Solvability for a system of Hadamard fractional multi-point boundary value problems

In this paper, we study a system of Hadamard fractional multi-point boundary value problems. We first obtain triple positive solutions when the nonlinearities satisfy some bounded conditions. Next, we also obtain a nontrivial solution when the nonlinearities can be asymptotically linear growth. Furthermore, we provide two examples to illustrate our main results.

Fractional-order equations, as a generalization of the case of integer order, can accurately characterize some complex phenomena in nature. It has been proved that there are many special advantages in some fields, such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, which has become a hot research topic of common concern in the world. For example, in [5], the authors investigated the following fractional-order advection-diffusion-reaction boundary value problem: where 1 < α 2, 0 < ε 1, γ ∈ R, C D α is the fractional derivative of Caputo sense, and S(t) is a spatially dependent source term. It has been observed that most of papers in the literature on the fractional-order equations involves either Riemann-Liouville-or Caputo-type fractional derivative. Apart from the two derivatives, Hadamard derivative is another kind of fractional derivative that was introduced by Hadamard [9]. This fractional derivative differs from the other ones in the sense that the kernel of the integral contains logarithmic function of arbitrary exponent. For detailed materials of Hadamard fractional derivative and integral, we refer to the papers [1-4, 8, 10, 11, 13-23, 25-29] and references therein. In [14], the authors studied the Riemann-Liouville fractional differential inclusion with Hadamard fractional integral boundary conditions RL D q x(t) ∈ F t, x(t) , 0 < t < T, 1 < q 2, In [29], Zhang et al. utilized the Guo-Krasnosel'skii fixed point theorem to obtain the multiple positive solutions for the Hadamard fractional integral boundary value problems where α ∈ (2, 3], β ∈ (1, 2], µ ∈ [0, β], ϕ p is the p-Laplacian, and the nonlinearity f grows (p − 1)-superlinearly and (p − 1)-sublinearly.
On the other hand, we note that coupled systems of fractional-order equations have also been investigated by many authors, we refer to [2, 4, 8, 11, 13, 17-21, 23, 25-27]. Ahmad and Ntouyas in [2] investigated some results for the system of Hadamard fractional differential equations where I γ is the Hadamard fractional integral with γ > 0. By using Leray-Schauder's alternative and Banach's contraction principle the authors obtained the existence and uniqueness of solutions, respectively. In [11], Jiang et al. adopted the fixed point index to study the existence of positive solutions for the system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions where the nonlinearities f i (i = 1, 2) can grow superlinearly and sublinearly. Inspired by the works above, in this paper, we use some fixed point methods to study the existence of solutions for (1). We first obtain triple positive solutions when the nonlinearities satisfy some bounded conditions. Next, we also obtain a nontrivial solution when the nonlinearities can be asymptotically linear growth. Finally, we offer two examples to illustrate our main results.
The outline of the paper is organized as follows. In Section 2, we give revelent definitions and lemmas, and some important properties of the corresponding Green's function are also obtained. In Section 3, we give the detailed proofs for the existence theorems. In Section 4, we present two examples to illustrate our main results.

Preliminaries
In this section, we only provide the definition of the Hadamard fractional derivative, for more details we refer the reader to [1]. Definition 1. (See [1].) The Hadamard derivative of fractional order q for a function g : [1, ∞) → R is defined as where n = [q] + 1, [q] denotes the integer part of the real number q, and log(·) = log e (·).
Proof. From Lemma 2(i) we have and This completes the proof.
Then from Lemma 1 we obtain that (1) is equivalent to the following system of Hammersteintype integral equations: Therefore, we can define an operator A : P × P → P × P as follows: Note that G i and f i (i = 1, 2) are nonnegative continuous functions, so the operators A i : P × P → P (i = 1, 2) and A: P × P → P × P are three completely continuous This implies that Consequently, if t ∈ [5/4, 3e/4], we obtain and then min On the other hand, using the method of Lemma 3, we also have G 2 (t, s) ω(t)G 2 (τ, s) for t, s, τ ∈ [1, e], and thus A 2 (P, P ) ⊂ P 0 . This completes the proof.
Lemma 6. (See [12].) Let E be a Banach space, and A: E → E be a completely continuous operator. Assume that T : E → E is a bounded linear operator such that 1 is not an eigenvalue of T and lim u →∞ Au − T u / u = 0. Then A has a fixed point in E.
This simple idea motivates our study in Lemma 1.
Combining the above, we do not need to construct new Green's functions to obtain the equivalent Hammerstein-type integral equations for our problem (1).

Main results
Now, we state our main theorems, and provide their proofs. Theorem 1. Let 0 < a < b < b /η < c , (H0)-(H1) and the following conditions hold: Then (1) has at least triple positive solutions.
Theorem 2. Suppose that (H0) and the following conditions hold: Then (1) has at least one nontrivial solution.
Proof. Define operators T i : E × E → E as follows: Now, we prove that 1 is not an eigenvalue of T i (i = 1, 2), and we only need to consider the case i = 1 (the case i = 2 can be dealt with a similar method). Argument by contrary. If let u + v = w, then we have and by Lemma 1 we obtain where q, δ, a i , ξ i (i = 1, 2, . . . , m − 1) satisfy (H0). We distinguish two cases.
Case 2. σ 1 = 0. From (7) we have This has a contradiction. Above all, 1 is not an eigenvalue of T i (i = 1, 2) as required. Hence, if we let the operator T : E × E → E × E as follows: we know 1 := (1, 1) is not an eigenvalue of T .
From (H5), for all ε > 0, there exist M i > 0 (i = 1, 2) such that Consequently, there are ζ 1 > 0, ζ 2 > 0 such that As a result, we have For the arbitrariness of ε, we have lim (u,v) →∞ A(u, v) − T (u, v) / (u, v) = 0. Note from (H6) that 0 = (0, 0) is not a fixed point of A. Hence, from Lemma 6 we have that A has a fixed point in E, and this fixed point is nontrivial, i.e., (1) has at least one nontrivial solution. This completes the proof.