Positive solutions of two-point boundary value problems of nonlinear fractional differential equation at resonance ∗

This paper deals with the existence of positive solutions for a kind of multi-point nonlinear fractional differential boundary value problem at resonance. Our main approach is different from the ones existed and our main ingredient is the Leggett-Williams norm-type theorem for coincidences due to O'Regan and Zima. The most interesting point is the acquisition of positive solutions for fractional differential boundary value problem at resonance. And an example is constructed to show that our result here is valid.


INTRODUCTION
This paper deals with positive solutions to the following boundary value problem: where c D α 0 + is the Caputo's fractional derivative of order α, 1 < α ≤ 2 is a real number, and Due to the fact that the fractional differential operator c D α 0 + is not inventible under Caputo's derivative, boundary value problems (in short:BVPs) of this type are referred to as problems at resonance.Recently, fractional differential equations (in short:FDE) have been studied extensively.
For an extensive collection of such results, we refer the readers to the monographs [1][2][3][4] and the reference therein.Some basic theory for the initial value problems of FDE involving Riemann-Liouville differential operator has been discussed [5][6][7][8][9][10].Also, there are some papers which deal with the existence of positive solutions for BVPs of nonlinear FDE by using techniques of topological degree theory [11][12][13][14][15][16].For example, the existence and multiplicity of positive solutions for the equation subject to the Dirichlet boundary condition have been studied by Bai and Lü [13] by means of the well-known Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem.D α 0 + is the standard Riemann-Liouville fractional derivative there.
In [14] and [15], Zhang also studied the existence of positive solutions of Eq.(1.3) under the boundary conditions and M. El-Shahed [16] established the existence of positive solutions to BVP by applying Krasnosel'skii fixed point theorem.
From above works, we can see a fact, although the BVPs of nonlinear FDE have been studied by some authors, to the best of our knowledge, all of existing works are limited to non-resonance boundary conditions.For the resonance case, as far as we know, no contributions exist.The aim of this paper is to fill the gap in the relevant literature.Our main tool is the recent Leggett-Williams norm-type theorem for coincidences due to O'Regan and Zima [17].

PRELIMINARIES
For the convenience of the reader, we demonstrate and study the definitions and some fundamental facts of Caputo's fractional derivative.
Definition 2.1.The Riemann-Liouville fractional integral of order α is defined by where Γ(α) is the Euler gamma function defined by for which, the reduction formula and formula where D n = d n dt n and n = [α] + 1, [α] denotes the integer part of α, and Remark 2.1.Under natural conditions on the function y(t), Caputo's derivative becomes a conventional m-th derivative of the function y(t) as α → m(see [2]).
From definitions 2.1 and 2.2, we can deduce the following statement.Lemma 2.1 [4] .The fractional differential equation Furthermore, for y ∈ AC n [0, 1], and (2.7) In the following, we review some standard facts on Fredholm operators and cones in Banach spaces.Let X, Y be real Banach spaces.Consider a linear mapping L : domL ⊂ X → Y and a nonlinear mapping Throughout we assume EJQTDE, 2011 No. 71, p. 4 The assumption 1 • implies that there exist continuous projections P : X → X and Q : Y → Y such that ImP = KerL and KerQ = ImL with X = KerL ⊕ KerP , Y = ImL ⊕ ImQ and dimImQ = dimKerL < ∞.And we can define an isomorphism J : ImQ → KerL.
Denote by L p the restriction of L to KerP ∩ domL.Clearly, L p is an isomorphism from KerP ∩ domL to ImL, we denote its inverse by K p : ImL → KerP ∩ domL.It is known (see [19]) that the coincidence equation Lx = Nx is equivalent to A nonempty closed convex set C ⊂ X is said to be a cone in X provided that: It is well known that C induces a partial order in X by x y if and only if y − x ∈ C.
We will write x y for y − x ∈ C.Moreover, for every u ∈ C \ {0} there exists a positive number σ(u) such that Let γ : X → C be a retraction, that is, a continuous mapping such that γ(x) = x for all We make use of the following result due to O'Regan and Zima [17].
Theorem 2.1.Let C be a cone in X and let Ω 1 , Ω 2 be open bounded subsets of X with Assume that the following conditions hold. where Then the equation Lx = Nx has a solution in the set C ∩ (Ω 2 \ Ω 1 ).
For simplicity of notation, we set Remark 2.2.The computation of the function G(t, s) is shown in the proof of Theorem 3.1.

MAIN RESULTS
In order to prove the existence result, we present here a definition.Definition 3.1.We say that the function f : In this paper, we consider the Banach spaces and N : X → Y by Nx(t) = f (t, x(t)).
In order to obtain our main results, we firstly present and prove the following lemma.
Lemma 3.1.L : domL ⊂ X → Y is a Fredholm operator of index zero, and the linear operator K p : ImL → domL ∩ KerP can be written as where Proof.It is clear that We will show that ImL = {y ∈ Y : Since the problem has solution x(t) satisfies boundary conditions (1.2) if and only if In fact, if (3.2) has solution x(t) satisfies (1.2), then from (3.2) we have In view of x ′ (0) = x ′ (1), we can obtain that On the other hand, if (3.3) holds, setting where C is arbitrary constant, then x(t) is a solution of (3.2), and x(0) = 0, x ′ (0) = x ′ (1).Hence (3.1) holds.
Next, we define P : X → X by (P x)(t) = α(α − 1)t It is easy to see that the operators P and Q are all projections.In fact, for t ∈ [0, 1], The same to the operator Q.
In the sense of isomorphism, ImP = KerL and KerQ = ImL.So dimKerL = 1 = dimImQ = codimImL.Notice that ImL is closed, L is a Fredholm operator of index zero.
For y ∈ ImL, the inverse K p : ImL → domL ∩ KerP of L p can be given by In fact, for x ∈ domL ∩ KerP , we have y(t) = − c D α 0+ x(t) ∈ ImL and .
We can solve that where k(t, s) is given by (3.4).
To prove 5 which is a contradiction.In addition, if λ = 0, then B = 0, which is impossible.Thus,  x * (t) is not a trivial solution due to the fact that f (t, 0) ≡ 0 for t ∈ [0, 1].This completes the proof of Theorem 3.1.
) respectively.Due to the fact that the BVPs based on Riemann-Liouville derivative with non-zero boundary conditions can't be converted into an equivalent integral equation, while the Caputo's derivative is to meet the requirements.The conditions (1.5) and (1.6) are not zero boundary value, so the author investigated the BVPs (1.3)-(1.5)and (1.3)-(1.6)by involving the Caputo's fractional derivative.EJQTDE, 2011 No. 71, p. 2