Multiple Positive Solutions for (n-1, 1)-type Semipositone Conjugate Boundary Value Problems for Coupled Systems of Nonlinear Fractional Differential Equations *

In this paper, we consider (n-1, 1)-type conjugate boundary value problem for coupled systems of the nonlinear fractional differential equation          D α 0+ u + λf (t, v) = 0, 0 < t < 1, λ > 0, D α 0+ v + λg(t, u) = 0, u (i) (0) = v (i) (0) = 0, 0 ≤ i ≤ n − 2, u(1) = v(1) = 0, where λ is a parameter, α ∈ (n − 1, n] is a real number and n ≥ 3, and D α 0+ is the Riemann-Liouville's fractional derivative, and f, g are continuous and semipositone. We give properties of Green's function of the boundary value problem, and derive an interval on λ such that for any λ lying in this interval, the semipositone boundary value problem has multiple positive solutions.


Introduction
We consider the (n-1, 1)-type conjugate boundary value problem for nonlinear fractional differential equation involving Riemann-Liouville's derivative where λ is a parameter, α ∈ (n − 1, n] is a real number, n ≥ 3, D α 0+ is the Riemann-Liouville's fractional derivative, and f, g are sign-changing continuous functions.As far as we know, there are few papers which deal with the boundary value problem for nonlinear fractional differential equation. Because of fractional differential equation's modeling capabilities in engineering, science, economics, and other fields, the last few decades has resulted in a rapid development of the theory of fractional differential equations, see [1]- [7] for a good overview.Within this development, a fair amount of the theory has been devoted to initial and boundary value problems problems (see [9]- [20]).In most papers, the definition of fractional derivative is the Riemann-Liouville's fractional derivative or the Caputo's fractional derivative.For details, see the references.
In this paper, we give sufficient conditions for the existence of positive solution of the semipositone boundary value problems (1.1) for a sufficiently small λ > 0 where f, g may change sign.Our analysis relies on nonlinear alternative of Leray-Schauder type and Krasnosel'skii's fixed-point theorems.

Preliminaries
For completeness, in this section, we will demonstrate and study the definitions and some fundamental facts of Riemann-Liouville's derivatives of fractional order which can been founded in [3].
From the definition of the Riemann-Liouville derivative, we can obtain the statement.As examples, for µ > −1, we have , where N is the smallest integer greater than or equal to α.
Lemma 2.1 Let α > 0; then the differential equation . . ., n, as unique solutions, where n is the smallest integer greater than or equal to α.
As D α 0+ I α 0+ u = u.From the lemma 2.1, we deduce the following statement.
Lemma 2.2 Let α > 0, then for some c i ∈ R, i = 1, 2, . . ., n, n is the smallest integer greater than or equal to α.
Lemma 2.3 [16] Let h(t) ∈ C[0, 1] be a given function, then the boundary-value problem has a unique solution where where The following a nonlinear alternative of Leray-Schauder type and Krasnosel'skii's fixed-point theorems, will play major role in our next analysis.
Theorem 2.5 [12] Let X be a Banach space with Ω ⊂ X be closed and convex.Assume U is a relatively open subsets of Ω with 0 ∈ U , and let S : U → Ω be a compact, continuous map.Then either 1. S has a fixed point in U , or 2. there exists u ∈ ∂U and ν ∈ (0, 1), with u = νSu.Theorem 2.6 [8] Let X be a Banach space, and let P ⊂ X be a cone in and let S : P → P be a completely continuous operator such that, either Then S has a fixed point in P ∩ (Ω 2 \Ω 1 ).
In fact, we only consider the boundary value problem We will show there exists a solution (x, y) for the boundary value problem (3.1) with x(t) ≥ w(t) and y(t) ≥ w(t) for t ∈ [0, 1].If this is true, then u(t) = x(t) − w(t) and v(t) = y(t) − w(t) is a nonnegative solution (positive on (0, 1)) of the boundary value problem (1.1).Since for any t ∈ (0, 1), we have ).As a result, we will concentrate our study on the boundary value problem (3.1).
We note that (3.1) is equal to From (3.2) we have For our constructions, we shall consider the Banach space E = C[0, 1] equipped with standard norm x = max 0≤t≤1 |x(t)|, x ∈ X.We define a cone P by Define an integral operator T : P → X by Notice, from Lemma 2.3, we have T x(t) ≥ 0 on [0, 1] for x ∈ P and On the other hand, we have Thus, T (P ) ⊂ P .In addition, standard arguments show that T is a compact, completely continuous operator.Theorem 3.1 Suppose that (H 1 ) and (H 2 ) hold.Then there exists a constant λ > 0 such that, for any 0 < λ ≤ λ, the boundary value problem (1.1) has at least one positive solution.
Since condition (H 1 ) implies conditions (H * 1 ) and (H 4 ), then from proof of Theorem 3.1 and 3.2, we immediately have the following theorem: