Existence and Stability of Solutions for Linear and Nonlinear Damping of q-Fractional Duffing–Rayleigh Problem

In this current paper, using q-fractional calculus, we study the Duffing–Rayleigh type problem with sequential fractional q-derivative of the Caputo type. We investigate the existence and uniqueness of solutions by applying some classical fixed point theorems. Also we define and study the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of solutions for our problem. An example is presented to illustrate the main results.


Introduction
The study of nonlinear differential equations involving q-calculus has gained intensive interest in the last years, these types of equations have several applications in diverse fields and thus have evolved into multidisciplinary subjects, see for instance [7,8,16]. Also nonlinear differential equations with fractional q-calculus have been investigated by several scholars, see for example [9,13,20,22,23,30] and the references cited therein. Recently, many scientific researchers, have studied the uniqueness, existence and Hyers-Ulam stability (H-US) of solutions for some differential equations with fractional quantum calculus, see the papers [10,18,26]. Considerable attention has been given to the study of the existence and uniqueness of solutions of sequential q-differential equations, see the research works [1][2][3]15,16].
In Sect. 2, we recall some essential definition of fractional quantum calculus. Section 3 contains our main results in this work, while an example is presented to support the validity of our obtained results. stability results are extensively discussed in Sect. 4. An illustrative example with some needed algorithms for the problem are given in Sect. 5. Finally, in Sect. 6, conclusion are presented.

Preliminaries Regarding q-Fractional Calculus
The operator D κ q is the fractional q-derivative of the Caputo type of order κ is given by is the smallest integer greater than or equal to κ. The fractional q-integral of the Riemann-Liouville type [4,18], defined by (κ−1) s (r) d q r, κ > 0, , n ∈ N. Algorithm 1 shows MATLAB function to obtain q-gamma function. Using Algorithm 2, we can calculate this type q-integral. We recall the following lemmas [4,18].
such that n is the smallest integer greater than or equal to κ.

Lemma 2.3.
For κ ∈ R + and σ > −1, we have Lemma 2.4. [28] Assume that W is a completely continuous self operator on a Banach space F and the set To study the problem (1.1), we need the following space endowed with the norm Then it is well known that (E, . E ) is a Banach space.

Existence of Solutions for q − FD-RP
In this part, we demonstrate the existence and uniqueness of solution of problem (1.1).

2)
Proof. Applying Lemma 2.2, we can write Now, using the operator I μ q , we get It follows that where e i ∈ R, (i = 0, 1, 2). Using the boundary conditions, we get Hence, we obtain (3.2).
Before stating and proving the main results, we impose the following hypotheses. (H 1 ) : The functions ϕ and ψ are continuous over Ω × R 2 , φ is continuous over Ω × R and p is continuous over Ω. (H 2 ) : There exists constant a j > 0, j = 1, 2, 3 such that for all τ ∈ Ω and s i , t i ∈ R, (i = 1, 2), we have

Theorem 3.2.
Assume that (H 1 ) and (H 2 ) hold and that , Then, the q − FD-RP (1.1) has a unique solution.
Proof. Let us define , (3.14) Then, thanks to (3.13) and (3.14), we can write Also, one can observe that Using (3.7), we can write In consequence, we get So, we obtain On the other hand, we have By (3.7), we get Consequently, we obtain Then for all s ∈ B ε and thanks to (H 3 ) and (3.7), we can write On the other hand, for any s ∈ B ε , we get Thanks to (3.7), we have From the above inequalities, it follows that the operator W is uniformly bounded. Next, we prove that W is equicontinuous. Let s ∈ B ε and τ 1 , τ 2 ∈ Ω, with τ 2 < τ 1 . Then by (H 3 ) , we have On the other hand, we have Thanks to the above inequalities, we can state that W s (τ 1 ) − W s (τ 2 ) E → 0 as τ 1 → τ 2 . By the Arzelà-Ascoli theorem, we conclude that W is a completely continuous operator. Finally, it will be proved that the set Ψ, given is bounded. Let s ∈ Ψ, then s = ςW (s) for some 0 < ς < 1. Thus for any τ ∈ Ω, we can write s (τ ) = ςW (s). Then by (H 3 ) , we get On the other hand, Using (3.7), we can obtain It follows from above inequalities, that where Θ, Θ * , Π * and Φ are given by (3.10) and (3.12). This shows that the set Ψ is bounded. Thanks to Lemma 2.4, we deduce that W has at least one fixed point, which is a solution of q − FD-RP (1.1).

Ulam-Hyers Stability of q − FD-RP
In this part, the Ulam-stability type of the q − FD-RP (1.1) will be discussed.
for each τ ∈ Ω, there exists a solution s of the q − FD-RP (1.1) with for each τ ∈ Ω, there exists a solution s of the q − FD-RP (1.1) with Remark 4.2. A function t ∈ E is a solution of the inequality (4.1) iff there exists a function h : Ω → R (which depend on t) such that |h (τ )| ≤ d, for each τ ∈ Ω, and Proof. Let us denote by s ∈ E the unique solution of the problem and |h (τ )| ≤ d, τ ∈ Ω. By Lemma 3.1 and Remark 4.2, we have
On the other hand, we have Thanks to (4.4) and (H 2 ) , we can write Then , (4.5) if we put , Proof. From Remark 4.2, we have where |h (τ )| ≤ dm (τ ) , τ ∈ Ω and t ∈ E is a solution of the inequality (4.2). Let us denote by s ∈ E the unique solution of (4.3), then using Lemma 3.1, we get' From these relations, we have Thanks to (H 2 ) and by (3.7), we get Then, we have If we take we can obtain t − s E ≤ D ϕ * ,ψ * ,φ * ,p dm (τ ), for τ ∈ Ω. Therefore, the q−FD-

Conclusion
The q−FD-RP has been investigated in this work in details. The investigation of this particular equation provides us with a powerful tool in modeling most scientific phenomena without the need to remove most parameters which have an essential role in the physical interpretation of the studied phenomena. q − FD-RP (1.1) has been studied under some B.Cs. An example has been provided to support our results' validity and applicability in fields of physics and engineering.