On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria

In this paper, we consider a nonlinear sequential q-difference equation based on the Caputo fractional quantum derivatives with nonlocal boundary value conditions containing Riemann–Liouville fractional quantum integrals in four points. In this direction, we derive some criteria and conditions of the existence and uniqueness of solutions to a given Caputo fractional q-difference boundary value problem. Some pure techniques based on condensing operators and Sadovskii’s measure and the eigenvalue of an operator are employed to prove the main results. Also, the Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated. We examine our results by providing two illustrative examples.

By virtue of developments in fractional quantum calculus (q-FC), a number of scientists and researchers [19,20] were attracted to a study of fractional q-difference equations, beginning in the nineteenth century, and wide interest lately [21][22][23].
In 2007, Atici et al. [24] studied some notions in relation to fractional q-calculus on time scales. Then in 2012, Annaby and Mansour presented their investigations by pub-lishing a book on equations and BVPs in the context of fractional q-calculus [25]. Jarad et al. [26] turned to the stability notion on q-fractional non-autonomous systems and after that, Abdeljawad et al. [27] introduced Gronwall-type inequality in q-operator settings. By combining the two above notions, Butt et al. [28] investigated Ulam stability for a Caputo delay q-difference equation by means of q-Gronwall-type inequality. Also, some fascinating insights concerning IVPs and BVPs containing q-difference equations can be found in [29][30][31][32][33][34][35] and the references therein. Ahmad, Nieto, Alsaedi and Al-Hutami [36] turned to the q-difference FBVP with nonlocal integral conditions and implemented an existence analysis on the solutions of the proposed q-BVP which takes the format where f ∈ C([0, 1] × R, R), ρ, β ∈ (0, 1], q ∈ (0, 1), b, d 1 , d 2 ∈ R and C D ρ q , C D β q denote the q-fractional derivatives in Caputo sense of orders ρ and β.
In continuation to the investigation of the q-variant of fractional problems and inspired by the aforementioned work, we aim to examine this area from another angle. Several known methods of functional analysis are used to establish required results on the existence of solutions for a class of q-difference problem. More specifically, we consider the sequential four-point Caputo fractional q-difference boundary value problem (q-CFBVP) of the format where D μ q is the μth-q-difference derivative in the Caputo structure with μ ∈ {γ , β, α} such that 0 < α, β ≤ 1, 0 < γ ≤ 1 and I θ q is the θ th-q-difference integral in the Riemann-Liouville structure with θ > 0 subject to θ ∈ {σ 1 , σ 2 } and also f , g : J × R − → R are continuous functions. a 1 , a 2 , b 1 , b 2 , λ 1 , λ 2 are suitably chosen constants in R + .
Regarding to the novelty of the paper, in comparison to above q-problems, our supposed sequential q-CFBVP is more general. Under the given boundary value conditions, we have used both Caputo and Riemann-Liouville q-fractional operators in four different points of domain of the unknown solution function u simultaneously, in which the linear combinations of the unknown function and its fractional derivative is corresponding to a multiple of q-Riemann-Liouville integral in two mid-points. In this paper, we have designed an extended form of Langevin equations by providing a nonlinear function g in the left-side hand of the given boundary value problem (3). Also, to prove the existence of solutions for such an applied q-problem, we shall utilize some pure notions of functional analysis based on the measure of non-compactness, condensing operators and eigenvalue of the operator, which have been used in papers limited in this regard so far and this distinguishes our research from the work of others. Moreover, we here emphasize that this paper may have useful and effective applications in physics and quantum mechanics such as Langevin systems in the context of quantum operators.
The remaining part of this paper is organized as follows: Sect. 2 is devoted to the primitive notions of q-FC. At first, in Sect. 3, we give an auxiliary lemma which provides the solution of the supposed q-CFBVP (3) and then based on the obtained integral equation, by using fixed point theorems due to Sadovski, Krasnoselskii-Zabreiko and O'Regan, we establish the existence of solutions for the q-CFBVP (3) and also for its uniqueness, we utilize the famous Banach principle. In Sect. 4, the stability criteria of Ulam-Hyers type and its generalized type are checked. Additionally, in Sect. 5, we provide two examples which ensure the usability of the results presented in Sect. 3. The manuscript is ended by our conclusions in Sect. 6.

Preliminaries regarding q-operators
We collect some important basic notions of q-FC in this section. For details, we refer to [19,21,38,39]. Let q ∈ (0, 1). A q-real number is denoted by [m] q and is defined as The q-power function (mn) k with m, n ∈ R is On the other side, [c(mn)] (β) = c β (mn) (β) holds for c ∈ R and also notice that m (β) = m β if n = 0. The q-Gamma function is given by and satisfies q (α + 1) = [α] q q (α). The 1st-q-derivative of an arbitrary mapping φ is defined by the following rule: and for the higher orders, it becomes The 1st-q-integral of an arbitrary mapping φ given on the interval [0, n] is given by Similarly, for the higher orders, it becomes For two first order q-operators D q and I q , we have Here, we assemble some definitions about such q-operators from the fractional point of view.

Definition 2 ([19])
The Caputo αth-q-derivative for an absolutely continuous mapping φ is formulated by where [α] denotes the integer part of α.
For more information on the fractional q-operators, we refer the reader to [38].

Results regarding existence property
In the present section, before moving to our fundamental results, we define · on X = C(J, R) as u = sup t∈J |u(t)|, which in this phase, X transforms into a Banach space. Now, in the first place, we provide the next auxiliary lemma.
By categorizing similar terms, we obtain the expressions Therefore, from (9) and (10), we get and, by inserting k 0 into (9), we obtain Substituting the value of k 0 , k 1 in (8), we get (5), which completes the proof.
Note that, for simplicity, we set g(t, u(t)) = g u (t) and f (t, u(t)) = f u (t) throughout the manuscript.

The first existence criterion
In this subsection, we prove an existence result for the sequential four-point q-CFBVP (3) by making use of Sadovskii's fixed-point theorem. Before moving towards it, we would like to recall several auxiliary facts which are our main tools. X is supposed as a Banach space.
in which α is introduced above.
ii. K 2 is compact.
Theorem 11 ([43]) Consider the bounded, closed and convex subset B of X and the condensing mapping : B → B. Then has a fixed point.
From now on, we put Theorem 12 Consider the following assertions: Proof Consider a bounded, closed and convex subset B r = {u ∈ X : u ≤ r} of X = C(J, R) for a fixed constant r. With regard to Lemma 7, define K : X → X as follows: We split the operator K on the set and We want to prove that the operators K 1 and K 2 follow all the assertions of Theorem 11. We proceed to implement the proof in four steps.
Step 2: K 2 is compact In view of Step 1, we observe that the operator K 2 is uniformly bounded; indeed for any u ∈ B r : Now, take t 1 , t 2 ∈ J by assuming t 1 < t 2 and u ∈ B r . Hence we have The right-hand side of (16) tends to zero (not depending upon u) as t 2 → t 1 . This shows that K 2 is equicontinuous. From the above reasons, it is clear that K 2 is relatively compact on B r . Application of the Arzelà-Ascoli theorem proves the compactness of K 2 on B r .
Step 3: K 1 is Q-contractive. From (B 1 ) and (B 2 ) and for each u, v ∈ B r , we have Step 4: K is condensing. As K 1 and K 2 are continuous Q-contraction and compact, respectively, thus by Lemma 10, K : B r → B r with K = K 1 + K 2 is a condensing map on B r . From the above arguments, by Theorem 11, we conclude that the map K has a fixed point, which leads to the existence of at least one solution for the sequential four-point q-CFBVP (3) in X.

The second existence criterion
We now use another fixed point result due to Krasnoselskii-Zabreiko to demonstrate the following existence criterion for the sequential four-point q-CFBVP (3).
(H2) The function g : J × R → R is continuous and ∃A ∈ R + so that Then there exists at least one solution for the sequential four-point q-CFBVP (3) on J such that where 1 and 2 are, respectively, given by (11) and (12).
Proof Consider a sequence {u n } ⊂ B r which converges to u. We know that f and g are continuous, so, by letting n → ∞, we get Thus, for t ∈ J, we write Therefore the right-hand side of (19) tends to zero. Therefore, the continuity of K is established. Now, for r > 0, we set N = {u ∈ C(J, R); u ≤ r} and f * = sup (t,u)∈J×N |f u (t)|. Thus, which yields Ku ≤ 1 f * + A 2 r. This shows the uniformly boundedness of K. We now claim that K is equicontinuous. Let t 1 , t 2 ∈ J via t 1 < t 2 . Then, by setting f * = sup (t,u)∈J×N |f u (t)|, we obtain It is clear that |Ku(t 2 ) -Ku(t 1 )| → 0 as t 2 → t 1 independent of u. In consequence, from the above arguments, K is relatively compact on N . Application of the Arzelà-Ascoli theorem proves the compactness of K on N .

Now, by considering the sequential four-point q-CFBVP (3) to be linear by taking f u (t) = f (t, u(t)) = λ(t)u(t), the operator L, by Lemma 7, is formulated by
Our next claim is that 1 is not an eigenvalue of L. If it is so, by (18), we estimate which is not possible. Hence we established our claim. Finally, we show that K(u) -L(u) / u vanishes as u → ∞. For t ∈ J, one may write

This means that
By (17) and letting u → ∞, it is concluded that | f u (·) uλ(·)| → 0. Thus we obtain Consequently, by Theorem 13, the supposed sequential four-point q-CFBVP (3) admits a solution in X. The proof is ended.

The third existence criterion
We now present our last existence criterion based on the O'Regan theorem [45].

Then there exists a solution for the supposed sequential four-point q-CFBVP (3) on J.
Proof We consider K : X → X defined by (13) as where the operators K 1 and K 2 are, respectively, given in (14) and (15). By (D3), ∃ε > 0 so that and take B ε = {u ∈ X : u < ε}. We demonstrate the continuity and complete continuity of K 1 . Before this, we prove the uniform boundedness of K 1 . Taking any u ∈B ε , we have in which b * = sup t∈J |b(t)|. Thus K 1 is uniformly bounded. Let t 1 , t 2 ∈ J such that t 1 < t 2 . Then which tends to zero as t 2 → t 1 free of u. This gives the equicontinuity of K 1 . Application of the Arzelà-Ascoli theorem proves the compactness of K 1 and consequently its complete continuity. Furthermore, the continuity of K 1 can be deduced from that of f by the hypothesis. We now show that K 2 is a nonlinear contraction. By (D2) and for u, v ∈ B ε , we have By setting ϒ(u) = 2 κu, note that ϒ(0) = 0 and ϒ(u) = 2 κu < u for u > 0 since κ 2 < 1. Thus Hence K 2 is a nonlinear contraction. Now again, by (D2), for arbitrary u ∈ B ε , we estimate g u (t) = g(t, u) ≤ g(t, u)g(t, 0) + g(t, 0) ≤ φ 1 u + g(t, 0) ≤ κε + l.
where l = sup t∈J |g(t, 0)|. Hence, we get which confirms the boundedness of K 2 . Thus, K = K 1 + K 2 is bounded.
In the final step, we prove that the assumption (C2) of Theorem 15 does not hold. To prove this, consider the existence of μ ∈ (0, 1) and u ∈ ∂B ε such that u = μKu. So u = ε and Taking the supremum for all t ∈ J yields Hence, we get which contradicts (D3). Thus K 1 and K 2 satisfy all the assertions of Theorem 15. Therefore, a fixed-point of K in B ε exists, which is the same solution of the sequential four-point q-CFBVP (3). The proof is finished.

The uniqueness property
Finally, we investigate the uniqueness property for the solutions of the sequential fourpoint q-CFBVP (3) by referring to the Banach principle.

Theorem 17
Let Then the sequential four-point q-CFBVP (3) has a unique solution on J if where 1 , 2 are given in (11) and (12), respectively.
Proof To prove the relevant result, define the ball B r = {u ∈ X : u ≤ r} for some r > 0 satisfying where g * 0 = sup t∈J |g(t, 0)| and f * 0 = sup t∈J |f (t, 0)| and 1 and 2 are, respectively, given by (11) and (12). Now, we prove KB r ⊂ B r in which the operator K : X → X is illustrated as (13). Similar to Step 1 in Theorem 12, for u ∈ B r , we get which implies K(u) ≤ r. Thus, K maps B r into itself. Next, we prove that K is a contraction. For u, v ∈ X, and applying (11) and (12), we have Consequently, we get Since 1 + a 2 < 1, the above inequality proves that K is a contraction. Thus application of the Banach principle shows that K has a unique fixed point, corresponding to unique solution of the sequential four-point q-CFBVP (3) on J. This ends the proof.

The criterion of Ulam-Hyers stability
Due to the importance of the notion of stability for possible solutions of different dynamical systems, in this section, we review two Ulam-Hyers and generalized Ulam-Hyers stabilities for solutions of the sequential four-point q-CFBVP (3). For more information, see [46][47][48]. (3) is Ulam-Hyers stable if ∃c * ∈ R + such that ∀ε > 0 and ∀u * (t) ∈ C(J, R) as a solution function satisfying

Theorem 20 If (H 4 ) and (H 5 ) are fulfilled, then the sequential four-point q-CFBVP (3) is Ulam-Hyers stable on J and accordingly is generalized Ulam-Hyers stable whenever
where 1 , 2 are in the same forms given in (11) and (12), respectively.

Two examples
Here, we aim to present some examples to examine the obtained results.

Conclusions
In the present research, we considered a new boundary problem in the context of the quantum fractional operators. In other words, we defined a sequential q-fractional system of q-difference equation in which boundary conditions are designed as a linear combination of an unknown function and its q-derivative corresponding to a multiple of qintegrals in four points. The main focus of this research is on the solution's existence and its uniqueness with the help of some methods inspired by several pure concepts in functional analysis. We used three different fixed-point methods for this aim relying on the measure of non-compactness and condensing operators and compact operators. The existence of a unique solution is investigated based on the Banach criterion. The investigation of stability of the given q-CFBVP system in two formats based on Ulam-Hyers' conditions is implemented. Lastly, two examples are provided to ensure the findings. It is evident that this structure is more general and has many special applied cases. By assuming g(t, u(t)) = -μ ∈ R and a 1 = b 1 = a 2 = b 2 = 1 and σ 1 = σ 2 = 1 and by letting q → 1, our proposed sequential four-point q-CFBVP (3)  which is considered as one of the most important equations in mathematical physics. Therefore, one can observe that the research study presented in the manuscript is not only new in the existing structure, but will also lead to other various quantum fractional problems as special cases. In future studies, we can generalize our boundary conditions to multi-point ones and investigate similar results in the context of newly-defined fractional (p, q)-operators in both cases of difference equations and inclusions.