Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Systems with Nonlocal Coupled Fractional Integral Boundary Conditions

: In this paper, we initiate the study of existence of solutions for a fractional differential system which contains mixed Riemann–Liouville and Hadamard–Caputo fractional derivatives, complemented with nonlocal coupled fractional integral boundary conditions. We derive necessary conditions for the existence and uniqueness of solutions of the considered system, by using standard ﬁxed point theorems, such as Banach contraction mapping principle and Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.


Introduction
Fractional differential equations have played a very important role in almost all branches of applied sciences because they are considered a valuable tool to model many real world problems. For details and applications, we refer the reader to monographs [1][2][3][4][5][6][7][8][9][10][11]. The study of coupled systems of fractional differential equations is important as such systems appear in various problems in applied sciences, see [12][13][14][15][16].
On the other hand, multi-term fractional differential equations also gained considerable importance in view of their occurrence in the mathematical models of certain real world problems, such as behavior of real materials [17], continuum and statistical mechanics [18], an inextensible pendulum with fractional damping terms [19], etc.
Fractional differential equations have several kinds of fractional derivatives, such as Riemann-Liouville fractional derivative, Caputo fractional derivative, Hadamard fractional derivative, and so on. In the literature, there are many papers studying existence and uniqueness results for boundary value problems and coupled systems of fractional differential equations and used mixed types of fractional derivatives, see [20][21][22][23][24][25][26][27][28][29]. In [23], the following boundary value problem is considered: x(t))] = f (t, x(t)), 0 < t < T, where RL D q , C D r are Riemann-Liouville and Caputo fractional derivatives of orders q, r ∈ (0, 1), respectively, I p is the Riemann-Liouville fractional integral of order p > 0, f , g : J × R → R are given continuous functions and φ, h : C(J, R) → R are two given functionals.
In [24], the authors initiated the study of a coupled system of sequential mixed Caputo and Hadamard fractional differential equations supplemented with coupled separated boundary conditions. To be more precisely, in [24], existence and uniqueness results are established for the following couple system: H D q 2 C D p 2 y(t) = g(t, x(t), y(t)), t ∈ [a, b], where C D p i and H D q i are notations of the Caputo and Hadamard fractional derivatives of orders p i and q i , respectively, 0 In [25], the existence and uniqueness of solutions for neutral fractional order coupled systems containing mixed Caputo and Riemann-Liouville sequential fractional derivatives were studied, complemented with nonlocal multi-point and Riemann-Stieltjes integral multi-strip conditions of the form: where RL D p , RL D p 1 , and c D q , c D q 1 denote the Riemann-Liouville and Caputo fractional derivatives of order p, p 1 and q, q 1 , respectively, 0 < p, p 1 , q, q 1 ≤ 1, with 1 < p + q ≤ 2, 1 < p 1 + q 1 ≤ 2, f , f 1 and g, g 1 are given continuous functions, 0 < ξ i < η i < 1, 0 < θ j < ζ j < 1, α i , β j ∈ R, i = 1, 2, . . . , n, j = 1, 2, . . . , m, a, a 1 , b, b 1 ∈ R, and H(·) is a function of bounded variation.
To the best of the authors' knowledge, there are some papers dealing with sequential mixed type fractional derivatives, but we not find in the literature papers dealing with coupled systems with sequential Riemann-Liouville and Hadamard-Caputo fractional differential equations. Motivated by this fact, and to fill this gap, in the present paper, we investigate the existence and uniqueness of solutions for the following coupled system of sequential Riemann-Liouville and Hadamard-Caputo fractional differential equations supplemented with nonlocal coupled fractional integral boundary conditions where RL D p r and HC D q r are the Riemann-Liouville and Hadamard-Caputo fractional derivatives of orders p r and q r , respectively, 0 < p r , q r < 1, r = 1, 2, the nonlinear continuous functions f , g : [0, T] × R 2 → R, RL I φ is the Riemann-Liouville fractional integral of orders φ > 0, φ ∈ {β i , δ j } and given constants α i , λ j ∈ R, ξ i , η j ∈ (0, T), i = 1, . . . , m, j = 1, . . . , k.
Let us compare the coupled system (4) with the coupled system (2) studied in [24].
(i) In (2), we studied a coupled system consisting by mixed Caputo and Hadamard fractional derivatives, while, in (4), we consider mixed Riemann-Liouville and Hadamard-Caputo fractional derivatives. (ii) In (2), the coupled system was subjected to coupled separated boundary conditions, while, in (4), the coupled system is subjected to nonlocal coupled fractional integral boundary conditions. (iii) In both problems (4) and (2), the same method is used to establish the existence and uniqueness results, and based on standard fixed point theorems, but their presentation in the framework of mixed coupled Caputo and Hadamard and Riemann-Liouville and Hadamard-Caputo fractional derivatives is new.
We also notice that the conditions HC D q 1 x(0) = 0 and HC D q 2 y(0) = 0 are necessary for the well-posedness of the problem.
By using standard tools from fixed point theory in the present study, we establish existence and uniqueness results for the coupled system (4). The Banach contraction mapping principle is used to obtain the existence and uniqueness result, while an existence result is derived via the Leray-Schauder alternative.
The rest of the paper is organized as follows. In Section 2, some basic definitions and lemmas from fractional calculus are recalled. In addition, an auxiliary lemma, concerning a linear variant of (4), which plays a key role in obtaining the main results, is proved. The main results are presented in Section 3, which also include examples illustrating the basic results. We emphasize that our results are new and significantly enhance the existing literature on the topic, and, as far as we know, they are the first results concerning a coupled system with sequential mixed Riemann-Liouville and Hadamard-Caputo fractional derivatives.

Preliminaries
In this section, we introduce some notations and definitions of fractional calculus [2,30] and present preliminary results needed in our proofs later.
provided the right side is pointwise defined on R + .

Definition 3.
For an at least n-times differentiable function g : (0, ∞) → R, the Hadamard-Caputo derivative of fractional order q > 0 is defined as where δ = t d dt and log(·) = log e (·).
Definition 4. The Hadamard fractional integral of order q > 0 is defined as provided the integral exists.

holds.
Next, the integral equations are obtained by transformation of a linear variant of problem (4). For convenience in computation, we set some constants , , R) be two given functions. Then, the linear system equivalent to problem (4) of sequential Riemann-Liouville and Hadamard-Caputo fractional differential equations can be written into integral equations as and Proof. For t ∈ [0, T] and by taking the Riemann-Liouville fractional integral of order p 1 to the first equation of (5), we obtain Similarly, for the second equation of (5), we have Since 0 < p r < 1, r = 1, 2, the conditions HC D q 1 x(0) = 0 and HC D q 2 y(0) = 0 imply c 1 = 0 and d 1 = 0, respectively. Applying the Hadamard fractional integral of orders q 1 and q 2 to (8) and (9), respectively, and substituting the values of c 1 , d 1 , we get and Now, we consider the terms and Consequently, by (10)- (13) and boundary fractional integral conditions in (5), it follows that Substituting the values of c 0 and d 0 in (10) and (11), we obtain integral equations in (6) and (7), respectively, as desired. The converse follows by direct computation. This completes the proof.
Next, we establish formulas for multiple fractional integrals of Riemann-Liouville and Hadamard types.
Lemma 5. Let a, b, c > 0 be constants. Then, we have (i) Proof. Since RL I a (1) = t a Γ(a + 1) , we have by using Lemma 3, and (i) is proved. To prove (ii), taking the Riemann-Liouville fractional integral of order c > 0 in (14), we have from RL I c t a = Γ(a + 1) Γ(a + c + 1) t a+c . The proof is completed.

Corollary 1.
Let constants p r , q r , r = 1, 2, β i , ξ i , δ j , η j be defined in problem (4). Then, from Lemma 5, we have , , which will be used in the next section.
In the following, for brevity, we use the subscript notation in fractional integral as where φ ∈ {t, T, ξ i , η j }. In addition, we use it in multiple fractional integrations.
In view of Lemma 4, we define the operator P : and λ j RL I δ j H I q 1 RL I p 1 f x,y (η j ) + H I q 2 RL I p 2 g x,y (t).
For computational convenience, we set In the first result, Banach's contraction mapping principle is used to prove existence and uniqueness of solutions of system (4). Theorem 1. Suppose that f , g : [0, T] × R 2 → R are continuous functions. In addition, we assume that f , g satisfies the Lipchitz condition: for all t ∈ [0, T] and u i , v i ∈ R, i = 1, 2. Then, the system (4) has a unique solution on [0, T], if Proof. Let us define sup t∈[0,T] f (t, 0, 0) = N 1 < ∞ and sup t∈[0,T] g(t, 0, 0) = N 2 < ∞. Choose a constant r > 0 satisfying At first, we shall show that the set P B r ⊂ B r , where a ball B r = {(x, y) ∈ X × Y : (x, y) ≤ r}. For (x, y) ∈ B r , and using and |g x,y | ≤ |g x,y − g 0,0 | + |g 0,0 | ≤ n 1 x + n 2 y + N 2 , we get relations |α i | RL I β i H I q 2 RL I p 2 1 (ξ i )(n 1 x + n 2 y + N 2 ) |λ j | RL I δ j H I q 1 RL I p 1 1 (η j )(m 1 x + m 2 y + N 1 ) Therefore, we deduce that In a similar way of computation, we get Then, we conclude that which leads to P B r ⊂ B r . In the next step, we will show that the P is a contraction operator. For any (x 1 , y 1 ), (x 2 , y 2 ) ∈ X × Y, we have |α i | RL I β i H I q 2 RL I p 2 |g x 1 ,y 1 − g x 2 ,y 2 | (ξ i ) Then, we get the result that In addition, we have which yields The above results in (18) and (19) imply Since (M 1 + M 3 )(m 1 + m 2 ) + (M 2 + M 4 )(n 1 + n 2 ) < 1, then the operator P is a contraction. From the benefits of Banach's fixed point theorem, the operator P has a unique fixed point, which is the unique solution of (4) on [0, T]. The proof is completed.
The Leray-Schauder alternative is applied to our second existence result.

Lemma 6.
(Leray-Schauder alternative) [31]. Let Q : U → U be a completely continuous operator. Let Then, either the set µ(Q) is unbounded, or Q has at least one fixed point.

Theorem 2.
Suppose that there exist constants a r , b r ≥ 0 for r = 1, 2 and a 0 , b 0 > 0. In addition, for any u, v ∈ R, we assume that Proof. The first task of the proof is to show that the operator P : X × Y → X × Y is completely continuous. The continuity of the functions f , g on [0, T] × R × R can be used to claim that the operator P is continuous. Now, we let Φ be the bounded subset of X × Y.
Therefore, we obtain which lead to Thus, the following inequality holds: where Hence, the set µ is a bounded set. Then, by using Lemma 6, the operator P has at least one fixed point. Therefore, we conclude that problem (4) has at least one solution on [0, T]. The proof is complete.
Next, we present examples to illustrate our results.

Conclusions
In this paper, we studied a new system of sequential fractional differential equations which consists of mixed fractional derivatives of Riemann-Liouville and Hadamard-Caputo types, supplemented with nonlocal coupled fractional integral boundary conditions. To the best of our knowledge, this is the first system of this type that appeared in the literature. After proving a basic lemma, helping us to transform the considered system into a fixed point problem, we use the standard tools from functional analysis to establish existence and uniqueness results. We use a Banach contraction mapping principle to derive the uniqueness result and Leray-Schauder alternative to obtain an existence result. The obtained results are well illustrated by numerical examples. The obtained results enrich the existing literature on sequential systems of fractional differential equations. Other cases of fractional systems with other types of mixed fractional derivatives or other types of boundary conditions can be studied using the methodology of this paper.