Multiple Solutions of Nonlocal Boundary Value Problems for Fractional Differential Equations on the Half-line *

In this paper, we study the existence of multiple solutions of nonlocal boundary value problems for fractional differential equations with integral boundary conditions on the half-line. Applying the fixed point theory and the upper and lower solutions method, some new results on the existence of at least three nonnegative solutions are obtained. An example is presented to illustrate the application of our main results.


Introduction
In this paper, we consider the following nonlocal boundary value problem for fractional differential equations with integral boundary condition on the half-line where C D α is the standard Caputo derivative, 0 < α < 1 is a constant, f , g, p and q are given functions.
Boundary value problems (BVPs) of differential equation have received much attention in recent years due to their broad applications in applied mathematics and physics.There are many papers concerning the existence of solutions, positive solutions or multiple solutions of two point BVPs, three point BVPs, m-point even nonlocal boundary conditions such as integral boundary conditions about the integer order differential equation.For details we can refer to [6, 10, 13, 16-21, 24, 26].
Boundary value problems on the half-line have been applied in unsteady flow of gas through a semiinfinite porous medium, the theory of drain flows, etc.In the paper [1], Agarwal and O'Regan gave infinite interval problems modeling phenomena which arise in the theory of plasma and electrical potential theory.In [6,10,11,13,19,26], authors studied two-point or multipoint boundary value problems on the half-line by using different method.The papers [20,21] studied the existence of positive solutions for second-order boundary value problems of differential equations system with integral boundary condition on the half-line.
It is well known that fractional order differential equations have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering, and they also have been of great interests, see [9,15].Recently, there are some papers which deal with the existence of the solutions of the boundary values problems for fractional differential equations on finite intervals.For details, see [2,4,5,7,8,12,14,23,25,27,28] and the references therein.
In [9] and [15], the basic theories for the fractional calculus and the fractional differential equations were discussed.In [5], Benchohra, Hamania and Ntouyas investigated the existence and uniqueness of solutions for problem: By using Schauder's fixed point theorem combined with the diagonalization method, Arara and co-authors (see [4]) studied the existence of solutions for boundary value problems for fractional order differential equation of the form In [2], Ahmad and Nieto stuided some existence results for a boundary value problem involving a nonlinear integrodifferential equation of fractional order 1 < q ≤ 2 with integral boundary conditions by using contraction mapping principle and Krasnoselskií's fixed point theorem.
However, researches for the multiple solutions of the fractional differential equations with nonlocal boundary condition on infinite intervals are few.In this paper, we aim to discuss the multiple solutions for fractional differential equations with integral boundary condition on the half-line.
Applying the well-known Amann theorem and the upper and lower solutions method, we obtain a new result on the existence of at least three distinct nonnegative solutions under some conditions.
An example is presented to illustrate the application of our main result.EJQTDE, 2011 No. 56, p. 2 In this section, we introduce preliminary facts which are used throughout this paper.We denote that R = (−∞, +∞) and R + = [0, +∞).Definition 2.1 (See [9,15]) Let α > 0. The fractional (arbitrary) order integral of the function y : R + → R of order α is defined by provided the integral exists, where Γ is the Gamma function.
Definition 2.2 (See [9,15]) The Caputo fractional order derivative of the function y of order α is defined by provided the right side is pointwise defined on (0, +∞), where n = [α] + 1 and [α] denotes the integer part of α.
Throughout the paper, we suppose that the following hypotheses are satisfied: It is obvious that 0 < 1 − ||g|| 1 ≤ 1 if (H1) holds.From (H2), we can get that k(s) ≥ 0 and lim s→+∞ k(s) = 0.So k(s) is bounded, which implies that there exists a constant K 0 > 0 such that We define that and By (H1), (H2), (2.1) and (2.2), we can easily get that K and G satisfy the following lemma.
Let E be a Banach space, P ⊂ E be a cone in E. A cone P is called solid if it contains interior points, i.e., P = Ø.Every cone P in E defines a partial ordering in E given by x y iff y − x ∈ P .
If x y and x = y, we write x y; if a cone P is solid and y − x ∈ P , we write x ≪ y.A cone P is said to be normal if there exists a constant N > 0 such that 0 x y implies ||x|| ≤ N ||y||.If P is normal, then every order interval [x, y] = {z ∈ E|x z y} is bounded.
The following Lemma 2.4 is the well-known Amann three-solution theorem (see [3,22]), which will be used in the later proof of our main results about the multiple solutions of the boundary value problem.
Lemma 2.4 Let E be a Banach space, and P be a normal solid cone.Suppose that there exist α 1 , and T : [α 1 , β 2 ] −→ E is a completely continuous strongly increasing operator such that Then the operator T has at least three fixed points x 1 , x 2 , x 3 such that 3 Multiple solutions of the boundary value problem In order to obtain the results, we suppose the following conditions hold: (H3) q ∈ L 1 (R + ), q(t) is nonnegative on R + and q > 0 a.e.Proof.First of all, let us show the operator T is well defined, and T : P −→ P .
For any fixed u ∈ P , it implies that u is bounded, by Lemma 2.1, (H3) and (H4), and we can get (T u)(t) ≥ 0 for t ∈ R + .And there exists a constant Thus, T : P −→ P is well defined.
Secondly, we show that T is continuous.EJQTDE, 2011 No. 56, p. 7 Let {u n } ⊂ P , u ∈ P , and u n → u 0 as n → ∞.So, there exists a constant f M1 > 0, such that 0 ≤ f (t, u n (t)), f (t, u 0 (t)) ≤ f M1 for any t ∈ R + .By (H3), (H4) and Lemma 2.1, we can see According to the Lebesgue's dominated convergence theorem, we can show Therefore, the operator T is continuous.
Finally, we will prove that the operator T maps bounded sets into relatively compact sets.
For the bounded set Ω ⊂ P , there exists a constant M 2 > 0, such that ||u|| ≤ M 2 for any u ∈ Ω.
Thus there exists a constant Thus, the set T (Ω) is uniformly bounded.
Using Lemma 3.1, we can obtain that the set T (Ω) is a relatively compact set.Hence, the operator T maps bounded sets into relatively compact sets.
Therefore, we can get that the operator T is completely continuous.Let us prove that T is a strongly increasing operator.
For any w 1 , w 2 ∈ P , with w 1 w 2 , that is to say that w 1 (t) ≤ w 2 (t) for all t ∈ R + , and there exists [a 0 , b 0 ] ⊂ R + such that w 1 (t) < w 2 (t) for any t ∈ [a 0 , b 0 ].
Hence, we conclude that T is a strongly increasing operator.
Let us now prove that x 1 T x 1 .
We denote x = T x 1 − x 1 .
Noting that x 1 is the lower solution of boundary value problems (1.1) and applying the definition EJQTDE, 2011 No. 56, p. 9 of the operator T , we have It follows from Lemma 2.3 that Then Similarly, we can get that Since x 2 is an lower solution of (1.1) and not a solution of (1.1), we have (T x 2 ) = x 2 .Thus Using the same method, we can also get that T y 1 y 1 , T y 2 y 2 .
We can easily verify the condition (H4) holds.

Definition 3 . 1 .
u = u(t) is called an upper (lower) solution of boundary value problem (1.1), if

+∞ 0 e
−2t x i (t)dt, y i (∞) = +∞ 0 e −2t Obviously, P is a normal solid cone in E, and u v ∈ E if and only if u(t) ≤ v(t) for t ∈ R + .
[11] + .EJQTDE, 2011 No. 56, p. 6 (H4) f : R +× R + −→ R + is a Carathéodory function, that is to say, f (•, u) is measurable for any u ∈ R + and f (t, •) is continuous for almost every t ∈ R + .f(t,u) is bounded for t ∈ R + when u is bounded, and f (t, u 1 ) < f (t, u 2 ) with u 1 < u 2 ∈ R + , for almost every t ∈ R + .Let E = {u ∈ C(R + ) | lim t→+∞ u(t) exists } be endowed with the norm u := sup t∈R + |u(t)|, thenE is a Banach space.We define the cone P ⊂ E byP := {u ∈ E | u(t) ≥ 0, t ∈ R + }. u v ∈ E if and only if u(t) ≤ v(t)and u(t) ≡ v(t), which implies that there exists an interval[a 0 , b 0 ] ⊂ R + such that u(t) < v(t) for t ∈ [a 0 , b 0 ].Lemma 3.1 (See[11]) Let E be defined as before and D ⊂ E. Then D is relatively compact in E if the following conditions hold: Theorem 3.3 Suppose that (H1)-(H4) hold, and there exist two lower solutions x 1 , x 2 and two upper solutions y 1 , y 2 of boundary value problem (1.1) such that x 2 , y 1 are not the solutions of the boundary value problem (1.1) withx 1 y 1 x 2 y 2 .Then the boundary value problem (1.1) has at least three distinct nonnegative solutions u 1 , u 2 , u 3 which satisfy that for t∈ R + x 1 (t) ≤ u 1 (t) < y 1 (t), x 2 (t) < u 2 (t) ≤ y 2 (t), x 2 (t) u 3 (t) y 1 (t).Proof.It is obvious that the boundary value problem (1.1) has nonnegative solutions if and only if the operator T has fixed points on P .It follows from Lemma 3.2 that T : [x 1 , y 2 ] → P is completely continuous.