Kuratowski MNC method on a generalized fractional Caputo Sturm–Liouville–Langevin q-difference problem with generalized Ulam–Hyers stability

In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations.


Introduction
The topics of fractional calculus and (quantum) q-calculus in general and fractional differential equations, especially, have appeared extensively, and they are one of the applied branches in mathematical analysis which have enormous impact in exact description of existing real phenomena. In the meantime, in 1910, q-difference equations were introduced by Jackson [1]. Next, the research on such q-difference equations was implemented in most of the works, more specifically, by Carmichael and Al-Salam in [2,3]. Regarding several earlier manuscripts on this topic, we cite [4,5], while the initial ideas on q-FC can be observed in [6]. To review other applications on this field, see for instance [7][8][9]. Also, some new papers in this regard are [10,11].
By making use of techniques of nonlinear analysis, many researchers have turned to the existence and uniqueness of solutions to nonlinear fractional differential equations equipped with a variety of boundary conditions as special cases, since they are considered as accurate procedures to describe the actual processes . On the other hand, some other imperative and extra extremely good studies, which have these days attracted greater interest, have been dedicated to the qualitative investigation of differential equations of non-integer orders [35][36][37][38][39]. The primary effort was made by Ulam in 1940 and later by Hyers [40], and the notion of stability was developed by Rassias. Finally, in comparison to other works, Obloza [41] was the first researcher who investigated the U-H stability for a given differential equation in 1993.
The Langevin equation (designed in 1908 by Langevin for elaborate interpretation of Brownian motions) is introduced to be an applied mathematical model to describe the cases of the evolution of every physical phenomenon in the environments having fluctuations [42]. On the other side, the Sturm-Liouville problems involve various applications in some fields of engineering and science [43]. Recently, Kiataramkul along with a group of mathematicians presented a new structure of Hadamard FBVP in the fractional settings by combining the Langevin and Sturm-Liouville equations [44]. This combination would be an appropriate description in relation to each dynamical process defined in a fractal medium in which both properties of the fractal and memory via a dissipative memory kernel are incorporated.
Regarding the novelty of the present manuscript, no contributions exist, as far as we know, concerning the existence theory on the Caputo q-difference equations with the help of the technique of Kuratowski measure on noncompactness (KMNC-method) combined with Mönch's fixed point theorem. As a result, the goal of this paper is to enrich this academic area via new techniques based on a special notion of Kuratowski measure. Our proposed method is essentially based on the result given by Banas et al. [45]. Some authors utilized similar methods via the KMNC technique to different types of FBVPs, including [46][47][48][49][50][51][52][53][54]. Therefore, it is emphasized that the KMNC technique is implemented for the first time on the generalized fractional q-Caputo Sturm-Liouville-Langevin q-difference problems.
Our suggested model in the context of quantum operators is not only new in the existing structure but also is equivalent to some known physical models pertinent to the specific values of the functions and parameters involved in q-FBVP (1). In spite of some similar research implemented by Berhail et al. [55] based on Hadamard operators and also by Kiataramkul et al. [44] based on anti-periodic boundary conditions, more precisely, our q-FBVP (1) is formulated in a generalized form which combines both Langevin equations and Sturm-Liouville problems in the context of q-operators for the first time. Indeed, we have: • By choosing r(t) ≡ 0 and q → 1, the nonlinear generalized fractional q-Caputo Sturm-Liouville-Langevin q-difference BVP (1) is converted to the standard nonlinear Sturm-Liouville equation • By choosing ρ(t) ≡ 1 and r(t) = -μ, μ ∈ R and q → 1, the nonlinear generalized fractional q-Caputo Sturm-Liouville-Langevin q-difference BVP (1) is converted to the standard nonlinear Langevin equation In fact, in this paper, we aim to show that one can model some known equations with respect to q-operators, because these operators have discrete structures and give accurate numerical results in different simulations. Here is a brief outline of the arrangement of the paper. The next section provides definitions and preliminary lemmas that will be needed to prove the main theorems. In Sect. 3, we establish existence and uniqueness of solutions to the given problem of the generalized Sturm-Liouville-Langevin q-difference equation by following the KMNC-method. In Sect. 4, we discuss some types of fractional Ulam stability. In Sect. 5, we give an example to illustrate numerical findings. In the last part of the paper, we conclude our exposition with some final remarks and observations.

Preliminary notions
We follow the present section by recalling and assembling some required notions for further arguments and developments.
Consider the Banach space of all real-valued continuous functions U = C(I, E) (here E is assumed to be the space of real numbers) with the supremum norm w ∞ = sup w(t) : t ∈ I , and M U represents the class of all bounded mappings in U .
Consider L 1 (I, E) as a Banach space of all measurable Bochner integrable mappings like w : I → E which are furnished with the integral norm In what follows, we recollect some elementary definitions and properties related to fractional q-calculus. Refer to [1,2]. Let q ∈ (0, 1) be a real number. For each c ∈ R, we define

Definition 4 ([5, 6])
The integral of a function σ : I → R given as and is named the Riemann-Liouville q-integral of order δ ∈ R + , if the integral exists.

Definition 7 ([56])
The δ th -q-Caputo derivative for an absolutely continuous mapping σ is given by where [δ] is the integer part of δ and the integral exists.
Remark 1 Note that if q → 1, then both Definition 6 and Definition 7 are converted to the standard Riemann-Liouville and Caputo fractional derivatives.

Lemma 9 ([57])
Let σ be a function defined on I and suppose that δ, β are two real nonnegative numbers. Then the following hold: Now we review some properties of the concept of Kuratowski measure of noncompactness (KMNC).
Proposition 11 ([45, 46]) The Kuratowski MNC satisfies the following: , where C and conv(C) are the closure and the convex hull of C, respectively, Notation 12 Let be the set of functions w : Theorem 13 ([48, 54]) Let the subset W = ∅ be convex and bounded, and in the Banach then σ has a fixed point.

Lemma 14 ([48])
Let the subset W be convex, bounded, and closed in the Banach space U , G ∈ C(I × I), σ : I × W → U be Caratheodory, and there exists p ∈ L 1 (I, R + ) such that, for each t ∈ I and every bounded set B ⊂ W , By assuming as an equicontinuous set of mappings I → W , we have

Main theorems
In this section, we are concerned with the existence of solutions of the given generalized Sturm-Liouville-Langevin q-difference FBVP (1).

Definition 15 By a solution of the generalized Sturm
In what follows, we present the characterization of solutions in relation to suggested generalized Sturm-Liouville-Langevin q-difference FBVP (1).

Lemma 16
Let K(t) ∈ U , 0 < α, β ≤ 1, ρ ∈ C(I, R\{0}), and r ∈ C(I, R). Then the solution of the following linear generalized Sturm-Liouville-Langevin q-difference FBVP is given by Proof Taking the α th -q-Riemann-Liouville integral to the FDEq of (3), we get where c 0 ∈ R. The second BCs of system (3) gives Taking the β th -q-Riemann-Liouville integral to (5), we obtain where c 1 ∈ R. Using the condition w(0) = 0 of (3), we have Substituting the obtained value for c 1 , we derive the q-integral equation (4), and the proof is completed.
Note that, on the other side, if we apply the Caputo β th -q-derivative and α th -q-derivative to both sides of (4) and use Lemma 9, then the given system (3) immediately is established. Now, consider the nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1). On the basis of Lemma 16, the solutions of (1) correspond to q-integral equation in the following form:

Existence result via the KMNC-method
We further will use the following hypotheses.
Proof Firstly, for w ∈ U , we consider the operator G : U → U defined by (12) Evidently, the fixed points of G are solutions of the nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1). We take D R is convex, closed, and bounded. We shall follow the proof in three steps. STEP 1: G is sequentially continuous: Let {w n } n be a sequence with w n → w in U . Then, for each t ∈ I, one may write Since the function σ is continuous and satisfies (H1), so σ (t, w n (t)) tends uniformly to σ (t, w(t)). In accordance with Lebesgue's dominated convergence theorem, {G(w n )(t)} tends uniformly to G(w)(t), that is, Gw n → Gw. Hence G : D R → D R is sequentially continuous. STEP 2: G(D R ) ⊆ D R : Take w ∈ D R . By (H2) and for each t ∈ I, let G(w)(t) = 0. Then Hence we get STEP 3: G(D R ) is equicontinuous: By considering STEP 2, it is known that G(D R ) ⊂ U is bounded uniformly. In relation to the equicontinuity of G(D R ), we take t 1 , t 2 ∈ I, t 1 < t 2 , and w ∈ D R . Then As t 1 → t 2 , the right-hand side of (14) goes to 0 independent of w, and thus |Gw(t 2 ) -Gw(t 1 )| → 0. The equicontinuity of G is confirmed. The implication (2) is proved in the last step: Let ⊂ D R be such that = conv(G( ) ∪ {0}). Since is equicontinuous and bounded, the mapping t → w(t) = κ( (t)) has the continuity property on I. From (H2) and some given properties of the Kuratowski MNC κ, for any t ∈ I, we get This means that By (9) it follows that w = 0, that is, w(t) = 0 for any t ∈ I, so κ( ) = 0, and then (t) is relatively compact in U . From the Ascoli-Arzela theorem, has the relative compactness in D R . By Theorem 13, we find out that G has a fixed point, which is the same solution of the nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1).

Uniqueness criterion
Then the nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1) has a unique solution on I such that where μ and ν are given by Equations (10) and (11), respectively.
Proof In the first place, we show GB ω ⊂ B ω , where the operator G : U → U is defined by Equation (7), and for ω > 0, B ω = {w ∈ U, w ≤ ω} such that ω ≥ μσ 0 1 -, and σ 0 = sup 0≤t≤T |σ (t, 0)|. For any w ∈ B ω , using (G2), we write which implies G(w) ≤ ω after taking the supremum on I. Thus, G corresponds B ω to itself. Next, we investigate that G(w) is a contraction. For w, v ∈ U , and by utilizing the notations of (10) and (11), we have Consequently, we get which states that G is a contraction by (16). By the Banach contraction principle, G has a unique fixed point, which is the unique solution of the nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1) on I.

Stability results
In the recent section, we are interested in studying UH and GHR stability of the given nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1).

Definition 19
The nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1) is UH stable if there is c σ ∈ R + such that, for each ∈ R + and for each w ∈ U satisfying a unique solution w ∈ U of (1) exists with w -w ≤ c σ .
Remark 2 A function w ∈ C(I, R) is a solution of (19) if and only if there exists ψ ∈ C(I, R) (which depends on w) such that Here we review the UH and GUH stability of solutions to the nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1).

Theorem 21 Let (G2) and (16) be fulfilled. Then the nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1) is UH and GUH stable.
Proof Let ε > 0, let w ∈ C(I, R) satisfy (17), and let w ∈ C(I, R) be the unique solution of the generalized Sturm-Liouville-Langevin q-difference FBVP By Lemma 16, we have Since we have supposed that w satisfies (19), hence, by Remark 2, we get Again by Lemma 16, we have For each t ∈ I, In view of part 1 of Remark 2 and (G2), we obtain in which is illustrated in (16). In accordance with the above, it gives w -w ≤ μ 1 -.

Example
In this part, two examples are presented to show the validity and also the effectiveness of theoretical results.
Example 1 Based on the nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1), we fix ρ(t) ≡ 1 and r(t) = -λ, λ ∈ R. In this case, (1) is reduced to Langevin where The function σ is continuous on I. On the other hand, ∀w ∈ R and t ∈ I, we get |w|.

Conclusion
In this paper, we provided required criteria for the existence/uniqueness of solutions to a new category of nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1).
To arrive at such an aim, we dealt with a technique involving the Kuratowski measure of noncompactness (KMNC) along with a fixed point theorem of Mönch type. Although the used method based on KMNC-Mönch is considered as a standard method, its application in the current context is new yet, while the uniqueness property was derived with the help of BCP. Subsequently, the Ulam-Hyers (UH) and generalized Ulam-Hyers (GUH) stability were established for the proposed nonlinear generalized Sturm-Liouville-Langevin q-difference FBVP (1). Moreover, two examples were presented for the illustration of the obtained theory. Regarding next research projects, we are going to continue the analysis of such combined structures of physical and mathematical models by using nonsingular fractional operators which give more accurate numerical results. In particular, the Caputo-Fabrizio derivative is clearly well known in the field of fractional differential equations. The appearance of this derivative helps us to deal with some complicated phenomena. We will follow this study on the Caputo-Fabrizio derivative.