Positive Solutions of Nonlinear Eigenvalue Problems for a Nonlocal Fractional Differential Equation

By using the fixed point theorem, positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equation D 0 u t λa t f t, u t 0, 0 < t < 1, u 0 0, u 1 Σi 1αiu ξi are considered, where 1 < α ≤ 2 is a real number, λ is a positive parameter, D 0 is the standard Riemann-Liouville differentiation, and ξi ∈ 0, 1 , αi ∈ 0,∞ with Σi 1αiξ i < 1, a t ∈ C 0, 1 , 0,∞ , f t, u ∈ C 0,∞ , 0,∞ .


Introduction
Fractional differential equations have been of great interest recently.This is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering.For details, see 1-6 and references therein.
Recently, many results were obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations by the use of techniques of nonlinear analysis, see 7-21 and the reference therein.Bai and L ü 7 studied the existence of positive solutions of nonlinear fractional differential equation where 1 < α ≤ 2 is a real number, D α 0 is the standard Riemann-Liouville differentiation, and f : 0, 1 × 0, ∞ → 0, ∞ is continuous.They derived the corresponding Green function and obtained some properties as follows.

1.3
It is well known that the cone plays a very important role in applying Green's function in research area.In 7 , the authors cannot acquire a positive constant taken instead of the role of positive function γ s with 1 < α < 2 in 1.2 .In 9 , Jiang and Yuan obtained some new properties of the Green function and established a new cone.The results can be stated as follows.

Proposition 1.2. Green's function G t, s defined by 1.3 has the following properties
Proposition 1.3.The function G * t, s : t 2−α G t, s has the following properties: In this paper, we study the existence of positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equation where 1 < α ≤ 2 is a real number, λ is a positive parameter, D α 0 is the standard Riemann-Liouville differentiation, and We assume the following conditions hold throughout the paper:

The Preliminary Lemmas
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory.These definitions can be found in the recent literature.
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, ∞ .
Definition 2.2.The fractional derivative of order α > 0 of a function y : 0, ∞ → R is given by where n α 1, provided the right side is pointwise defined on 0, ∞ .
Lemma 2.3.Let α > 0. If one assumes u ∈ C 0, 1 ∩ L 0, 1 , then the fractional differential equation where N is the smallest integer greater than or equal to α, as unique solutions.

2.10
Proof.By applying Lemmas 2.4 and 2.5, we have

The Main Results
Let

3.1
Then u t is the solution of BVP 1.6 if and only if Tu t u t , where T is the operator defined by Tu t : λ 1 0 G 0 t, s a s f s, u s ds.

3.2
By similar arguments to Proposition 1.3, we obtain the following result.
Let E C 0, 1 be endowed with the ordering u ≤ v if u t ≤ v t for all t ∈ 0, 1 , and the maximum norm u max 0≤t≤1 |u t |.Define the cone P ⊂ E by P {u ∈ E | u t ≥ 0}, and where q t is defined by 3.3 .
It is easy to see that P and K are cones in E. For any 0 For convenience, we introduce the following notations: 3.5 By similar arguments to Lemma 4.1 of 9 , we obtain the following result.

Lemma 3.2. Assume that (H1)-(H3) hold. Let T : K → E be the operator defined by
Tu t : λ 1 0 G t, s a s f s, s α−2 u s ds.

3.6
Then T : K → K is completely continuous.
Proof of Theorem 3.3.Let λ be given as in 3.7 , and choose ε > 0 such that 3.9 Beginning with g 0 , there exists an

3.10
Thus, Tu ≥ u .So, if we let

3.12
It remains to consider g ∞ .There exists an H 2 such that g u ≤ g ∞ ε u, for all u ≥ H 2 .There are the two cases, a , where g is bounded, and b , where g is unbounded.
Case a. Suppose N > 0 is such that g u ≤ N, for all 0 < u < ∞.
Let H 2 max{2H 1 , Nλ 1 0 Ψ s a s q 2 s ds}.Then, for u ∈ K with u H 2 , we have Ψ s a s q 2 s g u s ds ≤ λN 1 0 Ψ s a s q 2 s ds ≤ H 2 u .

3.13
So, if we let and so Tu ≤ u .For this case, if we let
Proof of Theorem 3.4.Let λ be given as in 3.8 , and choose ε > 0 such that 0 Ψ s a s q 2 s ds g 0 ε .

3.22
Beginning with g 0 , there exists an H 1 > 0 such that g u ≤ g 0 ε u, for 0 < u ≤ H G t, s a s f s, s α−2 u s ds.

3.29
By similar method to Theorem 3.3, we can get y 0 0, then y t t α−2 u t is solution of 1.6 for t ∈ 0, 1 .We complete the proof.