Existence and Uniqueness for a System of Caputo-Hadamard Fractional Differential Equations with Multipoint Boundary Conditions

In this paper, we study existence and uniqueness of solutions for a system of Caputo-Hadamard fractional differential equations supplemented with multi-point boundary conditions. Our results are based on some classical fixed point theorems such as Banach contraction mapping principle, Leray-Schauder fixed point theorems. At last, we have presented two examples for the illustration of main results.


Introduction
In recent years, fractional differential equations (FDE) gain enormous attention among scientists due to the applications which were not possible with ordinary or partial differential equations of integer order. FDEs becomes a very successful tool in modeling anomalous diffusion and fractal-like nature. Agrawal discusses diffusion and heat equations of fractional order in [1][2][3]. Agrawal et al., Baleanu, and others investigated the boundary value problems for fractional differential equations [4]. Fractional dynamic models, fractional control systems, fractional population dynamics models, and fractional fluid dynamics all involve at least one ordinary or partial fractional derivative.
Fractional differential equations have several kinds of fractional derivatives, such as Riemann-Liouville fractional derivative, Caputo fractional derivative, and Grunwald-Letnikov fractional derivative. Another kind of fractional derivative is Hadamard type which was introduced in 1892 [5]. This derivative differs from various derivatives in the sense that the kernel of the integral in the definition of Hadamard derivative contains logarithmic function of arbitrary exponent. A detailed description of Hadamard fractional derivative and integral can be found in [6]. The readers who are interested in the subject of fractional calculus is referred to the books by Kilbas et al. [7], Podlubny [8], Miller and Ross [9], Samko et al. [10], Diethelm [11], and Zhou [12] and the references therein.
In [35], the authors investigated the existence and uniqueness of solutions for the coupled system of nonlinear fractional differential equations with three-point boundary conditions Liouville fractional derivative and f , g : ½0, 1 × R × R → R are given continuous functions.
Recently, Alsulami et al. [36] established the existence and uniqueness results for a nonlinear coupled system of Caputo type fractional differential equations supplemented with nonseparated coupled boundary conditions.
where c D α , c D β denote the Caputo fractional derivatives of order α and β, respectively, f , g : ½0, T × R × R → R are appropriately chosen functions, and λ i , Motivated by the research going on in this direction, in this paper, we study existence and uniqueness of solutions for a coupled system of Caputo-Hadamard fractional differential equations.
with multipoint boundary conditions The paper is organized as follows. In Sect. 2, we present some preliminary concepts of fractional calculus. Sect. 3 contains main results concerning the existence and uniqueness of solutions for the given problem (3), (4). The Leray-Schauder alternative theorem is applied to prove existence, while the uniqueness result was obtained via the Banach contraction mapping principle. Finally, we also discuss some examples for illustration of the existence-uniqueness results.

Preliminaries
For the convenience of the reader, we present some concepts of Hadamard type fractional calculus to facilitate the analysis of system (3), (4).
Definition 1 [7]. The Hadamard fractional integral of order q > 0 of a function xðtÞ for all t > a > 0 is defined by where ΓðqÞ = Ð ∞ 0 t q−1 e −t dt is the gamma function,s provided the right side is pointwise defined on R + . Definition 2 [7]. The Hadamard fractional derivative of order q > 0 of a function xðtÞ for all t > a > 0 is defined by where n = ½q + 1 with ½q denotes the integral part of the real number q and ln ð·Þ = ln e ð·Þ.
Definition 3 [38]. Let q ≥ 0 and n = ½q The Caputo type modification of the Hadamard fractional derivative of order q is defined by Theorem 4 [38]. Let q ≥ 0, and n = ½q (ii) if q ∈ N 0 , then C D q a + yðtÞ = δ n yðtÞ Lemma 6 [38]. Let q ≥ 0 and n = ½q + 1. If xðtÞ ∈ AC n δ ½a, b, then the Caputo-Hadamard fractional differential equation C D q a + xðtÞ = 0 has a solution: and the following formula holds: where c k ∈ R, k = 1, 2, ⋯, n − 1. Now, we present an auxiliary lemma for boundary value problem of linear fractional differential equation with Caputo-Hadamard derivative.
Then, the solution of the linear Caputo-Hadamard fractional differential system is equivalent to the system of integral equations where Proof. We apply Lemma [6] that the general solution of the Caputo-Hadamard fractional differential equation in (13) can be written as: where c i , d i , i = 0, 1, are arbitrary real constants. From (17) and (18) we have

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Using the boundary conditions uðaÞ = λ 1 vðbÞ and vðaÞ = μ 1 uðbÞ from (17) and (18), we have Using the boundary conditions λ 2 Solving the resulting equations for c 1 and d 1 , we find that substituting c 1 and d 1 in (23) and (24), we have and Inserting the values of c i , d i , i = 0, 1 in (17) and (18), which leads to the solution system (14), (15). The converse follows by direct computation. The proof is completed.

Existence and Uniqueness Results
This section is concerned with the main results of the paper. First of all, we fix our terminology. Let C = Cð½a, b, R Þ, a > 0 be the Banach space of all continuous functions from ½a, b to R . Space X = fuðtÞ: uðtÞ ∈ C 2 ð½a, b, R Þg endowed with the norm kuk = sup f|uðtÞ|,t ∈ ½a, bg is a Banach space. In addition, let Y = fvðtÞ: vðtÞ ∈ C 2 ð½a, b, RÞg with the norm kvk = sup f|vð tÞ|,t ∈ ½a, bg: It is obvious that product space ðX × Y, kðu, vÞ kÞ is a Banach space with the norm kðu, vÞk = kuk + kvk: In view of Lemma 7, we introduce an operator T : X × Y → X × Y as follows: where Journal of Function Spaces Here, For computational convenience, we set Now, we are in a position to present our main results. The methods used to prove the existence and uniqueness solutions of boundary value problem (3), (4) via Banach's contraction principle.
By assumption ðH1Þ, for ðu, vÞ ∈ B r , t ∈ ½a, b, we have that which leads to Hence, In the same way, we can obtain that Consequently, it follows that which implies T B r ⊂ B r . Next, we show that operator T is contraction mapping.
For any ðu 1 , v 1 Þ, ðu 2 , v 2 Þ ∈ X × Y and for any t ∈ ½a, b, we obtain Journal of Function Spaces Therefore, we get the following inequality Similarly, From inequalities (43) and (44), it yields Since ðK 1 + K 3 Þðm 1 + m 2 Þ + ðK 2 + K 4 Þðn 1 + n 2 Þ < 1, therefore, T is a contraction operator. So, by applying Banach's fixed point theorem, the operator T has a unique fixed point in B r . Hence, there exists a unique solution of problem (3), (4) on ½a, b. Now, we prove our second existence result via the Leray-Schauder alternative.
Lemma 9 (Leray-Schauder alternative [39]). Let F : E → E be a completely continuous operator (i.e., a map restricted to any bounded set in E is compact). Let Then, either the set εðFÞ is unbounded or F has at least one fixed point.
Proof. By the continuity of functions f , g on ½a, b × R × R, the operator T is continuous. Now, we show that the operator T : X × Y → X × Y is completely continuous. Let Ω ⊂ X × Y be bounded. Then, there exist two positive constants, M 1 and M 2 , such that