Solvability for a Coupled System of Fractional Integrodifferential Equations with m-Point Boundary Conditions on the Half-Line

and Applied Analysis 3 Proof. By Lemma 5, the solution of (12) can be writen as u (t) = c1t α−1 + c2t α−2 + c3t α−3 − I α h (t) . (19) Using the boundary conditions (13), we find that c2 = c3 = 0 and D α−1 u (t) = c1Γ (α) − I 1 h (t) . (20) Now considering the second boundary condition, we have


Introduction
The theory of derivatives and integrals of fractional order has undergone rapid development over the years and played a very important role in modern applied mathematical models of real processes arising in phenomena studied in physics, mechanics, engineering, and so on [1][2][3].Recently, the existence of solutions for coupled systems involving fractional differential equations is one of the theoretical fields investigated by many authors [4][5][6][7][8][9][10][11][12][13].

< Γ(𝑞).
Motivated by [10], in this paper, we consider a coupled system of nonlinear fractional integrodifferential equations on an unbounded domain and more general boundary conditions: where 2 < ,  ≤ 3, 0 <  Integrodifferential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences.In particular, some physical phenomena involving certain type of memory effects are represented by integrodifferential equations [14][15][16][17][18].
However, to the best of our knowledge, no work has been reported on the existence results for coupled system of nonlinear fractional integrodifferential equations on an unbounded domain.
The paper is organized as follows.In Section 2, we recall some basic definitions, notations, and preliminary facts.Section 3 is devoted to the existence results for system of nonlinear fractional integrodifferential equations on an unbounded domain.In Section 4, an example is given to demonstrate the applicability of our results.

Preliminaries
In this section, we will first recall some basic definitions and lemmas which are used in what follows and can be found in [2,19].
Definition 1.The Riemann-Liouville fractional integral of order  > 0 of a function  : (0, ∞) → R is given by provided that the right-hand side is pointwise defined.
Definition 2. The Riemann-Liouville fractional derivative of order  > 0 of a continuous function  : (0, ∞) → R is given by where  = [] + 1, provided that the right-hand side is pointwise defined.
Remark 3. The following properties are well known: − ,  > −1,  > 0. ( Lemma 4. For  > 0, the equation   () = 0 is valid if and only if where  is the smallest integer greater than or equal to .

Main Results
In this section, we prove the existence results for the boundary value problem (2).For convenience we use the following notation: By replacing ,   ,   with ,   ,   , respectively, we can define Δ  .
Lemma 6.Let ℎ ∈ [0, ∞) and Δ  > 0; then, the unique solution of is given by where   (, ) is Green's function given by with Proof.By Lemma 5, the solution of ( 12) can be writen as Using the boundary conditions (13), we find that  2 =  3 = 0 and Now considering the second boundary condition, we have Therefore, the unique solution of the boundary value problem ( 12)-( 14) is where   (, ),   (, ), and (, ) are defined by ( 16), (17), and (18), respectively.The proof is complete.Now, we introduce the following function: where Remark 7. From the definition of   (, ) and   (, ), for any where Let an operator  :   ×   →   ×   be defined by From the definition of operator where (, ) is the beta-function.
Then, the system (2) has a solution. Proof.Take and define a ball At the first step, we prove that the operator  transforms the ball   into itself.For any (, V) ∈   we have In a similar way, we can get Hence, ‖(, V)‖ , ≤  and this shows that   ⊂   .
Next, we show that  :   →   is completely continuous.First, Let (  , V  ) → (, V) as  → ∞ in   .From (32) we have Then, by the Lebsegue dominated convergence theorem and continuity of , we obtain as  → ∞.Therefore, by Remark 7, we have as  → ∞.Similar process can be repeated for In view of (37), by the similar process used in [20], we can easily prove that operator  1 is equicontinuous.Similar process can be repeated for

Conclusion
In the current paper, we have studied the existence results for a coupled system of nonlinear fractional integrodifferential equations with m-point fractional boundary conditions on an unbounded domain.The result obtained in this paper is based on Schauder's fixed point theorem.In order to show the validity of the assumptions made in our result, we also include an illustrative example.
, the problem (2) has a solution if and only if the operator  has a fixed point.