New results on modified intuitionistic generalized fuzzy metric spaces by employing E.A property and common E.A property for coupled maps

Abstract

In this paper, we introduce the notions of E.A and common E.A on modified intuitionistic generalized fuzzy metric space. We utilize our new notions to introduce and formulate some common fixed point theorems on modified intuitionistic generalized fuzzy metric space for coupled maps.

Keywords

Coupled maps complete subspace property (E.A)fuzzy metric space (FM-space)common E.A property modified intuitionistic generalized fuzzy metric space (MIGFM-space)

1 Introduction

Initial credit of propounding and representing the pragmatic concept of fuzzy set goes to Zadeh [46]. His contribution proved to be a turning point in the progression and further advancement of Mathematics. This theory in FM-space is a hot range of mathematical research that has varied applications inside and in addition outside the mathematics. Several authors have presented FM-space in various notions. For example, George and Veeramani [14] changed the idea of a FM-space presented by Kramosil and Michalek [23]. Moreover, Sun and Yang [43] gave the idea of $G$ -fuzzy metric space.

Definition 1.2. [30] A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is a t-norm if it satisfies the following axioms for ι₁, ι₂, ι₃, ι₄ ∈ [0, 1]:

∗ is commutative and associative;

ι₁ ∗ 1 = ι₁;

∗ is continuous;

ι₁ ∗ ι₂ ≤ ι₃ ∗ ι₄, whenever ι₁ ≤ ι₃ and ι₂ ≤ ι₄.

Definition 1.2. [46] A fuzzy set ψ in Γ is characterized by its membership function Ω_ψ : Γ → [0, 1] .

Definition 1.3. [43] Let $G$ be a fuzzy set on Γ³ × (0, + ∞), where Γis non empty set. A 3- tuple $(Γ, G, *)$ is said to be $G$ - fuzzy metric space if it holds the following axioms:

For each μ, ϑ ∈ Γ and ℏ, s > 0

$G (μ, μ, ϑ, ℏ) > 0$ with μ ≠ ϑ;

$G (μ, μ, ϑ, ℏ) \geq G (μ, ϑ, ρ, t)$ for all μ, ϑ, ρ ∈ Γ with ϑ ≠ ρ;

$G (μ, ϑ, ρ, ℏ) = 1$ if and only if μ = ϑ = ρ;

$G (μ, ϑ, ρ, ℏ) = G (p (μ, ϑ, ρ), t))$ , where p is a permutation function;

$G (μ, a, a, ℏ) * G (a, ϑ, ρ, s) \leq G (μ, ϑ, ρ, ℏ + s);$

$G (μ, ϑ, ρ, \cdot) : (0, \infty) \to [0, 1]$ is continuous.

The idea of an intuitionistic fuzzy set was initiated by Atanassov [4]. Utilizing this theory and with the assistance of the continuous t-norm and co-norm, Alaca et al. [1] characterized the idea of intuitionistic FM-space. Saadati et al. [28] introduced the modified intuitionistic FM-space by introducing the idea of continuous $T$ - representable and demonstrated fixed point results for compatible and weakly compatible maps. The paper [28] is the motivation for a substantial number of eminent mathematicians to utilize the concept of modified intuitionistic fuzzy metric space (MIF-metric space) and its applications. For more related significant results, we refer to [2 , 16].

Lemma 1.4. [11] Consider the set $L^{*}$ and operation $\leq_{L^{*}}$ defined as $\begin{matrix} L^{*} = {(u, v) : (u, v) \in [0, 1]^{2} and u + v \leq 1}, \end{matrix}$ $(u, v) \leq_{L^{*}} (w, z)$ ⇔ u ≤ w and v ≥ z for every $(u, v), (w, z) \in L^{*} .$ Then $(L^{*}, \leq_{L^{*}})$ is a complete lattice.

Definition 1.5. [12] A triangular norm on $L^{*}$ is a mapping $T : (L^{*})^{2} \to L^{*}$ satisfying the following conditions for all $μ, ϑ, z, μ^{'}, ϑ^{'} \in L^{*}$ :

$T (μ, 1_{L^{*}}) = μ;$

$T (μ, ϑ) = T (ϑ, μ);$

$T (μ, T (ϑ, z)) = T (T (μ, ϑ), z);$

$μ \leq_{L^{*}} μ^{'}$ and $ϑ \leq_{L^{*}} ϑ^{'}$ ⇒ $T (μ, ϑ) \leq_{L^{*}} T (μ^{'}, ϑ^{'})$ .

Definition 1.6. [28] Let ψ and ζ be fuzzy sets such that ψ (μ, ϑ, ℏ) + ζ (μ, ϑ, ℏ) ≤1. The term $(Γ, Δ_{ψ, ζ}, T)$ is called a MIF-metric space if Γ ≠ φ, continuous ℏ- representable $T$ and Δ_ψ,ζ is a function $Γ \times Γ \times (0, \infty) \to L^{*}$ such that for all μ, ϑ, ρ ∈ Γ, ℏ, s > 0 it satisfying the following conditions:

$Δ_{ψ, ζ} (μ, ϑ, ℏ) >_{L^{*}} 0_{L^{*}}$ ;

$Δ_{ψ, ζ} (μ, ϑ, ℏ) = 1_{L^{*}} \Leftrightarrow μ = ϑ$ ;

Δ_ψ,ζ (μ, ϑ, ℏ) = Δ_ψ,ζ (ϑ, μ, ℏ);

$Δ_{ψ, ζ} (μ, ϑ, ℏ + s) \geq_{L^{*}} T (Δ_{ψ, ζ} (μ, ρ, ℏ), F_{ψ, ζ} (ρ, ϑ, s))$ ;

$Δ_{ψ, ζ} (μ, ϑ, \cdot) : (0, \infty) \to L^{*}$ is continuous.

Here, $\begin{matrix} Δ_{ψ, ζ} (μ, ϑ, ℏ) = (ψ (μ, ϑ, ℏ), ζ (μ, ϑ, ℏ)) . \end{matrix}$ With the help of previous literature, Gupta and Kanwar [20] presented the idea of modified intuitionistic generalized fuzzy metric space (say MIGFM-space) and discussed its fundamental properties.

Definition 1.7. [20] Let ψ and ζ be fuzzy sets defined on Γ × Γ × Γ × (0, ∞) such that ψ (μ, ϑ, ρ, ℏ) + ζ (μ, ϑ, ρ, ℏ) ≤1 . The triplet $(Γ, Δ_{ψ, ζ}, T)$ is a MIGFM-space if Γ ≠ φ, continuous ℏ- representable $T$ and Δ_ψ,ζ is a function $Γ \times Γ \times Γ \times (0, \infty) \to L^{*}$ such that for all μ, ϑ, ρ, ϖ ∈ Γ, ℏ , s > 0 it satisfying the following axioms:

$Δ_{ψ, ζ} (μ, ϑ, ρ, ℏ) >_{L^{*}} 0_{L^{*}}$ ;

$Δ_{ψ, ζ} (μ, ϑ, ρ, ℏ) = 1_{L^{*}} \Leftrightarrow μ = ϑ = ρ$ ;

$Δ_{ψ, ζ} (μ, ϑ, ρ, ℏ) \leq_{L^{*}} Δ_{ψ, ζ} (μ, μ, ρ, ℏ)$ ;

Δ_ψ,ζ (μ, ϑ, ρ, ℏ) = Δ_ψ,ζ (p (μ, ϑ, ρ) , ℏ), where p is permutation function;

$Δ_{ψ, ζ} (μ, ϑ, ρ, s + ℏ) \geq_{L^{*}} T (Δ_{ψ, ζ} (μ, ϖ, ϖ, s), Δ_{ψ, ζ} (ϖ, ϑ, ρ, ℏ))$ ;

$Δ_{ψ, ζ} (μ, ϑ, ρ, \cdot) : (0, \infty) \to L^{*}$ is continuous.

In this case, Δ_ψ,ζ is said to be an MIGF-metric and $\begin{matrix} Δ_{ψ, ζ} (μ, ϑ, ρ, ℏ) = (ψ (μ, ϑ, ρ, ℏ), ζ (μ, ϑ, ρ, ℏ)) . \end{matrix}$

Lemma 1.8. [20] Let $(Γ, Δ_{ψ, ζ}, T)$ be a MIGFM-space.

If $Δ_{ψ, ζ} (μ_{r}, μ_{r}, μ_{r + 1}, ℏ) \geq_{L^{*}} Δ_{ψ, ζ} (μ_{0}, μ_{0}, μ_{1}, k^{r} ℏ)$ for some k > 1, r ∈ N. Then {μ_r} is a Cauchy sequence.

Definition 1.9. [20] The mappings E : Γ × Γ → Γ and M : Γ → Γ are said to be compatible mappings defined on an MIGFM-space $(Γ, Δ_{ψ, ζ}, T)$ if $\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (ME (μ_{r}, ϑ_{r}), ME (μ_{r}, ϑ_{r}), \\ E (M μ_{r}, M ϑ_{r}), ℏ) = 1_{L^{*}}; \end{matrix}$ and $\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (ME (ϑ_{r}, μ_{r}), ME (ϑ_{r}, μ_{r}), \\ E (M ϑ_{r}, M μ_{r}), ℏ) = 1_{L^{*}}, \end{matrix}$ whenever {μ_r} and {ϑ_r} are sequences in Γ such that for all μ, ϑ ∈ Γ, ℏ>0, $\begin{matrix} lim_{r \to \infty} M (μ_{r}) = lim_{r \to \infty} E (μ_{r}, ϑ_{r}) = μ; \\ lim_{r \to \infty} M (ϑ_{r}) = lim_{r \to \infty} E (ϑ_{r}, μ_{r}) = ϑ . \end{matrix}$ Aamri and Moutawakil [5] and Liu et al. [24] respectively defined the property (E.A.) and common property (E.A.) and utilize the same to prove common fixed point theorems in metric spaces. Similar results are also proved by Imdad et al. [22] via common property (E.A). For more fixed point results, we refer to [7 , 31–45].

The idea of characterizing intuitionistic fuzzy set as generalized fuzzy set is genuinely intriguing and valuable in numerous application zones like sale analysis, new product marketing, financial services, negotiation process, psychological investigations and so forth. This work looks to feature the utilization of the idea of modified intuitionistic generalized fuzzy metric spaces for demonstrating some fixed point results. Initially, we define E.A property as well as common E.A property on modified intuitionistic generalized fuzzy metric spaces. These new outcomes will further help to comprehend the idea of fixed point theory on modified intuitionistic generalized fuzzy metric spaces.

2 Main results

Here, the notion of E.A and common E.A property on MIGFM-space is presented. Moreover, utilizing these properties, common fixed point theorems on MIGFM-space are proved.

Definition 2.1. The functions E : Γ × Γ → Γ and M : Γ → Γ are said to have E.A property on a MIGFM-space $(Γ, Δ_{ψ, ζ}, T)$ if there exist sequences {μ_r} and {ϑ_r} in Γ as $\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (E (μ_{r}, ϑ_{r}), E (μ_{r}, ϑ_{r}), ϖ, ℏ) \\ = 1_{L^{*}} = lim_{r \to \infty} Δ_{ψ, ζ} (M (μ_{r}), M (μ_{r}), ϖ, ℏ); \end{matrix}$ and $\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (E (ϑ_{r}, μ_{r}), E (ϑ_{r}, μ_{r}), κ, ℏ) \\ = 1_{L^{*}} = lim_{r \to \infty} Δ_{ψ, ζ} (M (ϑ_{r}), M (ϑ_{r}), κ, ℏ) \end{matrix}$ for some ϖ, κ ∈ Γ, ℏ>0.

Example 2.2. Let $(Γ, Δ_{ψ, ζ}, T)$ be a MIGFM-space with Γ = [-1, 1]. Define $\begin{matrix} Δ_{ψ, ζ} (β_{1}, β_{2}, β_{3}, ℏ) \\ = (\frac{ℏ}{ℏ + G (β_{1}, β_{2}, β_{3})}, \frac{G (β_{1}, β_{2}, β_{3})}{ℏ + G (β_{1}, β_{2}, β_{3})}), \end{matrix}$ where $G (β_{1}, β_{2}, β_{3})$ is $G$ -metric space defined as $G (β_{1}, β_{2}, β_{3}) = | β_{1} - β_{2} | + | β_{2} - β_{3} | + | β_{3} - β_{1} |$ for all β₁, β₂, β₃ ∈ Γ, ℏ>0 .

Suppose the functions E : Γ × Γ → Γ and M : Γ → Γ define as $E (μ, ϑ) = \frac{μ^{2}}{2} + \frac{ϑ^{2}}{2}$ and $M (μ) = \frac{μ}{2}$ . Consider the sequences {μ_r}= ${\frac{1}{2} + \frac{1}{r}}$ and {ϑ_r}= ${\frac{1}{2} - \frac{1}{r}}$ .

Then, $E (μ_{r}, ϑ_{r}) = \frac{1}{4} + \frac{1}{r^{2}} = E (ϑ_{r}, μ_{r})$ and $M (μ_{r}) = \frac{1}{4} + \frac{1}{2 r}$ , $M (ϑ_{r}) = \frac{1}{4} - \frac{1}{2 r} .$

As r→ ∞, the pair (E, M) satisfy E.A property on MIGFM-space $(Γ, Δ_{ψ, ζ}, T) .$

Definition 2.3. Let functions E, H : Γ × Γ → Γ and M, N : Γ → Γ be defined on a MIGFM-space $(Γ, Δ_{ψ, ζ}, T) .$ The pairs (E, M) and (N, H) in Γ are supposed to share common E.A property if there exist sequences {μ_r} and {ϑ_r} in Γ such that $\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (H (μ_{r}, ϑ_{r}), H (μ_{r}, ϑ_{r}), ϖ, ℏ) \\ = lim_{r \to \infty} Δ_{ψ, ζ} (N (μ_{r}), N (μ_{r}), ϖ, ℏ) = 1_{L^{*}} \\ = lim_{r \to \infty} Δ_{ψ, ζ} (E (μ_{r}, ϑ_{r}), E (μ_{r}, ϑ_{r}), ϖ, ℏ) \\ = lim_{r \to \infty} Δ_{ψ, ζ} (M (μ_{r}), M (μ_{r}), ϖ, ℏ) \end{matrix}$ and $\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (H (ϑ_{r}, μ_{r}), H (ϑ_{r}, μ_{r}), κ, ℏ) \\ = lim_{r \to \infty} Δ_{ψ, ζ} (N (ϑ_{r}), N (ϑ_{r}), κ, ℏ) = 1_{L^{*}} \\ = lim_{r \to \infty} Δ_{ψ, ζ} (E (ϑ_{r}, μ_{r}), E (ϑ_{r}, μ_{r}), κ, ℏ) \\ = lim_{r \to \infty} Δ_{ψ, ζ} (M (ϑ_{r}), M (ϑ_{r}), κ, ℏ), \end{matrix}$ for some ϖ, κ ∈ Γ, ℏ>0.

Now let Φ be the set of all continuous $Θ : L^{*} \to L^{*}$ and $Θ (ℏ) >_{L^{*}} ℏ$ for all $ℏ \in L^{*} / {0_{L^{*}}, 1_{L^{*}}}$ .

Theorem 2.4. Let $(Γ, Δ_{ψ, ζ}, T)$ be a MIGFM- space and the functions E, H : Γ × Γ → Γ and M, N : Γ → Γ are such that

E (Γ × Γ) ⊂ N (Γ) , H (Γ × Γ) ⊂ M (Γ);

for some k > 1 and ∀μ, ϑ ∈ Γ, ℏ>0: $\begin{matrix} Δ_{ψ, ζ} (E (μ, ϑ), E (μ, ϑ), H (μ, ϑ), ℏ) \geq_{L^{*}} \\ Θ {min (\begin{matrix} Δ_{ψ, ζ} (M μ, M μ, Nu, k ℏ), \\ Δ_{ψ, ζ} (E (μ, ϑ), E (μ, ϑ), M μ, k ℏ), \\ Δ_{ψ, ζ} (H (μ, ϑ), H (μ, ϑ), Nu, k ℏ), \\ Δ_{ψ, ζ} (E (μ, ϑ), E (μ, ϑ), Nu, k ℏ) \end{matrix})}, \end{matrix}$

where Θ ∈ Φ;

the pair (E, M) or (H, N) holds E.A property;

if one of E (Γ × Γ), H (Γ × Γ), N (Γ) and M (Γ) is a complete subspace of Γ.

Then the pairs (E, M) and (H, N) have coupled coincident point.

Moreover, weakly compatibility of the pairs (E, M) and (H, N) implies that the maps E, H, N and M have unique common fixed point in Γ.

Proof. Assume that the pair (H, N) holds E.A property, then there exist sequences {μ_r} and {ϑ_r} in Γ such that for some ϖ, κ ∈ Γ, ℏ>0: $\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (H (μ_{r}, ϑ_{r}), H (μ_{r}, ϑ_{r}), ϖ, ℏ) \\ = 1_{L^{*}} = lim_{r \to \infty} Δ_{ψ, ζ} (N (μ_{r}), N (μ_{r}), ϖ, ℏ); \end{matrix}$

$\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (H (ϑ_{r}, μ_{r}), H (ϑ_{r}, μ_{r}), κ, ℏ) \\ = 1_{L^{*}} = lim_{r \to \infty} Δ_{ψ, ζ} (N (ϑ_{r}), N (ϑ_{r}), κ, ℏ) . \end{matrix}$ (1)

From the condition (a), H (Γ × Γ) ⊂ M (μ) , therefore there exist sequences {ɛ_r} and {γ_r} in Γ such that

$H (μ_{r}, ϑ_{r}) = M (ɛ_{r}); H (ϑ_{r}, μ_{r}) = M (γ_{r}) .$ (2) From (refeq19) and (refeq20), and letting r→ ∞, we have

$lim_{r \to \infty} M (ɛ_{r}) = ϖ; lim_{r \to \infty} M (γ_{r}) = κ .$ (3) With the help of (b),

$\begin{matrix} Δ_{ψ, ζ} (E (ɛ_{r}, γ_{r}), E (ɛ_{r}, γ_{r}), H (μ_{r}, ϑ_{r}), ℏ) \geq_{L^{*}} \\ Θ {min (\begin{matrix} Δ_{ψ, ζ} (M ɛ_{r}, M ɛ_{r}, N μ_{r}, k ℏ), \\ Δ_{ψ, ζ} (E (ɛ_{r}, γ_{r}), E (ɛ_{r}, γ_{r}), M ɛ_{r}, k ℏ), \\ Δ_{ψ, ζ} (H (μ_{r}, ϑ_{r}), H (μ_{r}, ϑ_{r}), N μ_{r}, k ℏ), \\ Δ_{ψ, ζ} (E (ɛ_{r}, γ_{r}), E (ɛ_{r}, γ_{r}), N μ_{r}, k ℏ) \end{matrix})} . \end{matrix}$

By taking r→ ∞ and $Θ (ℏ) >_{L^{*}} ℏ$ , we get $\begin{matrix} Δ_{ψ, ζ} (E (ɛ_{r}, γ_{r}), E (ɛ_{r}, γ_{r}), ϖ, ℏ) \\ \geq_{L^{*}} Θ {min (Δ_{ψ, ζ} (E (ɛ_{r}, γ_{r}), E (ɛ_{r}, γ_{r}), ϖ, k ℏ))} \\ >_{L^{*}} Δ_{ψ, ζ} (E (ɛ_{r}, γ_{r}), E (ɛ_{r}, γ_{r}), ϖ, k ℏ) . \end{matrix}$ This gives, $\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (E (ɛ_{r}, γ_{r}), E (ɛ_{r}, γ_{r}), ϖ, ℏ) = 1_{L^{*}}; \\ lim_{r \to \infty} Δ_{ψ, ζ} (E (γ_{r}, ɛ_{r}), E (γ_{r}, ɛ_{r}), ϖ, ℏ) = 1_{L^{*}} . \end{matrix}$ (4) Let the completeness property of M (Γ) implies that there exist ℓ₁, ℓ ₂ ∈ Γ, as $lim_{r \to \infty} M (ɛ_{r}) = M (ℓ_{1}); lim_{r \to \infty} M (γ_{r}) = M (ℓ_{2}) .$ (5) From (b),

$\begin{matrix} Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), H (μ_{r}, ϑ_{r}), ℏ) \\ \geq_{L^{*}} Θ {min (\begin{matrix} Δ_{ψ, ζ} (M (ℓ_{1}), M (ℓ_{1}), N (μ_{r}), k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), M ℓ_{1}, k ℏ), \\ Δ_{ψ, ζ} (H (μ_{r}, ϑ_{r}), H (μ_{r}, ϑ_{r}), N μ_{r}, k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), N (μ_{r}), k ℏ) \end{matrix})} . \end{matrix}$

Taking r→ ∞ and using (3-5), we have $\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), M (ℓ_{1}), ℏ) = 1_{L^{*}}; \\ lim_{r \to \infty} Δ_{ψ, ζ} (E (ℓ_{2}, ℓ_{1}), E (ℓ_{2}, ℓ_{1}), M (ℓ_{2}), ℏ) = 1_{L^{*}} . \end{matrix}$ (6) The definition of weakly compatible functions (E, M) implies that

$\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (ME (ℓ_{1}, ℓ_{2}), ME (ℓ_{1}, ℓ_{2}), \\ E (M ℓ_{1}, M ℓ_{2}), ℏ) = 1_{L^{*}}; \\ lim_{r \to \infty} Δ_{ψ, ζ} (ME (ℓ_{2}, ℓ_{1}), ME (ℓ_{2}, ℓ_{1}), \\ E (M ℓ_{2}, M ℓ_{1}), ℏ) = 1_{L^{*}} . \end{matrix}$ (7) From (a), E (Γ × Γ) ⊂ N (Γ), therefore there exist ℓ₃, ℓ ₄ ∈ Γ such that $E (ℓ_{1}, ℓ_{2}) = N (ℓ_{3}); E (ℓ_{2}, ℓ_{1}) = N (ℓ_{4}),$ (8) and from condition (b), we have

$\begin{matrix} Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), H (ℓ_{3}, ℓ_{4}), ℏ) \geq_{L^{*}} \\ Θ (min (\begin{matrix} Δ_{ψ, ζ} (M (ℓ_{1}), M (ℓ_{1}), N ℓ_{3}, k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), M ℓ_{1}, k ℏ), \\ Δ_{ψ, ζ} (H (ℓ_{3}, ℓ_{4}), H (ℓ_{3}, ℓ_{4}), N ℓ_{3}, k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), N ℓ_{3}, k ℏ) \end{matrix})) . \end{matrix}$

Taking (refeq24), (refeq25) and (refeq26) and $Θ (ℏ) >_{L^{*}} ℏ$ , one can get $\begin{matrix} Δ_{ψ, ζ} (N ℓ_{3}, N ℓ_{3}, H (ℓ_{3}, ℓ_{4}), ℏ) = 1_{L^{*}}; \\ Δ_{ψ, ζ} (N ℓ_{4}, N ℓ_{4}, H (ℓ_{4}, ℓ_{3}), ℏ) = 1_{L^{*}} . \end{matrix}$ So,

$\begin{matrix} E (ℓ_{1}, ℓ_{2}) = M (ℓ_{1}) = N (ℓ_{3}) = H (ℓ_{3}, ℓ_{4}) = ν_{1}; \\ E (ℓ_{2}, ℓ_{1}) = M (ℓ_{2}) = N (ℓ_{4}) = H (ℓ_{4}, ℓ_{3}) = ν_{2} . \end{matrix}$ (9) This implies that,

$\begin{matrix} E (ν_{1}, ν_{2}) = M (ν_{1}); E (ν_{2}, ν_{1}) = M (ν_{2}) . \end{matrix}$ (10) The weak compatibility of (H, N) implies that $\begin{matrix} Δ_{ψ, ζ} (NH (ℓ_{3}, ℓ_{4}), NH (ℓ_{3}, ℓ_{4}), \\ E (M ℓ_{3}, M ℓ_{4}), ℏ) = 1_{L^{*}}; \end{matrix}$ and

$Δ_{ψ, ζ} (NH (ℓ_{4}, ℓ_{3}), NH (ℓ_{4}, ℓ_{3}), E (M ℓ_{4}, M ℓ_{3}), ℏ) = 1_{L^{*}} .$

This gives,

$N (ν_{1}) = H (ν_{1}, ν_{2}); N (ν_{2}) = H (ν_{2}, ν_{2}) .$ (11) With the help of (b) and (refeq29), $\begin{matrix} Δ_{ψ, ζ} (E (ν_{1}, ν_{2}), E (ν_{1}, ν_{2}), H (ℓ_{3}, ℓ_{4}), ℏ) \geq_{L^{*}} \\ Θ (min (\begin{matrix} Δ_{ψ, ζ} (M (ν_{1}), M (ν_{1}), N ℓ_{3}, k ℏ), \\ Δ_{ψ, ζ} (E (ν_{1}, ν_{2}), E (ν_{1}, ν_{2}), M ν_{1}, k ℏ), \\ Δ_{ψ, ζ} (H (ℓ_{3}, ℓ_{4}), H (ℓ_{3}, ℓ_{4}), N ℓ_{3}, k ℏ), \\ Δ_{ψ, ζ} (E (ν_{1}, ν_{2}), E (ν_{1}, ν_{2}), N ℓ_{3}), k ℏ) \end{matrix})) . \end{matrix}$ Hence, $\begin{matrix} E (ν_{1}, ν_{2}) = ν_{1} = M (ν_{1}); \\ E (ν_{2}, ν_{1}) = ν_{2} = M (ν_{2}) . \end{matrix}$ (12) From condition (b), $\begin{matrix} Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), H (ν_{1}, ν_{2}), ℏ) \geq_{L^{*}} \\ Θ (min (\begin{matrix} Δ_{ψ, ζ} (M ℓ_{1}, M ℓ_{1}, N ν_{1}, k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), M ℓ_{1}, k ℏ), \\ Δ_{ψ, ζ} (H (ν_{1}, ν_{2}), H (ν_{1}, ν_{2}), N ν_{1}, k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{1}, ℓ_{2}), E (ℓ_{1}, ℓ_{2}), N ν_{1}, k ℏ) \end{matrix})) . \end{matrix}$ This gives,

$H (ν_{1}, ν_{2}) = ν_{1} = N ν_{1}; H (ν_{2}, ν_{1}) = ν_{2} = N ν_{2} .$ (13) Thus (refeq30) and (refeq31) shows that (E, M) and (H, N) have common coupled fixed point.

By using (2) and $Θ (ℏ) >_{L^{*}} ℏ$ , we have

$\begin{matrix} Δ_{ψ, ζ} (ν_{1}, ν_{1}, ν_{2}, ℏ) = \\ Δ_{ψ, ζ} (E (ν_{1}, ν_{2}), E (ν_{1}, ν_{2}), H (ν_{2}, ν_{1}), ℏ) \geq_{L^{*}} \\ Θ (min (\begin{matrix} Δ_{ψ, ζ} (M ν_{1}, M ν_{1}, N ν_{2}, k ℏ), \\ Δ_{ψ, ζ} (E (ν_{1}, ν_{2}), E (ν_{1}, ν_{2}), M ν_{1}, k ℏ) \\ Δ_{ψ, ζ} (H (ν_{2}, ν_{1}), H (ν_{2}, ν_{1}), N ν_{2}, k ℏ), \\ Δ_{ψ, ζ} (E (ν_{1}, ν_{2}), E (ν_{1}, ν_{2}), N ν_{2}, k ℏ) \end{matrix})) >_{L^{*}} \\ Θ (Δ_{ψ, ζ} (ν_{1}, ν_{1}, ν_{2}, k ℏ)) >_{L^{*}} \\ Δ_{ψ, ζ} (ν_{1}, ν_{1}, ν_{2}, k ℏ) . \end{matrix}$

Thus ν₁ = ν₂. Therefore E, H, N and M have a common fixed point in Γ. Moreover, the uniqueness of fixed point can proved with the help of condition (b).

Theorem 2.5. Let $(Γ, Δ_{ψ, ζ}, T)$ be a MIGFM- space and functions E, H : Γ × Γ → Γ and M, N : Γ → Γ are such that for all μ, ϑ ∈ Γ, some k > 1 and ℏ>0:

$\begin{matrix} Δ_{ψ, ζ} (E (μ, ϑ), E (μ, ϑ), H (μ, ϑ), ℏ) \geq_{L^{*}} \\ Θ {min (\begin{matrix} Δ_{ψ, ζ} (M μ, M μ, N μ, k ℏ), \\ Δ_{ψ, ζ} (E (μ, ϑ), E (μ, ϑ), M μ, k ℏ), \\ Δ_{ψ, ζ} (H (μ, ϑ), H (μ, ϑ), N μ, k ℏ), \\ Δ_{ψ, ζ} (E (μ, ϑ), E (μ, ϑ), N μ, k ℏ) \end{matrix})}, \end{matrix}$

where Θ ∈ Φ;

the pairs (E, M) and (H, N) share common E.A property;

M (Γ) and N (Γ) are closed subset of Γ.

Then the pairs (E, M) and (H, N) have coupled coincident point.

Moreover, weakly compatibility of the pairs (E, M) and (H, N) implies that the maps E, H, N and M have unique common fixed point in Γ.

Proof. Assume that the pairs (E, M) and (H, N) share the common E.A property, then there exist sequences {μ_r} and {ϑ_r} in Γ such that for some ϖ, κ ∈ Γ, ℏ>0,

$\begin{matrix} lim_{r \to \infty} Δ_{ψ, ζ} (H (μ_{r}, ϑ_{r}), H (μ_{r}, ϑ_{r}), ϖ, ℏ) = 1_{L^{*}} \\ = lim_{r \to \infty} Δ_{ψ, ζ} (N (μ_{r}), N (μ_{r}), ϖ, ℏ) \\ = lim_{r \to \infty} Δ_{ψ, ζ} (E (μ_{r}, ϑ_{r}), ϖ, ℏ) \\ = lim_{r \to \infty} Δ_{ψ, ζ} (M (μ_{r}), M (ϑ_{r}), ϖ, ℏ); \\ lim_{r \to \infty} Δ_{ψ, ζ} (H (ϑ_{r}, μ_{r}), H (ϑ_{r}, ϑ_{r}), ϖ, ℏ) = 1_{L^{*}} \\ = lim_{r \to \infty} Δ_{ψ, ζ} (N (ϑ_{r}), N (ϑ_{r}), κ, ℏ) \\ = lim_{r \to \infty} Δ_{ψ, ζ} (E (ϑ_{r}, μ_{r}), κ, ℏ) \\ = lim_{r \to \infty} Δ_{ψ, ζ} (M (ϑ_{r}), M (ϑ_{r}), ϖ, ℏ) . \end{matrix}$ (14) Due to the closed subset N (Γ) of Γ, there exist ℓ₁, ℓ ₂ ∈ Γ such that

$N (ℓ_{1}) = ϖ; N (ℓ_{2}) = κ .$ (15) By using (i), $\begin{matrix} Δ_{ψ, ζ} (E (μ_{r}, ϑ_{r}), E (μ_{r}, ϑ_{r}), H (ℓ_{1}, ℓ_{2}), ℏ) \geq_{L^{*}} \\ Θ (min (\begin{matrix} Δ_{ψ, ζ} (M μ_{r}, M μ_{r}, M ℓ_{1}, k ℏ), \\ Δ_{ψ, ζ} (E (μ_{r}, ϑ_{r}), E (μ_{r}, ϑ_{r}), M μ_{r}, k ℏ), \\ Δ_{ψ, ζ} (H (ℓ_{1}, ℓ_{2}), H (ℓ_{1}, ℓ_{2}), N ℓ_{1}, k ℏ), \\ Δ_{ψ, ζ} (E (μ_{r}, ϑ_{r}), E (μ_{r}, ϑ_{r}), N ℓ_{1}, k ℏ) \end{matrix})) . \end{matrix}$ Letting r→ ∞ and using (refeq32) and (refeq33), we have $N (ℓ_{1}) = ϖ = H (ℓ_{1}, ℓ_{2}); N (ℓ_{2}) = κ = H (ℓ_{2}, ℓ_{1}) .$ (16) Using closed subset property for M (Γ) of Γ, there exist ℓ₃, ℓ ₄ ∈ Γ such that $M (ℓ_{3}) = ϖ; M (ℓ_{4}) = κ .$ (17) By using condition (i), we get $\begin{matrix} Δ_{ψ, ζ} (E (ℓ_{3}, ℓ_{4}), E (ℓ_{3}, ℓ_{4}), H (μ_{r}, ϑ_{r}), ℏ) \geq_{L^{*}} \\ Θ (min (\begin{matrix} Δ_{ψ, ζ} (M ℓ_{3}, M ℓ_{3}, N μ_{r}, k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{3}, ℓ_{4}) E (ℓ_{3}, ℓ_{4}), M ℓ_{3}, k ℏ), \\ Δ_{ψ, ζ} (H (μ_{r}, ϑ_{r}) H (μ_{r}, ϑ_{r}), N μ_{r}, k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{3}, ℓ_{4}), E (ℓ_{3}, ℓ_{4}), N μ_{r}, k ℏ) \end{matrix})) . \end{matrix}$ As r→ ∞ and using (refeq32) - (refeq34), we have $\begin{matrix} E (ℓ_{3}, ℓ_{4}) = ϖ = M (ℓ_{3}); E (ℓ_{4}, ℓ_{3}) = κ = M (ℓ_{4}) . \end{matrix}$ Since the pair (H, N) are weakly compatible, then one can have

$\begin{matrix} Δ_{ψ, ζ} (NH (ℓ_{1}, ℓ_{2}), NH (ℓ_{1}, ℓ_{2}), E (M ℓ_{1}, M ℓ_{2}), ℏ) = 1_{L^{*}}; \\ Δ_{ψ, ζ} (NH (ℓ_{2}, ℓ_{1}), NH (ℓ_{2}, ℓ_{1}), E (M ℓ_{2}, M ℓ_{1}), ℏ) = 1_{L^{*}} . \end{matrix}$

From (i), we have $\begin{matrix} Δ_{ψ, ζ} (E (ℓ_{3}, ℓ_{4}), E (ℓ_{3}, ℓ_{4}), H (ϖ, κ), ℏ) \geq_{L^{*}} \\ Θ (min (\begin{matrix} Δ_{ψ, ζ} (M ℓ_{3}, M ℓ_{3}, N ϖ, k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{3}, ℓ_{4}) E (ℓ_{3}, ℓ_{4}), M ℓ_{3}, k ℏ), \\ Δ_{ψ, ζ} (H (ϖ, κ) H (ϖ, κ), N ϖ, k ℏ), \\ Δ_{ψ, ζ} (E (ℓ_{3}, ℓ_{4}), E (ℓ_{3}, ℓ_{4}), N ϖ, k ℏ) \end{matrix})) . \end{matrix}$ This implies, $\begin{matrix} H (ϖ, κ) = ϖ = N (ϖ); H (κ, ϖ) = κ = N (κ) . \end{matrix}$ (18) The notion of weakly compatibly mapping (E, M) implies $\begin{matrix} Δ_{ψ, ζ} (ME (ℓ_{3}, ℓ_{4}), ME (ℓ_{3}, ℓ_{4}), E (M ℓ_{3}, M ℓ_{4}), ℏ) = 1_{L^{*}}; \\ Δ_{ψ, ζ} (ME (ℓ_{4}, ℓ_{3}), ME (ℓ_{4}, ℓ_{3}), E (M ℓ_{4}, M ℓ_{3}), ℏ) = 1_{L^{*}} . \end{matrix}$

By using (i), we have

$\begin{matrix} Δ_{ψ, ζ} (E (ϖ, κ), E (ϖ, κ), H (ℓ_{1}, ℓ_{2}), ℏ) \\ \geq_{L^{*}} Θ (min (\begin{matrix} Δ_{ψ, ζ} (M ϖ, M ϖ, N ℓ_{1}, k ℏ) \\ Δ_{ψ, ζ} (E (ϖ, κ), E (ϖ, κ), M ϖ, k ℏ), \\ Δ_{ψ, ζ} (H (ℓ_{1}, ℓ_{2}), H (ℓ_{1}, ℓ_{2}), N ℓ_{1}, k ℏ), \\ Δ_{ψ, ζ} (E (ϖ, κ), E (ϖ, κ), N ℓ_{1}, k ℏ) \end{matrix})) . \end{matrix}$

This gives,

$E (ϖ, κ) = ϖ = M (ϖ); E (κ, ϖ) = κ = M (κ) .$ (19) Equations (refeq36) and (refeq37) show that (E, M) and (H, N) have a common coupled fixed point.

Again, $\begin{matrix} Δ_{ψ, ζ} (ϖ, ϖ, κ, ℏ) \\ = Δ_{ψ, ζ} (E (ϖ, κ), E (ϖ, κ), H (κ, ϖ), ℏ) \geq_{L^{*}} \\ Θ (min (\begin{matrix} Δ_{ψ, ζ} (M ϖ, M ϖ, N κ, k ℏ), \\ Δ_{ψ, ζ} (E (ϖ, κ), E (ϖ, κ), M ϖ, k ℏ), \\ Δ_{ψ, ζ} (H (κ, ϖ), H (κ, ϖ), N κ, k ℏ) \\ Δ_{ψ, ζ} (E (ϖ, κ), E (ϖ, κ), N κ, ℏ) \end{matrix})) \\ \geq_{L^{*}} Θ (Δ_{ψ, ζ} (ϖ, ϖ, κ, k ℏ)) \\ >_{L^{*}} Δ_{ψ, ζ} (ϖ, ϖ, κ, k ℏ) . \end{matrix}$ This shows, E, H, N and M have a common fixed point. The uniqueness of the fixed point can proved with help of condition (i).

Example 2.6. Let $(Γ, Δ_{ψ, ζ}, T)$ be a MIGFM-space with Γ = [0, 1] and $T (a, b) = (a_{1} b_{1}, \min (a_{2} + b_{2}, 1)$ for all a = (a₁, a₂) and $b = (b_{1}, b_{2}) \in L^{*}$ $\begin{matrix} Δ_{ψ, ζ} (β_{1}, β_{2}, β_{3}, ℏ) = \\ (\frac{ℏ}{ℏ + G (β_{1}, β_{2}, β_{3})}, \frac{G (β_{1}, β_{2}, β_{3})}{ℏ + G (β_{1}, β_{2}, β_{3})}), \end{matrix}$ where $G (β_{1}, β_{2}, β_{3})$ is $G$ -metric space defined as $\begin{matrix} G (β_{1}, β_{2}, β_{3}) = | β_{1} - β_{2} | + | β_{2} - β_{3} | + | β_{3} - β_{1} | \end{matrix}$ for all β₁, β₂, β₃ ∈ Γ, ℏ>0 and $Θ (t_{1}, t_{2}) = (\sqrt{t_{1}}, 0)$ for every $t = (t_{1}, t_{2}) \in L^{*}$ .

Suppose the functions E, H : Γ × Γ → Γ and M, N : Γ → Γ are defined as $E (μ, ϑ) = \frac{μ + ϑ}{24}, H (μ, ϑ) = \frac{μ^{2}}{12} + \frac{ϑ^{2}}{12}$ and $M (μ) = μ^{2}, N (μ) = \frac{μ}{2} .$

Consider the sequences {μ_r}= ${\frac{1}{r}}$ and {ϑ_r}= ${\frac{1}{\sqrt{r}}}$ .

Clearly, the pair (H, N) satisfy E.A property. All the conditions (a-d) of Theorem 2.4 are satisfied. Hence, 0 is the unique common fixed point of E, H, M and N.

Example 2.7. Let $(Γ, Δ_{ψ, ζ}, T)$ be a MIGFM-space with Γ = [2, 15]. Define $\begin{matrix} Δ_{ψ, ζ} (β_{1}, β_{2}, β_{3}, ℏ) = \\ (\frac{ℏ}{ℏ + G (β_{1}, β_{2}, β_{3})}, \frac{G (β_{1}, β_{2}, β_{3})}{ℏ + G (β_{1}, β_{2}, β_{3})}), \end{matrix}$ where $G (β_{1}, β_{2}, β_{3})$ is $G$ -metric space defined as $\begin{matrix} G (β_{1}, β_{2}, β_{3}) = | β_{1} - β_{2} | + | β_{2} - β_{3} | + | β_{3} - β_{1} | \end{matrix}$ for all β₁, β₂, β₃ ∈ Γ, ℏ>0 and $Θ (t_{1}, t_{2}) = (\sqrt{t_{1}}, 0)$ for every $t = (t_{1}, t_{2}) \in L^{*}$ .

Suppose the functions E, H : Γ × Γ → Γ and M, N : Γ → Γ define as $E (μ, ϑ) = {\begin{matrix} 2 & if μ and ϑ \in 2 \cup (9, 15] \\ 21 & if μ and ϑ \in (2, 9], \end{matrix}$

$H (μ, ϑ) = {\begin{matrix} 2 & if μ and ϑ \in 2 \cup (9, 15] \\ 8 & if μ and ϑ \in (2, 9], \end{matrix}$

$M (μ) = {\begin{matrix} 2 & if μ \in [2, 9] \\ \frac{μ + 1}{5} & if μ \in (9, 15], \end{matrix}$

$N (μ) = {\begin{matrix} 2 & if μ = 2 \\ 10 & if μ \in (2, 9] \\ μ - 7 & if μ \in (9, 15] . \end{matrix}$

Consider the sequences {μ_r}= ${9 + \frac{1}{r}}$ and {ϑ_r}=2. Here, the pairs (E, M) and (H, N) share common E.A property and the other conditions of Theorem 2.5 are also satisfied. Hence, 2 is the unique common fixed point of E, H, M and N.

3 Conclusion

The intuitionistic fuzzy metric space is a accommodating instrument to represent uncertain and imprecise information and procedures. This paper presents an idea of E.A property and common E.A property on MIGFM-space with its application in form of fixed point results. Some other existing concepts can be utilized for proving fixed point results with the understanding of E.A property and common E.A property on MIGFM-space. Moreover, One can apply these properties to obtain new fixed point results on modified intuitionistic generalized fuzzy metric space.

Footnotes

Acknowledgment

The authors thank the reviewers and the editor for their valuable remarks and comments on the paper.

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