Uniqueness and Existence of Solution for a System of Fractional q-Difference Equations

and Applied Analysis 3 Definition 3 (see [48]). The fractional q-derivative of the Riemann-Liouville type of order α ≥ 0 is defined by ( RLD 0 q f)(x) = f(x) and ( RLD α q f) (x) = (D [α] q I [α]−α q f) (x) , α > 0, (15) where [α] is the smallest integer greater than or equal to α. Definition 4 (see [48]). The fractional q-derivative of the Caputo type of order α ≥ 0 is defined by ( C D α q f) (x) = (I [α]−α q D [α] q f) (x) , α > 0, (16) where [α] is the smallest integer greater than or equal to α. Lemma 5. Let α, β ≥ 0 and let f be a function defined on [0, 1]. Then the next formulas hold: (1) (I q I α q f)(x) = (I α+β q f)(x), (2) (D q I α q f)(x) = f(x). Lemma 6 (see [42]). Let α ≥ 0 and n ∈ N. Then the following equality holds: ( RLI α q RL D n q f) (x) = RLD n q RL I α q f (x) − α−1 ∑ k=0 x α−n+k Γ q (α + k − n + 1) (D k q f) (0) . (17) Lemma 7 (see [48]). Let α > 0 and n ∈ R \ N. Then the following equality holds: (I α q C D α q f) (x) = f (x) − [α]−1 ∑ k=0 x k Γ q (k + 1) (D k q f) (0) . (18) For convenience, one introduces the following notations: b 1 = γ 1 (α 2 − γ 2 ) Δ , b 2 = γ 1 (α 2 + β 2 − γ 2 η 2 ) Δ , b 3 = (α 1 − γ 1 ) Δ , b 4 = (β 1 + γ 1 η 1 ) Δ , Δ = (γ2 − α2) (β1 + γ1η1) + (γ1 − α1) (α2 + β2 − γ2η2) . (19) From Lemmas 5 and 7, we can obtain the following lemma. Lemma 8. Let h ∈ C[0, 1] and Δ ̸ = 0; then the unique solution of the linear fractional boundary value problem C D α q u (t) = h (t) , 1 < α ≤ 2, t ∈ [0, 1] , α 1 u (0) − β1Dqu (0) = γ1u (η1) , α 2 u (1) + β2Dqu (1) = γ2u (η2) (20) is given by u (t) = ∫ t 0 (t − qs) (α−1)

In the last few years, fractional differential equations (in short FDEs) have been studied extensively.The motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering.For an extensive collection of such results, we refer the readers to the monographs by Kilbas et al. [1], Miller and Ross [2], Oldham and Spanier [3], Podlubny [4], and Samko et al. [5].
Some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator has been discussed by ), Babakhani and Daftardar-Gejji ( [9][10][11]), Bai [12], and so on.Also, there are some papers which deal with the existence and multiplicity of solutions (or positive solution) for nonlinear FDE of BVPs by using techniques of nonlinear analysis (fixed-point theorems, Leray-Shauder theory, topological degree theory, etc.)-see ( [13][14][15][16][17][18]) and the references therein.The study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications.The reader is referred to the papers ( [19][20][21][22]) and the references cited therein.
The pioneer work on -difference calculus or quantum calculus dates back to Jackson's papers ( [23,24]), while a systematic treatment of the subject can be found in [25,26].For some recent existence results on -difference equations, see [27][28][29] and the references therein.
There has also been a growing interest on the subject of discrete fractional equations on the time scale Z.Some interesting results on the topic can be found in a series of papers [30][31][32][33][34][35][36][37][38].Fractional -difference equations have recently attracted the attention of several researchers.For some earlier work on the topic, we refer to [39,40], whereas some recent work on the existence theory of fractional 2 Abstract and Applied Analysis -difference equations can be found in [41][42][43][44][45].However, the study of boundary value problems of fractional -difference equations is at its infancy and much of the work on the topic is yet to be done.
From the above works, we can see a fact, although the fractional boundary value problems have been investigated by some authors.To the best of our knowledge, there have been few papers which deal with problem (1) for nonlinear fractional differential equation.Motivated by all the works above, in this paper we discuss problem (1).Using nonlinear alternative of Leray-Schauder type, we will give the existence and uniqueness of solution for a system of fractional differential equations with Riemann-Liouville integral boundary conditions of different order (1).
The paper is organized as follows.In Section 2, we give some preliminary results that will be used in the proof of the main results.In Section 3, we establish the uniqueness and existence of a solution for the nonlinear fractional differential equation boundary value problem (1).In last section, we give two examples to illustrate our work.

Preliminaries and Lemmas
In this section, we cite some definitions and fundamental results of the -calculus as well as of the fractional -calculus ( [46,47]).We also give a lemma that will be used in obtaining the main results of the paper.
The -derivative of a function  is here defined by and -derivatives of higher order by The -integral of a function  defined in the interval [0, ] is given by If  ∈ [0, ] and  is defined in the interval [0, ], its integral from  to  is defined by Similarly, as done for derivatives, an operator    can be defined, namely, by The fundamental theorem of calculus applies to these operators   and   ; that is, and if  is continuous at  = 0, then Basic properties of the two operators can be found in the book that is mentioned in [8].We now point out three formulas that will be used later (    denotes the derivative with respect to variable ) [43]: Remark 1.We note that if  > 0 and  ≤  ≤ , then ( − ) () ≥ ( − ) () [43].
Definition 4 (see [48]).The fractional -derivative of the Caputo type of order  ≥ 0 is defined by where [] is the smallest integer greater than or equal to .
Lemma 8. Let ℎ ∈ [0, 1] and Δ ̸ = 0; then the unique solution of the linear fractional boundary value problem is given by The following lemma is fundamental in the proofs of our main result.
Lemma 9 (see [49]; nonlinear alternative of Leray-Schauder type).Let  be a Banach space with  ⊆  closed and convex.Assume that  is a relatively open subset of  with 0 ∈  and  :  →  is continuous, compact (i.e., () is a relatively compact subset of ) map.Then either (i)  has a fixed point in  or (ii) there exist  ∈  and  ∈ (0, 1) with  = .

Main Results
In this section, we will discuss the uniqueness and existence of solutions for boundary value problem (1).
First of all, we define the Banach space For convenience, we set and let  ̸ = 0. Note Employing Lemma 8, system (1) can be expressed as where  1 ,  2 ,  3 ,  4 are given by ( 21), and b 1 , b 2 , b 3 , b 4 are given by ( 22) From Lemma 8 in Section 2, we can obtain the following lemma.
Let (, V) ∈ ×; define an operator  :  ×  →  ×  as where then, by Lemma 10, the fixed point of operator  coincides with the solution of system (1).
In the first result, we prove uniqueness of solution of the boundary value problem (1) via Banach's contraction principle.
Theorem 11.Assume that ,  : [0, 1] × R → R are continuous functions and the following conditions hold: In addition, assume that where where Δ and  are given by (19) and (22), respectively.Then system (1) has a unique solution.
In order to show that  is a contraction, let , V,  1 , V 1 ∈ , and, for any  ∈ [0, 1], we get which, in view of  1 < 1 and (31), implies that Similarly, we have Since  1 < 1,  2 < 1, therefore, the operator  is a contraction.Hence, by Banach's contraction principle, the operator  has a unique fixed point, which is the unique solution of the system (1).This completes the proof.
(H3) There exists a constant  > 0 such that where (1 − ) As before, it can be shown that Similarly, we have Thus,  maps bounded sets into bounded sets in  × .
In the sequel we present two examples which illustrate Theorems 11 and 12.