Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions

This paper ismotivated by some papers treating the fractional hybrid differential equations with nonlocal conditions and the system of coupled hybrid fractional differential equations; an existence theorem for fractional hybrid differential equations involving Caputo differential operators of order 1 < α ≤ 2 is proved under mixed Lipschitz and Carathéodory conditions. The existence and uniqueness result is elaborated for the system of coupled hybrid fractional differential equations.


International Journal of Differential Equations
Fractional differential equations are a generalization of ordinary differential equations and integration to arbitrary noninteger orders.The origin of fractional calculus goes back to Newton and Leibniz in the seventeenth century.It is widely and efficiently used to describe many phenomena arising in engineering, physics, economy, and science.There are several concepts of fractional derivatives, some classical, such as Riemann-Liouville or Caputo definitions.For noteworthy papers dealing with the integral operator and the arbitrary fractional order differential operator, see [1][2][3][4][5][6][7].
The quadratic perturbations of nonlinear differential equations have attracted much attention.We call such fractional hybrid differential equations.There have been many works on the theory of hybrid differential equations, and we refer the readers to the articles [8][9][10][11][12].
Dhage and Lakshmikantham [11] discussed the following first order hybrid differential equation where  ∈ ( × R, R \ {0}) and  ∈ C( × R, R).They established the existence, uniqueness results, and some fundamental differential inequalities for hybrid differential equations initiating the study of theory of such systems and proved, utilizing the theory of inequalities, the existence of extremal solutions and comparison results.Zhao et al. [13] have discussed the following fractional hybrid differential equations involving Riemann-Liouville differential operators: where The authors of [13] established the existence theorem for fractional hybrid differential equation and some fundamental differential inequalities.They also established the existence of extremal solutions.Hilal and Kajouni [14] have studied boundary fractional hybrid differential equations involving Caputo differential operators of order 0 <  < 1 as follows: where  ∈ ( × R, R \ {0}),  ∈ C( × R, R) and , , and  are real constants with  +  ̸ = 0.They proved the existence result for boundary fractional hybrid differential equations under mixed Lipschitz and Carathéodory conditions.Some fundamental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions.Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions.
The nonlocal condition is a condition attached to the main equation; it replaces the classic nonlocal condition in order to model physical phenomena of the fashion nearest from reality.The nonlocal condition involves the function where   ,  = 1, 2, . . ., , are given constants and 0 < Let us observe that Cauchy problems with nonlocal conditions were initiated by Byszewski and Lakshmikantham [2] and, since then, such problems have also attracted several authors including A. Aizicovici, K. Ezzinbi, Z. Fan, J. Liu, J. Liang, Y. Lin, T.-J.Xiao, G. N'Guérékata, E. Hernàndez, and H. Lee (see [2,15]).

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
By  = (,R) we denote the Banach space of all continuous functions from  = [0, ] into R with the norm And let C( × R, R) denote the class of functions  :  × R → R such that (i) the map   → (, ) is measurable for each  ∈ R, (ii) the map   → (, ) is continuous for each  ∈ .
The class C( × R, R) is called the Carathéodory class of functions on  × R which are Lebesgue integrable when bounded by a Lebesgue integrable function on .
By  1 (; R) we denote the space of Lebesgue integrable real-valued functions on  equipped with the norm ‖ ⋅ ‖  1 defined by Definition 1.The fractional integral of the function ℎ ∈  1 ([, ], R + ) of order  ∈ R + is defined by where Γ is the gamma function.
Definition 2. For a function ℎ given on the interval [, ], the Caputo fractional order derivative of ℎ is defined by where  = [] + 1 and [] denotes the integer part of .

Existence Result
In this section, we prove the existence results for the boundary value problems for hybrid differential equations with fractional order (1) on the closed and bounded interval  = [0, ] under mixed Lipschitz and Carathéodory conditions on the nonlinearities involved in it.We defined the multiplication in  by () () =  ()  () , for ,  ∈ .
Clearly,  = (; R) is a Banach algebra with respect to above norm and multiplication in it.
We prove the existence of solution for the BVPHDEFNL (1) by a fixed point theorem in Banach algebra due to Dhage [10].
Lemma 6 (see [10]).Let  be a nonempty, closed convex, and bounded subset of the Banach algebra  and let  :  →  and  :  →  be two operators such that for all  ∈  and ,  ∈ R.
As a consequence of Lemmas 3 and 4 we have the following result which is useful in what follows.
Proof.We defined a subset  of  by where  =  0 ((2  /Γ( + 1))‖ℎ‖ It is clear that  satisfies hypothesis of Lemma 6.By an application of Lemma 7, ( 1) is equivalent to the nonlinear hybrid integral equation Then the hybrid integral equation ( 25) is transformed into the operator equation as We will show that the operators  and  satisfy all the conditions of Lemma And since L is a continuous function lim for all  ∈ .This shows that  is a continuous operator on .
Thus, all the conditions of Lemma 6 are satisfied and hence the operator equation  =  has a solution in .As a result, BVPHDEFNL (1) has a solution defined on .This completes the proof.
In view of Lemma 7, we define an operator Φ : where International Journal of Differential Equations 7 In the sequel, we need the following assumptions: (H  1 ): the functions   are continuous and bounded; that is, there exist positive numbers (H  2 ): there exist real constants  0 ,  0 > 0 and   ,   ≥ 0 For brevity, let us set Now we present our result for the existence and uniqueness of solutions for problem (49).This result is based on Banach's contraction mapping principle.
In our second result, we discuss the existence of solutions for problem (49) by means of Leray-Schauder alternative.
Lemma 10 (see [17]).Let F : J → J be a completely continuous operator (i.e., a map that is restricted to any bounded set in  is compact).Let P(F) = { ∈ J :  = F   0 <  < 1}.Then either the set P(F) is unbounded or F has at least one fixed point.

Theorem 11. Assume that conditions (𝐻
where  1 and  2 are given by (52).Then the boundary value problem (49) has at least one solution.
Proof.We will show that the operator Φ : K × R → K × R satisfies all the assumptions of Lemma 10.In the first step, we prove that the operator Φ is completely continuous.Clearly, it follows by the continuity of functions  1 ,  2 ,  1 , and  2 that the operator Φ is continuous.
Let S ⊂ K × R be bounded.
which yields In a similar manner, We deduce that the operator Φ is uniformly bounded.Now we show that the operator Φ is equicontinuous.We take  1 ,  2 ∈ [0, 1] with  1 <  2 and obtain International Journal of Differential Equations 9 which tend to 0 independently of (, ).This implies that the operator Φ(, ) is equicontinuous.Thus, by the above findings, the operator Φ(, ) is completely continuous.In the next step, it will be established that the set P = {(, ) ∈ K × R/(, ) = Φ(, ), 0 ≤  ≤ 1} is bounded.
6.Claim 2 (we show that  is continuous in ).Let (  ) be a sequence in  converging to a point  ∈ .Then by Lebesgue dominated convergence theorem, for all  ∈ .Taking supremum over , we obtain      −      ≤       −      ,(29)for all ,  ∈ .