Application of some special operators on the analysis of a new generalized fractional Navier problem in the context of q-calculus

The key objective of this study is determining several existence criteria for the sequential generalized fractional models of an elastic beam, fourth-order Navier equation in the context of quantum calculus (q-calculus). The required way to accomplish the desired goal is that we first explore an integral equation of fractional order w.r.t. q-RL-integrals. Then, for the existence of solutions, we utilize some fixed point and endpoint conditions with the aid of some new special operators belonging to operator subclasses, orbital α-admissible and α-ψ-contractive operators and multivalued operators involving approximate endpoint criteria, which are constructed by using aforementioned integral equation. Furthermore, we design two examples to numerically analyze our results.


Introduction
With the passing of years and even decades, people need to be more and more aware of details of various natural phenomena. The logical tools and notions available in mathematics and especially mathematical operators are one of possible ways to achieve this aim in modeling various processes. In this direction, many researchers developed numerous fractional operators such that their applicability and usefulness become more and more evident to researchers each day. As a result, using fractional operators, different processes are modeled and examined from all aspects in the mathematical structures such as boundary value problems. In broad fields such as chemistry, biology, physics, economics, engineering, and so on fractional calculus, related differential equations and BVPs are commonly used [1][2][3][4][5]. In a vast domain of papers, scientists have examined numerous mathematical procedures across different facets of fractional differential equations [6][7][8][9][10][11][12][13].
In recent years, there has been a great deal of interest in the analysis of q-difference equations. These equations have been found to be applicable in various fields of physics and mechanics and thus have been developed into multidisciplinary topics. Fractional q-calculus is considered as a special fractional variant of calculus, originally it was suggested by Jackson [14]. and then further investigations were performed by Al-Salam and Agarwal [15,16]. Some fascinating studies into IVPs and BVPs with equations involving q-operators are available in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].
More specifically, Ferreira [32] considered the following nonlinear fractional terminal q-BVP and discussed the existence of a nontrivial solution: where t ∈ O = [0, 1], D 1 q is the standard Riemann-Liouville fractional q-derivative, and M : O × R → R is a continuous function.
An elastic beam is considered as an essential feature in constructions like ships, bridges, building structures, and aviation industry. In this direction and in mathematical point of view, the following fourth-order BVP of Navier differential equation can be used in modeling deformation of the beam (see [34]): (4) (t) = M(t, μ(t), μ (t)), where M : O × R 2 → R is continuous, and t ∈ O := [0, 1]. By transforming (1) into the second-order integro-differential equation with bounded M, Aftabizadeh [35] utilized Schauder's fixed-point theorem and discussed the existence and uniqueness of solutions for (1). The upper and lower solution method was used by Ma et al. [36] for problem (1). In 2004, Bai et al. [37] extended a monotone method to upper and lower solutions of the beam model (1). In the context of fractional calculus, Bachar and Eltayeb [38] proposed the fractional variant of the Riemann-Liouville model (1) and explored the existence, unique-ness, and positivity for the solutions of a system designed by the following format: where 1 ∈ (1, 2], 2 ∈ (1, 2), RL D 1 and RL D 2 are the fractional derivatives in the Riemann-Liouville sense, and M : O × R 2 → R is continuous. In the case 1 = 2 = 2, problem (2) reduces to (1).
Inspired by aforesaid ideas given in the papers mentioned, in terms of the standard Navier equation, we review and discuss a new sequential generalized fractional q-Navier BVP along with its inclusion version given by where 1 ∈ (1, 2], 2 ∈ (1, 2), and γ , δ, λ, β ∈ R + . Moreover, the operator C D (·) q is the qderivative of given fractional orders in the Caputo sense. Furthermore, a continuous single-valued function M : O × R 2 → R and a multivalued function M : O × R 2 → P(R) are assumed to be arbitrary equipped with some required specifications explained subsequently.
The novelty of our paper is that the above suggested structure for Navier problem is unique and novel, which can be regarded as a generalized fractional model of the standard Navier problem in the context of quantum operators. Indeed, by taking 1 = 2 = 2, q → 1, and γ = δ = λ = β = 1 we have the standard Navier BVP (1). Also, we establish our results by new techniques involving some special operators.
We have organized the remaining sections of the paper as follows. The upcoming section is assigned to the basic ideas of fractional q-calculus. Section 3 starts with a lemma, which specifies the solution of our proposed Navier BVPs (3)-(4) in terms of an integral equation of noninteger order. After that, we follow the well.-known fixed-point methods due to Krasnoselskii [39] and new operators introduced by Samet et al. [40] to obtain the existence of solutions for single-valued maps. In Sect. 4, we consider the inclusion variant (4) of the Navier BVP and explore the existence of solutions using the methods presented by Mohammadi et al. [41] and approximated end-point property. Section 5 provides illustrations of the results given in Sects. 3 and 4. In the last section, we present the concluding remarks and future proposals.

Basic preliminaries
We assemble and examine supplementary and fundamental concepts concerning qcalculus in the light of our approaches to this research.

Notation 2.2
Let (A * , · A * ) be a normed space. By P B (A * ), P CL (A * ), P CM (A * ), and P CX (A * ) we denote the classes of all bounded, closed, compact, and convex sets in A * , respectively.

Definition 2.4 ([52])
(1) A member μ ∈ A * is called an end-point of a multivalued function M : where H d is the Pompeiu-Hausdorff metric.
We recall some necessary fixed-point results in connection with the suggested boundary problem.
(3) for any sequence {μ n } in A * such that μ n → μ and α(μ n , μ n+1 ) ≥ 1 for all n ≥ 1, we have α(μ n , μ) ≥ 1 for all n ≥ 1. Then there is a fixed-point for M. Theorem 2.7 ([39], Krasnoselskii) Let G = ∅ be a closed bounded convex set contained in a Banach space A * , and let M 1 and M 2 be such that: (2) M 1 is compact and continuous; 3 the space A * has the property that for each sequence {μ n } in A * such that α(μ n , μ n+1 ) ≥ 1 and μ n → μ for all n ∈ N, there exists a subsequence {μ n r } of {μ n } such that α(μ n r , μ) ≥ 1 for all r ∈ N. Then M has a fixed point.

Then a unique endpoint of M exists iff M has an approximate end-point criterion.
3 Results for q-Navier FBVP (3) The following lemma presents a solution to the proposed problem (3) in the form of an integral equation, which is important in determining our key findings.
if and only if it satisfies the q-integral equation Proof First, let a function μ * be a solution of the nonlinear sequential generalized q-Navier FBVP (9). Then C D 1 q ( C D 2 q μ * )(t) = η(t). Since 1 ∈ (1, 2), taking the th 1 -q-integral in the Riemann-Liouville setting, we obtain where we need to find the constants m 0 , m 1 ∈ R. By the third condition λ C D 2 q μ(0) = 0 we obtain m 0 = 0. So On the other hand, by (11) and the fourth condition β C D 2 q μ(1) = 0 we get and thus In view of (12), relation (11) becomes Again, since 2 ∈ (1, 2), taking the th 2 -q-integral in the Riemann-Liouville setting in (13), we obtain where the constants m * 0 , m * 1 ∈ R are to find. The first condition γ μ(0) = 0 gives m * 0 = 0. In consequence, At last, the second condition δμ(1) = 0 implies that Consequently, Inserting m * 1 into (14), we obtain which yields that μ * settles q-integral equation (10). On the other hand, we can simply prove the converse by direct computation, and ultimately the arguments are finished. Now consider the operator N : We can easily infer that μ * is a solution of fractional q-Navier BVP (3) iff μ * is a fixed point of the operator N. For simplicity, set 1 = 2 q ( 1 + 2 + 1) and 1 = 1 q ( 1 + 2 + 1) . (16) and a nondecreasing function ψ ∈ such that: for all n and t ∈ O, we have Then the generalized q-Navier BVP (3) has a solution.
Then if where k = sup t∈O |k(t)| and 1 , 2 are defined in (16), then the generalized q-Navier FBVP (3) has a solution.
Proof Define = sup t∈O | (t)| and choose an appropriate constant ε > 0 such that where 1 and 2 are defined in (15). Define the set Y ε = {μ ∈ A * : μ A * ≤ ε}. It is a nonempty, closed, bounded, and convex set contained in A * . Define N 1 and N 2 on Y ε as Therefore N 1 μ 1 + N 2 μ 2 A * ≤ô ( 1 + 2 ) ≤ ε, which implies that From the continuity of the single-valued function M it is evident that N 1 is continuous on its domain. Now, for all μ ∈ Y ε , we get that This clearly shows the uniform boundedness of the operator N 1 on Y ε . To ensure the compactness of N 1 on Y ε , consider t 1 , t 2 ∈ O such that t 1 < t 2 . Then we get the following inequalities: Thus, as t 1 goes to t 2 , |(N 1 μ)(t 2 ) -(N 1 μ)(t 1 )| tends to zero independently of μ. Also, we find that Thus |( C D 2 q N 1 μ)(t 2 ) -( C D 2 q N 1 μ)(t 1 )| goes to zero as t 1 approaches to t 2 independently of μ. Therefore (N 1 μ)(t 2 ) -(N 1 μ)(t 1 ) A * → 0 as t 1 → t 2 . Consequently, the equicontinuity of the operator N 1 is confirmed. Therefore by the Arzelà-Ascoli theorem N 1 is a compact operator on Y ε . At last, we show that N 2 is a contraction mapping. Let μ 1 , μ 2 ∈ Y ε . Then Thus where the constant L < 1. Therefore N 2 is a contraction on Y ε . Hence Theorem 2.7 implies the existence of a solution for the generalized q-Navier FBVP (3).
By (X 9 ) there exists a subsequence {μ n r } r≥1 of {μ n } such that U μ n r (t), C D 2 q μ n r (t) , μ(t), C D 2 q μ(t) ≥ 0 for all t ∈ O. This implies that α(μ n r , μ) ≥ 1 for all r. Hence all the assumptions of Theorem 2.8 are fulfilled. This confirms the existence of a fixed-point of the operator Z. Therefore it follows that the generalized q-Navier FBVP (4) has a solution.

Examples
We provide a few illustrative numerical examples to our theoretical and analytical findings in the previous sections.
Next, we consider the set-valued map Z : Ultimately, by Theorem 4.2 we find that the generalized q-Navier BVP (24) has a solution.

Conclusion
In this paper, we modeled the standard Navier equation to q-fractional Navier BVP and explored the existence of solutions by making use of the well-known results from functional analysis due to some techniques introduced by Krasnoselskii, Samet, Mohammadi, and Amini-Harandi based on special operators. In fact, by deriving an integral equation we defined some operators based on it, and then by utilizing a subclass of special operators such as orbital α-admissible maps, α-ψ-contractions, the multifunctions having approximate endpoint criterion, and so on we proved the required results. Finally, we gave illustrations by two examples to explain the consistency of the findings for the proposed sequential generalized Navier q-BVP. As a possible future plan, some other operators may be considered in the next papers to discuss the existence of solutions, stability, and other qualitative aspects of solutions of the generalized Navier fractional model in two singular or nonsingular formats.