Multiple positive solutions for (n-1, 1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations, Electronic Journal of Qualitative Theory of Differential Equations

In this paper, we consider (n-1, 1)-type conjugate boundary value problem for the nonlinear fractional differential equation D0+u(t) + λf(t, u(t)) = 0, 0 < t < 1, λ > 0, u(0) = 0, 0 ≤ j ≤ n − 2, u(1) = 0, where λ is a parameter, α ∈ (n − 1, n] is a real number and n ≥ 3, and D0+ is the Riemann-Liouville’s fractional derivative, and f is continuous and semipositone. We give properties of Green’s function of the boundary value problems, and derive an interval of λ such that any λ lying in this interval, the semipositone boundary value problem has multiple positive solutions.

Because of fractional differential equation's modeling capabilities in engineering, science, economy, and other fields, the last few decades has resulted in a rapid development of the theory of fractional differential equation; 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 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 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 be found in [3].Definition 1.1 [3] The integral where α > 0, is called Riemann-Liouville fractional integral of order α.Definition 1.2 [3] For a function f (x) given in the interval [0, ∞), the expression where n = [α] + 1, [α] denotes the integer part of number α, is called the Riemann-Liouville fractional derivative of order s.
From the definition of Riemann-Liouville's 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.
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 2.3 ([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 2.4 ( [8]) Let X be a Banach space, and let P ⊂ X be a cone in X. Assume Ω 1 , Ω 2 are open subsets of X with 0 ∈ Ω 1 ⊂ Ω 1 ⊂ Ω 2 , 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 ).EJQTDE, 2010 No. 36, p. 2

Green's Function and Its Properties
In this section, we derive the corresponding Green's function for boundary value problem (1.1), and obtained some properties of the Green's function.First of all, we find the Green's function for boundary-value problem (1.1).

Lemma 3.1 Let h(t) ∈ C[0, 1] be a given function, then the boundary-value problem
has a unique solution where Here G(t, s) is called the Green's function of boundary value problem (3.1).
(H 4 ) 0 < 1 0 q(s)g(s)ds < +∞ and 1 0 q(s)f (s, y)ds < +∞ for any y ∈ [0, R], R > 0 is any constant.In fact, we only consider the boundary value problem where ds, which is the solution of the boundary value problem EJQTDE, 2010 No. 36, p. 4 We will show there exists a solution x for the boundary value problem (4.1) with is a nonnegative solution (positive on (0, 1)) of the boundary value problem (1.1).Since for any t ∈ (0, 1), As a result, we will concentrate our study on the boundary value problem (4.1).
We note that x(t) is a solution of (4.1) if and only if 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 ) is satisfied and define the integral operator T : P → E by Then T : P → P is completely continuous.
Proof First, we prove that T : P → P .Notice from (4.3) and Lemma 3.2 that, for x ∈ P , T x(t) ≥ 0 on [0, 1] and )ds.On the other hand, we have Thus, T (P ) ⊂ P .In addition, from f is continuous it follows that T is continuous.Next, we show T is uniformly bounded.Let D ⊂ P be bounded, i.e. there exists a positive constant L > 0 such that y ≤ L, for all y ∈ D. Let M = max Hence, T (D) is bounded.Finally, we show T is equicontinuous.
For all ε > 0, each u Thus, we obtain By means of the Arzela-Ascoli theorem, T : P → P is completely continuous.
If condition (H 1 ) is replaced by (H * 1 ), let where . By repeating the similar proof above, we get that T n is a completely continuous operator on P for each n ≥ 2. Furthermore, for any R > 0, set Ω R = {u ∈ P : u ≤ R}, then T n converges uniformly to T on Ω n as n → ∞.In fact, for R > 0 and EJQTDE, 2010 No. 36, p. 6 So we conclude that T n converges uniformly to T on Ω n as n → ∞.Thus, T is completely continuous.The proof is completed.
Let x ∈ P and ν ∈ (0, 1) be such that x = νT (x), we claim that x = R 0 .In fact, if x = R 0 , then which implies that x = R 0 .Let U = {x ∈ P : x < R 0 }.By the nonlinear alternative theorem of Leray-Schauder type, T has a fixed point x ∈ U .Moreover, if we combine (4.3), (4.4) and R 0 < ε, we obtain The proof of the theorem is completed.
Then for any x ∈ P ∩ ∂Ω 1 , then x = R 1 and x(s) − v(s) ≤ x(s) ≤ x , we have On the other hand, choose a constant N > 0 such that By the assumptions (H 3 ), for any t ∈ [θ 1 , θ 2 ], there exists a constant B > 0 such that EJQTDE, 2010 No. 36, p. 8 Choose , then for any x ∈ P ∩∂Ω 2 , we have And then Condition (2) of Krasnoesel'skii's fixed-point theorem is satisfied.So T has a fixed point x with r < x < R 2 such that x(1) = 0. EJQTDE, 2010 No. 36, p. 9 Since r < x ,

Examples
To illustrate the usefulness of the results, we give some examples.Example 1.Consider the boundary value problem where a > 1.Then, if λ > 0 is sufficiently small, then (5.1) has a positive solutions u with u(t) > 0 for t ∈ (0, 1).
y) + e(t)| + 1, then for x ∈ D, from the Lemma 3.1, one has