On a fractional cantilever beam model in the q-difference inclusion settings via special multi-valued operators

The fundamental goal of the study under consideration is to establish some of the existence criteria needed for a particular fractional inclusion model of cantilever beam in the setting of quantum calculus using new arguments of existence theory. In this way, we investigate a fractional integral equation that corresponds to the aforementioned boundary value problem. In a more concrete sense, we design new multi-valued operators based on this integral equation, which belong to the certain subclasses of functions, called α-admissible and α-ψ-contractive multi-functions, in combination with the AEP-property. Also, we use some inequalities such as Ω-inequality and set-valued version inequalities. Moreover, we add a simulative example for a numerical analysis of our results obtained in this study.


Introduction
Fractional calculus and its corresponding differential equations and BvPs have been widely utilized in the vast fields of science, including biology, chemistry, economy, physics, engineering, etc. [1][2][3]. Fractional derivatives do not merely represent a generalization of ordinary derivatives but also precisely and accurately describe the complex behavior, in contrast to integer order derivatives, of diverse physical structures. Several investigators have examined differential equation of arbitrary order starting from the existence and uniqueness of solutions to the analytical and computational approaches in search of solutions. A number of monographs and articles are available concerning the developments of theory of fractional differential equations and inclusions .
On the other hand, the quantum calculus is a field without the concept of limit that corresponds to the traditional infinitesimal one. Regardless of their vast background, both theories are in the domain of mathematical analysis, working on their properties did not emerge two ages later. Quantum difference operators (q-DiffOper) were first exhibited and introduced by Jackson [32] and have been widely analyzed in order to explain complex physical structures with a number of non-differentiable functions. In early nineties, numerous academician [33,34] came forward with the studies on q-difference equations which lately received great interest and attention [32,35,36]. There are some intriguing insights into IVPs and BVPs coupled with q-difference equations in [37][38][39][40][41][42][43][44][45][46][47][48].
Note that the above inclusion q-FBvP (2) is an extension of the standard practical model of the cantilever beam to the fractional q-analogue structure. By assuming κ = 3, ω = 1, and q → 1 and (·) = {ψ(·)}, we obtain the above fourth-order differential inclusion arising in the cantilever beam model (1). One can state some physical interpretations for the q-FBvP model (2) by assuming such assumptions over κ and ω as follows: u(ς) stands for the deformation function, C D 3 q→1 ( C D 1 q→1 u) is the load density stiffness, C D 3 q→1 u denotes the stiffness of the function u under the shear force, C D 2 q→1 u denotes the stiffness of the function u in the bending moment, and C D 1 q→1 u represents the slope [54,55]. About the novelty of this work, one can state that such a fractional model of cantilever beam based on q-difference operators has not been studied in any research paper so far, and on this new structure, we derive our mathematical and analytical results ensuring the solution's existence by means of some special subclasses of multi-functions. For our applied technique, we here use α-ψ-contractive multi-functions for the confirmation of the existence of a fixed point and also the confirmation of the existence of an end point by making use of another family of multi-functions having the AEP-property.
The rest of the manuscript is structured as follows: Sect. 2 is dedicated to the fundamental ideas of q-analogue of fractional calculus. In the beginning of Sect. 3, we provide a lemma which presents the solution of the cantilever beam q-FBvP (2) in the form of an integral equation, and then, by making use of the α-admissible multi-valued mappings with control function and approximate end point theory, we guarantee the solutions' existence for the cantilever beam q-FBvP (2). Section 4 is assigned to the illustration of the results presented in Sect. 3 with the aid of an example. Finally, Sect. 5 describes the concluded remarks.

Preliminaries
We compile and study, in the light of our approaches used in this investigation, some auxiliary and primitive definitions regarding q-calculus.
Regarding (X , · ) as a normed space, the classes P CL (X ) (all closed sets), P BN (X ) (all bounded sets), P CP (X ) (all compact sets), and P CV (X ) (all convex sets) involve the respective form of subsets of X .
The multi-function admits an approximate end point property (AEP) if Mohammadi et al. [62] introduced a new notion given as the subclass of all nondecreasing functions like so that ∞ n=1 ψ n (ς) < ∞ for any ς > 0. Now, by making use of such a category, we define a new family of multi-functions.
where H d is the Pompeiu-Hausdorff metric.
Next we recall requisite theorems concerning the investigation of the proposed q-FBvP (2).

Theorem 7 ([61]) Let (X , d) be a metric space of complete type and consider (1) an upper semi-continuous map
(2) a multi-function : X → P CL,BN (X ) such that Then a unique end point of exists iff has the AEP-property.

Existence results
} as a Banach space of all real-valued continuous functions on S equipped with a sup-norm In the following proposition, the solution to the proposed fractional cantilever qproblem (2) is presented in the form of an integral equation, which will be helpful in establishing our main findings. 3], ω ∈ (0, 1], and G ∈ C(S, R). Then the solution of the linear q-FBvP is presented in the following form: where [ω + 2] q = 1-q ω+2 1-q .
Proof Assume that u * is a solution of the given q-FBvP (5).
Proof Evidently, the fixed point of the multi-function U : X → P(X ) given by (13) is identified as the solution for the cantilever beam inclusion q-FBvP (2), so we try to check the assumptions of Theorem 6 on this multi-valued operator. Since the compact-valued set-valued map ς → (ς, u(ς), C D ω q u(ς), C D ω+1 q u(ς), C D ω+2 q u(ς)) is measurable as well as closed-valued for any u ∈ X , so the map has a measurable selection and (SEL) ,u = ∅. At first, we verify that the subset U(u) of X is closed ∀u ∈ X . For this purpose, consider a sequence {u n } n≥1 in U(u) via u n → u. Now, for each n ≥ 1, we have a memberφ n ∈ (SEL) ,u satisfying for almost all ς ∈ S. Since admits compact values, we take a subsequence of {φ n } n≥1 (following the same symbol) that tends to someφ ∈ L 1 (S). Thus,φ ∈ (SEL) ,u and for all ς ∈ S. This leads to the conclusion that u ∈ U(u) and the multi-function U is closedvalued. Since is compact-valued, it is easy to ensure the boundedness of U(u) for every u ∈ X . Next, we prove that U is an α-ψ-contractive multi-function. In view of this intention, we take a function α on X × X with nonnegative values which is defined by and otherwise, it is defined to be zero for all u,ū ∈ X . Suppose u,ū ∈ X and ν 1 ∈ U(ū) and selectφ 1 ∈ (SEL) ,ū such that for all ς ∈ S. Utilizing (16), we get for almost all ς ∈ S. Thus, a member exists such that Now, consider a map ℵ : S → P(X ) which is characterized by for any ς ∈ S. Sinceφ 1 and so that, for all ς ∈ S, we have Consider ν 2 ∈ U(u) given by for any ς ∈ S. Then we obtain the following inequalities: Also, we have Thus, for any u,ū ∈ X which indicates that U is an α-ψ-contractive multi-function. Now, suppose that u ∈ X andū ∈ U(u) satisfy α(u,ū) ≥ 1, and so Then, from the hypothesis, a member r ∈ U(ū) exists such that It implies that α(ū, r) ≥ 1, and so we deduce that U is α-admissible. Now, take u 0 ∈ X and u ∈ U(u 0 ) so that for all ς ∈ S. Then it follows that α(u 0 ,ū) ≥ 1. Let us assume that {u n } n≥1 is a sequence in X such that u n → u and α(u n , u n+1 ) ≥ 1 for all n. Then we obtain Utilization of the assumption (T 4 ) leads to the existence of a subsequence {u n t } t≥1 of {u n } such that for all ς ∈ S. This directly implies that α(u n t , u) ≥ 1 for all t. Hence, Theorem 6 is settled and the multi-function U possesses a fixed point which is regarded as a solution for the fractional cantilever beam inclusion q-BvP (2). Now, we utilize the notion of end points for other subclass of multi-functions to achieve the desired aim.
Proof To fulfill Theorem 7, we have to prove that the multi-map U : X → P(X ), given by (13) has an end point. At first, since is a closed-valued as well as measurable map, so has a measurable selection and (SEL) ,u = ∅ for each u ∈ X . Using the same method as given in Theorem 9, one can deduce that U(u) has closed values. Since the multi-function is compact, so U(u) is bounded for any u ∈ X . This time, we only try to show that H d (U(u), U(r)) ≤ ψ( ur ). Let us assume that u, r ∈ X and ν 1 ∈ U(r). Selectφ 1 ∈ (SEL) ,r such that for almost all ς ∈ S. Since, for any ς ∈ S, we have there exists a member for any ς ∈ S. Now, the map : S → P(X ) is considered which is given by the following way: The multi-function (·) ∩ (·, u(·), C D ω q u(·), C D ω+1 q u(·), C D ω+2 q u(·)) is measurable becausé ϕ 1 and is selected such that for all ς ∈ S. Choose ν 2 ∈ U(u) such that, for all ς ∈ S, By employing the same methodology used in the proof of Theorem 9, we reach ≤ ( 1 + 2 + 3 + 4 )ψ ur = ψ ur .
Accordingly, we have H d (U(u), U(r)) ≤ ψ( ur ) for any u, r ∈ X . From assumption (T 10 ) implying the existence of the AEP-property for the multi-function U, the application of Theorem 7 leads to the existence of u * ∈ X satisfying U(u * ) = {u * }. This indicates that u * is a solution for the cantilever beam inclusion q-FBvP (2).

An example
Based on the fractional cantilever beam inclusion q-BvP (2), we here present some examples in this framework to confirm the validity of the results.

Conclusion
A variety of complex natural phenomena that arise from science and technology are modeled by fractional operators. In the present study, we considered a fractional inclusion model of cantilever beam in the context of quantum calculus. We therefore, specified several operators based on the special classes of α-admissible and α-ψ-contractive multifunctions, relying on the equivalent integral equation. We studied the existence of solutions and, in addition, for such operators, we explored the AEP-property. Lastly, an example was given to examine the results regarding the proposed cantilever beam inclusion q-FBvP. As a future proposal, one can consider some other fractional operators to discuss the existence of solutions and approximating them for different generalized fractional models of the cantilever beam equation.