1. Introduction

In recent years, the theory of differential fractional equations has become an interesting field to explore. It is to be noted that such theory has many applications in several events existing in the real world, and also in many sciences such as: engineering, physics, chemistry, biology, etc ..., (¹³). Moreover, the study of the systems of fractional differential equations has become more and more popular tool for controlling and modeling different systems (²), (⁷), (¹⁵)-(¹⁷). Thus the fixed point theory is a powerful mathematical tool in the study of the existence, uniqueness, positivity and stability of solutions, see (¹), (³)- (⁶), (⁹)-(¹⁴).

In this work, we consider the following system of fractional differential equations with boundary conditions:

where denotes the Reimann-Liouville fractional derivative, 2<α <3, u= ( ^{_{u1, u2,}} ..., ^_un ) ^T is an unknown function with

This paper is organized as follows: in Section 2, some preliminary materials to be used later are stated. In Section 3, we present and prove our main results consisting of the existence and uniqueness of the solution of (FS). Finally our study is ended by an example illustrating the obtained results.

2. Preliminaries

In this section, we recall the basic definitions and lemmas from the fractional calculus theory, see (¹³).

Definition 1. The Riemann-Liouville fractional integrals of order α of a function h is defined as

Definition 2. The Riemann-Liouville derivative of fractional order α >0 for a function h is defined as

where n= α +1 ( α denotes the integer part of the real number α ).

Lemma 3. For α >0, the general solution of the homogeneous equation

is given by

where ^_ci , i=1,2,...,n-1, are arbitrary real constants.

Lemma 4. Let p, q ≥ 0, ʄ ∈ L ₁ a,b . Then

3. Main results

Lemma 5. Let y ∈ C ( 0,1 ,R). Assume that a, b ∈ R such that a - b (α - 1) ≠ 0, then for i ∈ {1,..,n} , the linear nonhomogeneous problem

has the following solution

where

Proof. Let ^_ui be a solution of the fractional boundary value problem (FS). Using Lemma 3, we obtain

then, by multiplying ( 3.4) by ^_t3-α , it yields

According to the condition u(0) =0, we obtain C = 0. Therefore, differentiating (3.4), we have

Multiplying (3.5) by ^_t3-α , we obtain

From condition ^_u´i (0) =0, it follows B = 0, thus,

Since ^_aui (1) - ^_bu´i (1) =0, then

By substituting A in (3.7), we get

^_ui (t) =∫₀ ^{1 _Gi} (t, s) y (s) ds.

Lemma 6. If a>0 and b<0, then the functions ^_Gi are nonnegative, continuous

and

Proof. The proof is direct, we omit it.

Let X be the Banach space of all functions u ∈ _^Cn 0,1 =C [0,1 ×...×C 0,1 with the norm ||.|| defined by ||u|| = Define the integral operator T : X → X by T (u) = ( ^{_{T1u, T2u,...,Tnu}} ), where

Lemma 7. The function u ∈ X is a solution of the system (FS) if and only if ^_Tiu (t) = u (t), for all t ∈ 0,1 , ∀i ∈ {1,...,n} .

The first main statement in this work is the uniqueness of solution of the boundary problem (FS).

Theorem 8. Assume that

i) ^_ʄi ∈ C (R ⁿ, R) ,g ∈ L ¹ ( 0,1 , R)

ii) There exists a constant L>0 such that

and

for all t ∈ 0,1 and for all ^{_{xi, yi}} ∈ }, i =1,...n. Then, the boundary value problem (FS) has a unique solution in X.

Proof. We will use the Banach contraction principle to prove that the operator T has a fixed point. Using the properties of the function ^_Gi , it yields

then by taking the maximum over t ∈ 0,1 , it follows

Summing the n inequalities in ( 3.13), it yields

Since <1, then T is a contraction. As a consequence of Banach fixed-point theorem, we deduce that T has a fixed point that is the unique solution of the (FS), this achieves the proof.

The second mains statement of this work is an existence result for the boundary problem (FS).

Theorem 9. Assume that ^_ʄi (0) ≠ 0, i ∈ {1,..,n} ,there exist η >0 and a nonnegative function ψ ∈ C (R ⁿ, (0, ∞)) satisfying ψ( ^_x1,...,xn ) ≤ ψ ( ^_y1,...,yn ) for 0≤ ^_xi ≤ ^_yi , i =1,...,n. If

for all t ∈ 0,1 and all u ∈ R ⁿ and

then, the problem (FS) has at least one nontrivial solution u ^* ∈ X.

For the proof of Theorem we need the nonlinear alternative of Leray-Schauder:

Lemma 10. Let F be a Banach space and Ω a bounded open subset of F, 0 ∈ Ω . Let T: Ω → F be a completely continuous operator. Then, either there exists x ∈ ∂Ω, λ >1 such that T(x) = λ x, or there exists a fixed point x ∈ Ω of T.

Proof. of Theorem 9. The continuity of the operator T follows from the continuity of ʄ. Set ^_Bη = {u ∈ X: ||u|| ≤ η} . Let us prove that T: ^_Bη → X is a completely continuous operator. From (3.14), we have for each t ∈ 0,1

Taking the supremum over 0,1 , then summing the obtained inequalities according to i from 1 to n, we get

which implies that T( ^_Bη ) is uniformly bounded.

Let us show that (Tu) is equicontinuous, u ∈ ^_Bη . Let ^_t1 , ^_t2 ∈ 0,1 , ^_t1 < ^_t2 , then

then

As ^_t1 → ^_t2 , the right-hand side of the above inequality tends to zero. By Ascoli-Arzela theorem, we conclude that the operator T :X → X is completely continuous.

Now we apply the nonlinear alternative of Leray-Schauder. Let u ∈ ∂ ^_Bη , such that u = λ Tu for some 0<λ <1. We have

Taking the supremum over 0,1 , then summing the obtained inequalities according to i from 1 to n, we get

taking into account (3.15) we conclude

that contradicts the fact that u ∈ ∂ ^_Bη . So, we conclude that T has at least one fixed point u ^* ∈ ^_Bη and then the (FS) has a nontrivial solution ^_u* ∈ ^_Bη .

4. Examples

In this section, we give examples to illustrate the usefulness of our main results.

Example 1. Consider the following two-dimensional fractional order system

Then, according to the Theorem 9, the boundary value problem (4.1) has at least one fixed point ^_u* ∈ ^_B2 .

Example 2. Consider the following two-dimensional fractional order system

Theorem 8 are satisfied. So, the boundary value problem (4.2) has a unique solution u ∈ X.