On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems

: In this work, we initiate the study of a new class of impulsive boundary value problems consisting of mixed type fractional quantum and Hadamard derivatives. We will establish existence and uniqueness results by using tools from the functional analysis. We prove the uniqueness result via Banach ’ s contraction mapping principle, while we will use the Leray - Schauder nonlinear alternative to establish an existence result. We also present examples to illustrate the obtained results.


Introduction
Fractional calculus is a generalization of classical calculus to an arbitrary real order and has evolved as an interesting and important area of research. Fractional differential equations have gained much attention in the literature because some real-world problems in physics, mechanics, engineering, game theory, stability, optimal control, and other fields can be described better with the help of fractional differential equations. Fractional differential equations constitute a significant branch of nonlinear analysis. The theory and applications of the fractional differential equations have been greatly developed; for more details, one can see the monographs [1][2][3][4][5][6][7][8] and references therein.
Impulsive differential equations arise when at certain moments they change their state rapidly and have many applications in physics, engineering, medicine, population dynamics, pharmacology, biotechnology, and economics. There are two types of impulses: the instantaneous impulses in which the duration of these changes is relatively short and the noninstantaneous impulses in which an impulsive jump starts abruptly at any fixed point and continues on a finite interval of time. For the results of instantaneous impulses, see, e.g., the monographs [9][10][11], the papers [12][13][14], and the references cited therein. Nonin-stantaneous impulsive differential equation was introduced by Hernández and O'Regan in [15], who pointed out that the instantaneous impulses cannot characterize some processes such as evolution processes in pharmacotherapy. In noninstantaneous impulses, impulsive action starts at an arbitrary fixed point and remains active on a finite time interval, which is much different from classical instantaneous impulsive that changes are relatively short compared to the overall duration of the whole process. Let us consider the hemodynamic equilibrium of a person. The introduction of the drugs in the bloodstream and the consequent absorption for the body are a gradual and continuous process. In fact, this situation should be characterized by a new case of impulsive action, which starts at an arbitrary fixed point and stays active on a finite time interval. For more details, see the monograph [16].
In [21], the notions of q k -derivative and q k -integral were introduced, and their properties were investigated. New concepts of fractional quantum calculus involving a new q k -shifting operator were introduced in [22]. On the other hand, the analysis of fractional differential equations involving Hadamard fractional derivatives has increased interest in the mathematical analysis; see, for example, the recent monograph [8].
Motivated by the aforementioned papers, in this investigation, we initiate the study of a new class of boundary value problems consisting of impulses and mixed type fractional quantum and Hadamard derivatives. More precisely, we study the following impulsive boundary value problem, in which we combine fractional quantum and Hadamard derivatives of the form are the fractional quantum difference and Hadamard fractional derivative in the sense of Caputo type of orders α β , 0,1 , and J J J T = < < < <⋯< < < = + are given. We establish existence and uniqueness results for the impulsive mixed fractional quantum and Hadamard boundary value problem (1) by using tools from the functional analysis. The main results are presented in Section 3, where Banach's contraction mapping principle is applied for the uniqueness result, while the Leray-Schauder nonlinear alternative is used to establish an existence result. In Section 2, some basic concepts from quantum calculus and Hadamard derivatives are recalled, and also an auxiliary result concerning a linear variant of the problem (1) is proved. This result is pivotal to transform the problem (1) into a fixed point problem. Illustrative numerical examples are also presented.

Preliminaries
In this section, we give some basic concepts of fractional quantum and Hadamard calculus such as derivatives and integrals. For more details, see [2,[22][23][24]. Staring, the q-shifting operator is defined by where x ∈ , a 0 ≥ , and q 0 1 < < . For any positive integer k, we have Impulsive fractional quantum and Hadamard differential equations  1599 The power function of x y − involving q-shifting operator is given by In general, if α ∈ , then we have ] can be defined in the term of q-shifting by ] is defined as follows: Now, we present the formulas of Riemann-Liouville fractional q-integral and Caputo fractional q-derivative over the interval a b , [ ] as follows.  , we have  and α ∈ . Then, for a 0 > , we have Next, let us introduce the future used notation as follows: , since the inequality in the subscript is not true. In addition, we assume the constants The next lemma deals with a linear variant of the boundary value problem (1).
and λ 3 be given constants, which satisfy problem (1). Then, the linear boundary value problem of mixed type quantum and Hadamard fractional derivatives of the form: has a unique solution x on J presented by Proof. In the first equation in (6) by taking the fractional q 0 -integral of order α 0 from s 0 to t s t , In the second interval t s , 1 1 [ ), we can get x t ( ) by Hadamard fractional integration of order β 0 1 > as follows: for t t s , . In the third interval s t , 1 2 [ ), it is the fractional q 1 -difference of an unknown x t ( ). Then, we apply the fractional q 1 -integral of order α 1 by To claim that our formula (10) is true, we use the mathematical induction by putting i 0 = and i 1 = in the first and second parts of (10), respectively. Then, the initial step holds by (8) and (9). The inductive step will be proved by assuming that the first part of (10) is true for i n = , that is, [ )  [ ) ∈ + + . In addition, suppose that the second part of (10) is fulfilled when i n = , that is, t t s , . Thus, in the consecutive interval s t , By substituting the constant c 0 , (12), into (10), we obtain a unique solution of linear mixed fractional quantum and Hadamard impulsive boundary value problem (6).
Conversely, for t s t , Replacing i by i 1 − and putting t by t i − in the first equation of (10), we get x t i ( ) + after multiplying γ i and adding η i , respectively. Then, x s i ( ) + is obtained by putting t s i = − , and respectively, multiplying ξ i , adding θ i , in the second equation of (10). The value of λ 3 can be computed by putting i 0 = with multiplying λ 1 and i m = , t T = with multiplying λ 2 , in the first equation of (10). Therefore, the proof is completed. □

Existence and uniqueness results
In the first step of this section, we define the spaces of functions and the operator, which are related to the investigated problem (1). Let J be an interval defined in (1) and let PC J, ( ) and PC J, 1 ( ) be the spaces of piecewise continuous functions defined by PC J R x J R , : , respectively. By applying the Banach contraction mapping principle, and Leray-Schauder nonlinear alternative, we are in the position to prove the existence and uniqueness of solutions of the problem (1). The following constants  f t x f t y L x y t J and g t x g t y L x y t J , , , < , then the mixed fractional quantum and Hadamard derivatives impulsive boundary value problem (1) has a unique solution on J .
Proof. Let B r be a ball of radius r 0 > , subset of E, defined by B x E x r : We will prove that B B .
∈ , respectively, we have , and < , then, is a contraction. Hence, we can conclude, by using Banach's contraction mapping principle, that the operator has a fixed point, which is the solution of the mixed fractional quantum and Hadamard derivatives impulsive boundary value problem (1). This completes the proof. □ The next theorem of Leray-Schauders nonlinear alternative will be used to prove our existence result.  which implies that x Φ 5 ‖ ‖ ≤ . Hence, B ρ ( ) is a uniformly bounded set. To prove the equicontinuity of the set B ρ ( ), let t t T , 0, such that t t 1 2 < . Then, for any x B ρ ∈ , we obtain  Finally, we consider the existence criteria for the initial vale problem. If we replace λ 1 1 = and λ 0 2 = in problem (1), that is, then we obtain an impulsive initial value problem and we also get constants Ω 1 = , λ ξγ Φ

Examples
In this section, we give some examples to illustrate the usefulness of our main results.