FractionalHybridDifferentialEquationsandCoupledFixed-Point Results for α-Admissible F ( ψ 1 , ψ 2 ) − Contractions in M − Metric Spaces

Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Mathematics, Ankaya University, 06790 Etimesgut, Ankara, Turkey Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt Department of Mathematics, College of Science and Arts, Qassim University, Unaizah, Qassim, Saudi Arabia Department of Mathematics, Texas A & M University-Kingsville, 700 University Blvd., MSC 172, Kingsville, Texas 78363-8202, USA Florida Institute of Technology, 150 W. University Blvd, Melbourne, FL 32901, USA


Introduction
Fixed-point theory is an outstanding source which gives responsible techniques for the existence of fixed points for self-mappings under different conditions. One of the newest branches of fixed-point theory concerned with the study of coupled fixed points, brought by Guo and Lakshmikantham [1]. In [2], Bhaskar and Lakshmikantham established some fixed and coupled fixed-point theorems for contractions in two variables defined on partially ordered metric spaces with applications to ordinary differential equations. ereafter, these results were extended by several authors (see [3][4][5][6]).
Inspired by the notion of partial metric (or, p−metric) which is one of the vital generalizations of the standard metric, Asadi et al. [7] proposed the concept of M−metric which refines the p−metric and produces useful basic topological concepts. For some fixed-point results and various contractive definitions that have been employed in M−metric space, we refer the reader to [8][9][10][11][12].
Hybrid differential equations have been of great interest as they include several dynamic systems as special cases. e papers [17,18] discussed the existence and uniqueness results and some fundamental differential inequalities for firstorder hybrid differential equations with perturbations of 1st and 2nd type, respectively.
Fractional calculus is a field of mathematics that deals with the derivatives and integrals of arbitrary order. Indeed, it is found to be more realistic in describing and modeling several natural phenomena than the classical one. In fact, fractional differential equations (FDEs) play a major role in modeling many real-life problems such as physical phenomena, computer networking, medicine (the modeling of human tissue), mechanics (theory of viscoelasticity), electrical engineering (transmission of ultrasound waves) and many others (see [19][20][21]).
Fractional hybrid differential equations (FHDEs) can be employed in modeling and describing nonhomogenous physical phenomena that take place in their form. FHDEs have been studied using a Riemann-Liouville differential operator of order α > 0 in many literature studies (see [22][23][24][25][26]).
In [27], Shaob et al. used Bashiri fixed-point theorem [22] to prove the existence only of a solution to a three-point boundary value problem for a coupled system of FHDEs in Banach spaces.
In line with the above studies, our purpose in this paper is to introduce the notion of α−admissible mapping with two variables and generalize eorem 1 to coupled fixed-point version. en, we apply our main results to prove the existence and uniqueness of a solution to the following system of FHDEs involving Riemann-Liouville fractional derivative: and g ∈ C(J × R 2 ).

Preliminaries
In 1994, Matthews [28] introduced the notion of a p−metric space as a part of the study of denotational semantics of dataflow networks. In p−metric spaces, self-distance of an arbitrary point need not be equal to zero.
Definition 1 (see [28]). A p−metric on a nonempty set X is a mapping p: X × X ⟶ [0, ∞) such that, for all x, y, z ∈ X, en, (X, p) is called a p−metric space. Notice that, every metric space can be defined to be p−metric space with zero self-distance. After that, Asadi et al. generalized the above definition by relaxing the axiom (p 2 ) as follows.
Definition 2 (see [7]). For a nonempty set X, a function μ: X × X ⟶ [0, ∞) is called an M−metric if it fulfils the following: en, the pair (X, μ) is called an M−metric space.
Here, we give an example to show that the converse might not be held.
us, the class of M−metric spaces is effectively larger than that of both ordinary metric and p−metric spaces.

(5)
Hence, μ w is an ordinary metric induced by the M−metric μ.
Each M−metric μ on X generates a T 0 topology τ μ on X formed by the set where 2 Discrete Dynamics in Nature and Society e notions of convergent sequence, Cauchy sequence, and complete M−metric space (X, μ) are given as follows: exist and are finite.
Lemma 2 (see [7]). Let (X, μ) be an M−metric space; then, Lemma 3 (see [7]). Assume that x n ⟶ x and y n ⟶ y in an M−metric space (X, μ); then, As a consequence of Lemma 3, we have Definition 3 (see [29]). A mapping F: [0, ∞) 2 ⟶ R is called a C−class function if it is continuous and satisfies the following axioms: Definition 4 (see [20,21]). e fractional integral of order α > 0 of a function x: [0, ∞) ⟶ R is given by provided that the right side is pointwise defined on [0, ∞).

Fixed-Point Results
First, we introduce the following concepts that generalize the corresponding ones used in [13] and will be beneficial in the sequel.
Note that, if equation (16) holds, then we have α(T(y, x), T(v, u)) ≥ 1 too. Consider the following classes of functions: Discrete Dynamics in Nature and Society Theorem 2. Let (X, μ) be a complete M−metric space and T: X × X ⟶ X be an α−admissible mapping for which there exist F ∈ C, ϕ ∈ Φ, ψ 1 ∈ Ψ 1 , and ψ 2 ∈ Ψ 2 such that where Suppose that either (a) T is continuous.
Proof. Starting with x 0 , y 0 ∈ X, define the sequences x n , y n ⊂ X by x n+1 � T x n , y n , y n+1 � T y n , x n , By induction methodology for n ∈ N 0 , we shall prove that α x n , x n+1 ≥ 1, α y n , y n+1 ≥ 1, ∀n.
4 Discrete Dynamics in Nature and Society Hence, where w n � μ(x n , x n+1 ) + μ(y n , y n+1 ). Similarly, we have Adding (25) and (26) and using properties on F and ϕ, we obtain Since ψ 1 is strictly increasing, then w n ≤ w n−1 , ∀n. Hence, the sequence w n is monotone decreasing and bounded as follows. erefore, there exist some w ≥ 0 such that lim n⟶∞ w n � w.
In what follows, we prove that x n and y n are μ−Cauchy sequences in (X, μ). Since we have On the other hand, we have erefore, (33) and (34) imply that x n is an μ−Cauchy sequence. In a similar way, we can show that y n is also a μ−Cauchy sequence. By the completeness of the space (X, μ), there exist x, y ∈ X such that  With respect to the sequence x n , we obtain μ x n , x n ⟶ 0⟹m x n ,x ⟶ 0⟹μ x n , x , M x n ,x ⟶ 0, as n ⟶ ∞, but M x n ,x � max μ x n , x n , μ(x, x) ⟶ μ(x, x). (37) us, the uniqueness of the limit implies that Now, suppose that (a) holds. According to Lemma 2, since x n and y n are Cauchy sequences in a complete M−metric space (X, μ), then they converge to some x, y in the metric space (X, μ w ). Also, as F is continuous, F(x n , y n ) converges to F(x, y) in (X, μ w ), that is, lim n⟶∞ μ w (F(x n , y n ), F(x, y)) � 0 which is equivalent to μ F x n , y n , F(x, y) − m F x n ,y n ( ),F(x,y) ⟶ 0, M F x n ,y n ( ),F(x,y) − m F x n ,y n ( ),F(x,y) ⟶ 0, as n ⟶ ∞.
For the uniqueness of the coupled fixed point in eorem 2, we consider the following condition: if (x, y)and (u, v)are two coupled fixed points of T, (53) □ Theorem 3. Adding condition (53) to the hypotheses of eorem 2, we obtain that T has a unique coupled fixed point.
Proof. eorem 2 asserts that T has at least one coupled fixed point. Assume that (x, y) and (u, v) are two coupled fixed points of T, then α(x, u) ≥ 1 or α(y, v) ≥ 1. Now, we apply (18) and use the properties of ϕ, ψ 1 , ψ 2 , and F to obtain (54) Hence, we have Similarly, we have Hence, by (m 1 ) x � u and y � v, i.e., (x, y) is the unique coupled fixed point of T.
If we define F(s, t) � s − t and then we get the following corollary which is a generalization of the main results in [31].
□ Corollary 1. Let (X, μ) be an ordered complete M−metric space and T: X × X ⟶ X be an increasing mapping for which there exist ϕ ∈ Φ, ψ 1 ∈ Ψ 1 , and ψ 2 ∈ Ψ 2 such that ψ 1 (t, t) ≤ ϕ(t) and for all (x, y), (u, v) ∈ X 2 with x ≺ u and y ≺ v; we have where Suppose that either (a) T is continuous.
(b) For a convergent sequence x n in (X, μ), we have Discrete Dynamics in Nature and Society x n ≺ x n+1 ⟹x n ≺ x, ∀n, If there exist x 0 , y 0 ∈ X such that x 0 ≺ T(x 0 , y 0 ) and y 0 ≺ T(y 0 , x 0 ), then T has a coupled fixed point. Now, we introduce the following classes of functions Ψ and Φ by If we consider ϕ(t) � ψ(t), ψ 1 (s, t) � ψ(t) for some ψ ∈ Ψ and ψ 2 (s, t) � φ(t) for some φ ∈ Φ, then we obtain an extension of the main result in [13].

Corollary 2.
Let (X, μ) be a complete M−metric space and T: X × X ⟶ X be an α−admissible mapping such that x n ⟶ y⟹α(x, y) ≥ 1.

(63)
If there exist x 0 , y 0 ∈ X such that α(x 0 , T(x 0 , y 0 )) ≥ 1 and α(y 0 , T(y 0 , x 0 )) ≥ 1, then T has a coupled fixed point. Remark 1. Notice that in [32,33], it was shown that each coupled fixed-point theorem can be observed from the analogue of single/standard fixed-point theorems. On the other hand, for the usage of it in application, the coupled fixed-point theorem can be used to handle the problem. erefore, in this paper, we consider the coupled fixed-point results, eorem 2 and eorem 3.

Fractional Differential Equations
In this section, we present sufficient conditions for the existence and uniqueness of the solution of coupled systems (2) and (3). Before starting and proving the main results, we need to fix the analytical framework of our considered problem.
Consider the complete M−metric space (X, μ), where X � C(J, R) and μ is defined by In addition, define the operator T: X × X ⟶ X as where Now, we claim that whenever (x, y) ∈ X 2 is a coupled fixed point of the operator T, it follows that x(t) and y(t) solve (2) and (3).

if and only if (x, y) is a solution of FHIE systems (76) and (77).
Proof. Let x and y be a solution of (2) and (3). en, by Lemma 5, we gain that the general solution of (2) has the integral form presented in (76) and the solution of (3) has the form presented in (77). us, x and y satisfy (76) and (77).
As a consequence of Lemma 6, the coupled fixed point of the operator T coincides with the solution of (76) and (77) and then with the solution of (2) and (3).

Concluding Remarks
In this work, we proved some coupled fixed-point results for α−admissible mappings which are F(ψ 1 , ψ 2 )-contractions in a larger structure such as M−metric spaces. Furthermore, we applied aforesaid fixed-point results to investigate the existence of a unique solution for a coupled system of higherorder fractional hybrid differential equations which are equipped with three-point boundary conditions. e respective results have been verified by providing a suitable example.
In fact, the results dealing with solutions of the general systems of fractional differential equations are useful in applications to various problems which are simply modelled by means of these systems.
It is believed that several recent studies (see, for example, [35][36][37][38][39][40][41][42]) on fractional calculus and its widespread applications will possibly motivate further research studies on mathematical modeling and analysis of applied problems along the lines which we have developed in this article.

Data Availability
No data were used to support this study.