Eigenvalue Problem for Nonlinear Fractional Differential Equations with Integral Boundary Conditions

and Applied Analysis 3 Lemma 6. Let G be the Green function, which is given by the expression (7). For 0 < λ < 2, the following property holds: t G (1, s) ≤ G (t, s) ≤ 2α ξ (α − 2) G (1, s) , ∀t, s ∈ (0, 1) . (14) The proof is similar to that of Lemma 2.4 in [7], so we omit it. Consider the Banach space X = C[0, 1] with general norm ‖u‖ = sup t∈[0,1] |u (t)| . (15) Define the cone P = {u ∈ X : u(t) ≥ (ξ(α − 1)/2α)t‖u‖}. Suppose u is a solution of (1). It is clear from Lemma 5 that u (t) = λ∫ 1 0 G (t, s) f (s, u (s)) ds, ∀t ∈ [0, 1] . (16) Define the operator S λ : P → X as follows: (Sλu) (t) = λ∫ 1 0 G (t, s) f (s, u (s)) ds, ∀t ∈ [0, 1] . (17) Lemma 7. S λ : P → P is completely continuous. Proof. Since 0 < ξ < 2, it is obvious that G(t, s) ≥ 0. So we have 󵄩󵄩󵄩󵄩Sλu 󵄩󵄩󵄩󵄩 = sup t∈[0,1] λ∫ 1 0 G (t, s) f (s, u (s)) ds


Introduction
Fractional calculus has been receiving more and more attention in view of its extensive applications in the mathematical modelling coming from physical and other applied sciences; see books [1][2][3][4][5].Recently, the existence of solutions (or positive solutions) of nonlinear fractional differential equation has been investigated in many papers (see  and references cited therein).However, in terms of the eigenvalue problem of fractional differential equation, there are only a few results [29][30][31][32][33].
Our proof is based upon the properties of the Green function and Guo-Krasnoselskii's fixed point theorem given in [34].Our purpose here is to give the eigenvalue interval for nonlinear fractional differential equation with integral boundary conditions.Moreover, according to the range of the eigenvalue , we establish some sufficient conditions for the existence and nonexistence of at least one positive solution of the problem (1).

Preliminaries
For the convenience of the readers, we first present some background materials.Definition 1.For a function  : [0, ∞) → R, the Caputo derivative of fractional order  is defined as where [] denotes the integer part of the real number .
Definition 2. The Riemann-Liouville fractional integral of order  for a function  is defined as provided that such integral exists.
is given by the expression where Proof.It is well known that the equation    () + () = 0 can be reduced to an equivalent integral equation: for some   ∈ R ( = 0, 1, 2, . . ., ).
Suppose  is a solution of (1).It is clear from Lemma 5 that Define the operator   :  →  as follows: Lemma 7.   :  →  is completely continuous.

Main Result
For convenience, we list the denotation: Next, we will establish some sufficient conditions for the existence and nonexistence of positive solution for problem (1).Theorem 8. Let  ∈ (0, 1) be a constant.Then for each problem (1) has at least one positive solution.
Theorem 10.Let  ∈ (0, 1) be a constant.Then for each problem (1) has at least one positive solution.