Investigating a Coupled Hybrid System of Nonlinear Fractional Differential Equations

Fractional calculus is found to bemore practical and effective than the classical calculus in the mathematical modeling of several phenomena. Fractional differential equations are very important and significant part of the mathematics and have various applications in viscoelasticity, electroanalytical chemistry, and many physical problems [1–6]. A systematic presentation of the applications of fractional differential equations can be found in the book of Balachandran and Park [7]. In recent years, many works have been devoted to the study of the mathematical aspects of fractional order differential equations [8–12]. There are numerous advanced and efficient methods, which have been focusing on the existence of solution to fractional differential equations. One of the powerful tools for obtaining the existence of solutions to such equations is the fixed point methods. Many authors use fixed point theorems to prove the existence and uniqueness of solution to nonlinear fractional differential equations; see, for example, [13–17]. On the other hand, the study for coupled systems of fractional differential equations is also important as such systems occur in various problems of applied nature, for instance, [18–25]. Additionally, fixed point theory can be used to develop the existence theory for the coupled systems of fractional hybrid differential equations [13, 16, 17]. Bashiri et al. [17] discussed the existence of solution to the following system of fractional hybrid differential equations of order p ∈ (0, 1): Dp [x (t) − f (t, x (t))] = g (t, y (t) , Iαy (t)) , a.e. t ∈ [0, T] , T > 0, Dp [y (t) − f (t, y (t))] = g (t, x (t) , Iαx (t)) , a.e. t ∈ [0, T] , T > 0, x (0) = 0, y (0) = 0, (1)


Introduction
Fractional calculus is found to be more practical and effective than the classical calculus in the mathematical modeling of several phenomena.Fractional differential equations are very important and significant part of the mathematics and have various applications in viscoelasticity, electroanalytical chemistry, and many physical problems [1][2][3][4][5][6].A systematic presentation of the applications of fractional differential equations can be found in the book of Balachandran and Park [7].In recent years, many works have been devoted to the study of the mathematical aspects of fractional order differential equations [8][9][10][11][12].There are numerous advanced and efficient methods, which have been focusing on the existence of solution to fractional differential equations.One of the powerful tools for obtaining the existence of solutions to such equations is the fixed point methods.Many authors use fixed point theorems to prove the existence and uniqueness of solution to nonlinear fractional differential equations; see, for example, [13][14][15][16][17].

Preliminaries
provided that the right side is pointwise defined on (0, ∞).

Definition 2.
Let  be a positive real number, such that −1 ≤  < ,  ∈ N, and   () exists, a function of class .Then Caputo fractional derivative of  is defined as provided that the right side is pointwise defined on (0, ∞), where  = [] + 1 and [] represents the integer part of .
Proof.Set  = ([0, 1], R) and a subset  of  defined by Clearly  is a closed, convex, and bounded subset of the Banach space .Now, since () is a solution of the FHDEs system (2) if and only if () satisfies the system of integral equations in Lemma 6, to show the existence solution of system (2) it is enough to show the existence solution of the integral equations in Lemma 6.For this, define two operators  :  →  and  :  →  by −1  (,  () ,    ()) . ( Then the operators form of system (2) is We have to show that the operators  and  satisfy all the conditions of Theorem 4. For this, let ,  ∈ ; then we have Taking the supremum over  and using (7) Thus  satisfies condition ( 1 ) of Theorem 4 with  = 1/2 and () = 3/4( + ) +  2 /4( +  2 ).
Next, we show that  is compact and continuous operator on .Let {  } be a sequence in S such that {  } →  ∈ .Then for all  ∈ [0, 1], we have lim −1  (,  () ,    ())  =  () . ( Thus the map  is continuous on .Let  ∈ ; then we have Taking the supremum over , we get Thus  is uniformly bounded on .Now, let  1 ,  2 ∈ [0, 1] such that  1 ̸ =  2 ; then for any  ∈ , we have Since   is uniformly continuous on [0, 1] for  − 1 <  < , for any  > 0 we can find  * 1 > 0 such that Taking the supremum over  on [0, 1], we can write That is,  ∈ .Thus condition ( 3 ) of Theorem 4 holds.Therefore, all the conditions of Theorem 4 are satisfied; hence the operator (, ) = + has a coupled fixed point on S.
To illustrate Theorem 7, we construct the following example.

Conclusion
We have successfully developed appropriate conditions for existence of at least one solution to a complicated higher order coupled system of nonlinear hybrid fractional differential equations.The respective conditions have been derived by using coupled fixed point theorem of Krasnoselskii type.
The obtained results were also demonstrated by a suitable example.