,𝑡)− 1) } and

𝑁_𝑟(𝜉,η, 𝑡) = 𝑁_𝑟(A𝜉, Bη, 𝑡)

(10)

2326-9865

≤ { 𝑎(𝑁₀(ℎ𝜉, gη, 𝑡)) + 𝑏 (𝑁₀(ℎ𝜉 , Bη, 2𝑡)◊𝑁₀(gη, 𝐴𝜉, 2𝑡) 𝑁₀(ℎ𝜉, gη, 𝑡) )

+𝑐(𝑁_𝑟(ℎ𝜉, 𝐴𝜉, 𝑡) + 𝑁_𝑟(gη, Bη, 𝑡)) + 𝑑(𝑁_𝑟(gη, 𝐴𝜉, 𝑡) + 𝑁_𝑟(ℎ𝜉, 𝐵η, 𝑡)) }

≤ {𝑎(𝑁₀(𝜉,η, 𝑡)) + 𝑏 (^𝑁⁰^{(𝜉,𝜉,𝑡)◊𝑁}⁰^{(𝜉 ,η,𝑡)◊𝑁}_𝑁 ⁰^{(η,η,𝑡)◊𝑁}⁰^{(η,𝜉,𝑡)}

0(𝜉,η,𝑡) )

+2𝑑(𝑁_𝑟(η, 𝜉, 𝑡)) }

(27)

After simplification, we obtain

1

𝑀_𝑟(𝜉,η,𝑡)− 1 ≤ (𝑎 + 𝑏 + 2𝑑) ( ¹

𝑀_𝑟(𝜉,η,𝑡)− 1) ⟹ (1 − 𝑎 − 𝑏 − 2𝑑) ( ¹

𝑀_𝑟(𝜉,η,𝑡)− 1) ≤ 0. for t ≫θ, and

𝑁_𝑟(𝜉,η, 𝑡) ≤ (𝑎 + 𝑏 + 2𝑑)(𝑁_𝑟(𝜉,η, 𝑡)) ⟹ (1 − 𝑎 − 𝑏 − 2𝑑)(𝑁_𝑟(𝜉,η, 𝑡)) ≤ 0. for t ≫θ, (28)

Since (1 − a − b − 2d) ≠ 0, therefore, 𝑀_𝑟(ξ,η, t) = 1 ⇒ ξ = η and 𝑁_𝑟(ξ,η, t) = 0 ⇒ ξ = η for t ≫ θ. This completes the proof.

Corollary 11.

1

𝑀₀(𝐴𝜆,𝐵𝜇,𝑡)− 1 ≤ {

𝑎 ( ¹

𝑀₀(ℎ𝜆,𝑔𝜇,𝑡)− 1) + 𝑏 ( ^𝑀⁰^{(ℎ𝜆,𝑔𝜇,𝑡)}

𝑀₀(ℎ𝜆,𝐵𝜇,2𝑡)∗𝑀₀(𝑔𝜇,𝐴𝜆,2𝑡)− 1) +𝑐 ( ¹

𝑀_𝑟(ℎ𝜆,𝐴𝜆,𝑡)− 1 + ¹

𝑀_𝑟(𝑔𝜇,𝐵𝜇,𝑡)− 1) } and 𝑁₀(𝐴𝜆, 𝐵𝜇, 𝑡) ≤ {𝑎(𝑁₀(ℎ𝜆, 𝑔𝜇, 𝑡)) + 𝑏 (^𝑁⁰(ℎ𝜆,𝐵𝜇,2𝑡)◊𝑁₀(𝑔𝜇,𝐴𝜆,2𝑡)

𝑁₀(ℎ𝜆,𝑔𝜇,𝑡) )

+𝑐(𝑁_𝑟(ℎ𝜆, 𝐴𝜆, 𝑡) + 𝑁_𝑟(𝑔𝜇, 𝐵𝜇, 𝑡)) } (29)

∀ λ,μ ∈ U, t ≫ θ, and 0 ≤ a, b, c < 1 with (a + b + 2c) < 1. If A(U) ⊆ g(U), B(U) ⊆ h(U) and consider that

(1) h is a continuous self-mapping (2) A pair (A, h) is compatible, and (3) A pair (B, g) is weakly-compatible

Then, the mappings A, B, g, and h have a unique CFP in U.

Corollary 12.

1

𝑀₀(𝐴𝜆,𝐵𝜇,𝑡)− 1 ≤ {

𝑎 ( ¹

𝑀₀(ℎ𝜆,𝑔𝜇,𝑡)− 1) + 𝑏 ( ^𝑀⁰^{(ℎ𝜆,𝑔𝜇,𝑡)}

𝑀₀(ℎ𝜆,𝐵𝜇,2𝑡)∗𝑀₀(𝑔𝜇,𝐴𝜆,2𝑡)− 1) +𝑑 ( ¹

𝑀_𝑟(𝑔𝜇,𝐴𝜆,𝑡)− 1 + ¹

𝑀_𝑟(ℎ𝜆,𝐵𝜇,𝑡)− 1) } and 𝑁₀(𝐴𝜆, 𝐵𝜇, 𝑡) ≤ {𝑎(𝑁₀(ℎ𝜆, 𝑔𝜇, 𝑡)) + 𝑏 (^𝑁⁰(ℎ𝜆,𝐵𝜇,2𝑡)◊ 𝑁₀(𝑔𝜇,𝐴𝜆,2𝑡)

𝑁₀(ℎ𝜆,𝑔𝜇,𝑡) )

+𝑑(𝑁_𝑟(𝑔𝜇, 𝐴𝜆, 𝑡) + 𝑁_𝑟(ℎ𝜆, 𝐵𝜇, 𝑡)) } (30)

∀ λ,μ ∈ U, t ≫θ, and 0 ≤ a, b, c < 1 with (a + b + 2d) < 1. If A(U) ⊆ g(U), B(U) ⊆ h(U) and consider that

(1) h is a continuous self-mapping (2) A pair (A, h) is compatible, and

(11)

2326-9865

(3) A pair (B, g) is weakly-compatible

Corollary 13.

1

𝑀₀(𝐴𝜆,𝐵𝜇,𝑡)− 1 ≤ {

𝑎 ( ¹

𝑀₀(ℎ𝜆,𝑔𝜇,𝑡)− 1) + 𝑐 ( ¹

𝑀_𝑟(ℎ𝜆,𝐴𝜆,𝑡)− 1 + ¹

𝑀_𝑟(𝑔𝜇,𝐵𝜇,𝑡)− 1) +𝑑 ( ¹

𝑀_𝑟(𝑔𝜇,𝐴𝜆,𝑡)− 1 + ¹

𝑀_𝑟(ℎ𝜆,𝐵𝜇,𝑡)− 1) } and 𝑁₀(𝐴𝜆, 𝐵𝜇, 𝑡) ≤ {𝑎(𝑁₀(ℎ𝜆, 𝑔𝜇, 𝑡)) + 𝑐(𝑁_𝑟(ℎ𝜆, 𝐴𝜆, 𝑡) + 𝑁_𝑟(𝑔𝜇, 𝐵𝜇, 𝑡))

+𝑑(𝑁_𝑟(𝑔𝜇, 𝐴𝜆, 𝑡) + 𝑁_𝑟(ℎ𝜆, 𝐵𝜇, 𝑡)) } (31)

∀ λ,μ ∈ U, t ≫ θ, and 0 ≤ a, b, c < 1 with (a + 2c + 2d) < 1. If A(U) ⊆ g(U), B(U) ⊆ h(U) and consider that

(1) h is a continuous self-mapping (2) A pair (A, h) is compatible, and (3) A pair (B, g) is weakly-compatible

Example 14.

Assume that U = [0, ∞), ∗,◊ be a continuous t-norm, t-conorm and 𝑀₀, 𝑁₀ ∶ U × U × (0, ∞) ⟶ [0, 1] be written as

𝑀₀(𝜆, 𝜇, 𝑡) = ^𝑡

𝑡+|𝜆−𝜇|, 𝑁₀(𝜆, 𝜇, 𝑡) = ^{|𝜆−𝜇|}

𝑡+|𝜆−𝜇|, ∀𝜆, 𝜇 ∈ 𝑈, 𝑡 ≫ 𝜃:

(32)

Then, it is easy to verify that IFCM 𝑀₀, 𝑁₀ is triangular and (𝑈, 𝑀₀, 𝑁₀,∗,◊) is a complete IFCM space. Now, the mappings, A, g, h, B ∶ U ⟶ U, be defined by, (for all λ ∈ U);

𝐴( λ ) = 𝐵( λ ) = (

1 3(^{3 λ}

4 +¹

8) , 𝑖𝑓 λ ≠ 0

o, if λ = 0 (33)

And

ℎ( λ ) = 𝑔( λ ) = ((^{3 λ}

4 +¹

8) , 𝑖𝑓 λ ≠ 0

o, if λ = 0 (34)

Since, from the above equation, A(U) = B(U) and g(U) = h(U), so that we conclude that A(U) ⊆ g(U) or B(U) ⊆ h(U). Then

1

𝑀₀(ℎ𝜆,𝑔𝜇,𝑡)− 1 =^{|ℎ𝜆−𝑔𝜇|}

𝑡 =^{3|𝜆−𝜇|}

4𝑡 for 𝑡 ≫ 𝜃: and 𝑁₀(ℎ𝜆, 𝑔𝜇, 𝑡) =^{|ℎ𝜆−𝑔𝜇|}

𝑡 = ^{3|𝜆−𝜇|}

4𝑡 for 𝑡 ≫ 𝜃:

(35) And

1

𝑀₀(𝐴𝜆,𝐵𝜇,𝑡)− 1 =|𝐴(𝜆)−𝐵(𝜇)|

𝑡 =¹

3( ¹

𝑀₀(ℎ𝜆,𝑔𝜇,𝑡)− 1) for 𝑡 ≫ 𝜃: and 𝑁₀(𝐴𝜆, 𝐵𝜇, 𝑡) =|𝐴(𝜆)−𝐵(𝜇)|

𝑡 =¹

3(𝑁₀(ℎ𝜆, 𝑔𝜇, 𝑡)) for 𝑡 ≫ 𝜃:

(36)