Existence and Uniqueness Results for a Coupled System of Caputo-Hadamard Fractional Differential Equations with Nonlocal Hadamard Type Integral Boundary Conditions

In this paper, we study a coupled system of Caputo-Hadamard type sequential fractional differential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. The sufficient criteria ensuring the existence and uniqueness of solutions for the given problem are obtained. We make use of the Leray-Schauder alternative and contraction mapping principle to derive the desired results. Illustrative examples for the main results are also presented.


Introduction
Fractional calculus has emerged as an important area of investigation in view of its extensive applications in mathematical modeling of many complex and nonlocal nonlinear systems. An important characteristic of fractional-order operators is their nonlocal nature that accounts for the hereditary properties of the underlying phenomena. The interactions among macromolecules in the damping phenomenon give rise to a macroscopic stress-strain relation in terms of fractional differential operators. For the fractional law dealing with the viscoelastic materials, see [1] and the references cited therein. In [2], transport processes influenced by the past and present histories are described by the Caputo power law. For the details on dynamic memory involved in the economic processes, see [3,4].
In 1892, Hadamard [5] suggested a concept of fractional integro-differentiation in terms of the fractional power of the type (t d dt ) q in contrast to its Riemann-Liouville counterpart of the form ( d dt ) q . The Hadamard fractional derivative contains a logarithmic function of an arbitrary exponent in the kernel of the integral appearing in its definition. For the details of Hadamard fractional calculus, we refer the reader to the works [6][7][8][9]. Fractional differential equations involving Hadamard derivative attracted significant attention in recent years, for instance, see ( [10][11][12][13][14][15][16][17][18][19][20]) and the references cited therein.
More recently, Jarad et al. [21] introduced Caputo modification of Hadamard fractional derivative which is more suitable for physically interpretable initial conditions as in case of Caputo fractional differential equations. One can find some recent results on Caputo-Hadamard type fractional differential equations in [22][23][24][25][26][27][28] and the references cited therein.
In this paper, we introduce a new class of boundary value problems consisting of Caputo-Hadamard type fractional differential equations and Hadamard type fractional integral boundary conditions. In precise terms, we investigate the following boundary value problem: , v(t), C D ξ v(t)), 1 < α ≤ 2, 0 < ξ < 1, λ > 0, ( C D β + λ C D β−1 )v(t) = g(t, u(t), C Dξ u(t), v(t)), 1 < β ≤ 2, 0 <ξ < 1, where C D (.) and I (.) respectively denote the Caputo-Hadamard fractional derivative and Hadamard fractional integral (to be defined later), f , g : [1, T] × R 3 → R are given appropriate functions and The rest of the paper is organized as follows. In Section 2, we recall the background material related to the topic under investigation and prove an auxiliary lemma which plays a key role in deriving the desired results. Section 3 contains the main results.

Preliminaries
In this section, we recall some preliminary concepts of Hadamard and Caputo-Hadamard fractional calculus related to our work. We also prove an auxiliary lemma, which plays a key role in converting the given problem into a fixed point problem.
) ∈ AC[a, b]}. The Hadamard derivative of fractional order q for a function g ∈ AC n δ [a, b] is defined as where n − 1 < q < n, n = [q] + 1 and [q] denotes the integer part of the real number q and log(·) = log e (·).
In particular, .

Lemma 2 ([21]
). Let g ∈ AC n δ [a, b] or C n δ [a, b] and q ∈ C, then Now we present an auxiliary lemma dealing with the linear variant of the problem (1).

Lemma 3.
Let h 1 , h 2 ∈ AC n δ [1, T]. Then the solution of the linear system of fractional differential equations: supplemented with the boundary conditions: is given by and where Proof. In view of Theorem 1 and lemma 2, the general solution of the system (2) can be written as where c i , d i (i = 0, 1) are unknown arbitrary constants. Using the data u(1) = 0, v(1) = 0 given by (3) in (9) and (10), we find that c 0 = 0 and d 0 = 0. Thus (9) and (10) take the form: Using the nonlocal integral boundary conditions: (11) and (12), we obtain where A i and B i (i = 1, 2) are respectively given by (7) and (8), and Solving the system (13) for c 1 and d 1 , we find that where ∆ is given by (6). Substituting the values of c 1 and d 1 in (11) and (12), we obtain the solution (4)- (5). This completes the proof.

Existence and Uniqueness Results
This section is concerned with the main results of the paper. First of all, we fix our terminology.
Using Lemma 3, we introduce an operator T : X × Y → X × Y as follows: where Next we enlist the assumptions that we need in the sequel.
(H 2 ) There exist positive constants l, l 1 such that For computational convenience, we set Now, we are in a position to present our first existence result for the boundary value problem (1), which is based on Leray-Schauder alternative.
Proof. In the first step, we establish that the operator T : X × Y → X × Y is completely continuous. By continuity of the functions f and g, it follows that the operators T 1 and T 2 are continuous. In consequence, the operator T is continuous. In order to show that the operator T is uniformly bounded, let Ω ⊂ X × Y be a bounded set. Then there exist positive constants L 1 and L 2 such that | f (t, u(t), v(t), C D ξ v(t))| ≤ L 1 , |g(t, u(t), C Dξ u(t), v(t))| ≤ L 2 , ∀(u, v) ∈ Ω. Then, for any (u, v) ∈ Ω, we have which, on taking the norm for t ∈ [1, T] and using (22), (24) and (26) yields Since 0 <ξ < 1, we use Remark 1 to get where Θ 1 , M 1 and M 2 are respectively given by (22), (25) and (27). Hence Similarly, using (23), (28) and (30), we obtain As before, one can find that where Θ 2 , N 1 and N 2 are respectively given by (23), (29) and (31).
In consequence, we get From the inequalities (35) and (36), we deduce that T 1 and T 2 are uniformly bounded, which implies that the operator T is uniformly bounded.
Next, we show that T is equicontinuous. Let t 1 , t 2 ∈ [1, T] with t 1 < t 2 . Then we have independent of (u, v). In a similar manner, one can obtain that as t 2 → t 1 independent of (u, v) on account of the boundedness of f and g. Thus the operator T is equicontinuous in view of equicontinuity of T 1 and T 2 . Therefore, by Arzela-Ascoli's theorem, it follows that the operator T is compact (completely continuous). Finally, it will be shown that the set ε( which, on taking the norm for t ∈ [1, T], yields Similarly one can find that Consequently, we have Likewise, we can derive that From (37) and (38), we get This shows that ε(T) is bounded. Thus, Lemma 4 applies and that T has at least one fixed point. This implies that the boundary value problem (1) has at least one solution on [1, T]. The proof is completed.
The next result deals with the uniqueness of solutions for the problem (1) and relies on Banach contraction mapping principle. For computational convenience, we introduce the notations: Theorem 3. Assume that (H 2 ) holds. Then the boundary value problem (1) has a unique solution on [1, T], provided that where Ψ i and Ψ i (i = 1, 2) are given by (42).
If we take a 1 = 0 = a 2 in the results of this paper, we obtain the ones for a coupled system of Caputo-Hadamard fractional differential equations and uncoupled Dirichlet boundary conditions. We emphasize that the main results as well as the special cases presented in this paper are new and enrich the existing literature on the topic.