On a Riemann–Liouville Type Implicit Coupled System via Generalized Boundary Conditions †

: We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder ﬁxed point theorem. Furthermore, Hyers–Ulam type stabilities are discussed. An example is presented to illustrate our main result. The suggested system is the generalization of fourth-order ordinary differential equations with anti-periodic, classical, and initial boundary conditions.


Introduction
The generalization of ordinary derivatives leads us to the theory of fractional derivatives. The concept of fractional derivatives was established in 1695, after the well-known conversation of Leibniz and L'Hospital [1]. Mathematicians like Riemann, Liouville, Caputo, Hadamard, Fourier, and Laplace contributed a lot and made the area more interesting for researchers. A fractional-order derivative is a global operator, which may act as a tool to modify or modernize different physical phenomena like control theory [2], dynamical process [3], electro-chemistry [4], mathematical biology [5], image and signal processing [6], etc. For more applications of the fractional differential equations (FDES), we refer the reader to the works in [7][8][9][10][11]. Furthermore, the theory of coupled systems of differential equations is referred to as an important theory in the applied sciences envisaging different areas of biochemistry, ecology, biology, and classical fields of physical sciences and engineering. For details see in [12][13][14].
The theory regarding the existence of solutions of FDES, drew significant attention of the researchers working on different boundary conditions, e.g., classical, integral, multipoint, non-local, periodic, and anti-periodic [15][16][17][18]. Among the qualitative properties of FDES, the stability property of the solution is the central one, particularly the Hyers-Ulam (HU) stability [19][20][21][22][23][24][25][26]. Stability theory in the sense of HU was first discussed by Ulam [27] in the form of a question in 1940 and the following year, Hyers [28] answered his question in the context of Banach spaces. Recently, generalized HU stability was discussed by Alqifiary et al. [29] for linear differential equations. Razaei et al. [30] presented Laplace transform and HU stability of linear differential equations. Wang et al. [31] studied HU stability for two types of linear FDES. Shen et al. [32] worked on the HU stability of linear FDES with constant coefficients using Laplace transform method. Liu et al. [33] proved the HU stability of linear Caputo-Fabrizio FDES. Liu et al. [34] studied the HU stability of linear Caputo-Fabrizio FDES with the Mittag-Leffler kernel by Laplace transform method.
Higher-order ordinary differential equations (ODES) can be used to model problems arising from the field of applied sciences and engineering [35,36]. The generalization of fourth-order ODES are FDES (1) if α = κ = 4. Fourth-order differential equations have important applications in mechanics, thus have attracted considerable attention over the last three decades. The problem of static deflection of a uniform beam, which can be modeled as a fourth-order initial value problem is a good example of a real problem in engineering [37,38].
This problem has been extensively analyzed, some new techniques were developed and numerous general and impressive results regarding the existence of solutions were established in [39][40][41][42]. Sometimes, mathematical modeling of the various physical phenomena may arise as a coupled system of the forgoing ODES. Furthermore, for η i = −1 (i = 1, 2, . . . , 8), we can obtain anti-periodic boundary conditions which are applicable in several mathematical models, some are given in [43,44].
The manuscript is categorized as follows. For our main results, we establish some basic notations, definitions, and lemma in Section 2. In Section 3, we present existence, uniqueness, and at least one solution of system (1) by applying the Banach contraction fixed point theorem and Leray-Schauder fixed point theorem. In Section 4, we discuss definitions of HU type stabilities, which help us to show that system (1) has HU type stabilities by two different approaches. In Section 5, by a particular example of the system (1), we show that our results are applicable.

Background Materials
In this fragment, we present basic notations with Banach spaces, definitions of the considered derivative and integral, and lemma, which will be utilized in the next sections.
Similarly, (v, u) S = v S 1 + u S 2 is the norm defined on the product space, where S = S 1 × S 2 . Obviously S, (v, u) S is a Banach space. Definition 1. [45] For a continuous function v : R + → R, the Riemann-Liouville integral of order α > 0 is defined as such that the integral is pointwise defined on R + . Definition 2. [45] For a continuous function v : R + → R, the Riemann-Liouville derivative of order α > 0 is defined as where [α] represents the integer part of α and n = [α] + 1. We note that for > −1, where k i (i = 1, 2, 3, . . . , n) are unknowns.

Existence Theory
This section is devoted to the equivalent integral form of the proposed problem.

Remark 1.
Let µ ∈ C(J), the following κ ∈ (3, 4] order FDE with boundary conditions has the solution where G κ (t, τ) is given by gives Green's function G α (t, τ) of fourth-order ODE with anti-periodic boundary conditions.
For the reason of advantage, we set the following notations: and We use the following notations for convenience: Then, the fixed point of F and the solution of system (1) coincided, i.e., Using Banach contraction theorem in the following, we prove the uniqueness of solution of system (1).

Theorem 1.
Let the functions χ 1 , χ 2 : J × R × R → R are continuous and satisfy the hypothesis: In addition, suppose that where Q α and Q κ are defined by Equations (6) and (7), respectively. Furthermore, 0 ≤ L χ 1 , L χ 2 < 1 (through out the paper). Then, the solution of system (1) is unique. Consider Substituting (10) in (9), we get Therefore, On the same way, we can write Inequalities (11) and (12) combined give For any t ∈ J, and (v 1 , u 1 ), (v 2 , u 2 ) ∈ S, we get and thus we get Similarly, From the inequalities (13) and (14), we get that Therefore, F is a contraction operator. Therefore, by Banach's fixed point theorem, F has a unique fixed point, so the solution of the problem (1) is unique.
The next result is based on the following Leray-Schauder alternative theorem. Theorem 2. [46] Let F : S → S be an operator which is completely continuous (i.e., a map that restricted to any bounded set in S is compact). Suppose Then, either the operator F has at least one fixed point or the set B(F) is unbounded.
Similarly, for every t ∈ J and v, X : J → R, there are ϕ i (i = 1, 2, 3) : J → R + , such that 1, 2, 3) . In addition, it is assumed that Then, the system (1) has at least one solution.
Proof. First, we prove that F is completely continuous. In view of continuity of χ 1 , χ 2 , the operator F is also continuous. For any (v, u) ∈ B r , we have Now by H 2 , we have Therefore, (16) implies which implies that Similarly, we get Thus, it follows from the inequalities (18) and (19) that F is uniformly bounded. Now, we prove that F is equicontinuous. Let 0 ≤ t 2 ≤ t 1 ≤ t. Then, we have Therefore, we get Similarly Therefore, F(v, u) is equicontinuous. Thus, we proved that the operator F(v, u) is continuous, uniformly bounded, and equicontinuous, concluding that F(v, u) is completely continuous. Now, by using Arzela-Ascoli theorem, the operator F(v, u) is compact.
Finally, we are going to check that Then, and Therefore, from (20) and (21), we have Consequently, we get , for any t ∈ J, where Q 0 is defined by (15), which infer that B is bounded. Therefore, by Theorem 2, F has at least one fixed point. Thus, the system (1) has at least one solution.

Stability Results
Let us recall some definitions related to HU stabilities: Suppose the functions Θ α , Θ κ : J → R + are nondecreasing and α , κ > 0. Consider the inequalities given below.

Method (I)
Theorem 4. If hypothesis H 1 and Proof. Let (v, u) ∈ S be the solution of (22) and (w, ζ) ∈ S be the solution of following system: Then in view of Lemma 1, for t ∈ J the solution of (33) is given by: Consider Applying Lemma 3 in (35), we get Using H 1 of Theorem 1 and (6) in (36) Similarly, we can get We write (37) and (38) as From the above, we get where . (40), then by Definition 4 the problem (1) is generalized HU stable.

Conclusions
This paper concluded that the solution of coupled implicit FDES (1) is unique and exists by using the Banach contraction theorem and Leray-Schauder fixed point theorem. Under some assumptions, the aforesaid coupled system has at least one solution. Besides this, the considered coupled system is HU, generalized HU, HU-Rassias and generalized HU-Rassias stable. An example is presented to illustrate our obtained results. The proposed system (1) gives the following well-known system of ODES, which has wide applications in applied sciences [5] • η i = −1 (i = 1, 2, . . . , 8) and α, κ = 4, then we get fourth-order ODES system with anti-periodic boundary conditions. • η i = 0 (i = 1, 2, . . . , 8) and α, κ = 4, then we get fourth-order ODES system with initial conditions. Funding: Not applicable.

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