POSITIVE SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATION BOUNDARY VALUE PROBLEMS AT RESONANCE∗

In this article, we study a class of fractional differential equations with resonant boundary value conditions. Some sufficient conditions for the existence of positive solutions are considered by means of the spectral theory of linear operator and the fixed point index theory.

When ηξ α−β−1 = 1, FBVP (1.1) is non-resonant. Li et al. [15] established the existence of positive solutions for the non-resonant case of (1.1) by means of the fixed point theorem. Xu and Fei [33] obtained the existence of positive solutions for the non-resonant case of (1.1) by using the Schauder fixed-point theorem.
Some recent papers have studied the existence of positive solutions to nonresonant FBVPs under conditions concerning the first eigenvalue of the relevant linear operator, for example [5,17,18,29,36,37]. By the theory of u 0 -positive operator, Cui [5] investigated the uniqueness results of solution to Zhang and Zhong [37] established the uniqueness results of solution to (1.3) by using the Banach contraction map principle and and the theory of u 0 -positive linear operator. The main novelty of [5] and [37] was that the Lipschitz constant was associated with the first eigenvalues of the relevant linear operator.
While there are a lot of works dealing with non-resonant FBVPs, the results considering resonant FBVPs are relatively scarce. The main tool used to seek solutions for resonant boundary value problems is the coincidence degree theory (see [11][12][13][14]22,34]). To the best of our knowledge, research on positive solutions for resonant boundary value problems is quite rarely seen. For the case that α is an integer, Liang [16] and Webb [31] established existence results of positive solutions for second order boundary value problems at resonance by considering equivalent non-resonant perturbed problems. In [24], we obtained necessary conditions of the existence of positive solutions to the following fractional boundary value problem at resonance: In [26], we established the existence and uniqueness results of positive solutions to the following resonant FBVP: The cone which is usually deduced from properties of the Green's function plays an important role in seeking positive solutions. It should be noted that the cone used in [26] does not suitable for the method of [31].
Motivated by the above works, in this paper we aim to establish the existence of positive solutions to the FBVP (1.1). Our work presented in this paper has the following features. Firstly, we consider the case that the resonant boundary condition involves an arbitrary fractional derivative which has been seldom studied due to the difficulties in direct sum decomposition for the relevant linear space. Secondly, some new properties of the Green function for this case have been obtained to deduce a suitable cone. Thirdly, the results on the existence of positive solutions concern the behavior of the nonlinearity at 0 and at ∞.

Basic definitions and preliminaries
provided that the right-hand side is point-wise defined on (0, +∞).
Clearly, we know the function has a unique positive root M * , that is, g(M * ) = 0.
It is obvious that (1.1) is equivalent to the following FBVP: Throughout this article, we always assume that the following assumption holds: For convenience, we use the following notations: , x k Γ(kτ +ν) is the Mittag-Leffler function. It is easy to check that (2.1) has a unique solution written as Proof. By 0 < ξ < 1 and ηξ α−β−1 = 1, we can get From [21], the solution of (2.3) can be expressed by It follows from u(0) = 0 that c 2 = 0. Hence Noticing (2.1) and (2.2), by direct calculation, we have Therefore, Substituting into (2.4), we have the unique solution of (2.3) is Lemma 2.3. The function K 0 (t, s) satisfies the following properties: Proof. By the notation of H(t), we can get Noticing g is strictly increasing on (0, +∞) and g(M * ) = 0, by direct calculation, we obtain Then we have that H(t) is strictly increasing on [0, 1] and H (t) is strictly decreasing on (0, 1]. In addition, it is clear that H (t) is strictly increasing on (0, 1].
Since the proof is similar to the Theorem 3.1 in [27], we omit it here.
Lemma 2.4. The function K(t, s) admits the following properties: Proof. By the notations, the functions K 0 and K can be expressed by Let s 0 ∈ [0, 1] be fixed. If t ∈ [s 0 , 1], we have If t ∈ [0, s 0 ], it is obvious that . This combing with (2.1) implies that Then (i), (ii) and (iii) can be deduced from Lemma 2.3 directly.
Lemma 2.5. The function G(t, s) has the following properties: , .
Hence (i) holds. Noticing it follows from (iv) of Lemma 2.3 and (iii) of Lemma 2.4 that Therefore (ii) holds. By the notations of K and K 0 , it is easy to get that Then So (iii) holds. From Lemma 2.5, we have the following Lemma: Lemma 2.6. The function G * (t, s) := t 2−α G(t, s) satisfies: Let E = C[0, 1] be endowed with the maximum norm u = max 0≤t≤1 |u(t)|, θ is the zero element of E, B r = {u ∈ E : u < r}. Define two cones P and Q by Define four operators as follows:  It is clear that the fixed point of the operator A is a solution of the resonant FBVP (1.1). By Lemma 2.5 and Arzela-Ascoli theorem, we can get A : P → P is completely continuous, L : P → P is a completely continuous linear operator. By Lemma 2.6, we can get L * : Q → Q is a completely continuous linear operator.
Noticing that λ = 0 is the eigenvalue of the linear problem (1.2) and t α−1 is the corresponding eigenfunction, we have that the first eigenvalue of L is λ 1 = M * , and ϕ 1 (t) = t α−1 is a positive eigenfunction corresponding to λ 1 , that is ϕ 1 = M * Lϕ 1 . Therefore, the first eigenvalue of L * is λ * 1 = M * , and ϕ * 1 (t) = t is a positive eigenfunction corresponding to λ * 1 , that is, It follows from [30] that the following Lemma holds.

Main results
Theorem 3.1. Assume that the following assumptions hold: Then (1.1) has at least one positive solution.
Proof. The proof is similar to the Theorem 4.1 in [26], we omit it here.
Theorem 3.2. Assume that the following assumptions hold: Then (1.1) has at least one positive solution.
Proof. By Lemma 2.6 and Arzela-Ascoli theorem, we can get A * : Q → Q is completely continuous. It follows from (3.1) that there exists r 1 > 0 such that For any u ∈ ∂B r1 ∩ Q, we have In the following, we will show that A * has at least one fixed point on Q.
Suppose that A * has no fixed points on ∂B r1 ∩ Q. Next, we will show that where ϕ * 1 is the positive eigenfunction of L * . In fact, if there exist µ 0 > 0 and u 1 ∈ ∂B r1 ∩ Q such that It follows from the definition of Q that µ * < +∞. Clearly we have µ * ≥ µ 0 and u 1 ≥ µ * ϕ * 1 . Therefore, Hence contradicts with the definition of µ * , that is (3.3) holds. By Lemma 2.8, we have From (3.2), there exist ∈ (0, M * ) and r 2 > r 1 such that Next, we will show that W is bounded. For u ∈ W , we have hereũ(t) = min{u(t), r 2 }. It is easy to see that Therefore Then Therefore A * has a fixed point u * ∈ (B R \B r1 ) ∩ Q, that is, It is easy to see that t α−2 u * (t) is a positive solution of FBVP (1.1).
Then (1.1) has at least one positive solution.
Proof. It follows from (3.7) that there exists r 3 > 0 such that Suppose that A * has no fixed points on ∂B r3 ∩ Q (otherwise, the proof is finished).
Hence µ n 0 u 1 ≤ T n u 1 ≤ T n u 1 . By the Gelfand's formula, we have r(T ) = lim n→+∞ n T n ≥ µ 0 > 1, contradicts with r(T ) = M * r(L * ) = 1. Therefore, it follows from Lemma 2.9 that i(A * , B r3 ∩ Q, Q) = 1. (3.10) From (3.8), there exist > 0 and r 4 > r 3 such that By Lemma 2.7, there exists m large enough such that For any u ∈ ∂B R ∩ Q, we have By virtue of the Krein-Rutmann theorem, we know that there exists a positive eigenfunction ψ m corresponding to the first eigenvalue of L * m , that is, L * m ψ m = r(L * m )ψ m . For any u ∈ ∂B R ∩ Q. It follows from (3.11) that . We may suppose that A * has no fixed points on ∂B R ∩ Q (if otherwise, the proof is finished). Next, we will proof that If otherwise, there exist u 1 ∈ ∂B R ∩ Q and µ 0 > 0 such that It is clear that µ 0 ≤ µ * < +∞ and u 1 ≥ µ * ψ m . Then contradicts with the definition of µ * . Hence (3.12) holds. By Lemma 2.9, we have   where

Conclusions
In this paper, we discuss a class of FBVPs at resonance by considering equivalent non-resonant perturbed problems with the same boundary conditions. The main novelty of the problem we discussed lies in that an arbitrary fractional derivative is involved in the resonant boundary condition. Some interesting properties of the Green's function have been deduced to construct a suitable cone. By using the fixed point index theory on the cone, some existence results of positive solutions to the resonant FBVPs are obtained. The main results of this paper is valid to the multi-point boundary value problem at resonance: where 0 < ξ 1 < · · · < ξ m−2 < 1, η i > 0 with