A novel fractional structure of a multi-order quantum multi-integro-differential problem

In the present research manuscript, we formulate a new generalized structure of the nonlinear Caputo fractional quantum multi-integro-differential equation in which such a multi-order structure of quantum integrals is considered for the first time. In fact, in the light of this type of boundary value problem equipped with the multi-integro-differential setting, one can simply study different cases of the existing usual integro-differential problems in the literature. In this direction, we utilize well-known analytical techniques to derive desired criteria which guarantee the existence of solutions for the proposed multi-order quantum multi-integro-differential problem. Further, some numerical examples are considered to examine our theoretical and analytical findings using the proposed methods.


Introduction
As years and even decades go by, the human beings need to be acquainted with different natural phenomena more and more. One possible way to achieve this purpose is to apply the logical techniques and tools available in mathematics, and particularly the mathematical operators, in the modeling of different processes. Various fractional operators have been formulated by different researchers, and their applicability is becoming increasingly apparent to researchers every day. In consequence, it is necessary that we derive and investigate various models of processes from all aspects by utilizing the fractional operators in boundary value problems. Some instances of the application of these operators can be found in applied sciences such as electrical circuits, medicine, biomathematics, etc. [1][2][3][4][5][6]. Moreover, the importance of this field implies that the researchers are interested in finding different aspects of the structure of the general fractional BVPs and some dynamical properties of their solutions. In this context, a lot of researchers have been studying many modern and general fractional models and relevant dynamical behaviors of this type of fractional BVPs (see, for example, [7][8][9][10][11][12][13][14][15][16][17][18]).
Inspired by the aforementioned ideas given in the above-cited papers, we formulate a new generalized structure of the nonlinear Caputo fractional quantum multi-integrodifferential equation furnished with fractional multi-order quantum integrals conditions: such that z ∈ [0, 1], σ ∈ (1, 2), q ∈ (0, 1), δ * 1 , δ * 2 , γ * 1 , γ * 2 ∈ (0, 1), θ * 1 , θ * 2 , θ * 3 > 0, and η * , μ * are nonzero real positive constants and λ * 1 , λ * 2 ∈ R ≥0 . Moreover, two operators, C D (·) q and RL I (·) q , stand for the Caputo quantum derivative and the Riemann-Liouville quantum integral of given fractional orders, respectively. Also, both real-valued functionŝ h * ,f * : [0, 1] × R → R are supposed to be continuous. It is necessary that all researchers pay attention to that the proposed multi-order Caputo quantum multi-integro-differential equation has a novel and unique structure. In other words, the formulated structure for given fractional multi-integro-differential problem (1) includes one quantum derivative in the Caputo sense and also seven quantum integrals of the Riemann-Liouville type. This combined boundary problem covers many different special cases of various nonlinear integro-differential equations. Therefore, we emphasize that this kind of the Caputo quantum multi-integro-differential problem has not been investigated in the literature so far. In this direction, we apply well-known analytical techniques to derive desired criteria which guarantee the existence of solutions for the proposed Caputo quantum multiintegro-differential boundary problem (1).
The organization of the contents of the current manuscript is as follows. In the next section, some required notions in the context of the quantum calculus are assembled. Section 3 is devoted to establishing the main theorems in which the existence criteria can be obtained under some necessary conditions. In Sect. 4, numerical examples are considered to examine our theoretical and analytical findings by using the proposed methods.

Preliminaries
In this part of the present research manuscript, some required notions in the context of the quantum calculus are assembled. Let us assume that q ∈ (0, 1). For the given power function (m 1m 2 ) n with n ∈ N 0 , its q-analogue is defined by (m 1m 2 ) (0) = 1 and (m 1m 2 ) (n) = n-1 k=0 m 1m 2 q k , such that m 1 , m 2 ∈ R and N 0 := {0, 1, 2, . . .} [52]. Here, the constant n = σ is supposed to be an arbitrary real number. In this case, one can define the q-analogue of mentioned power function (m 1m 2 ) n in the q-fractional setting as follows: for m 1 = 0. Notice that if we take m 2 = 0, then we reach an equality m (σ ) 1 = m σ 1 immediately [52]. For the given real number m 1 ∈ R, a q-number [m 1 ] q is considered as The quantum Gamma function, or simply the q-Gamma function, is provided by the following rule:

Algorithm 2
The pseudo-code to compute different values of (D q )(z) Require: q ∈ (0, 1), (z), z [19,52]. It is notable that q (z + 1) = [z] q q (z) is true [19]. In Algorithm 1, we provide a pseudo-code based on relations (2) and (3) to compute different values of the Gamma function in the quantum setting.
In the following, the quantum derivative of a real-valued continuous function is defined by and also (D q )(0) = lim z→0 (D q )(z) [22]. One can simply extend the quantum derivative of a function to arbitrary higher order by (D n q )(z) = D q (D n-1 q )(z) for any n ∈ N [22]. It is obvious that (D 0 q )(z) = (z). Similar to above, a pseudo-code based on (4) is provided to compute the quantum derivative of a function in Algorithm 2.
The quantum integral of a real-valued continuous function defined on [0, m 2 ] is formulated by provided that the series is absolutely convergent [22]. Similar to a quantum derivative, we can extend the quantum integral of a function to arbitrary higher order by iterative rule (I n q )(z) = I q (I n-1 q )(z) for all n ≥ 1 [22]. In addition, it is evident that (I 0 q )(z) = (z). Note that a pseudo-code based on (5) is provided to compute the quantum integral of a function in Algorithm 3. At this moment, let us assume that m 1 ∈ [0, m 2 ]. In this case,

Main results
Let W = C R ([0, 1]) be the space of all real-valued continuous functions on [0, 1]. One can simply verify that the set W will be a Banach space if we define the sup norm W = sup z∈[0,1] | (z)| for all members ∈ W. At this point, we first provide the following structural lemma which characterizes the construction of solutions for the equivalent quantum integral equation related to the proposed quantum multi-integro-differential problem (1). Lemma 3.1 Let * ∈ W, σ ∈ (1, 2), δ * j ∈ (0, 1), θ * i > 0 for j = 1, 2 and i = 1, 2, 3. Also, let η * , μ * be nonzero real positive constants and consider the following nonzero positive constant: Then the function * is a solution to the nonlinear Caputo quantum fractional problem if and only if * is a solution to the fractional quantum integral equation Proof At first, we regard the given function * as a solution for the Caputo quantum fractional problem (8). Then we get Taking fractional quantum integral in the Riemann-Liouville sense of order σ on both sides of the latter equation, we reach whereα 0 ,α 1 ∈ R are some constants that we need to find. It is immediately deduced that α 0 = 0 by the first boundary condition and (10). On the other hand, by considering the properties of the Riemann-Liouville quantum integral, we have Then the second boundary condition (8) implies where˜ * = 0 is provided by (7). Eventually, we substitute both obtained values ofα 0 and α 1 into (10). In this case, we observe that the function * satisfies the quantum integral equation (9), and so * is a solution for the mentioned integral equation. In the opposite direction, it is simple to confirm that * is a solution for the given nonlinear Caputo quantum fractional boundary problem (8) whenever * is regarded as a solution for the quantum integral equation (9). This completes the proof.
Also, hypothesis (HK2) implies that |f * (z)| = |f * (z, )| ≤ ϒ(z) for z ∈ [0, 1]. Then for any elements 1 , 2 ∈ Br, one can write The latter inequality demonstrates that A 1 1 + A 2 2 ≤r and thus A 1 1 + A 2 2 ∈ Br for each 1 , 2 ∈ Br. This also means that condition (i1) of Theorem 2.2 holds for both operators A 1 and A 2 . At this point, we proceed to verify that A 1 is a contraction. For arbitrary elements 1 , 2 ∈ Br and z ∈ [0, 1], and in view of hypothesis (HK1), we have In the light of the given hypothesis, we know that˜ (1) * +b * ˜ (2) * < 1. Thus we conclude that A 1 is a contraction and so condition (i2) of Theorem 2.2 is valid for the operator A 1 .
In the sequel, we intend to verify the continuity of A 2 . To reach this goal, let us assume that { n } n≥1 is a convergent sequence belonging to the given ball Br such that n approaches . Then for any z ∈ [0, 1], we obtain But by the hypothesis, we know that the functionf * is continuous on [0, 1] × W, thus we find that A 2 n -A 2 approaches zero whenever n → . Therefore, we conclude that A 2 is a continuous operator defined on Br. In the subsequent stage, we claim that the operator A 2 is compact. To confirm this claim, we first check the uniform boundedness of A 2 . For given member ∈ Br and z ∈ [0, 1], we may write which illustrates that A 2 ≤˜ (3) * ϒ and A 2 is uniformly bounded. Besides, we establish that A 2 is an equicontinuous operator. To establish this result, we consider two elements z, x ∈ [0, 1] such that z < x. In fact, we shall verify that bounded sets are mapped to equicontinuous sets by the operator A 2 . Hence for every ∈ Br, we get We find that the right-hand side of the obtained inequality is not dependent on ∈ Br and also approaches 0 when z tends to x. In consequence, we realize that A 2 is equicontinuous. Hence, it is concluded that A 2 is a relatively compact operator on ∈ Br and thus the Arzelá-Ascoli theorem implies that A 2 is completely continuous, and eventually A 2 is a compact operator on the given ball ∈ Br. Therefore condition (i3) of Theorem 2.2 is valid for the operator A 2 . In consequence, all three hypotheses of Theorem 2.2 are valid for both single-valued operators A 1 and A 2 . Therefore Theorem 2.2 implies that the given nonlinear Caputo fractional quantum multi-integro-differential problem (1) has at least one solution on the interval [0, 1], and so this completes the proof.
Leray-Schauder nonlinear alternative theorem is another analytical tool by which we will be able to derive our desired existence criteria for the mentioned nonlinear Caputo fractional quantum multi-integro-differential problem (1).
Proof To reach the desired conclusion, we check all the hypotheses of Leray-Schauder nonlinear alternative (Theorem 2.3) in the subsequent steps. At first, we are going to show that the operator A defined by (11) maps bounded sets (i.e., balls) into bounded sets in W. For a positive real numberR, construct a bounded ball BR = { ∈ W : ≤R} in W.
Then for any z ∈ [0, 1] and in view of hypothesis (HK3), we can write Hence, the above inequality yields This indicates that the operator A is uniformly bounded. In the second stage, we proceed to verify that A maps bounded sets (i.e., balls) into equicontinuous subsets of W. To see We find that the right-hand side of the obtained inequality is not dependent on ∈ BR and also approaches 0 when z tends to x. In consequence, A is equicontinuous, and hence we have confirmed the complete continuity of A : W → W by the Arzelá-Ascoli theorem. In this case, we get In the light of hypothesis (HK4), we can find a real number N > 0 so that We simply see that A : O → W is an operator which is continuous and completely continuous. In view of this choice of O, we cannot find * ∈ ∂O which satisfies an equation * =ĉ(A * ) for someĉ ∈ (0, 1). Finally, by the nonlinear alternative of Leray-Schauder type, we realize that the operator A has a fixed point belonging to O. In consequence, there is at least one solution on [0, 1] for the nonlinear Caputo fractional quantum multiintegro-differential problem (1).
In the following part of the present section, the uniqueness criterion for solutions of the given nonlinear Caputo fractional quantum multi-integro-differential problem (1) is checked with the aid of Banach contraction principle (Theorem 2.4). (HK5) there is a real constant K > 0 so that for any 1 , 2 ∈ W, we have Then there exists a unique solution on [0, 1] for the nonlinear Caputo fractional quantum multi-integro-differential problem (1) such that˜ (1) * are given by (12).

Examples
In the current section of this manuscript, three illustrative numerical examples are considered to examine our theoretical and analytical findings by using the proposed methods.
In consequence, we conclude that all the hypotheses of Theorem 3.5 hold for this problem. Therefore, by Theorem 3.5, the nonlinear multi-order Caputo fractional quantum multiintegro-differential boundary problem (15) has at least one solution on [0, 1].   Table 1. Indeed, we only calculated required numerical data of above examples for q = 0.5.

Conclusion
As years and even decades go by, the human beings need to be acquainted with different natural phenomena more and more. One possible way to achieve this purpose is to study the mathematical structures of these processes by means of the logical techniques and tools available in mathematics. In the present framework of this research manuscript, we formulate a new generalized structure of the nonlinear Caputo fractional quantum multiintegro-differential equation in which such multi-order structure of quantum integrals are considered for the first time. In fact, in the light of this type of boundary value problem equipped with the multi-integro-differential setting, one can simply study different cases of the existing usual integro-differential problems in the literature. In this direction, we utilize well-known analytical techniques to derive desired criteria which guarantee the existence of solutions for the proposed multi-order quantum multi-integro-differential problem. Further, some numerical examples are provided to examine our theoretical and analytical findings based on the proposed methods.