Nonlinear Fractional q-Difference Equation with Fractional Hadamard and Quantum Integral Nonlocal Conditions

Department of Mechanical Engineering Technology, College of Industrial Engineering Technology, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand Department of General Educations, Merchant Marine Training Centre, Bang Nang Kreng, Sukhumvit Rd., Samutprakarn 10270, Thailand Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Intelligent and Nonlinear Dynamic Innovations, Department of Mathematics, Faculty of Applied Science, KingMongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand


Introduction
The aim of this paper is to investigate the existence and uniqueness of solutions for a nonlinear fractional q-difference equation subject to fractional Hadamard and quantum integral condition of the form: where D α q is the fractional q-derivative of order α, with a quantum number q ∈ ð0, 1Þ, f : ½0, T × ℝ ⟶ ℝ is a nonlinear continuous function, I fractional differential equations, see [2][3][4][5][6][7][8][9][10][11] and references cited therein.
In the present paper, the novelty lies in the fact that we combine in boundary conditions both Hadamard and quantum integrals. To the best of our knowledge, this type of boundary condition appears for the first time in the literature. It is important to notice that we are combining in our work, fractional calculus, and quantum calculus. The key tool for this combination is the Property 2.25 of [1].
Some special cases of the second condition of (1) can be seen by reducing m = n = 1 as which is mixed quantum and Hadamard calculus. If p 1 = 1, then we have which is also mixed Riemann-Liouville and Hadamard fractional integral condition. If μ 1 = σ 1 = 1, we have integral condition of the form: which is a variety used in physical boundary value problems. We establish existence and uniqueness results by using standard fixed point theorems. We prove two existence and uniqueness results with the help of the Banach contraction mapping principle and a fixed point theorem on nonlinear contractions due to Boyd and Wong. Moreover, we prove two existence results, one via Leray-Schauder nonlinear alternative and another one via Krasnosel'ski i's fixed point theorem.
The paper is organized as follows: in Section 2, we recall some preliminary facts that we need in the sequel. In Section 3, we prove our main results. Some examples to illustrate our results are presented in Section 4.

Preliminaries
To present the preliminary, we suggest the basic quantum calculus in the book of Kac and Cheung [22], fractional quantum calculus in [23][24][25], and the Hadamard fractional calculus in [1]. Let a fixed constant q ∈ ð0, 1Þ be a quantum number. The q-number is defined by For example, ½3 q = 1 + q + q 2 . The q-power function for any a, b ∈ ℝ, a ≠ 0, is defined as For example, ða − bÞ ð3Þ q = ða − bÞða − q bÞða − q 2 bÞ. The notation of q-power function is appeared in kernels of fractional q-calculus as Definitions 1 and 2. Now, the q-gamma function Γ q ðtÞ is defined by Now, we observe that Γ q ðt + 1Þ = ½t q Γ q ðtÞ. Next, we discuss about the q-derivative of a function f : ½0, ∞Þ ⟶ ℝ which is defined by If f ′ ðtÞ exists, then lim q→1 D q f ðtÞ = f ′ ðtÞ. The q-integral formula can be presented as The higher order of q-derivative and q-integral operators is with ðD 0 q f ÞðtÞ = f ðtÞ and ðI 0 q f ÞðtÞ = f ðtÞ. Next, the fundamental theorem of calculus for operators D q and I q can be stated as formulas and if f is continuous at the point t = 0, then

Journal of Function Spaces
Let us give the definitions of fractional quantum calculus of the Riemann-Liouville type fractional derivative and also integral operators.
Definition 1 [24]. Let a constant α ≥ 0 and f be the function on ½0, ∞Þ. The Riemann-Liouville fractional q -integral of f order α is defined by and ðI 0 q f ÞðtÞ = f ðtÞ.
Definition 2 [24]. The Riemann-Liouville fractional q -derivative of order α ≥ 0 of a function f : ½0,∞Þ ⟶ ℝ is given by and ðD 0 q f ÞðtÞ = f ðtÞ, where n is the smallest integer greater than or equal to α. Now, for t, s > 0, the q-beta function is presented by which is related to the q-gamma function by The fundamental formulas for fractional quantum calculus are in the following lemma.
Lemma 3 [24,26]. Let α, β ≥ 0, n be a positive integer and f be a function defined in ½0, ∞Þ. Then, the following formulas hold The fractional q-integration of the two deferent quantum numbers is given by lemma.
Lemma 4 [27]. Let constants α, β > 0 and 0 < p, q < 1 be quantum numbers. Then, for η ∈ ℝ + , we have The Hadamard fractional calculus is the subject of fractional derivative and integral which have logarithm kernels inside the singular integral formulas as in the definitions.
Definition 5 [1]. The Hadamard derivative of fractional order α for a function f : ½0,∞Þ ⟶ ℝ is defined as where the notation ½α denotes the integer part of the real number α, log ð·Þ = log e ð·Þ, and Γ is the usual Gamma function.
Definition 6 [1]. The Hadamard fractional integral of order α for a function f : ½0,∞Þ ⟶ ℝ is defined by provided the integral in right hand side exists.
The key tool for combining the two type of fractional calculus in our work is the following lemma.
To accomplish our main purpose, we will use the fixed point theory for considering an operator equation x = Qx. For finding the operator Q, let us see the following lemma.

Journal of Function Spaces
where h : ½0, T ⟶ ℝ, and subject to mixed fractional integrals of Hadamard and quantum boundary conditions is equivalent to the linear integral equation Proof. Since α ∈ ð1, 2, then (23) can be written as Applying the fractional q-integral of order α and using Lemma 3, we obtain which yields where k 1 , k 2 ∈ ℝ. The first boundary condition of (24) implies that k 2 = 0: Then, (28) is reduced to Now, we apply the fractional quantum integral of Riemann-Liouville of order μ i with quantum number p i to (29) as Using Lemma 7 for taking the Hadamard fractional integral of order σ j to (29), we get From the second boundary condition of (24) and above two equations, it follows that and consequently where the nonzero constant Ω is defined by (22). Substituting the constant k 1 in (29), then, we obtain (25), which is the solution of BVP (23) and (24). The converse can be obtained by a direct computation. The proof is completed.

Main Results
At first, we denote by C = Cð½0, T, ℝÞ the Banach space of all continuous functions from ½0, T to ℝ endowed with the sup norm as kxk = sup fjxðtÞj, t ∈ ½0, Tg. In view of Lemma 8 and replacing the function h by f ðt, xðtÞÞ, we define the oper- where I α q f x ðvÞ is denoted by while J σ j I α q f x ðη j Þ and I μ i p i I α q f x ðξ i Þ are the Hadamard and quantum fractional integrals of a function g as respectively. Now, we are going to prove the main results which are the existence criteria of solution for nonlocal mixed fractional integrals boundary value problem (1). The first, an existence and uniqueness result for (1), is given by using Banach's fixed point theorem.
ðH 1 Þ There exists a positive constant L such that |f ðt, xÞ − f ðt, yÞ | ≤Ljx − yj, for each t ∈ ½0, T and x, y ∈ ℝ. If Journal of Function Spaces where Φ is given by then the boundary value problem (1) has a unique solution on ½0, T.
Proof. The result allows from the operator equation x = Qx, where the operator Q is defined by (34). The Banach fixed point theorem is used to show that Q has a fixed point which is the unique solution of problem (1). Since the function f is continuous, then, we can set sup fjf ðt, 0Þj, t ∈ ½0, Tg = M < ∞. After that, we define the radius r satisfying of a ball B r = fx ∈ C : kxk ≤ rg: For any x ∈ B r , we see that in which we used the following fact: where v ∈ fT, ξ i , η j g. By applying Lemmas 4 and 2.3, we have Then, we obtain From this, we conclude that ∥Qx∥≤r which yields QB r ⊂ B r : Next, we will prove that the operator Q is a contraction. Let x, y ∈ C, and for each t ∈ ½0, T, then, we have Hence, we get the result that ∥Qx − Qy∥≤LΦ∥x − y∥: As LΦ < 1, from (37), the operator Q is a contraction. Applying the well known Banach fixed point theorem, it follows that Q has a fixed point which is the unique solution of the boundary value problem (1). This completes the proof.
Next, the nonlinear contraction theorem will be used to prove a second existence and uniqueness result.
Definition 10. Let E be a Banach space and let A : E ⟶ E be a mapping. The operator A is said to be a nonlinear contraction if there exists a continuous nondecreasing function Ψ : ℝ + ⟶ ℝ + such that Ψð0Þ = 0 and ΨðtÞ < t for all t > 0 with the property: Lemma 11 (see [28]). Let E be a Banach space and let A : E ⟶ E be a nonlinear contraction. Then, A has a unique fixed point in E: Suppose that a continuous function f : ½0, T × ℝ ⟶ ℝ satisfies the condition:

Journal of Function Spaces
T, x, y ∈ ℝ, where the function h : ½0, T ⟶ ℝ + is continuous, and a positive constant H * is defined by Then, the mixed fractional Hadamard and quantum integrals nonlocal problem (1) has a unique solution on ½0, T.
Proof. Let us consider the operator Q : C ⟶ C defined in (34) and define a continuous nondecreasing function Ψ : Then, we see that the function Ψ satisfies Ψð0Þ = 0 and ΨðλÞ < λ for all λ > 0.
Next, for any x, y ∈ C and for each t ∈ ½0, T, we obtain which implies that kQx − Qyk ≤ Ψðkx − ykÞ and also satisfies Definition 10. Therefore, Q is a nonlinear contraction. Thus, by applying Lemma 11, the operator Q has a unique fixed point which is the unique solution of the boundary value problem (1). The proof is finished.
Next, the first existence result will be obtained by applying the following theorem.
Theorem 13 (Nonlinear alternative for single valued maps) [29]. Let E be a Banach space, C a closed, convex subset of E, U be an open subset of C, and 0 ∈ U: Suppose that A : U ⟶ C is a continuous, compact (that is, Að UÞ is a relatively compact subset of C) map. Then, either (i) A has a fixed point in U, or (ii) There is a x ∈ ∂U (the boundary of U in C) and λ ∈ ð0, 1Þ with x = λAðxÞ: Theorem 14. Suppose that f : ½0, T × ℝ ⟶ ℝ is a nonlinear continuous function which satisfies the following conditions: ðH 3 Þ there exists a continuous nondecreasing function ψ : ½0, ∞Þ ⟶ ð0, ∞Þ and also a function p ∈ Cð½0, T, ℝ + Þ such that where Φ defined by (38). Then, the problem (1) has at least one solution on ½0, T: Proof. For a positive number ρ, we let B ρ = fx ∈ C : kxk ≤ ρg be a bounded ball in C. Now, we will prove that the set QB ρ is uniformly bounded. For t ∈ ½0, T, we can compute that which can be deduced that Then, the set QB ρ is uniformly bounded. Next, we will show that the set QB ρ is equicontinuous set of C: Journal of Function Spaces For any two points τ 1 , τ 2 ∈ ½0, T with τ 1 < τ 2 and x ∈ B ρ , we have As τ 2 − τ 1 ⟶ 0, the right hand side of the above inequality converses to zero, independently of x ∈ B ρ . Then, the set QB ρ is equicontinuous. Thus, we conclude that the set QB ρ is relatively compact. Therefore, by the Arzel a ′ -Ascoli theorem, the operator Q : C ⟶ C is completely continuous.
Finally, we show that the operator Q cannot be fulfilled the condition ðiiÞ in Theorem 13. Then, we have to claim that there exists an open set U ⊂ B ρ with x ≠ λQx for λ ∈ ð0, 1Þ and x ∈ ∂U: Then, for each t ∈ ½0, T, we apply the computation in the first step, that is which yields inequality The condition ðH 4 Þ implies that there exists a constant N such that kxk ≠ N: Now, we define the set From the previous results, we obtain that the operator Q~ U ⟶ C is continuous and completely continuous. Then, there is no x ∈ ∂U such that x = λQx for some λ ∈ ð0, 1Þ: By applying the nonlinear alternative of the Leray-Schauder type, we get that the operator Q has a fixed point x ∈ U which is a solution of the nonlinear fractional q-difference equation with fractional Hadamard and quantum integral nonlocal conditions. This finishes the proof.
The next existence result is based on Krasnosel'ski i's fixed point theorem which can be used to relax the condition in Theorem 9.

If inequality
holds, then the nonlocal problem (1) has at least one solution on ½0, T: Proof. Now, we define sup f|κðtÞ| : t ∈ ½0, Tg = kκk and choose a positive constant r such that where Φ is defined by (38), to be a radius of the ball B r = fx ∈ C : kxk ≤ rg. Furthermore, we set the operators Q 1 and Q 2 on B r as A and B in Theorem 15, respectively, by