On solutions of some fractional m-point boundary value problems at resonance ∗

In this paper, the following fractional order ordinary differential equation boundary value problem: D � ( �)


Introduction
The subject of fractional calculus has gained considerable popularity and importance during the past decades or so, due mainly to its demonstrated applications in numerous seemingly and widespread fields of science and engineering.It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables.For details, see [1-9, 13-18, 21-25] and the references therein.
Recently, m-point integer-order differential equation boundary value problems have been studied by many authors, see [4,12,13,14].However, there are few papers consider the nonlocal boundary value problem at resonance for nonlinear ordinary differential equations of fractional order.In [6] we investigated the nonlinear nonlocal non-resonance problem where 1 < α ≤ 2, 0 < βη α−1 < 1.In [7], we investigated the boundary value problem at resonance ) + e(t), 0 < t < 1, where In this paper, the following fractional order ordinary differential equation boundary value problem: is considered, where 1 < α ≤ 2 is a real number, D α 0+ and I α 0+ are the standard Riemann-Liouville differentiation and integration, and The m-point boundary value problem (1.1), (1.2) happens to be at resonance in the sense that its associated linear homogeneous boundary value problem The purpose of this paper is to study the existence of solution for boundary value problem (1.1), (1.2) at resonance case, and establish an existence theorem under nonlinear growth restrictions of f .Our method is based upon the coincidence degree theory of Mawhin [22].Finally, we also give an example to demonstrate our result.
The main tool we use is the Theorem 2.4 of [22].
Theorem 1.1 Let L be a Fredholm operator of index zero and let N be L-compact on Ω. Assume that the following conditions are satisfied: Then the equation Lx = N x has at least one solution in dom(L) ∩ Ω.
The rest of this paper is organized as follows.In section 2, we give some notations and lemmas.In section 3, we establish a theorem of existence of a solution for the problem (1.1), (1.2).In section 4, we give an example to demonstrate our result.

Background materials and preliminaries
For the convenience of the reader, we present here some necessary basic knowledge and definitions about fractional calculus theory.Which can be found in [6,16,24].
We use the classical Banach spaces For n ∈ N , we denote by AC n [0, 1] the space of functions u(t) which have continuous derivatives up to order n − 1 on [0, 1] such that u (n−1) (t) is absolutely continuous: EJQTDE, 2010 No. 37, p. 3 Definition 2.1 The fractional integral of order α > 0 of a function y : (0, ∞) → R is given by provided the right side is pointwise defined on (0, ∞).
It can be directly verified that the Riemann-Liouvell fractional integration and fractional differentiation operators of the power functions t µ yield power functions of the same form.For α ≥ 0, µ > −1, there are holds almost everywhere on [0, 1].Now, we define another spaces which are fundamental in our work.

Remark 2.1 By means of the linear functional analysis theory, we can prove that with the norm
EJQTDE, 2010 No. 37, p. 4 Definition 2.4 Let I α 0+ (L 1 (0, 1)), α > 0, denote the space of functions u(t), represented by fractional integral of order α of a summable function: In the following Lemma, we use the unified notation of both for fractional integrals and fractional derivatives assuming that Lemma 2.2 [16]The relation is valid in any of the following cases: and equicontinuous means that ∀ε > 0, ∃δ > 0, for all

Existence result
In this section, we always suppose that 1 < α ≤ 2 is a real number and m−2 EJQTDE, 2010 No. 37, p. 5 Combine with . Define L to be the linear operator from dom(L) ∩ Y to Z with and Then boundary value problem (1.1), (1.2) can be written as Lu = N u.
If y ∈ Im(L), then there exists a function u ∈ dom(L) such that y(t) = D α 0+ u(t).By Lemma 2.1, where EJQTDE, 2010 No. 37, p. 6 By the boundary condition and by Lemma 2.2, In view of the condition thus, we obtain (3.3).
On the other hand, suppose y ∈ Z and satisfies: Proof.Suppose it is not true, we have However, it is well known that the Vandermonde Determinant is not equal to zero, so there is a contradiction.¶ where And the linear operator K P : Im(L) → dom(L) ∩ Ker(P ) can be written by y 1 , for all y ∈ Im(L).
Proof.For y ∈ Z, let y 1 = y − Qy, then So, L is a Fredholm operator of index zero.
With definitions of P, K P , it is easy to show that the generalized inverse of L : Im(L) → dom(L) ∩ Ker(P ) is K P .In fact, for y ∈ Im(L), we have and for u ∈ dom(L) ∩ Ker(P ), we know where .
If λ = 1, then c 0 = 0. Otherwise, if |c 0 | > M * , in view of (3.12), one has If (3.13) holds, then define the set here V as in above.Similar to above argument, we can show that Ω 3 is bounded too.

EJQTDE, 2010
No. 37, p. 2 Now, we briefly recall some notation and an abstract existence result.Let Y, Z be real Banach spaces, L : dom(L) ⊂ Y → Z be a Fredholm map of index zero and P : Y → Y, Q : Z → Z be continuous projectors such that Im(P ) = Ker(L), Ker(Q) = Im(L) and Y = Ker(L) ⊕ Ker(P ), Z = Im(L) ⊕ Im(Q).It follows that L| dom(L)∩Ker(P ) : dom(L)∩Ker(P ) → Im(L) is invertible.We denote the inverse of the map by K P .If Ω is an open bounded subset of Y such that dom(L)∩ Ω = ∅, the map N : Y → Z will be called L-compact on Ω if QN (Ω) is bounded and

Lemma 3 . 3 L
: dom(L) ∩ Y → Z is a Fredholm operator of index zero.Furthermore, the linear continuous projector operators Q : Z → Z and P : Y → Y can be defined by Qu = C u t k , for every u ∈ Z, EJQTDE, 2010 No. 37, p. 7