Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative

: Riesz-Caputo fractional derivative refers to a fractional derivative that re ﬂ ects both the past and the future memory e ﬀ ects. This study gives su ﬃ cient conditions for the existence of solutions for a coupled system of fractional order hybrid di ﬀ erential equations involving the Riesz-Caputo fractional derivative. For this motive, the results are obtained via classical results due to Dhage.

As is known to all, Riesz-Caputo derivative refers to a fractional derivative that reflects both the future and the past memory effects; consequently, the Riesz fractional operator plays an important role in characterizing anomalous diffusion, owing to successful applications to subdiffusive, superdiffusive, and evolution problems.Space fractional quantum mechanics is a natural generalization of standard quantum mechanics, which arises when the Brownian trajectories in Feynman path integrals are replaced by Levy flights.The classical Levy flight is a stochastic processes, which in one dimension, is described by a jump length probability density function.The position space representation of the αth power of the momentum operator is given by: where | | D x α is the Riesz fractional derivative operator of order α.The Riesz fractional derivative is regarded as an effective tool for studying nonlocal and memory effects in physics, engineering, and applied sciences.Therefore, many scholars are engaged in the study of the solutions of differential equations with the Riesz fractional derivative.Many works carried out so far discuss the numerical solutions of diffusion equations, which contain the Riesz derivative, and fractional variational problems, which contain the Riesz-Caputo derivative, and only a few works report on the existence results of fractional boundary value problems, which contain the Riesz-Caputo derivative.
Gu et al. [12] discussed a new class of differential equations that contains the Riesz-Caputo fractional derivative: On the other hand, hybrid differential equation is a class of dynamical systems with quadratic perturbation.Nowadays, due to the wide range of application of hybrid differential equations in several areas of reallife problems, more researchers began to study the existence of solutions for hybrid differential equations with different perturbations (readers can refer to [13][14][15][16][17][18][19][20]).
Houas [21] studied the results for the coupled hybrid system that contains integral boundary conditions as follows: , Recently, Baleanu et al. [23] discussed the hybrid fractional coupled system as follows: , By virtue of the aforementioned documents, in this work, the authors present the solutions for the hybrid fractional coupled system that contains the Riesz-Caputo derivative as follows: (3) where , and and a a b , , 1 2 1 , and b 2 are the real constants.We will use "Dhage's fixed point theorem" to find the sufficient conditions for the existence of solutions.This is the first time for us to present the existence of solutions of Systems (1)-(4).

Some lemmas
for > α 0, and [ ] = n ω and [ ] ⋅ stands for the ceiling of a number.
[24] The Riesz-Caputo fractional derivative for a function ( ) ≤ ≤ x t t T , 0 , can be written as follows: Definition 2.2.[24] The right, left, and fractional Riemann-Liouville integrals of order α can be written as follows: Solutions of a system of hybrid fractional boundary value problems  3 From the aforementioned equations, thus we have Lemma 2.2.[25] Let S be a closed convex, bounded, and nonempty subset of a Banach algebra E, and let → A C E E , : and → B S E : be three operators such that (a) A and C are Lipschitzian with Lipschitz constants δ and ρ, respectively, (b) B is compact and continuous, Then, there is a solution for the operator equation

Main results
The Banach space [ ] C 0, 1 is recorded as X.The norm is defined as follows: is the solution of the boundary value problem: Proof.Integrating on both sides of equation ( 7) and considering ( 5), (8), and ( 9), we obtain , 0, , Consequently, we find that Therefore, (6) holds.□ Define the operator where , 0, , We give the hypotheses of this study.
( ) are continuous functions; ( ) , ¯, ¯¯¯, Solutions of a system of hybrid fractional boundary value problems  5 for all [ ] ∈ t x x y 0, 1 , , ¯, , and y ¯are the elements in R; ( ) and four continuous nondecreasing functions (17) for all [ ] ∈ t x 0, 1 , and y are the elements in R; ( ) H 4 The positive number r satisfies where and , 0, 0 , 0, 0 .
Then, S is a bounded convex closed subset of E. Define operators → A C E X , : (23) 0, , 0, , Then, operators T 1 and T 2 given by equations ( 11) and ( 12) are equivalent to (27) (28) Therefore, the operator T in equation ( 10) can be written as follows:

y t T x y t T x y t A x t y t B x t y t C x t y t
2 , and , . 1 2 In the following, we prove that the conditions in Lemma 2.2 can be satisfied.
First, we prove that A and C are Lipschitz on the space E. Given ) ( ) ∈ x y x y E , , , , and

A x y t A x y t g t x t y t g t x t y t ϕ t x t x t y t y t
, , Similarly, From equations ( 29) and (30), we obtain

A x y t A x y t A A x y A A x y ϕ t ϕ t x t x t y t y t
, , , , , , for all t on the interval [ ] 0, 1 .Taking the supremum on both sides of the aforementioned formula with respect to t, we have which implies that A is Lipschitz on the space E with Lipschitz constant We consider x y E : , , , , ,

C x y t C x y t f t x t y t f t x t y t ϕ t x t x t y t y t
, , , , , , . (32) From equations ( 31) and (32), we obtain  0, , 0, , (36) with Λ given in equation (21), which implies that the set ( ) B S is a uniformly bounded set in S.
Similarly, we obtain Taking the supremum on both sides of the aforementioned formula with respect to t, Similarly, we obtain (40) Thus, Therefore, we have Fourth, we show that x y S t , 0 , 1 using ( ) H , 4 we find Therefore, all the conditions of Lemma 2.2 are satisfied, so that ( )( ) ( ) ( ) ( ) = + T x y t A x y B x y C x y , , , , has a coupled fixed point in S; consequently, Problems (1)-( 4) have a solution defined on [0,1].□

Example
We consider the hybrid boundary value systems with the Riesz-Caputo derivative:    We choose = r 10; by calculation = = ρ Λ 0. So far, we have proved that all conditions in Theorem 3.2 are satisfied, which implies that Problems (42)-( 45) have a solution.

Conclusion
We have presented the existence results of a coupled system of hybrid boundary value problems with the Riesz-Caputo derivative with the aid of the Dhage fixed point theorem.The novelty of this kind of fractional derivative is Riesz-Caputo fractional derivative that can reflect both the past and the future memory effects.

0
stand for the Caputo fractional derivative.

μ
, and D ν stand for the Caputo's fractional derivative.