Common coupled fixed point theorem for two pairs compatible and sub-sequentially continuous mapping

Abu-Donia H.M.; S. Bakry Mona; Atia H.A.; M.A. Khater Omnia; A.M. Attia Raghda

doi:10.1016/j.joes.2022.04.009

Journal of Ocean Engineering and Science >

2024 , Vol. 9 >Issue 4: 401 - 407

DOI: https://doi.org/10.1016/j.joes.2022.04.009

Original article

Common coupled fixed point theorem for two pairs compatible and sub-sequentially continuous mapping

Abu-Donia H.M. ^a ,
S. Bakry Mona ^b ,
Atia H.A. ^a ,
M.A. Khater Omnia ^c ,
A.M. Attia Raghda ^,^d

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^a Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
^b Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
^c Department of Basic Science, Zagazig Higher Institute of Engineering and Technology, Zagazig, Egypt
^d Department of Basic Science, Higher Technological Institute 10th of Ramadan City, El Sharqia 44634, Egypt

Received date: 2022-03-27

Revised date: 2022-04-08

Accepted date: 2022-04-11

Online published: 2022-04-18

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Abstract

This study establishes a common coupled fixed point for two pairs of compatible and sequentially continuous mappings in the intuitionistic fuzzy metric space that satisfy the Ï• -contractive conditions. Many basic definitions and theorems have been used from some recent scientific papers about the binary operator, t-norm, t-conorm, intuitionistic fuzzy metric space, and compatible mapping for reaching to the paper's purpose.

Highlights

● Coupled Fixed Point Theorems

● Contractive Condition

● Intuitionistic Fuzzy Metric Spaces.

Key words： Intuitionistic fuzzy metric space; Ï• -contractive mapping; Compatible mappings; Couple fixed point

Cite this article

Abu-Donia H.M. , S. Bakry Mona , Atia H.A. , M.A. Khater Omnia , A.M. Attia Raghda . Common coupled fixed point theorem for two pairs compatible and sub-sequentially continuous mapping[J]. Journal of Ocean Engineering and Science, 2024 , 9(4) : 401 -407 . DOI: 10.1016/j.joes.2022.04.009

1. Introduction

In nonlinear analysis, fixed point theory is one of the most powerful and productive methods [1]. The fixed-point theory may be traced back to the Banach contraction principle [2]. Mathematicians often rely on it to solve existence issues in various fields [3]. The Banach contraction concept has been widely generalized [4]. By manipulating distances between points using a specific control function, it is possible to determine the existence and uniqueness of fixed points for self-maps on a metric space. Khan et al. first developed these control functions, which have since been used in a variety of papers, including one in which several fixed-point theorems were examined using these varying distance functions [5], [6]. Recent research suggests that studying metric spaces with partial ordering might help relax the contraction property constraints [7]. In the early days, Ran and Reurings pioneered this method by using it to solve matrix problems [8]. Periodic boundary value problems for ordinary differential equations were then included, and the technique was further developed (ODEs) [9], [10].

Functional analysis is a field of mathematics that deals with studying vector spaces with some limit-related structure (e. g., inner product, norm, topology, etc.) and linear functions created on these spaces that properly obey these structures [11], [12]. The history of this field may be traced back to the study of formulation characteristics of transformations, operators between function spaces and spaces of functions [13], [14], [15]. Integral and differential equations demonstrate the significance of this field of research [16], [17], [18], [19], [20].

It was Zadeh who initially coined the term “fuzzy sets” [21]. In many cases, rather than being stochastic, the nature of uncertainty in the behavior of given system processes is fuzzy [22]. Numerous researchers have recently shown an interest in studying fuzzy initial value issues in their theoretical framework [23]. Fuzzy derivatives were first proposed by Chang and Zadeh [24] and are now widely accepted. Dubosi and Prade [3] introduced the extension idea [25]. Recent work established the differential and integral calculus for fuzzy-set-valued functions [26].

In this context, our study proves some common coupled fixed point theorems for mappings under

ϕ

-contractive condition on intuitionistic fuzzy metric space under compatible and subsequently continuous mappings.

The rest sections in our study is ordered as following; Section 2 gives some basic definitions and examples of t-norm, t-conorm, intuitionistic fuzzy metric space (

Ξ

), and complete intuitionistic fuzzy metric space (

c Ξ

). Section 3 demonstrates the main results of our study. Section 4 gives the conclusion of the whole study.

2. Basic employed definitions

In this section, some basic definitions and examples of t-norm, t-conorm, intuitionistic fuzzy metric space (

Ξ

), and complete intuitionistic fuzzy metric space (

c Ξ

) are given as following

Definition 2.1

A binary operator

(*, ♦) : [0, 1] \times [0, 1] ⟶ [0, 1]

are continuous t-norm and t-conorm; respectively if they satisfy the following conditions [27]

{\begin{matrix} These two operator are commutative, associative, and continuous . \\ \forall ϱ \in [0, 1] \mapsto {\begin{matrix} ϱ * 1 = ϱ, \\ ϱ ♦ 0 = ϱ . \end{matrix} \\ \forall ϱ_{1}, ϱ_{2}, ϱ_{3}, ϱ_{4} \in [0, 1] where ϱ_{1} \leq ϱ_{3} & ϱ_{2} \leq ϱ_{4}, we get {\begin{matrix} ϱ_{1} * ϱ_{2} \leq ϱ_{3} * ϱ_{4}, \\ ϱ_{1} ♦ ϱ_{2} \leq ϱ_{3} ♦ ϱ_{4} . \end{matrix} \end{matrix}

Example 2.1

\forall ς, τ \in [0, 1]

where the following binary operators

(*, ♦) : [0, 1] \times [0, 1] ⟶ [0, 1]

, represent t-norm and t-conorm; respectively, we get

{\begin{matrix} ς * τ = {\begin{matrix} \min {ς, τ}, \\ ς \cdot τ, \\ \max {0, ς + τ - 1} . \end{matrix} \\ ς ♦ τ = {\begin{matrix} \max {ς, τ}, \\ ς + τ - ς \cdot τ, \\ \min {ς + τ, 1} . \end{matrix} \end{matrix}

Definition 2.2

Let

Ξ = (X, M, N, *, ♦)

is an intuitionistic fuzzy metric space where

X, (M, N), *, ♦

denote respectively an arbitrary set, fuzzy sets on

X \times X \times [0, \infty)

, a continuous

t

-norm, and a continuous

t

-conorm, satisfy the following conditions [28]

M (ζ_{1}, ζ_{2}, t) + N (ζ_{1}, ζ_{2}, t) \leq 1

\forall ζ_{1}, ζ_{2} \in X

and

t > 0

M (ζ_{1}, ζ_{2}, 0) = 0

\forall ζ_{1}, ζ_{2} \in X

M (ζ_{1}, ζ_{2}, t) = 1

\forall ζ_{1}, ζ_{2} \in X

and

t > 0

\Rightarrow

ζ_{1} = ζ_{2}

M (ζ_{1}, ζ_{2}, t) = M (ζ_{2}, ζ_{1}, t),

\forall ζ_{1}, ζ_{2} \in X

and

t > 0

M (ζ_{1}, ζ_{2}, t) * M (ζ_{2}, ζ_{3}, s) \leq M (ζ_{1}, ζ_{3}, t + s)

\forall ζ_{1}, ζ_{2}, ζ_{3} \in X

and

t, s > 0

M (ζ_{1}, ζ_{2}, \cdot) : [0, \infty) ⟶ [0, 1]

is left continuous,

\forall ζ_{1}, ζ_{2}, \in X

lim_{t ⟶ \infty} M (ζ_{1}, ζ_{2}, t) = 1

\forall ζ_{1}, ζ_{2} \in X

and

t > 0

N (ζ_{1}, ζ_{2}, 0) = 1

\forall ζ_{1}, ζ_{2} \in X

N (ζ_{1}, ζ_{2}, t) = 0

\forall ζ_{1}, ζ_{2} \in X

and

t > 0

\Leftrightarrow

ζ_{1} = ζ_{2}

10.

N (ζ_{1}, ζ_{2}, t) = N (ζ_{2}, ζ_{1}, t)

\forall ζ_{1}, ζ_{2} \in X

and

t > 0

11.

N (ζ_{1}, ζ_{2}, t) ♦ N (ζ_{2}, ζ_{3}, s) \geq N (ζ_{1}, ζ_{3}, t + s)

\forall ζ_{1}, ζ_{2}, ζ_{3} \in X

and

t, s > 0

12.

N (ζ_{1}, ζ_{2}, \cdot : [0, \infty) ⟶ [0, 1]

is right continuous,

\forall ζ_{1}, ζ_{2}, \in X

13.

lim_{t ⟶ \infty} N (ζ_{1}, ζ_{2}, t) = 0

\forall ζ_{1}, ζ_{2} \in X

and

t > 0

Consequently,

(M, N)

is called an intuitionistic fuzzy metric space. In this case, the functions

M = M (ζ_{1}, ζ_{2}, t), N = N (ζ_{1}, ζ_{2}, t)

represent the degree of nearness and non-nearness between

ζ_{1}, ζ_{2}

with respect to

t

Definition 2.3

let

Ξ

is complete if and only if every Cauchy sequence in

X

is convergent where [29]

\forall t > 0

and

α, β \in α_{o}

\exists

a sequence

{S_{α}}

X

is a Cauchy sequence if for any

ϵ > 0

there exists

α_{o} \in N

where

M (S_{α}, S_{β}, t) > 1 - ϵ

and

N (S_{α}, S_{β}, t) < ϵ

\forall t > 0

\exists

a sequence

{S_{α}}

X

is a convergent to a point

s \in X

where

lim_{α ⟶ \infty} M (S_{α}, s, t) = 1

and

lim_{α ⟶ \infty} N (S_{α}, s, t) = 0

Definition 2.4

A t-norm

Υ

is said to be of

H

-type if the family of functions

{Υ^{β} (t)}_{β = 1}^{\infty}

equicontinuous at

t = 1

[30], where

sup_{0 < t < 1} Υ (t, t) = 1, Υ^{1} (t) = t Υ t

Υ^{β + 1} (t) = t Υ (Υ^{β} (t)), β = 1, 2, \dots, t \in [0, 1]

. Additionally, the t-norm

Υ_{M} = \min

is an example of t-norm of

H

-type, but there are some other t-norms

Υ

of H-type.

Definition 2.5

A t-conorm

Λ

is said to be of

H

-type if the family of functions

{Λ^{β} (t)}_{β = 1}^{\infty}

equicontinuous at

t = 0

[31], where

inf_{0 < t < 1} Λ (t, t) = 0, Λ^{1} (t) = t Λ t

Λ^{β + 1} (t) = t Λ (Λ^{β} (t)), m = 1, 2, \dots, t \in [0, 1]

. Moreover, the t-conorm

Λ_{M} = \max

is an example of t-conorm of

H

-type.

Example 2.2

Let

X = {\frac{1}{α}; α = 1, 2, 3, \dots} ⋃ {0}

and let

*

be the continuous

t

-norm and

♦

be the continuous

t

-conorm defined by

ς * τ = ς τ

and

ς ♦ τ = \min {1, ς + τ}

\forall ς, τ \in [0, 1]

for each

ζ_{1}, ζ_{2} \in X and t > 0

, define

(M, N)

M (ζ_{1}, ζ_{2}, t) = {\begin{matrix} 0 & : t = 0 \\ \frac{t}{t + d (ζ_{1}, ζ_{2})} & : t > 0 \end{matrix}

N (x, y, t) = {\begin{matrix} 1 & : t = 0 \\ \frac{d (ζ_{1}, ζ_{2})}{t + d (ζ_{1}, ζ_{2})} & : t > 0 \end{matrix}

Thus,

(X, M, N, *, ♦)

is an intuitionistic fuzzy metric space.

Remark 2.1

It easy to prove that if

S_{α} ⟶ ζ_{1}

Q_{α} ⟶ ζ_{2}

t_{α} ⟶ t

, we get

lim_{α ⟶ \infty} M (S_{α}, Q_{α}, t_{α}) = M (ζ_{1}, ζ_{2}, t)

and

lim_{α ⟶ \infty} N (S_{α}, Q_{α}, t_{α}) = N (ζ_{1}, ζ_{2}, t) .

Definition 2.6

Let

Φ

contractive conditions where

Φ = {ϕ : R^{+} ⟶ R^{+}}

such that

ϕ \in Φ

satisfies the following conditions [32]:

ϕ

is non decreasing;

ii)

ϕ

is continuous;

iii)

\sum_{n = 0}^{\infty} ϕ^{n} (t) < \infty

\forall t > 0,

where

ϕ^{n + 1} (t) = ϕ (ϕ^{n} (t)), n \in N

Clearly if

ϕ \in Φ,

then

ϕ (t) < t

\forall t > 0 .

Lemma 2.1

Let

ϕ \in Φ

[32] . Thus

\forall t > 0

it holds that

lim_{n ⟶ \infty} ϕ^{n} (t) = 0

, where

ϕ^{n} (t)

denote the n-th iteration of

ϕ .

Lemma 2.2

Let

Ξ

be an intuitionistic fuzzy metric space where

*, ♦

respectively are continuous t-norm and continuous t-conorm of

H

-type [33] . If there exists

ϕ \in Φ

such that

M (ζ_{1}, ζ_{2}, ϕ (t)) \geq M (ζ_{1}, ζ_{2}, t)

and

N (ζ_{1}, ζ_{2}, ϕ (t)) \leq N (ζ_{1}, ζ_{2}, t)

for all

t > 0

. Then

ζ_{1} = ζ_{2} .

Definition 2.7

Let

Ξ

be an intuitionistic fuzzy metric space [34].

(M, N)

is said to be n-property on

x^{2} \times (0, \infty)

where

lim_{α ⟶ \infty} {M (ζ_{1}, ζ_{2}, k^{n} t)}^{n} = 1

and

lim_{n ⟶ \infty} {N (ζ_{1}, ζ_{2}, k^{n} t)}^{n} = 0

, whenever

ζ_{1}, ζ_{2} \in X, k > 1

and

n \in N

Definition 2.8

An element

(ζ_{1}, ζ_{2}) \in X \times X

is called a coupled fixed point (CFP), a coupled coincidence point (CCP), a common coupled fixed point (CCFP), and a common fixed point (CFP) if and only if satisfies the next conditions [35]

{\begin{matrix} A CFP of the mapping ↠ & F : X \times X \to X, \Rightarrow & {\begin{matrix} F (ζ_{1}, ζ_{2}) = ζ_{1}, \\ F (ζ_{2}, ζ_{1}) = ζ_{2} . \end{matrix} \\ A CCP of the mapping ↠ & {\begin{matrix} F : X \times X \to X, \\ G : X \to X, \end{matrix} \Rightarrow & {\begin{matrix} F (ζ_{1}, ζ_{2}) = G (ζ_{1}), \\ F (ζ_{2}, ζ_{1}) = G (ζ_{2}) . \end{matrix} \\ A CCFP of the mapping ↠ & {\begin{matrix} F : X \times X \to X, \\ G : X \to X, \end{matrix} \Rightarrow & {\begin{matrix} ζ_{1} = F (ζ_{1}, ζ_{2}) = G (ζ_{1}), \\ ζ_{2} = F (ζ_{2}, ζ_{1}) = G (ζ_{2}) . \end{matrix} \\ A CFP of the mapping ↠ & {\begin{matrix} F : X \times X \to X, \\ G : X \to X, \end{matrix} \Rightarrow & ζ_{1} = G (ζ_{1}) = F (ζ_{1}, ζ_{1}) . \end{matrix}

Definition 2.9

The mappings

F : X \times X ⟶ X

and

G : X ⟶ X

are called compatible when

\exists S_{α}, Q_{α} \in X

if and only if satisfy the following conditions [36]

\begin{matrix} {\begin{matrix} lim_{α ⟶ \infty} M (G F (S_{α}, Q_{α}), F (G S_{α}, G Q_{α}), t) = 1, \\ lim_{α ⟶ \infty} N (G F (S_{α}, Q_{α}), F (G S_{α}, G Q_{α}), t) = 0, \\ lim_{α ⟶ \infty} M (G F (Q_{α}, S_{α}), F (G Q_{α}, G S_{α}), t) = 1, \\ lim_{α ⟶ \infty} N (G F (Q_{α}, S_{α}), F (G Q_{α}, G S_{α}), t) = 0, \end{matrix} & where, & {\begin{matrix} lim_{α ⟶ \infty} F (S_{α}, Q_{α}) = lim_{α ⟶ \infty} G S_{α} = λ, \\ lim_{α ⟶ \infty} F (Q_{α}, S_{α}) = lim_{α ⟶ \infty} G Q_{α} = μ, \end{matrix} \end{matrix}

where

S_{α}, Q_{α}

are two sequence in

X

and

\forall λ, μ \in X

Definition 2.10

The mappings

F : X \times X ⟶ X

and

G : X ⟶ X

are sub-sequentially continuous if and only if there exist sequence

S_{α}, Q_{α} \in X

such that [37]

\begin{matrix} {\begin{matrix} lim_{α ⟶ \infty} F (S_{α}, Q_{α}) = lim_{α ⟶ \infty} G S_{α} = λ, \\ lim_{α ⟶ \infty} F (Q_{α}, S_{α}) = lim_{α ⟶ \infty} G Q_{α} = μ, \end{matrix} & where, & {\begin{matrix} lim_{α ⟶ \infty} F (G S_{α}, G Q_{α}) = F (λ, μ), \\ lim_{α ⟶ \infty} F (G Q_{α}, G S_{α}) = F (μ, λ), \\ lim_{α ⟶ \infty} G F (S_{α}, Q_{α}) = G λ, \\ lim_{α ⟶ \infty} G F (Q_{α}, S_{α}) = G μ . \end{matrix} \end{matrix}

3. Main results

In this section, we prescribe a common coupled fixed point in intuitionistic fuzzy metric space for two pairs of compatible and sequentially continuous mappings that fulfill the

ϕ

-contractive criteria.

Theorem 3.1

Let

Ξ

be an intuitionistic fuzzy metric space where

*

is continuous t-norm of

H

-type and

♦

is continuous t-conorm of

H

-type such that

lim_{t ⟶ \infty} M (ζ_{1}, ζ_{2}, t) = 1

and

lim_{t ⟶ \infty} N (ζ_{1}, ζ_{2}, t) = 0,

\forall ζ_{1}, ζ_{2} \in X

. Let

A, B : X \times X ⟶ X

and

S, T : X ⟶ X

be four mappings such that

a) The pairs

(A, S)

and

(B, T)

are compatible and sub-sequentially continuous.

b) There exists

ϕ \in Φ

such that

{\begin{matrix} M (A (ζ_{1}, ζ_{2}), B (υ, ω), ϕ (t)) \geq M (S ζ_{1}, T υ, t) * M (S ζ_{2}, T ω, t), \\ N (A (ζ_{1}, ζ_{2}), B (υ, ω), ϕ (t)) \leq N (S ζ_{1}, T υ, t) * N (S ζ_{2}, T ω, t), \end{matrix}

\forall ζ_{1}, ζ_{2}, υ, ω \in X

and

t > 0 .

Then there exist a unique point

φ \in X

such that

φ = S φ = T φ = A (φ, φ) = B (φ, φ)

Proof

Since the mappings

A

and

S

are sub-sequentially continuous and compatible. Thus, there exists sequence

{\begin{matrix} lim_{α ⟶ \infty} A (S_{α}, Q_{α}) = lim_{α ⟶ \infty} S S_{α} = φ, \\ lim_{α ⟶ \infty} A (S_{α}, Q_{α}) = lim_{α ⟶ \infty} S Q_{α} = ψ, \\ ⇓ ⇓ ⇓ \\ lim_{α ⟶ \infty} M (A (S S_{α}, S Q_{α}), S A (S_{α}, Q_{α}), t) = 1, \\ lim_{α ⟶ \infty} M (A (S Q_{α}, S S_{α}), S A (Q_{α}, S_{α}), t) = 1, \\ lim_{α ⟶ \infty} N (A (S S_{α}, S Q_{α}), S A (S_{α}, Q_{α}), t) = 0, \\ lim_{α ⟶ \infty} N (A (S Q_{α}, S S_{α}), S A (Q_{α}, S_{α}), t) = 0 . \end{matrix}

\forall φ, ψ \in X

. That grantees leading to

A (φ, ψ) = S φ

and

A (ψ, φ) = S ψ .

Using same technique with respect to

(B, T)

, there exist sequence

{S_{α}^{^{'}}}

and

{Q_{α}^{^{'}}}

X

leads to

(φ, ψ) \in X \times X

is couple coincidence point of

(A, S)

and

(φ^{^{'}}, ψ^{^{'}}) \in X \times X

is couple coincidence point of the pair

(B, T)

. □

Now. Suppose

(φ, ψ) = (φ^{^{'}}, ψ^{^{'}})

leads to

φ = φ^{^{'}}

and

ψ = ψ^{^{'}}

. Consequently for

S > 0, \exists T > 0

, where

{\underline{(1 - T) * (1 - T) * \dots * (1 - T)}}_{k} \geq 1 - S

and

{\underline{T ♦ T ♦ \dots ♦ T}}_{k} \leq S,

\forall k \in N .

Additionally,

M (ζ_{1}, ζ_{2}, \cdot), N (ζ_{1}, ζ_{2}, \cdot)

are continuous, thus, it satisfies the following conditions

{\begin{matrix} \begin{matrix} {\begin{matrix} lim_{t ⟶ \infty} M (ζ_{1}, ζ_{2}, t) = 1, \\ lim_{t ⟶ \infty} N (ζ_{1}, ζ_{2}, t) = 0, \end{matrix} & \forall ζ_{1}, ζ_{2} \in X \end{matrix} \\ ⇓ ⇓ \\ {\begin{matrix} \begin{matrix} M (φ, φ^{^{'}}, t_{o}) \geq 1 - S, \\ M (ψ, ψ^{^{'}}, t_{o}) \geq 1 - S, \\ N (φ, φ^{^{'}}, t_{o}) \leq S, \\ N (ψ, ψ^{^{'}}, t_{o}) \leq S . \end{matrix} & \forall t_{0} > 0 . \end{matrix} \end{matrix}

On the other hand, using the

Φ

contractive conditions (2.6), leads to

\sum_{α = 1}^{\infty} ϕ^{α} (t_{o}) < \infty

. Consequently,

\forall t > 0, \exists α_{0} \in N, t > \sum_{k = α_{o}}^{\infty} ϕ^{k} (t_{o}), ζ_{1} = S_{α}, ζ_{2} = Q_{α}, υ = S_{α}^{^{'}}, ω = Q_{α}^{^{'}} & ζ_{1} = Q_{α}, ζ_{2} = S_{α}, υ = Q_{α}^{^{'}}, ω = S_{α}^{^{'}}

which leads to

{\begin{matrix} M (A (S_{α}, Q_{α}), B (S_{α}^{^{'}}, Q_{α}^{^{'}}), ϕ (t_{o})) \geq M (S Q_{α}, T Q_{α}^{^{'}}, t) * M (S Q_{α}, T Q_{α}^{^{'}}, t), \\ M (A (Q_{α}, S_{α}), B (Q_{α}^{^{'}}, S_{α}^{^{'}}), ϕ (t_{o})) \geq M (S Q_{α}, T Q_{α}^{^{'}}, t_{o}) * M (S S_{α}, T S_{α}^{^{'}}, t_{o}), \\ N (A (S_{α}, Q_{α}), B (S_{α}^{^{'}}, Q_{α}^{^{'}}), ϕ (t_{o})) \leq N (S S_{α}, T S_{α}^{^{'}}, t) ♦ N (S Q_{α}, T Q_{α}^{^{'}}, t), \\ N (A (Q_{α}, S_{α}), B (Q_{α}^{^{'}}, S_{α}^{^{'}}), ϕ (t_{o})) \leq N (S Q_{α}, T Q_{α}^{^{'}}, t_{o}) ♦ N (S S_{α}, T S_{α}^{^{'}}, t_{o}) . \end{matrix}

While, for

α ⟶ \infty

, we get

{\begin{matrix} M (φ, φ^{^{'}}, ϕ (t_{o})) \geq M (φ, φ^{^{'}}, t_{o}) * M (ψ, ψ^{^{'}}, t_{o}), \\ N (φ, φ^{^{'}}, ϕ (t_{o})) \leq N (φ, φ^{^{'}}, t_{o}) ♦ N (ψ, ψ^{^{'}}, t_{o}), \\ M (ψ, ψ^{^{'}}, ϕ (t_{o})) \geq M (ψ, ψ^{^{'}}, t_{o}) * M (φ, φ^{^{'}}, t_{o}), \\ N (ψ, ψ^{^{'}}, ϕ (t_{o})) \leq N (ψ, ψ^{^{'}}, t_{o}) ♦ N (φ, φ^{^{'}}, t_{o}) . \end{matrix}

Consequently, we get

{\begin{matrix} M (φ, φ^{^{'}}, ϕ (t_{o})) * M (ψ, ψ^{^{'}}, ϕ (t_{o})) \geq {(M (φ, φ^{^{'}}, t_{o}))}^{2} * {(M (ψ, ψ^{^{'}}, t_{o}))}^{2}, \\ N (φ, φ^{^{'}}, ϕ (t_{o})) ♦ N (ψ, ψ^{^{'}}, ϕ (t_{o})) \leq {(N (φ, φ^{^{'}}, t_{o}))}^{2} ♦ {(N (ψ, ψ^{^{'}}, t_{o}))}^{2} . \end{matrix}

Using same technique

\forall α \in N

, we get

{\begin{matrix} {\begin{matrix} M (φ, φ^{^{'}}, ϕ^{α} (t_{o})) * M (ψ, ψ^{^{'}}, ϕ^{α} (t_{o})) & \geq {(M (φ, φ^{^{'}}, ϕ^{α - 1} (t_{o})))}^{2} * {(M (ψ, ψ^{^{'}}, ϕ^{α - 1} (t_{o})))}^{2} \\ \geq {(M (φ, φ^{^{'}}, t_{o}))}^{2^{α}} * {(M (ψ, ψ^{^{'}}, t_{o}))}^{2^{α}}, \end{matrix} \\ {\begin{matrix} N (φ, φ^{^{'}}, ϕ^{α} (t_{o})) ♦ N (ψ, ψ, ϕ^{α} (t_{o})) & \leq {(N (φ, φ^{^{'}}, ϕ^{α - 1} (t_{o})))}^{2} ♦ {(N (ψ, ψ^{^{'}}, ϕ^{α - 1} (t_{o})))}^{2} \\ \leq {(N (φ, φ^{^{'}}, t_{o}))}^{2^{α}} ♦ {(N (ψ, ψ^{^{'}}, t_{o}))}^{2^{α}} . \end{matrix} \end{matrix}

That leads to obtain

\begin{matrix} M (φ, φ^{^{'}}, t) * M (ψ, ψ^{^{'}}, t) & \geq [M (φ, φ^{^{'}}, \sum_{k = α_{o}}^{\infty} ϕ^{k} (t_{o}))] * [M (ψ, ψ^{^{'}}, \sum_{k = α_{o}}^{\infty} ϕ^{k} (t_{o}))] \\ \geq [M (φ, φ^{^{'}}, ϕ^{α_{o}} (t_{o}))] * [M (ψ, ψ^{^{'}}, ϕ^{α_{o}} (t_{o}))] \\ \geq {[M (φ, φ^{^{'}}, t_{o})]}^{2^{α_{o}}} * {[M (ψ, ψ^{^{'}}, t_{o})]}^{2^{α_{o}}} \\ \geq \underset{2^{α_{o}}}{\underline{(1 - T) * (1 - T) * \dots * (1 - T)}} \\ \geq 1 - S, \end{matrix}

\begin{matrix} N (φ, φ^{^{'}}, t) ♦ N (ψ, ψ^{^{'}}, t) & \leq [N (φ, φ^{^{'}}, \sum_{k = α_{o}}^{\infty} ϕ^{k} (t_{o}))] ♦ [N (ψ, ψ^{^{'}}, \sum_{k = α_{o}}^{\infty} ϕ^{k} (t_{o}))] \\ \leq [N (φ, φ^{^{'}}, ϕ^{α_{o}} (t_{o}))] ♦ [N (ψ, ψ^{^{'}}, ϕ^{α_{o}} (t_{o}))] \\ \leq {[N (φ, φ^{^{'}}, t_{o})]}^{2^{α_{o}}} ♦ {[N (ψ, ψ^{^{'}}, t_{o})]}^{2^{α_{o}}} \\ \leq \underset{2^{α_{o}}}{\underline{T ♦ T ♦ \dots ♦ T}} \\ \leq S . \end{matrix}

Additionally,

\forall S > 0, t > 0

we get

{\begin{matrix} M (φ, φ^{^{'}}, t) * M (ψ, ψ^{^{'}}, t) \geq 1 - S, \\ N (φ, φ^{^{'}}, t) ♦ N (ψ, ψ^{^{'}}, t) \leq S, \\ φ = φ^{^{'}}, \\ ψ = ψ^{^{'}} . \end{matrix}

Moreover, we obtain

\begin{matrix} A (φ, ψ) = S φ & & & A (ψ, φ) = S ψ, \\ B (φ, ψ) = T φ & & & B (ψ, φ) = T ψ . \end{matrix}

Now, we are going to prove

{\begin{matrix} S φ = T φ, \\ S ψ = T ψ, \end{matrix}

by employing the intuitionistic fuzzy metric space's characterizations as following:

\forall S > 0, k \in N \exists T > 0,

such that

{\begin{matrix} {\underline{(1 - T) * (1 - T) * \dots * (1 - T)}}_{k} \geq 1 - S, \\ {\underline{T ♦ T ♦ \dots ♦ T}}_{k} \leq S, \end{matrix}

where

M (ζ_{1}, ζ_{2}, \cdot), N (ζ_{1}, ζ_{2}, \cdot)

are continuous. Consequently, we get

{\begin{matrix} lim_{t ⟶ \infty} M (ζ_{1}, ζ_{2}, t) = 1, \\ lim_{t ⟶ \infty} N (ζ_{1}, ζ_{2}, t) = 0, \end{matrix}

\forall ζ_{1}, ζ_{2} \in X, \exists t_{o} > 0

such that

{\begin{matrix} M (S φ, T φ, t_{o}) \geq 1 - S & & & N (S φ, T φ, t_{o}) \leq S, \\ M (S ψ, T ψ, t_{o}) \geq 1 - S & & & N (S ψ, T ψ, t_{o}) \leq S . \end{matrix}

On the other hand, using the

ϕ

-contractive conditions leads to

\sum_{α = 1}^{\infty} ϕ^{α} (t_{o}) < \infty

and

t > \sum_{k = α_{o}}^{\infty} ϕ^{k} (t_{o})

. Additionally, for

ζ_{1} = υ = φ, ζ_{2} = ω = ψ & ζ_{1} = υ = ψ, ζ_{2} = ω = φ

respectively, we get

{\begin{matrix} M (A (φ, ψ), B (φ, ψ), ϕ (t_{o})) \geq M (S φ, T φ, t_{o}) * M (S ψ, T ψ, t_{o}), \\ M (A (ψ, φ), B (ψ, φ), ϕ (t_{o})) \geq M (S ψ, T ψ, t_{o}) * M (S φ, T φ, t_{o}) . \\ N (A (φ, ψ), B (φ, ψ), ϕ (t_{o})) \leq N (S φ, T φ, t_{o}) ♦ N (S ψ, T ψ, t_{o}), \\ N (A (ψ, φ), B (ψ, φ), ϕ (t_{o})) \leq N (S ψ, T ψ, t_{o}) ♦ N (S φ, T φ, t_{o}) . \\ M (S φ, T φ, ϕ (t_{o})) \geq M (S φ, T φ, t_{o}) * M (S ψ, T ψ, t_{o}), \\ M (S ψ, T ψ, ϕ (t_{o})) \geq M (S ψ, T ψ, t_{o}) * M (S φ, T φ, t_{o}) . \\ N (S φ, T φ, ϕ (t_{o})) \leq N (S φ, T φ, t_{o}) ♦ N (S ψ, T ψ, t_{o}), \\ N (S ψ, T ψ, ϕ (t_{o})) \leq N (S ψ, T ψ, t_{o}) ♦ N (S φ, T φ, t_{o}) . \end{matrix}

Based on the above inequalities, we can conclude

{\begin{matrix} M (S φ, T φ, ϕ (t_{o})) * M (S ψ, T ψ, ϕ (t_{o})) \geq {[M (S φ, T φ, t_{o})]}^{2} * {[M (S ψ, T ψ, t_{o})]}^{2}, \\ N (S φ, T φ, ϕ (t_{o})) ♦ N (S ψ, T ψ, ϕ (t_{o})) \leq {[N (S φ, T φ, t_{o})]}^{2} ♦ {[N (S ψ, T ψ, t_{o})]}^{2} . \end{matrix}

Repeating this process

α

-times where

α \in N

, we get

\begin{matrix} M (S φ, T φ, ϕ^{α} (t_{o})) * M (S ψ, T ψ, ϕ^{α} (t_{o})) & \geq {[M (S φ, T φ, ϕ^{α - 1} (t_{o}))]}^{2} * {[M (S ψ, T ψ, ϕ^{α - 1} (t_{o}))]}^{2} \\ \geq {[M (S φ, T φ, t_{o})]}^{2^{α}} * {[M (S ψ, T ψ, t_{o})]}^{2^{α}} . \end{matrix}

\begin{matrix} N (S φ, T φ, ϕ^{α} (t_{o})) ♦ N (S ψ, T ψ, ϕ^{α} (t_{o})) & \leq {[N (S φ, T φ, ϕ^{α - 1} (t_{o}))]}^{2} ♦ {[N (S ψ, T ψ, ϕ^{α - 1} (t_{o}))]}^{2} \\ \leq {[N (S φ, T φ, t_{o})]}^{2^{α}} ♦ {[N (S ψ, T ψ, t_{o})]}^{2^{α}} . \end{matrix}

Thus, we get

{\begin{matrix} {\begin{matrix} M (S φ, T φ, t) * M (S ψ, T ψ, t) & \geq [M (S φ, T φ, \sum_{k = α_{o}}^{\infty} ϕ^{k} (t_{o}))] * [M (S ψ, T ψ, \sum_{k = n_{o}}^{\infty} ϕ^{k} (t_{o}))] \\ \geq [M (S φ, T φ, ϕ^{α_{o}} (t_{o}))] * [M (S ψ, T ψ, ϕ^{n_{o}} (t_{o}))] \\ \geq {[M (S φ, T φ, t_{o})]}^{2^{^{α_{o}}}} * {[M (S ψ, T ψ, t_{o})]}^{2^{α_{o}}} \\ \geq (1 - T) * (1 - T) * \dots * (1 - T) \\ \geq 1 - S . \end{matrix} \\ {\begin{matrix} N (S φ, T φ, t) ♦ N (S ψ, T ψ, t) & \leq [N (S φ, T φ, \sum_{k = α_{o}}^{\infty} ϕ^{k} (t_{o}))] ♦ [N (S ψ, T ψ, \sum_{k = α_{o}}^{\infty} ϕ^{k} (t_{o}))] \\ \leq [N (S φ, T φ, ϕ^{α_{o}} (t_{o}))] ♦ [N (S ψ, T ψ, ϕ^{α_{o}} (t_{o}))] \\ \leq {[N (S φ, T φ, t_{o})]}^{2^{^{α_{o}}}} ♦ {[N (S ψ, T ψ, t_{o})]}^{2^{α_{o}}} \\ \leq T ♦ T ♦ \dots ♦ T \\ \leq S . \end{matrix} \end{matrix}

Consequently, we get for

(t > 0);

{\begin{matrix} M (S φ, T φ, t) * M (S ψ, T ψ, t) \geq 1 - S, \\ N (S φ, T φ, t) ♦ N (S ψ, T ψ, t) \leq S, \\ S φ = T φ = A (φ, ψ) = B (φ, ψ), \\ S ψ = T ψ = A (ψ, φ) = B (ψ, φ) . \end{matrix}

Now, we are going to prove the following conditions

{\begin{matrix} S φ = φ, \\ S ψ = ψ . \end{matrix}

Since,

M (ζ_{1}, ζ_{2}, \cdot), N (ζ_{1}, ζ_{2}, \cdot)

are continuous which grantee

{\begin{matrix} lim_{t ⟶ \infty} M (ζ_{1}, ζ_{2}, t) = 1, \\ lim_{t ⟶ \infty} N (ζ_{1}, ζ_{2}, t) = 0, \end{matrix}

\forall ζ_{1}, ζ_{2} \in X, \exists t_{o} > 0

, such that

\begin{matrix} M (S φ, φ, t_{o}) \geq 1 - S, & & & N (S φ, φ, t_{o}) \leq S, \\ M (S ψ, ψ, t_{o}) \geq 1 - S, & & & N (S ψ, ψ, t_{o}) \leq S . \end{matrix}

Employing the

Φ

contractive conditions (2.6) along with

ζ_{1} = φ, ζ_{2} = ψ, υ = S_{α}^{^{'}}, ω = Q_{α}^{^{'}} & ζ_{1} = ψ, ζ_{2} = φ, υ = Q_{α}^{^{'}} a n d ω = S_{α}^{^{'}}

leads to

\begin{matrix} {\begin{matrix} M (A (φ, ψ), B (S_{α}^{^{'}}, Q_{α}^{^{'}}), ϕ (t_{o})) \geq M (S φ, T S_{α}^{^{'}}, t_{o}) * M (S ψ, T Q_{α}^{^{'}}, t_{o}), \\ N (A (φ, ψ), B (S_{α}^{^{'}}, Q_{α}^{^{'}}), ϕ (t_{o})) \leq N (S φ, T S_{α}^{^{'}}, t_{o}) ♦ N (S ψ, T Q_{α}^{^{'}}, t_{o}) . \end{matrix} \end{matrix}

Suppose,

α ⟶ \infty

refers to

\begin{matrix} M (S φ, φ, ϕ (t_{o})) \geq M (S φ, φ, t_{o}) * M (S ψ, ψ, t_{o}), \\ N (S φ, φ, ϕ (t_{o})) \leq N (S φ, φ, t_{o}) ♦ N (S ψ, ψ, t_{o}) . \end{matrix}

Similarly, we can get.

\begin{matrix} M (S ψ, ψ, ϕ (t_{o})) \geq M (S ψ, ψ, t_{o}) * M (S φ, φ, t_{o}), \\ N (S ψ, ψ, ϕ (t_{o})) \leq N (S ψ, ψ, t_{o}) ♦ N (S φ, φ, t_{o}) . \end{matrix}

That obtains

\begin{matrix} M (S φ, φ, ϕ (t_{o})) * M (S ψ, ψ, ϕ (t_{o})) \geq {[M (S φ, φ, t_{o})]}^{2} * {[M (S ψ, ψ, t_{o})]}^{2}, \\ N (S φ, φ, ϕ (t_{o})) ♦ N (S ψ, ψ, ϕ (t_{o})) \leq {[N (S φ, φ, t_{o})]}^{2} ♦ {[N (S ψ, ψ, t_{o})]}^{2} . \end{matrix}

Repeating same style for all

α \in N

, gets

{\begin{matrix} {\begin{matrix} M (S φ, φ, ϕ^{α} (t_{o})) * M (S ψ, ψ, ϕ^{α} (t_{o})) & \geq {[M (S φ, φ, ϕ^{α - 1} (t_{o}))]}^{2} * {[M (S ψ, ψ, ϕ^{α - 1} (t_{o}))]}^{2} \\ \geq {[M (S φ, φ, t_{o})]}^{2^{α}} * {[M (S ψ, ψ, t_{o})]}^{2^{α}} . \end{matrix} \\ {\begin{matrix} N (S φ, φ, ϕ^{α} (t_{o})) ♦ N (S ψ, ψ, ϕ^{α} (t_{o})) & \leq {[N (S φ, φ, ϕ^{α - 1} (t_{o}))]}^{2} ♦ {[N (S ψ, ψ, ϕ^{α - 1} (t_{o}))]}^{2}, \\ \leq {[N (S φ, φ, t_{o})]}^{2^{α}} ♦ {[N (S ψ, ψ, t_{o})]}^{2^{α}} . \end{matrix} \end{matrix}

Consequently, we get

{\begin{matrix} {\begin{matrix} M (S φ, φ, t) * M (S ψ, ψ, t) & \geq [M (S φ, φ, \sum_{k = α_{0}}^{\infty} ϕ^{k} (t_{o}))] * [M (S ψ, ψ, \sum_{k = α_{0}}^{\infty} ϕ^{k} (t_{o}))] \\ \geq [M (S φ, φ, ϕ^{α_{0}} (t_{o}))] * [M (S ψ, ψ, ϕ^{α_{0}} (t_{o}))] \\ \geq {[M (S φ, φ, t_{o})]}^{2^{α_{0}}} * {[M (S ψ, ψ, t_{o})]}^{2^{α_{0}}} \\ \geq 1 - S, \end{matrix} \\ {\begin{matrix} N (S φ, φ, t) ♦ N (S ψ, ψ, t) \leq [N (S φ, φ, \sum_{k = α_{0}}^{\infty} ϕ^{k} (t_{o}))] & \leq [N (S ψ, ψ, \sum_{k = α_{0}}^{\infty} ϕ^{k} (t_{o}))] \\ \leq [N (S φ, φ, ϕ^{α_{0}} (t_{o}))] ♦ [N (S ψ, ψ, ϕ^{α_{0}} (t_{o}))] \\ \leq {[N (S φ, φ, t_{o})]}^{2^{α_{0}}} ♦ {[N (S ψ, ψ, t_{o})]}^{2^{α_{0}}} \\ \leq S . \end{matrix} \end{matrix}

Thus,

S φ = φ

and

S ψ = ψ

. Therefore

S φ = T φ = A (φ, ψ) = B (φ, ψ) = φ

and

S ψ = T ψ = A (ψ, φ) = B (ψ, φ) = ψ

Finally, we show

φ = ψ

Since

M (ζ_{1}, ζ_{2}, \cdot), N (ζ_{1}, ζ_{2}, \cdot)

are continuous which means

{\begin{matrix} lim_{t ⟶ \infty} M (ζ_{1}, ζ_{2}, t) = 1, \\ lim_{t ⟶ \infty} N (ζ_{1}, ζ_{2}, t) = 0, \end{matrix}

for all

ζ_{1}, ζ_{2} \in X

\exists t_{o} > 0

such that

\begin{matrix} M (φ, ψ, t_{o}) \geq 1 - S & & & N (φ, ψ, t_{o}) \leq S . \end{matrix}

Employing the

Φ

contractive conditions (2.6) along with

ζ_{1} = ω = φ, ζ_{2} = υ = ψ

, leads to

{\begin{matrix} M (A (φ, ψ), B (ψ, φ) ϕ (t_{o})) \geq M (S φ, T ψ, t_{o}) * M (S ψ, T φ, t_{o}), \\ M (φ, ψ, ϕ (t_{o})) \geq M (φ, ψ, t_{o}) * M (ψ, φ, t_{o}), \\ N (A (φ, ψ), B (ψ, φ) ϕ (t_{o})) \leq N (S φ, T ψ, t_{o}) ♦ N (S ψ, T φ, t_{o}), \\ N (φ, ψ, ϕ (t_{o})) \leq N (φ, ψ, t_{o}) ♦ N (ψ, φ, t_{o}) . \end{matrix}

Consequently, we obtain

{\begin{matrix} {\begin{matrix} M (φ, ψ, t) & \geq [M (φ, ψ, \sum_{k = α_{o}}^{\infty})] \\ \geq [M (φ, ψ, ϕ^{α_{o}} (t_{o}))] \\ \geq {[M (φ, ψ, t_{o})]}^{2^{α_{o}}} \\ \geq 1 - S, \end{matrix} \\ {\begin{matrix} N (φ, ψ, t) & \leq [N (φ, ψ, \sum_{k = α_{o}}^{\infty})] \\ \leq [N (φ, ψ, ϕ^{α_{o}} (t_{o}))] \\ \leq {[N (φ, ψ, t_{o})]}^{2^{α_{o}}} \\ \leq S . \end{matrix} \end{matrix}

This implies

φ = ψ

, therefore

φ = S φ = T φ = A (φ, φ) = B (φ, φ)

. then

φ

is a unique fixed point of

A, B, S

and

T

4. Conclusion

This study has successfully mentioned some fundamental theorem, lemma, and definitions for some essential mappings such as a binary operator, compatible and sequentially continuous mappings that are considered basic icons in intuitionistic fuzzy metric spaces. Additionally, we have used these icons for prescribing a common coupled fixed point in intuitionistic fuzzy metric space for two pairs of compatible and sequentially continuous mappings that fulfill the

ϕ

-contractive criteria.

Authors' contribution

All the study has been done by the author himself.

Funding

No fund has been received.

Availability of data and material

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The used code of this study is available from the corresponding author upon reasonable request.

Declaration of Competing Interest

There is no conflict of interest.

Acknowledgment

We greatly thank the journal stuff.

References

Publishing order | Descend order by publishing year | Descend order by cited within

[1]	X. Wu, T. Wang, P. Liu, G. Deniz Cayli, X. Zhang. arXiv e-prints (2021)

1.	arXiv:2111. 12677

2.	[ 2 J.-W. Cao, S. Mukherjee, T. Pham, Y. Wang, T. Wang, T. Zhang, X. Jiang, H.-J. Tang, K.A. Forrest, B. Space, M.J. Zaworotko, K.-J. Chen. Nat Commun, 12 (2021), p. 6507

[3]	Y.-L. Peng, T. Wang, C. Jin, C.-H. Deng, Y. Zhao, W. Liu, K.A. Forrest, R. Krishna, Y. Chen, T. Pham, B. Space, P. Cheng, M.J. Zaworotko,Z. Zhang. Nat Commun, 12 (2021), p. 5768

[4]	M. Shivanna, K.-i. Otake, B.-Q. Song, L.M. Wyk, Q.-Y. Yang, N. Kumar, W.K. Feldmann, T. Pham, S. Suepaul, B. Space, L.J. Barbour, S. Kitagawa, M.J. Zaworotko. Angewandte Chemie, 133 (37) (2021), pp. 20546-20553

[5]	S. Mukherjee, N. Kumar, A.A. Bezrukov, K. Tan, T. Pham, K.A. Forrest, K.A. Oyekan, O.T. Qazvini, D.G. Madden, B. Space, M.J. Zaworotko. Angewandte Chemie, 133 (19) (2021), pp. 10997-11004

[6]

Niu

, X.

Cui

, T.

Pham

, G.

Verma

, P.C.

Lan

, C.

Shan

, H.

Xing

, K.A.

Forrest

, S.

Suepaul

, B.

Space

, A.

Nafady

, A.M.

Al-Enizi

, S. Ma. Angewandte Chemie, 133 (10) (2021), DOI: 3. ange.202181062 4.

[ Z.

Niu

, X.

Cui

, T.

Pham

, G.

Verma

, P.C.

Lan

, C.

Shan

, H.

Xing

, K.A.

Forrest

, S.

Suepaul

, B.

Space

, A.

Nafady

, A.M.

Al-Enizi

, S. Ma. Angewandte Chemie, 133 (10) (2021), DOI: 3. ange.202181062 4. [7] Z. Niu, X. Cui, T. Pham, G. Verma, P.C. Lan, C. Shan, H. Xing, K.A. Forrest, S. Suepaul, B. Space, A. Nafady, A.M. Al-Enizi, S. Ma. Angewandte Chemie, 133 (10) (2021), pp. 5343-5348, DOI: ]

[8]	G. Verma, S. Kumar, H. Vardhan, J. Ren, Z. Niu, T. Pham, L. Wojtas, S. Butikofer, J.C. Echeverria Garcia, Y.-S. Chen, B. Space, S. Ma. Nano Res, 14 (2) (2021), pp. 512-517

[9]	V. Sreekanth, L.P. Clark, B. Bekbulat, M. Baum, S. Yang, L.M.P. Baylon, T.R. Gould, T. Larson, E. Seto, C.D. Space, J.D. Marshall.AGU Fall Meeting Abstracts, volume 2020 (2020), pp. A220-0003

[10]	S. Mukherjee, S. Chen, A.A. Bezrukov, M. Mostrom, V.V. Terskikh, D. Franz, S.-Q. Wang, A. Kumar, M. Chen, B. Space, Y. Huang, M.J. Zaworotko. Angewandte Chemie, 132 (37) (2020), pp. 16322-16328

[11]	M.M. Khater, S. Muhammad, A. Al-Ghamdi, M. Higazy. Journal of Ocean Engineering and Science (2022)

[12]	M.M. Khater. Journal of Ocean Engineering and Science (2022)

[13]	M.F. Alotaibi, M. Omri, E. Khalil, S. Abdel-Khalek, J. Bouslimi, M.M. Khater. Journal of Ocean Engineering and Science (2022)

[14]	M.M. Khater, S. Muhammad, A. Al-Ghamdi, M. Higazy. Journal of Ocean Engineering and Science (2022)

[15]	M.M. Khater. Journal of Ocean Engineering and Science (2022)

[16]	F. Wang, S.A. Salama, M.M. Khater. Journal of Ocean Engineering and Science (2022)

[17]	F. Wang, S. Muhammad, A. Al-Ghamdi, M. Higazy, M.M. Khater. Journal of Ocean Engineering and Science (2022)

[18]	F. Wang, M.M. Khater. Journal of Ocean Engineering and Science (2022)

[19]	M.M. Khater. Chaos, Solitons & Fractals, 157 (2022), p. 111970

[20]	M.M. Khater. Mod. Phys. Lett. B (2022), p. 2150614

[21]	Q. Kiran, H. Khatoon.International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2018), American Institute of Physics Conference Series, volume 2116 (2019), p. 190006

[22]	Y.-L. Peng, T. Pham, P. Li, T. Wang, Y. Chen, K.-J. Chen, K.A. Forrest, B. Space, P. Cheng, M.J. Zaworotko, Z. Zhang. Angewandte Chemie, 130 (34) (2018), pp. 11137-11141

[23]	Q.-Y. Yang, P. Lama, S. Sen, M. Lusi, K.-J. Chen, W.-Y. Gao, M. Shivanna, T. Pham, N. Hosono, S. Kusaka, I. Perry John J., S. Ma, B. Space, L.J. Barbour, S. Kitagawa, M.J. Zaworotko. Angewandte Chemie, 130 (20) (2018), pp. 5786-5791

[24]	M. Shivanna, Q.-Y. Yang, A. Bajpai, S. Sen, N. Hosono, S. Kusaka, T. Pham, K.A. Forrest, B. Space, S. Kitagawa, M.J. Zaworotko.Sci Adv, 4 (4) (2018), p. eaaq1636

[25]	H. He, Q. Sun, W. Gao, J.A. Perman, F. Sun, G. Zhu, B. Aguila, K. Forrest, B. Space, S. Ma. Angewandte Chemie, 130 (17) (2018), pp. 4747-4752

[26]	K.-J. Chen, Q.-Y. Yang, S. Sen, D.G. Madden, A. Kumar, T. Pham, K.A. Forrest, N. Hosono, B. Space, S. Kitagawa, M.J. Zaworotko. Angewandte Chemie, 130 (13) (2018), pp. 3390-3394

[27]	D.M. Franz, Z.E. Dyott, K.A. Forrest, A. Hogan, T. Pham, B. Space. Physical Chemistry Chemical Physics (Incorporating Faraday Transactions), 20 (3) (2018), pp. 1761-1777

[28]	M. Zhang, W. Zhou, T. Pham, K.A. Forrest, W. Liu, Y. He, H. Wu, T. Yildirim, B. Chen, B. Space, Y. Pan, M.J. Zaworotko, J. Bai. Angewandte Chemie, 129 (38) (2017), pp. 11584-11588

[29]	J.L. Belof, B. Space. arXiv e-prints (2017)

6.	[ 30 S. Arora, T. Kumar. Recent Advances in Fundamental and Applied Sciences: RAFAS2016, American Institute of Physics Conference Series, volume 1860 (2017), p. 020050

[31]	K.A. Forrest, T. Pham, B. Space. Physical Chemistry Chemical Physics (Incorporating Faraday Transactions), 19 (43) (2017), pp. 29204-29221

[32]	T. Pham, K.A. Forrest, D.M. Franz, Z. Guo, B. Chen, B. Space. Physical Chemistry Chemical Physics (Incorporating Faraday Transactions), 19 (28) (2017), pp. 18587-18602

[33]	T. Pham, K.A. Forrest, M. Mostrom, J.R. Hunt, H. Furukawa, J. Eckert, B. Space. Physical Chemistry Chemical Physics (Incorporating Faraday Transactions), 19 (20) (2017), pp. 13075-13082

[34]	K.-J. Chen, D.G. Madden, T. Pham, K.A. Forrest, A. Kumar, Q.-Y. Yang, W. Xue, B. Space, I. Perry John J., J.-P. Zhang, X.-M. Chen, M.J. Zaworotko. Angewandte Chemie, 128 (35) (2016), pp. 10424-10428

[35]	M.H. Mohamed, S.K. Elsaidi, T. Pham, K.A. Forrest, H.T. Schaef, A. Hogan, L. Wojtas, W. Xu, B. Space, M.J. Zaworotko, P.K. Thallapally. Angewandte Chemie, 128 (29) (2016), pp. 8425-8429

[36]	T. Pham, K.A. Forrest, B. Space. Physical Chemistry Chemical Physics (Incorporating Faraday Transactions), 18 (31) (2016), pp. 21421-21430

[37]	T. Pham, K.A. Forrest, B. Space, J. Eckert. Physical Chemistry Chemical Physics (Incorporating Faraday Transactions), 18 (26) (2016), pp. 17141-17158

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模态框（Modal）标题

Abstract

Cite this article

1. Introduction

2. Basic employed definitions

Definition 2.1

Example 2.1

Definition 2.2

Definition 2.3

Definition 2.4

Definition 2.5

Example 2.2

Remark 2.1

Definition 2.6

Lemma 2.1

Lemma 2.2

Definition 2.7

Definition 2.8

Definition 2.9

Definition 2.10

3. Main results

Theorem 3.1

Proof

4. Conclusion

Authors' contribution

Funding

Availability of data and material

Code availability

Acknowledgment

References

Links