Coupled Systems of Sequential Caputo and Hadamard Fractional Differential Equations with Coupled Separated Boundary Conditions

This paper studies the existence and uniqueness of solutions for a new coupled system of nonlinear sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions, which include as special cases the well-known symmetric boundary conditions. Banach’s contraction principle, Leray–Schauder’s alternative, and Krasnoselskii’s fixed-point theorem were used to derive the desired results, which are well-illustrated with examples.


Introduction
Fractional differential equations appear in the mathematical modeling of many real-world phenomena occurring in engineering and scientific disciplines, for instance, see References [1][2][3][4][5][6].Mathematical models based on fractional-order integral and differential operators yield more insight into the characteristics of the associated phenomena, as such operators are nonlocal in nature, in contrast to classical ones.In particular, coupled systems of fractional-order differential equations have received great attention in view of their great utility in handling and comprehending practical issues, such as the synchronization of chaotic systems [7,8], anomalous diffusion [9], and ecological effects [10].For recent theoretical results on the topic, we refer the reader to a series of papers [11][12][13][14][15][16][17][18] and the references cited therein.
Recently, in Reference [19], the authors discussed existence and the uniqueness of solutions for sequential Caputo and Hadamard fractional differential equations subject to separated boundary conditions as where C D p and H D q are the Caputo and Hadamard fractional derivatives of orders p and q, respectively, 0 < p, q ≤ 1, starting at a point a > 0, f : [a, b] × R → R is a continuous function and given constants α i , β i ∈ R, i = 1, 2.
In this paper, we established the existence criteria for a coupled system of sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions as: 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 < p i , q i ≤ 1, i = 1, 2, the nonlinear continuous functions f , g : 4. Meanwhile, the different definitions of Caputo and Hadamard fractional derivatives that appeared in System (2) are proposed to study the existence theory of solutions of a fractional differential system using a variety of fixed-point theorems.A special case, when p i = q i = 1, i = 1, 2, in differential Equation ( 2) can be presented as: which is mixed type of ordinary differential equations and boundary conditions.The rest of this paper is organized as follows: Section 2 aims to recall basic definitions and lemmas used in this paper.Section is devoted to the main results concerning the existence and uniqueness of solutions for for System (2).The Leray-Schauder alternative and Krasnoselskii's fixed-point theorem were applied to prove existence, while the uniqueness result was obtained via the Banach contraction mapping principle.Some illustrative examples are presented in Section 4.

Preliminaries
To ensure that readers can easily understand the results, we recall some notations and definitions of fractional calculus [3,20].Definition 1.The Caputo fractional derivative of order q for an at least n-times differentiable function g : [a, ∞) → R, starting at a point a > 0, is defined as: where [q] denotes the integer part of the real number q. Definition 2. The Riemann-Liouville fractional integral of order q of a function g : [a, ∞) → R, a > 0, is defined as: provided the right side of an integral exists.Definition 3. The Caputo-type Hadamard fractional derivative of order q for an at least n-times delta differentiable function g : [a, ∞) → R, starting at a point a > 0, is defined as where the delta derivative is defined by δ = t d dt and the natural logarithm log(•) = log e (•).
Definition 4. The Hadamard fractional integral of order q is defined as provided the integral exists.
Next, we transform Problem (2) to integral equations by using a linear variant of Problem (2).For convenience, we put constants and Then, the linear system of sequential Caputo and Hadamard fractional differential equations with coupled separated boundary value problem can be written as integral equations + H I q 1 RL I p 1 ω(t), and + RL I p 2 H I q 2 φ(t).
Proof.Taking the Riemann-Liouville fractional integral of order p 1 , p 1 ∈ (0, 1], to the first equation of Problem (5) and applying Problem (4), we obtain for t ∈ [a, b] In the above equation, we apply the Hadamard fractional integral of order q 1 , q 1 ∈ (0, 1], with (4) for t ∈ [a, b] and obtain Considering the second equation of Problem (5), and by using the Hadamard fractional integral of order q 2 , we get By taking the Riemann-Liouville fractional integral operator of order p 2 , we have In particular, for t = a in Equations ( 9) and (10), and applying the first condition of Problem (5), one has For t = b in Equations ( 9) and (10), it obtains by applying the second condition of Problem (5) as Substituting t = a in Equations ( 8) and (11) and applying the third condition of Problem (5), it leads to The fourth condition of Problem ( 5) can be applied when t = b in Equations ( 8) and (11) as Reduce the above Equations ( 12)-(15) in a system of constants by Computing for constants c 1 and c 2 and substituting it into Equations ( 12) and ( 14) for c 3 and c 4 , we have Substituting all obtained constants in Equations ( 9) and ( 11), we obtain integral Equations ( 6) and (7).By direct computation we can obtain the the converse.The proof is completed.

Let
Now, for brevity, we use the notations: and where φ ∈ {t, b}.We also use this one for a single fractional integral operator of the Riemann-Liouville and Hadamard types of orders p 1 and q 2 , respectively.In view of Lemma 3, we define two operators where and + RL I p 2 H I q 2 (g x,y )(t).
For computational convenience, we set (log(b/a)) q 2 Γ(q 2 + 1) (log(b/a)) q 2 Γ(q 2 + 1) Note that all information of Problem ( 2) is contained in constants M i , i = 1, 2, 3, 4, which are used to establish the following existence theorems.Banach's contraction mapping principle is applied in the first result to prove the existence and uniqueness of solutions of System (2).
In addition, we assume that: Then, System (2) has a unique solution on [a, b], if . Now, we show that the set KB r ⊂ B r , where B r = {(x, y) ∈ X × Y : (x, y) ≤ r}.For (x, y) ∈ B r , we have that Hence, By direct computation, we get Consequently, it follows that which implies KB r ⊂ B r .Next, we show that operator K is contraction mapping.For any (x 1 , y 1 ), (x 2 , y 2 ) ∈ X × Y, we obtain Therefore, we get the following inequality: In addition, we obtain From Inequalities (24) and (25), it yields Lemma 4. (Leray-Schauder alternative) [22].Let F : E → E be a completely continuous operator.Let Then, either set ξ(F) is unbounded, or F has at least one fixed point.
Theorem 2. Assume that there exist real constants u i , v i ≥ 0 for i = 1, 2 and u 0 , v 0 > 0, such that for any x i ∈ R, (i = 1, 2) we have Proof.By continuity of functions f , g on [a, b] × R × R, operator K is continuous.Now, we show that the operator K : X × Y → X × Y is completely continuous.Let Φ ⊂ X × Y be bounded.Then, there exist two positive constants, L 1 and L 2 , such that Then, for any (x, y) ∈ Φ, we have In addition, we obtain that Hence, from the above inequalities, we get that set KΦ is uniformly bounded.Next, we prove that set KΦ is equicontinuous.For any (x, y) ∈ Φ, and |(log(τ 2 /a)) q 1 − (log(τ 1 /a)) Therefore, we obtain Analogously, we can get the following inequality: Hence, set KΦ is equicontinuous.By applying the Arzelá-Ascoli theorem, set KΦ is relatively compact, which implies that operator K is completely continuous.Lastly, we show that set ξ = {(x, y) ∈ X × Y : (x, y) = λK(x, y), 0 ≤ λ ≤ 1} is bounded.Now, let (x, y) ∈ ξ, then we obtain (x, y) = λK(x, y), which yields, for any t ∈ [a, b], x(t) = λK 1 (x, y)(t), y(t) = λK 2 (x, y)(t).
Then, we have which imply that Thus, we get the inequality where The last-existence theorem is based on Krasnoselskii's fixed-point theorem.
Lemma 5. (Krasnoselskii's fixed-point theorem) [23] Let M be a closed, bounded, convex, and nonempty subset of a Banach space X.Let A, B be operators, such that (i) Ax + By ∈ M where x, y ∈ M, (ii) A is compact and continuous, and (iii) B is a contraction mapping.Then, there exists z ∈ M, such that z = Az + Bz.
In addition we suppose and there exist two positive constants P, Q such that for all t ∈ [a, b] and x i , then the problem (2) has at least one solution on [a, b].

Conclusions
We have proven the existence and uniqueness of solutions for a coupled system of sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions by applying the Banach fixed-point theorem, Leray-Schauder nonlinear alternative, and Krasnoselakki fixed-point theorem.We also provided examples to clarify our results.
therefore K is a contraction operator.By applying Banach's fixed-point theorem, operator K has a unique fixed point in B r .Hence, there exists a unique solution of Problem (2) on [a, b].The proof is completed.Now, we prove our second existence result via the Leray-Schauder alternative.
which shows that set ξ is bounded.Therefore, by applying Lemma 4, operator K has at least one fixed point in Φ.Therefore, we deduce that Problem (2) has at least one solution on [a, b].The proof is complete.