A coupled system of generalized Sturm–Liouville problems and Langevin fractional differential equations in the framework of nonlocal and nonsingular derivatives

In this paper, we study a coupled system of generalized Sturm–Liouville problems and Langevin fractional differential equations described by Atangana–Baleanu–Caputo (ABC for short) derivatives whose formulations are based on the notable Mittag-Leffler kernel. Prior to the main results, the equivalence of the coupled system to a nonlinear system of integral equations is proved. Once that has been done, we show in detail the existence–uniqueness and Ulam stability by the aid of fixed point theorems. Further, the continuous dependence of the solutions is extensively discussed. Some examples are given to illustrate the obtained results.


Introduction
The subject of fractional calculus is a generalization of ordinary differentiation and integration to an arbitrary order, which might be noninteger. Very recently it was recognized that fractional calculus arise naturally in various fields of science and engineering. Today we witness an increasing number of proposals for operators, both in the form of derivatives and integrals [1,2] with the extension of fractional calculus. In consequence, there are several contributions focusing on different definitions of fractional derivatives such as the Riemann-Liouville (RL), Hadamard, Grünwald-Letnikov, Riesz, Caputo, Marchaud, Weyl, and Hilfer derivatives; see [3][4][5][6][7][8][9][10][11][12] for some detailed information. All these derivatives are known to contain singular kernels and some generalized fractional derivatives are novel such as conformable fractional derivative [13], beta-derivative [14], or we have a new definition [15,16]. Generally, various definitions differ from one another in choosing spe-cial kernels and some form of differential operator. For example, for the kernel k(t, s) = ts and the differential operator d/dt, we obtain the Riemann-Liouville definition.
In the recent contribution, Caputo and Fabrizio [17] proposed a new formulation involving a fractional derivative whose kernel is an exponential function. Motivated by [17], Atangana and Baleanu in [18], introduced a new definition of the fractional derivative to answer some outstanding questions that were posed by many researchers within the field of fractional calculus based on fractional operators with Mittag-Leffler, nonsingular smooth kernel. Their derivative has a nonsingular and nonlocal kernel and accepts all properties of fractional derivatives. This new derivative has gained widely attention and attracted a large number of scientists in different scientific fields for the exploration of diverse topics. Afterward, many articles on this subject have been published in order to generalize the results of the fractional derivative without a singular kernel in many directions. To the best of our knowledge, few contributions associated with ABC-fractional derivatives have been published; see [19][20][21][22] and the references therein.
In addition, the Sturm-Liouville problem plays an important role in different areas of applied sciences and engineering; for example see [23]. A standard form of the linear Sturm-Liouville differential equation of second order is defined by with appropriate initial conditions, where the functions p(t) and u(t) are continuous on the interval [a, b] such that p(t) > 0 and u(t) > 0. D is the usual derivative and f : [a, b] × R → R + is a continuous function. The fractional Sturm-Liouville problems were developed by some researchers in theory and application; see [24]. On the other hand, in [25] Langevin introduced the classical Langevin equation as follows: (1. 2) The classical Langevin equation with various boundary conditions has been studied by many authors; see [26] and the references therein. Various generalizations of the Langevin equation have been offered to describe dynamical processes in a fractal medium. This gives rise to the study of the fractional Langevin equation; see [27]. The fractional Langevin equation was introduced by Mainardi and collaborators in the earlier 1990s. Several types of fractional Langevin equation were studied in [28][29][30][31][32][33]. Meanwhile, in the same year, research into fractional order systems has become a subject of focus because of many advantages of fractional derivatives. For more papers on fractional order systems, see [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] and the references therein.
More recently, the study of fractional Langevin equation in frame of Caputo derivative has comparably been of small scale; see [49,50] in which the authors discussed Sturm-Liouville and Langevin equations via Hadamard fractional derivatives and systems of fractional Langevin equations of Riemann-Liouville and Caputo types, respctively However, to the best of our knowledge, few of the relevant studies on coupled systems of fractional differential equations have been briefly reviewed for further information on this topic.
To conclude this introductory section, we introduce the coupled system involving ABC differential operators with nonsingular kernel, which are discussed throughout this paper, which take the form are some continuous functions.
Note that system (1.3) is a generalization of Sturm-Liouville and Langevin fractional differential systems. In the special case if q i (t) ≡ 0 then (1.3) is reduced to the Sturm-Liouville fractional differential equations. For the case p i (t) ≡ 1 and q i (t) ≡ λ i system (1.3) is reduced to the Langevin fractional differential equations. However, the theorems we present include and extend some previous results.
We arrange this paper as follows: In Sect. 2, we introduce some notations, properties, lemmas, definitions of fractional calculus. We present a slight generalization for the Ulam-Hyers theorem which was used in studying the stability. Section 3 contains main results and is divided into 6 subsections. In Sect. 3.1 we first solve the corresponding linear problem and show the equivalence between the nonlinear problem (1.3) and integral equation. In Sect. 3.2, we adopt Banach's contraction mapping principle In Sect. 3.3, we use Krasnoselskii's fixed point theorem to prove the existence and uniqueness of solutions for problem (1.3). Section 3.4 is devoted to the stable solution of the fractional coupled systems (1.3) which is provided by using the classical technique of nonlinear functional analysis investigated by Ulam. In Sect. 3.5, we look at the question as to how the solution u varies when we change the order of the ABC-differential operator or the initial values and the dependence on parameters of nonlinear term f is also established. Illustrative examples are presented in the last subsection. Finally, the paper is concluded in Sect. 4.

Preliminaries
In this subsection, we introduce some notations, definitions, properties and lemmas of fractional calculus, we present briefly the so-called operators with nonsingular kernel. and present preliminary results needed in our proofs later.
The left-sided RL-fractional derivative of order α ∈ (n -1, n], of a continuous function f : [0, ∞) − → R is given by provided that the right side is pointwise defined on R + . The corresponding left-sided RL-integral operator of order 0 < α ≤ 1, of a continuous function f : [0, ∞) − → R is given by provided that the right side is pointwise defined on R + .
Let us recall the well-known definition of the Caputo fractional derivative [3]. Given b > a, f ∈ H 1 (a, b) and 0 < α < 1, the Caputo fractional derivative of f of order α is given by By changing the kernel (tτ ) -α by the function , one obtains the new ABC-fractional derivative of order 0 < α < 1, where f ∈ H 1 (0, 1), 0 < α < 1 and B(α) is the known normalized positive function satisfying the properties B(0) = 1, B(1) = 1 and According to the ABC derivative, it is clear that, if f is a constant function, then D α t f (t) = 0 as in the usual Caputo derivative. The main difference between the usual Caputo derivative and ABC-derivative is that, contrary to the usual Caputo definition, the new kernel has no singularity for t = τ . This ABC-fractional derivative D α t is less affected by the past, compared with the c D α t which shows a slow stabilization. The term E α can be expressed as a single-or two-parameter Mittag-Leffler function defined by power series expansions where α > 0 and β ∈ C. When β = 1, we shortly write E α, The fractional integral associated to the ABC-fractional derivative with no-singular and non-local kernel is defined by where I α t is the left RL-fractional integral given in (2.2). We shall state some properties of the fractional integral and fractional differential operators.
(i) The RL-fractional integral operators I α τ satisfy the semigroup property (ii) The ABC-fractional derivative and ABC-fractional integral of a function f fulfill the semigroup property [51], (iii) The following statement holds: The following statements hold: (i) For any α ≥ 0 and β > 0, (ii) The RL-fractional integral and ABC-fractional integral of a function f fulfill the semigroup property In this paper, we take X = C(J, R) to be the Banach space of all continuous functions defined on J and endowed with the usual supremum norm. Obviously, the product space (X × X, (·, ·) ) is also a Banach space with the norm For completeness, we state the fixed point theorems and Ulam-Hyers stability theorem that will be employed therein.

Theorem 2.4 ([52])
Let B r be the closed ball of radius r > 0, centred at zero, in a Banach space X with Υ : B r → X a contraction and Υ (∂B r ) ⊆ B r . Then Υ has a unique fixed point in B r .

Theorem 2.5 ([52]) Let M be a closed, convex, non-empty subset of a Banach space X × X. Suppose that E and F map M into X and that
Definition 2.6 ([53]) Let X be a Banach space and Υ 1 , Υ 2 : X × X → X × X be two operators. Then the operational equations system provided by is called Ulam-Hyers stable if we can find σ j > 0, j = 1, . . . , 4, such that, for each ε 1 , ε 2 > 0, and each solution- (2.11)

and if the matrix
converges to zero. Then the operational equations system (2.12) is Ulam-Hyers stable.

Main results
This section contains our main results.

Fractional coupled system (1.3)
In order to study the nonlinear fractional coupled system (1.3), we first consider the associated linear problem and obtain its solution.

Linear fractional coupled system
In this subsection, we consider now the linear coupled system (3.1) Proof Assume u i (t) satisfies (3.1). By applying the fractional integral operators I α i and I β i successively to (3.1), we obtain for some real constants c 1 and c 2 . Using the first boundary condition u i (0) = 0 in (3.4), we have Using the second boundary condition in (3.3), we have Substituting the value of c 1 in (3.5), we obtain Substituting the values of c 1 and c 2 from (3.6) and (3.7), respectively, in (3.4), we end up with (3.2).
Conversely, it can be easily shown by direct computation that the integral equation (3.2) satisfies the boundary value problem (3.1). The proof is complete.
By a solution of problem (3.1) we mean a pair of functions (u 1 , u 2 ) ∈ X × X satisfying (3.2) for all t ∈ J, i = 1, 2.
Then the integral solution for the linear system of fractional differential equations (3.1) is given by the pair of functions (u 1 , u 2 ) ∈ X × X, with (3.2).

Nonlinear fractional coupled system
In this subsection, we consider a nonlinear coupled system of the form (1.3).
From problem (3.1) we get the fractional integral system which is equivalent to the initial value problem (1.3). By virtue of Lemma 3.2, we get the following.
Proof The proof is immediate from Lemma 3.1, so we omit it.
Since problem (1.3) and Eq. (3.8) are equivalent, it is enough to prove that there exists only one solution to (3.8).
In this paper, a closed ball with radius r centered on the zero function in X × X is defined by We define an operator Ψ : X × X → X × X by (Ψ u)(t) = Ψ (u 1 , u 2 )(t) = Ψ 1 (u 1 , u 2 )(t), Ψ 2 (u 1 , u 2 )(t) , ∀(u 1 , u 2 ) ∈ X × X, (3.9) where Observe that problem (3.8) has solutions if and only if the operator equation Ψ u = u has fixed points. We make use of the following notations: for i = 1, 2 where and (3.14) Throughout the remaining part of this paper, we assume the following conditions hold.
(A 1 ) Assume that f i : J × R × R − → R are continuous functions and there exist constants M i > 0 such that, for all t ∈ J and u i , v i ∈ R, i = 1, 2, we have (A 2 ) Assume that there exist real constants N i > 0 such that for all (t, u 1 , u 2 ) ∈ J × R × R. Also, let By our assumption, for (t, u 1 , u 2 ) ∈ J × R × R, we have Let us introduce the notation and

Existence and uniqueness of the solution of (3.8)
In this subsection, we apply Banach's fixed point theorem to establish existence and uniqueness of solutions of (3.8).

18)
then problem (1.3) has a unique solution u ∈ B r .

Proof
Step i. We show that Ψ (B r ) ⊆ B r . To see this, for u = (u 1 , u 2 ) ∈ B r , t ∈ J, i = 1, 2, we have We used the fact that These imply that Thus, we have From (3.15) and (3.20), we obtain In view of (3.20), we have Using the above estimate in inequality (3.19), we obtain where μ i (t) and 4,i (t) are given by (3.13) and (3.14), respectively.
Taking the maximum on both sides of the inequality (3.25), the following can be obtained: Choose a real constant r > 0 such that and taking into account that (3.28), we conclude that (3.27) holds.
Step ii. Next, we show that Ψ is a contraction mapping. To see this, let u = (u 1 , u 2 ), v = (v 1 , v 2 ) ∈ B r and for any t ∈ J, we get In view of (3.20), we have Similarly to the above argument, we can also obtain again from (3.31), we have In the same way, we obtain Using (3.29)-(3.34), we obtain with 4,i (t) as in (3.14), where γ * 1,β 1 (t) and μ i (t) are given by (3.12) and (3.13) respectively. Furthermore, for any t ∈ J, from inequality (3.36), we obtain implying that (3.37) holds, where Since L < 1, therefore, the operator Ψ is a contraction. Thus, by Theorem 2.4, problem (1.3) has a unique solution u ∈ B r . This completes the proof.
Proof We will prove the theorem in several steps. Clearly, B r is a closed, convex, nonempty subset of X × X.
Step 1: The first condition of Theorem 2.5 holds.
That is, For this purpose, take u = (u 1 , u 2 ) and v = (v 1 , v 2 ) in B r , t ∈ J, and consider Now taking the maximum on both sides of the inequality (3.47), we obtain Analogously, we obtain Therefore, from (3.42), (3.49) and (3.50), we get Similarly, taking the maximum on both sides of the inequality (3.51), the following can be obtained: Consequently, Hence, using (3.48) and (3.53), we can conclude that Choose a real constant r > 0 such that Thus, Eu + Fv ≤ r, this implying that (3.45) holds.
Step 2: F is a contraction mapping.
To see this, let u = (u 1 , u 2 ) and v = (v 1 , v 2 ) ∈ B r . Following the proof of Theorem 2.4, we have Taking the maximum on both sides of the inequality (3.59), we obtain where μ i (t) is defined in (3.13). So, from (3.60), we get where L = max{L 1 , L 2 }, with When L < 1, the operator F is a contraction.
Clearly, E is continuous in view of the continuity of u 1 and u 2 .
Step 5: E is uniformly bounded. It follows from (3.51) that E is uniformly bounded. Therefore, by the Arzelà-Ascoli theorem, we conclude that E is a compact operator. Thus, all the conditions of Theorem 2.5 are fulfilled. Hence, system (1.3) has at least one solution u ∈ B r . The proof is complete.

Ulam-type stability of solutions of (3.8)
In this subsection, we use Urs's [53] approach to establishing the Ulam-Hyers stability of solutions of (1.3). Thanks to Definition 2.6 and Theorem 2.7, the respective results are obtained. . (3.67) Further, assume the spectral radius of matrixH σ is less than one. Then the solutions of (1.3) are Ulam-Hyers stable.
Proof In view of Theorem 2.4, we have which implies that . (3.70) Since the spectral radius ofH σ is less than one, the solution of (1.3) is Ulam-Hyers stable.

Dependence of solution on the parameters
For f i Lipschitz in the second variables, the solution's dependence on the order of the differential operator, the boundary values, and the nonlinear term f i are also discussed. In the following, for any u i ∈ X, we let The dependence on parameters of the left-hand side of (3.8) In this subsection, we show that the solutions of two equations with neighboring orders will (under suitable conditions on their right-hand sides f i ) lie close to one another.
Proof Let u(t) and u (t) be the solutions of (1.3) and (3.86)-(3.87), respectively. Hence, by the above theorems, we can obtain the following results. Let