Well-posed conditions on a class of fractional q-differential equations by using the Schauder fixed point theorem

In this paper, we propose the conditions on which a class of boundary value problems, presented by fractional q-differential equations, is well-posed. First, under the suitable conditions, we will prove the existence and uniqueness of solution by means of the Schauder fixed point theorem. Then, the stability of solution will be discussed under the perturbations of boundary condition, a function existing in the problem, and the fractional order derivative. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Existence of solutions to fractional differential equations has received considerable interest in recent years. There are several papers dealing with the existence and uniqueness of solutions to initial and boundary value problem of fractional order in Caputo or Riemann-Liouville sense (for more details, see [50][51][52] and the references therein). Some authors have also investigated the existence and uniqueness solutions for a coupled system of multi-term fractional differential equations [53,54]. However, in general, the study of well-posed conditions for fractional differential equations is less considered in the literature.
In 2015, Houas et al. [55] investigated the existence and uniqueness of solutions for c D σ q [y](t) + w(y(t), c D ς q [y](t)) = 0, for t ∈ J 0 := [0, 1], where 2 < σ ≤ 3, ς ∈ J 0 := (0, 1), under the initial conditions y(0) = y 0 , y (0) = 0, y (1) = ηI ζ y(e), where c D σ is the Caputo fractional derivative, e ∈ J 0 , w is a continuous function on R 2 , and η is a real constant [55]. In [56], authors studied the existence and uniqueness of solution for the fractional differential equation D σ [y](t) = w(t, y(t), D ς [y](t)), where 2 < σ < 3, ς ∈ J 0 , via sum boundary conditions y(0) = 0, y (1) = 0, where, a i , e i ∈ J 0 and D σ q is the Caputo fractional derivative. In 2015, Akrami et al. [57] proved the conditions on which the following class of fractional differential equations C D σ [y](t) = w(y(t), D ς [y](t)) for t ∈ J 0 is well-posed, where 2 < σ ≤ 3 and ς ∈ J 0 , and C D σ q is the Caputo fractional derivative subject to the boundary value conditions y(0) = y (0) = 0, y(1) = ay(e), where e ∈ J 0 , 0 ≤ a < 1 e 2 . In this article, we investigate the conditions on which the fractional q-differential equation for t ∈ J 0 is well-posed, where 2 < σ ≤ 3, ς ∈ J 0 , and C D σ q is the standard Caputo qderivative subject to the boundary value conditions y(0) = y (0) = 0, y (1) = ay(e), where e ∈ J 0 with 0 ≤ a < 1 e 2 . We recall that a problem is said to be well-posed if it has a uniqueness solution and this solution depends on a parameter in a continuous way. This parameter, in the classical order differential equations, is dependent on the initial conditions and the function exists in the problem; whereas in the FDEs this dependency and the stability solution with respect to the perturbation of fractional order derivative should be taken into the account too [58].
The rest of the paper is organized as follows. We first prove the existence solution of (1) by means of the Schauder fixed point theorem on the interval J 0 in Sect. 3. Then, we prove the uniqueness by using the Banach contraction map theorem under a suitable condition in Sect. 3. Also, Sect. 3 is devoted to investigating the stability of solutions under the perturbations on boundary condition, the function exists in the problem and the fractional order derivative. Finally, in Sect. 4, we bring some examples to illustrate our results. Let us start with some basic preliminaries in Sect. 2 that we will use in the sequel.

Preliminaries and lemmas
This section is devoted to some notations and essential preliminaries that are acting as necessary prerequisites for the results of the subsequent sections. Throughout the context, we shall apply the notations of time scales calculus [59].
In fact, we consider the fractional q-calculus on the specific time scale T t 0 = {0} ∪ {t : t = t 0 q n } for n ∈ N, t 0 ∈ R, and q ∈ (0, 1). For brief, we shall denote T t 0 by T. Let a ∈ R.
whenever the series exists [61]. The operator I n q is given by [23]. It has been proved that D q ( , whenever the function ℘ is continuous at v = 0 [23]. The fractional Riemann-Liouville type q-integral of the function ℘ is defined by I 0 q ℘(v) = ℘(v) and for v ∈ [0, 1] and σ > 0 [24,29]. The Caputo fractional q-derivative of the function ℘ is defined by for v ∈ [0, 1] and σ > 0 [29,62]. It has been proved that I ν [29]. The authors in [61] presented all algorithms and MATLAB lines to simplify q-factorial functions (vw) ( (v), and some necessary equations. Now, we introduce some basic definitions, lemmas, and theorems which are used in the subsequent sections.

Main results
First, we consider the following important lemmas in our article.
Proof By Lemma (2.3) the solution of Eq. (3) can be written as Since y(0) = y (0) = 0, a simple calculation gives d 0d 1 = 0, and from the boundary condition, we get I σ q [v](1)d 2 = aI σ q [v](e)d 2 ae 2 . Hence, Thus, the solution of boundary value problem (3) is . This completes the proof. Now, in order to investigate the existence of solutions, we prove some properties of the function G q (t, ξ ).

Lemma 3.2
The functions G q (t, ·) and ∂ ∂t G q (t, ·) are integrable for each t ∈ J 0 and have the following properties: .
Proof Let t ∈ J 0 . Then we have .
Let C 1 (J 0 ) be the class of all continuous functions.
q is continuous for any y ∈ C 1 (J 0 ). Now, for y ∈ C 1 (J 0 ), we define the endowed with the maximum norm Therefore, using the convergence of uniformly on J 0 . On the other hand, we know This completes the proof.

Existence and uniqueness
According to the Schauder fixed point theorem, the existence result has been stated.
Theorem 3.4 Suppose that w : R 2 → R is a continuous function and there exist constants m 0 , m 1 ≥ 0, β 0 , β 1 ∈ J 0 such that one of the following conditions is satisfied: (A2) The function w satisfies Then boundary value problem (1) has at least one solution y(t).
Proof First, suppose that condition (A1) holds. Define the set B by and It is clear that B is a closed, bounded, and convex subset of Banach space A. Here, we prove that O : B → B. For any y ∈ B, we obtain Thus, for almost all ς ∈ J 0 , we have Clearly, Oy(t) and C D ς In the second case, suppose that condition (A2) holds. Choose Again, by a similar way, we get Oy ≤ δ, and therefore, in this case, O : B → B. Here, we need to show that O is a completely continuous operator. First, the equicontinuity of O will be shown as follows. Suppose that s 1 , s 2 ∈ J 0 with s 1 < s 2 and Then Oy(s 1 ) -Oy(s 2 ) .
Since the functions s 2 , and s 1 (s 2s 1 ) 1-ς are continuous, we conclude that Oy is an equicontinuous set. Obviously, Oy is uniformly bounded because O(B) ⊆ B. By means of the Arzelá-Ascoli theorem, O is a compact operator. Therefore, from the Schauder fixed point theorem, the operator O has a fixed point, i.e., the q-fractional boundary value problem (1) has a solution.
In what follows, we prove the uniqueness of solution for Eq. (1) based on application of the Banach fixed point theorem. i.e., Therefore Hence, by the Banach fixed point theorem, O has a unique fixed point which is a solution of problem (1).

Stability of solution
In this section, we study the stability analysis of problem (1) under various perturbations. Dependence solution on the boundary value condition is discussed in Theorem 3.6. Stability of the solution with respect to the perturbation of w is analyzed in Theorem 3.7. Finally, the perturbation effect of fractional order derivative on the solution is studied in Lemma 3.8 and Theorem 3.9.
This completes the proof.

Theorem 3.7 Suppose that the conditions of Theorem 3.5 hold, and letv(t) be the solution of the following perturbed problem on function w:
for t ∈ J 0 , 2 < α ≤ 3, and ς ∈ J 0 , with the boundary conditions y 0 = y 0 = 0, y 1 = ay(e) for e ∈ J 0 with 0 ≤ a < 1 e 2 . Then y -v = O( ).
Proof The solution of problem (13) iŝ Then, similar to the proof of the previous theorem . Indeed, .
This completes the proof.
For perturbation analysis on the fractional order of the q-derivative, we first state and prove the following lemma and then the main theorem will be discussed.
Proof According to the above discussion, the solution of problem (15) is given bŷ wherê , ξ < t, for t, ξ ∈ J 0 . Then Also, we have Therefore, According to the structure of G q (t, ξ ), we know that every term of |G q (t, ξ ) -Ĝ q (t, ξ )| and is in the form of Eq. (15). Hence, Lemma 3.8 implies Therefore, y -v = O( ) and the proof is complete.

Some illustrative examples
Herein, we give some examples to show the validity of the main results. In this way, we give a computational technique for checking problem (1). We need to present a simplified analysis that is able to execute the values of the q-gamma function. For this purpose, we provided a pseudo-code description of the method for calculation of the q-gamma function of order n [61].

Conclusion
The Schauder fixed point theorem has been applied in the research study to discuss the well-posed conditions for a class of q-fractional order boundary value problems As a result, we have proved the existence and uniqueness of solution by means of the Schauder fixed point and Banach contraction map theorems on the interval [0, 1]. We have also studied the perturbation on boundary condition on the function exists in the right-hand side of the problem and on the fractional order. To the leading of our information, the results have never been detailed in other works [11,12,61] that consider the problems. In this manner, it is very apparent that the solution of the problem is stable under the small perturbation.