Fractional p -Laplacian Coupled Systems with Multi-Point Boundary Conditions

: This article is allocated to the existence and uniqueness of solutions for a system of nonlinear differential equations consisting of the Caputo fractional-order derivatives. Our main results are proved via standard tools of ﬁxed point theory. Finally, the presented results are clariﬁed by constructing some examples.


Introduction
Fractional differential equations (FDEs) have been applied to qualify multiplex problems in various fields of natural sciences, as physics, biology, chemistry, image processing and so on, see [1][2][3][4][5].In the literature, different types of fractional derivatives and fractional integrals have been presented, where many of those definitions deal with the Riemann-Liouville and Caputo fractional derivatives.These definitions have been applied in many nonlocal boundary problems to present some physical phenomena (see [6][7][8][9]).Additionally, FDEs with a p-Laplacian operator appeared when Leibenson [10] was working to deduce an exact formula to construct a turbulent flow.Recently, many applications of FDEs with p-Laplacian operator have been obtained in the various areas such as glaciology and dynamics, see [11][12][13][14][15] and references therein.Hence, it is important to consider FDEs consisting of the p-Laplacian operator in different spaces.Recently, Srivastava et al. [16] have applied the p-Laplacian operator to consider the existence of solutions for a class of nonlinear differential equations of the form supplemented with coupled nonlocal boundary conditions in which κ : [c, d] × E −→ E is a given function, E is a Banach space with zero element θ, 0 < ρ ≤ 1, 1 < µ ≤ 2, λ i , i = 1, 2, . . ., m are real constants, C D ρ , denote the Caputo fractional derivatives, θ p indicates a p-Laplacian operator, that is The aim of this work is to extend the problem ( 1) and (2) on coupled system of differential equations by introducing a new coupled system of FDEs.The significance of coupled systems of FDEs is that such systems have appeared in various fields of sciences, see [17][18][19].In this paper, the existence of solutions for a system of nonlinear differential equations consisting of the Caputo fractional-order derivatives of the form supplemented with coupled nonlocal boundary conditions is investigated, in which 0 < ρ 1 , ρ 2 ≤ 1, 1 < µ 1 , µ 2 ≤ 2, C D δ denotes the Caputo fractional derivative of order δ ∈ {ρ 1 , ρ 2 , µ 1 , µ 2 }, θ p (p = p 1 , p 2 ) indicates a p-Laplacian operator, are continuous functions with some conditions, λ i , λ j , i = 1, 2, . . ., m, j = 1, 2, . . ., n are real constants and c 1 < η i , η j < d 1 .
The tools of fixed point theory will be applied to develop the existence theory for the problem (3) and (4).In fact, the Banach contraction mapping principle is applied to prove a uniqueness result, while two existence results are derived via Leray-Schauder alternative and Krasnosel'ski ȋ fixed point theorem.The presented system in this article is not only new, but some other systems are special cases of this system, for example, by taking µ 1 = µ 2 = p 1 = p 2 = 2 and ρ 1 = ρ 2 = 1, we have the third order system of the form (5) The structure of this article is as follows: In the next section, some definitions and basic Lemmas are collected, which will be required to prove the main results.In Section 3, an auxiliary lemma is proved to convert the problem (3) and ( 4) into a fixed point problem.The existence and uniqueness results are considered in Section 4. In Section 5, some examples are included to illustrate the obtained results.

Preliminaries
In this section, some results and concepts concerning the fractional calculus are presented.Definition 1 ([2,20]).The fractional integral of Riemann-Liouville type and order η > 0 of a function where Γ is the Gamma function.

An Auxiilliary
Then, the unique solution of the system supplemented with coupled nonlocal boundary conditions is given by where . Then, the system (7) can be divided into two problems: supplemented with coupled nonlocal boundary conditions The solution of the system (12) can be written as φ 1 (w) = RL I ρ 1 h 1 (w) and φ 2 (w) = RL I ρ 2 h 2 (w), respectively.On the other hand, by applying the Lemma 1 and the fractional integrals RL I µ 1 and RL I µ 2 on both sides of the equations in (13), we obtain Now, by applying the boundary conditions σ 1 (c 1 ) = σ 2 (c 1 ) = 0 in (15), we obtain e 0 = f 0 = 0. Hence, we have By the boundary conditions ) and ( 16), we obtain Consequently, By solving the above system, we obtain .
Replacing e 1 and f 1 in ( 16), we have the solution (9).The converse can be proved by direct computation.This completes the proof.

Existence and Uniqueness Results
Let X * = {w(q) : w(q) ∈ C([c 1 , d 1 ], R)} be the Banach space of all continuous functions from [c 1 , d 1 ] into R, which has been equipped with the norm The space (X * , . ) is a Banach space.In addition, the product space (X * × X * , . ) is also a Banach space with the norm (v, and For computational convenience, we set The following lemma is used in the sequel.

Uniqueness Result
In the following theorem, the Banach contraction mapping principle is applied to establish the existence and uniqueness result for the system (3) and (4).
Proof.The hypotheses of Banach's contraction mapping principle will be considered in the following steps: Step Step 2. B is a contraction.
For Step 1, let (v, w) ∈ B x .Then, we obtain Similarly, we can show that Consequently, we obtain Hence, Similarly, one can find that By ( 23) and ( 24), we infer that which implies that B is a contraction.Thus, by applying the Banach contraction mapping principle, the system (3) and ( 4) has a unique solution on [c 1 , d 1 ].The proof is completed.
Proof.In view of the continuity property of the functions κ 1 and κ 2 , we conclude that the operator B is continuous.Next, the completely continuous property of the operator B is showed.Let S be a bounded set of X * × X * .Then, there exist positive constant D 1 and D 2 such that for all (v, w) ∈ S we have |κ 1 (q, v(q), w(q))| ≤ D 1 and |κ 2 (q, v(q), w(q))| ≤ D 2 .
In consequence, for all (u, v) ∈ S, we have which yields Similarly, we have Hence, we obtain Consequently, the uniformly boundedness property of the operator B is obtained.Now the equicontinuous property of the operator B is verified.Let q1 , q2 ∈ [c 1 , d 1 ] with q1 < q2 , Then, we obtain Finally, it will be indicated that the set w) and for all q ∈ [c 1 , d 1 ] we have v(q) = λB 1 (v, w)(q) and w(q) = λB 2 (v, w)(q).Thus, we have and consequently Similarly, we obtain Hence, we have where Thus, the Leray-Schauder alternative implies that the operator B has at least one fixed point.Hence, the system (3) and ( 4) has at least one solution on [c 1 , d 1 ].The proof is completed.Now, Krasnosel'ski ȋ's fixed point theorem [22] is applied to prove the second existence result.
Then, the problem (3) and ( 4) has at least one solution on [c 1 , Proof.We split the operator B into four operator According to the proof of Theorem 2, we have Next, it will be proved that the operator (B 1,2 , B 2,2 ) is a contraction.As in the proof of Theorem 1, for (v 1 , w 1 ), (v 2 , w 2 ) ∈ B x we have and consequently In a similar way, we obtain By ( 28) and (29), we obtain which, by applying the condition (27), the contraction property of the operator (B 1,2 , B 2,2 ) is obtained.On the other hand, in view of the continuity property of κ 1 and κ 2 , the operator (B 1,1 , B 2,1 ) is continuous.In addition, Thus, (B 1,1 , B 2,1 ) is uniformly bounded.In the final step, we prove that the operator (B 1,1 , B 2,1 )B x is equicontinuous.For q1 , q2 ∈ [c 1 , d 1 ] with q1 < q2 and for all (v, w) ∈ B x , we have ) is compact on B x and hence by applying Krasnosel'ski ȋ's fixed point theorem, we conclude that the operator B has at least one fixed point, which is a solution of the system (3) and ( 4).The proof is finished.

Examples
In this section, we illustrate our main results by considering the following system of nonlinear differential equations consisting of the Caputo fractional-order derivatives of the form with coupled boundary conditions as Comparing problem (31) and ( 32) with ( 3) and ( 4), we have and Observe that both of them are unbounded and also satisfy the Lipschitz condition as

Conclusions
In this paper, the existence and uniqueness results have been established for a fractional p-Laplacian coupled system of nonlinear differential equations involving the Caputo fractional-order derivatives, subjected to multi-point boundary conditions.The fixed point technique has been applied to the given system by transforming it into a fixed point problem.Our problem is novel and some other problems are special cases of our problem.For example, by taking µ 1 = µ 2 = p 1 = p 2 = 2 and ρ 1 = ρ 2 = 1, our problem corresponds to the problem (5).For future work, we plan to extend the results of this paper to other kinds of boundary value problems.