Existence of solutions for q -fractional di ﬀ erential equations with nonlocal Erd´elyi-Kober q -fractional integral condition

: In this paper, we obtain su ﬃ cient conditions for the existence, uniqueness of solutions for a fractional q -di ﬀ erence equation with nonlocal Erd´elyi-Kober q -fractional integral condition. Our approach is based on some classical ﬁxed point techniques, as Banach contraction principle and Schauder’s ﬁxed point theorem. Examples illustrating the obtained results are also presented.

The q-calculus or quantum calculus is an old subject that was initially developed by Jackson [1], basic definitions and properties of q-calculus can be found in [2]. The fractional q-calculus had its origin in the works by Al-Salam [3] and Agarwal [4]. In recent years, considerable interest in qfractional differential equations has been stimulated due to its applicability in mathematical modeling in different branches like engineering, physics and technical,etc. There are many papers and books dealing with the theoretical development of q-fractional calcaulus and the existence of solutions of boundary value problems for nonlinear q-fractional differential equations, for examples and details, one can see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and references therein.
As we all know, few people solve the existence of solutions for a nonlinear Riemann-Liouville q-fractional differential equation subject to nonlocal Erdélyi-Kober q-fractional integral conditions. Inspired by the paper [23], we consider the existence and uniqueness for problem (1.1) by using Banach contraction principle and Schauder's fixed point theorem.

Preliminaries on q-calculus and Lemmas
Here we recall some definitions and fundamental results on fractional q-integral and fractional qderivative. See the references [4][5][6][7] for complete theory.
(a−bq k ), a, b ∈ R, If n is a positive integer. If ν is not a positive integer, then (a−b) (ν) = a ν ∞ n=0 a−bq n a−bq ν+n .
The q-derivative of a function f is defined by Some results about operator D q and I q can be found in references [4]. Let us define fractional q-derivative and q-integral and outline some of their properties [4,6,8]. ( , ν > 0, where l is the smallest integer greater than or equal to ν. Definition 3 ( [24]) For 0 < q < 1, the Erdélyi-Kober fractional q-integral of order µ > 0 with β > 0 and η ∈ R of a continuous function f : (0, ∞) → R is defined by provided the right side is pointwise defined on R + .
Remark 1 For β = 1 the above operator is reduced to the q-analogue Kober operator that is given in [4]. For η = 0 the q-analogue Kober operator is reduced to the Riemann-Liouville fractional q-integral with a power weight: Then the following equality holds Lemma 3 ( [4]) Let α > 0 and p be a positive integer. Then for t ∈ [0, b] the following equality holds

Main results
In this section, we will give the main results of this paper. Let the space It is known that the space E is a Banach space. To obtain our main results, we need the following lemma.
. Then for any t ∈ [0, T ], the solution of the following problem is given by Proof. Applying the operator I α q on both sides of the first equation of (3.1) for t ∈ (0, T ) and using Lemma 1 and Lemma 2, we have Applying the initial value condition x(0) = 0, we get c 2 = 0. By the boundary value condition, we have that is Substituting c 1 , c 2 to (3.3) , we obtain the solution (3.2). This completes the proof.
Using the Lemma 5, we can define an operator Q : E → E as follows: where and τ ∈ {t, T, ξ 1 , ξ 2 · · · , ξ n }. Then, the existence of solutions of system (1.1) is equivalent to the problem of fixed point of operator Q in (3.4).
In the following, we will use some classical fixed point techniques to give our main results.
for each x,x, y,ỹ ∈ R. Then problem (1.1) has an unique solution on [0, T ] if Proof. The conclusion will follow once we have shown that the operator Q defined (3.4) is contractively with respect to a suitable norm on E.
For any functions x, y ∈ E, we have on the other hand which implies that Thus the operator Q is a contraction in view of the condition (3.5). By Banach's contraction mapping principle, the problem (1.1) has an unique solution on [0, T ]. This completes the proof.
for each t ∈ [0, T ] and x,x, y,ỹ ∈ R. Then the problem (1.1) has an unique solution whenever In the following Theorem, the existence results for nontrivial solution for problem (1.1) are presented. For convenience, we denote where l i (t), i = 1, 2, 3 are defined in Theorem 2.
Proof. We shall use Schauder's fixed point theorem to prove our theorem.
. Note that E d is a closed, bounded and convex subset of the Banach space E. We now show that Q : E d → E d . In fact, for x ∈ E d we have that From the two inequalities above, we get Hence, Q maps E d into E d . Also, it is easy to check that Q is continuous, since f is continuous. For each x ∈ E d and each 0 < t 1 < t 2 < T , we have Let t 2 → t 1 , we get Qx(t 2 ) − Qx(t 2 ) → 0. Thus, Q is uniformly bounded and equicontinuous. The theorem of Arzelá-Ascoli implies that Q is completely continuous. By Schauder's fixed point theorem, Q has a fixed point in E d . Clearly x = 0 is not a fixed point because f (t, 0, 0) 0 for t ∈ [0, T ]. Hence, the problem (1.1) has at least one nontrivial solution. This proves the theorem.
Remark 2 In the Theorem 2, if r i > 1, i = 1, 2, we may choose l 2 (t), l 3 (t) and d such that For r i = 1, i = 1, 2, we have the following theorem.
Although Theorems 2 and Theorem 3 provide some simple conditions on the existence of solution of problem (1.1) and Theorem 1 provides a condition on the existence and uniqueness on the solution of problem (1.1), the following theorem provides an easily verifiable condition for the existence of a nontrivial solution for the problems (1.1).
holds. Then problem (1.1) has at least one nontrivial solution.
Proof. Choose a constant A such that By the condition (3.6), there exists a constant c 1 such that Then for any x ∈ E c , we have Ac. (3.7) Ac. (3.8) Thus (3.9) From (3.9), we obtain Q(E c ) ⊂ E c . By Schauder fixed point theorem, Q has at least one fixed point in E c . Clearly, x = 0 is not a fixed point because f (t, 0, 0) 0. Therefore, problem (1.1) has at least one nontrivial solution, which completes the proof.

Example
In this section, we illustrate the results obtained in the last section.
By computation, we deduce that then, the first condition is satisfied with L(t) = t 2 2 .

Conclusions
In this work, we utilize Banach contraction principle and Schauder's fixed point theorem to research the existence, uniqueness of solutions for a q-fractional differential equation with nonlocal Erdélyi-Kober q-fractional integral condition and in which the nonlinear term contains a fractional q-derivative of Rieman-Liouville type. Some existence and uniqueness results of solutions are obtained, we also provide an easily verifiable condition for the existence of nontrivial solution for the problem (1.1).