Positive Solutions for Multipoint Boundary Value Problem of Fractional Differential Equations

and Applied Analysis 3 where D 0 is the standard Riemann-Liouville fractional derivative of order α. Here, by a positive solution of BVP 1.6 , we mean a function which is positive on 0, 1 and satisfies the equation and boundary conditions in 1.6 . Throughout the paper, we will assume that the following conditions hold. H1 f : 0, 1 × 0,∞ → 0,∞ is continuous. H2 ai 0 for i 1, . . . , m − 2, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, and ρ ∑m−2 i 1 aiξ α−2 i with ρ < 1. There is a vast literature concerning the multipoint BVPs for the integer-order differential equations. An important recent paper was given by Webb and Infante 17 ; and they established a new unified method for the existence of multiple positive solutions to a large number of nonlinear nonlocal BVPs for the integer-order differential equations. While in the setting of the fractional-order derivatives, as far as we know, the existence of positive solutions for the multipoint BVP 1.6 has not been discussed in the literature. The rest of the paper is organized as follows. Section 2 preliminarily provides some definitions and lemmas which are crucial to the following discussion. In Section 3, we obtain the existence and multiplicity results of positive solutions for the BVP 1.6 by means of some fixed point theorems. Finally, we give a concrete example to illustrate the possible application of our analytical results. 2. Preliminaries In this section, we preliminarily provide some definitions and lemmas which are useful in the following discussion. Definition 2.1 see 3 . The fractional integral of order α > 0 of a function y : 0,∞ → R is given by I 0 y t 1 Γ α ∫ t 0 t − s α−1y s ds 2.1 provided the right side is pointwise defined on 0,∞ . Definition 2.2 see 3 . The standard Riemann-Liouville fractional derivative of order α > 0 of a continuous function y : 0,∞ → R is given by D 0 y t 1 Γ n − α ( d dt )n ∫ t 0 y s t − s α−n 1 ds, 2.2 where n α 1 provided the right side is pointwise defined on 0,∞ . Lemma 2.3 see 6 . Assume that u ∈ C 0, 1 ∩ L 0, 1 with a fractional derivative of order α > 0, then I 0 D α 0 u t u t c1t α−1 c2tα−2 · · · cNtα−N, 2.3 for some ci ∈ R, i 0, 1, . . . ,N, where N is the smallest integer greater than or equal to α. 4 Abstract and Applied Analysis By Lemma 2.3, we next present an integral presentation of the solution for the BVP of the linearized equation associated with the BVP 1.6 . Lemma 2.4. Let y ∈ C 0, 1 , then the BVP D 0 u t y t 0, 0 < t < 1, 2 < α < 3, u 0 u′ 0 0, u′ 1 m−2 ∑ i 1 aiu ′ ξi 2.4 has a unique solution u t ∫1 0 G t, s y s ds, 2.5 where the Green function G is given by G t, s G1 t, s G2 t, s , where G1 t, s ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − s α−2tα−1 − t − s α−1 Γ α , 0 s t 1, 1 − s α−2tα−1 Γ α , 0 t s 1, G2 t, s 1 ( 1 − ρΓ α ( ρ 1 − s α−2 − m−2 ∑ i 1 ai ξi − s χEi s ) tα−1. 2.6 Here, Ei 0, ξi , and χEi denotes the characteristic function of the set Ei for i 1, 2, . . . , m − 2. Proof. Lemma 2.3 yields u t −Iα 0 y t c1tα−1 c2tα−2 c3tα−3. 2.7 This, with the condition that u 0 0, gives c3 0. Furthermore, differentiating both sides of the expression of u t with respect to t, we obtain u′ t −Iα−1 0 y t c1 α − 1 tα−2 c2 α − 2 tα−3. 2.8 Now, by the conditions that u′ 0 0 and u′ 1 ∑m−2 i 1 aiu ′ ξi , we get that c2 0 and c1 1 α − 1 1 − ρ ( Iα−1 0 y 1 − m−2 ∑ i 1 aiI α−1 0 y ξi ) . 2.9 Abstract and Applied Analysis 5 Hence, u t − 1 Γ α ∫ t 0 t − s α−1y s ds t α−1 Γ α ∫1 0 1 − s α−2y s ds tα−1 ( 1 − ρΓ α ( ρ ∫1 0 1 − s α−2y s ds − m−2 ∑ i 1 ai ∫ ξi 0 ξi − s α−2y s ds )and Applied Analysis 5 Hence, u t − 1 Γ α ∫ t 0 t − s α−1y s ds t α−1 Γ α ∫1 0 1 − s α−2y s ds tα−1 ( 1 − ρΓ α ( ρ ∫1 0 1 − s α−2y s ds − m−2 ∑ i 1 ai ∫ ξi 0 ξi − s α−2y s ds ) − 1 Γ α ∫ t 0 t − s α−1y s ds t α−1 Γ α ∫1 0 1 − s α−2y s ds tα−1 ( 1 − ρΓ α ( ρ ∫1 0 1 − s α−2y s ds − ∫1 0 m−2 ∑ i 1 ai ξi − s χEi s y s ds ) ∫1 0 G t, s y s ds. 2.10 The proof is complete. The functions Gi t, s i 1, 2 have important properties as follows. Lemma 2.5. Assume (H2) holds, then Gi 1, s Gi t, s q t Gi 1, s 0 for all t, s ∈ 0, 1 and i 1, 2, where q t tα−1. Proof. The asserted relation for i 1 is a direct result of Lemma 2.8 in 14 . In addition, the case for i 2 follows from a direct application of the definition of G2 t, s and the fact that G2 t, s 1 ( 1 − ρΓ α ( m−2 ∑ i 1 aiξ α−2 i 1 − s α−2 − m−2 ∑ i 1 aiξ α−2 i ( 1 − s ξi )α−2 χEi s ) tα−1 1 ( 1 − ρΓ α ( m−2 ∑ i 1 aiξ α−2 i 1 − s α−2 − m−2 ∑ i 1 aiξ α−2 i 1 − s α−2 ) tα−1


Introduction
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order.It has been applied to almost every field of science, engineering, and mathematics in the last three decades 1-5 .But the investigation of the theory of fractional differential equations has only been started quite recently.
Among all the researches on the theory of the fractional differential equations, the study of the boundary value problems for fractional differential equations recently has attracted a great deal of attention from many researchers.And some results have been obtained on the existence of solutions or positive solutions of the boundary value problems for some specific fractional differential equations 6-11 .
More specifically, Bai 12 discussed the existence of positive solutions for the boundary value problem BVP for short D α 0 u t f t, u t 0, 0 < t < 1, 1 < α 2, u 0 0, u 1 βu η , 1.1 where 0 < βη α−1 < 1, 0 < η < 1, D α 0 is the standard Riemann-Liouville fractional derivative.Some existence results of at least one positive solution for the above-mentioned BVP are obtained by the use of fixed point index theory.
In 13 , Salem considered the existence of Pseudosolutions for the nonlinear m-point BVP, D α x t q t f t, x t 0, 0 < t < 1, α ∈ n − 1, n , n 2, where x takes values in a reflexive Banach space E. Here, 0 < η , and x k denotes the kth Pseudoderivative of x while D α denotes the Pseudofractional differential operator of order α.In light of the fixed point theorem given by O'Regan, the criteria for the existence of at least one Pseudo solution for the m-point BVP are established.
Very recently, M. El-Shahed 14 considered the existence and nonexistence of positive solutions to the fractional differential equation subject to the boundary conditions Their analysis relies on Krasnoselskii's fixed point theorem.Goodrich 15 then considered the BVP for the higher-dimensional fractional differential equation as follows: and a Harnack-like inequality associated with the Green's function related to the above problem is obtained improving the results in 16 .
Motivated by the aforementioned results and techniques in coping with those boundary value problems of the fractional differential equations, we then turn to investigate the existence and multiplicity of positive solutions for the following BVP: where D α 0 is the standard Riemann-Liouville fractional derivative of order α.Here, by a positive solution of BVP 1.6 , we mean a function which is positive on 0, 1 and satisfies the equation and boundary conditions in 1.6 .
Throughout the paper, we will assume that the following conditions hold.
There is a vast literature concerning the multipoint BVPs for the integer-order differential equations.An important recent paper was given by Webb and Infante 17 ; and they established a new unified method for the existence of multiple positive solutions to a large number of nonlinear nonlocal BVPs for the integer-order differential equations.While in the setting of the fractional-order derivatives, as far as we know, the existence of positive solutions for the multipoint BVP 1.6 has not been discussed in the literature.
The rest of the paper is organized as follows.Section 2 preliminarily provides some definitions and lemmas which are crucial to the following discussion.In Section 3, we obtain the existence and multiplicity results of positive solutions for the BVP 1.6 by means of some fixed point theorems.Finally, we give a concrete example to illustrate the possible application of our analytical results.

Preliminaries
In this section, we preliminarily provide some definitions and lemmas which are useful in the following discussion.Definition 2.1 see 3 .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 see 3 .The standard Riemann-Liouville fractional derivative of order α > 0 of a continuous function y : 0, ∞ → R is given by where n α 1 provided the right side is pointwise defined on 0, ∞ .
Lemma 2.3 see 6 .Assume that u ∈ C 0, 1 ∩ L 0, 1 with a fractional derivative of order α > 0, then for some c i ∈ R, i 0, 1, . . ., N, where N is the smallest integer greater than or equal to α.
By Lemma 2.3, we next present an integral presentation of the solution for the BVP of the linearized equation associated with the BVP 1.6 .
has a unique solution where the Green function G is given by G t, s G 1 t, s G 2 t, s , where

2.6
Here, E i 0, ξ i , and χ E i denotes the characteristic function of the set E i for i 1, 2, . . ., m − 2.
Proof.Lemma 2.3 yields This, with the condition that u 0 0, gives c 3 0. Furthermore, differentiating both sides of the expression of u t with respect to t, we obtain Now, by the conditions that u 0 0 and u 1 m−2 i 1 a i u ξ i , we get that c 2 0 and Hence,

2.10
The proof is complete.
The functions G i t, s i 1, 2 have important properties as follows.
Proof.The asserted relation for i 1 is a direct result of Lemma 2.8 in 14 .In addition, the case for i 2 follows from a direct application of the definition of G 2 t, s and the fact that holds for t, s ∈ 0, 1 .The proof is complete.
Remark 2.6.The definition of G t, s and Lemma 2.5 yield

2.12
Remark 2.7.It is necessary to mention that Bai and L ü 6 showed that their Green's function did not satisfy a classical Harnack-like inequality HLI for the homogenous two-point BVP of fractional differential equation with order α in 1, 2 .They proved that G t, s > γ s G s, s , where γ s → 0 as s → 0 , which is a challenge for our seeking positive solutions.
On the other hand, Goodrich 15 , for the homogenous BVP of fractional differential equation with order α in n − 1, n n > 3 , established a HLI: min t∈ 1/2,1 G t, s > γG 1, s , where γ is a positive constant.Using this inequality, the author obtained sufficient conditions on the existence of positive solutions.In Remark 2.6, we get a generalized HLI contrasting to the result in 15 .In fact, for a constant θ ∈ 0, 1 , it follows from Remark 2.6 that min t∈ θ,1 G t, s > θ α−1 G 1, s .
From the above discussion, we may infer that it is closely related to both the order of fractional differential equations and boundary conditions that whether or not the Green's function satisfies a traditional HLI, which needs more detailed and rigorous investigations.
Next, we introduce some fixed point theorems which will be adopted to prove the main results in the following section.

Main Results
In order to apply the fixed point theorems to the BVP 1.6 , we first import some notations and operator.Let B C 0, 1 be the classical Banach space with the norm u sup t∈ 0,1 |u t |.Furthermore, define the cone P by P u ∈ B | u t q t u for t ∈ 0, 1 .

3.1
Notice that u u 1 for each u ∈ P. For a positive number r, define the function space Ω r by Ω r {u ∈ C 0, 1 : u < r}.

3.2
Also define the operator A : P → B by Au t 1 0 G t, s f s, u s ds.

3.3
Next, we show some properties of the operator A.

Lemma 3.1. If (H1)-(H2) hold, then A P ⊂ P.
Proof.From the definition of the operator A and Remark 2.6, it follows that, for u ∈ P, Au t q t 1 0 G 1, s f s, u s ds q t max t∈ 0,1 1 0 G t, s f s, u s ds q t Au ,

3.4
Therefore, A P ⊂ P.This completes the proof.
Proof.Lemma 3.1 implies that A Ω r ∩ P ⊂ P.Moreover, the uniform continuity of the function f t, u on the compact set 0, 1 × 0, r yields that the operator A : Ω r ∩ P → P is continuous.We now show that A Ω r ∩ P is bounded.To this end, let l max{f t, u : 0 t 1, 0 u r}, then, for u ∈ Ω r ∩ P, it follows from Remark 2.6 that which implies that A Ω r ∩ P is bounded.

3.6
Now, using the fact that the functions t α−1 and t α are uniformly continuous on 0, 1 , we conclude that A Ω r ∩ P is an equicontinuous set on 0, 1 .It follows from the Arzelà-Ascoli Theorem that A Ω r ∩P is a relatively compact set.As a consequence, we complete the whole proof.
Lemma 3.3.Assume that (H1)-(H2) hold, then u ∈ C 0, 1 is a solution of the BVP 1.6 if and only if it is a fixed point of A in P.
Proof.If u ∈ P and Au u, then Thus, and u 0 0. Furthermore, differentiation of 3.7 with respect to t produces 3.9 This yields u 0 0 and Therefore, u is a positive solution of the BVP 1.6 .
On the other hand, if u is a positive solution of the BVP 1.6 , then Lemma 2.4 implies Au u.Moreover, in view of the proof of Lemma 3.1, we also get u t q t u for t ∈ 0, 1 .Hence, u is a fixed point of A in P. We consequently complete the proof.
In the following, fix η in 0,1 and set

3.11
We now present two main results on the existence of positive solutions for the BVP 1.6 .
Theorem 3.4.Assume that (H1)-(H2) hold.In addition, and suppose that one of the following two conditions holds: then the BVP 1.6 has at least one positive solution.
Proof.Notice that Lemma 3.2 guarantees that the operator A : Ω r ∩ P → P is completely continuous.Now, assume that condition H3 holds.Since lim u → 0 min t∈ 0,1 f t, u /u ∞, there exists an r 1 > 0 such that where the constant ε 1 > 0 is chosen, so that Thus, f t, u t ε 1 u t for u ∈ P ∩ ∂Ω r 1 , t ∈ 0, 1 .

3.14
This, together with the definitions of A and Remark 2.6, implies that for any u ∈ P ∩ ∂Ω r 1 ,

3.19
Thus, the operator A satisfies condition B2 of Lemma 2.8.Consequently, the operator A has at least one fixed point u ∈ P ∩ Ω 2 \ Ω 1 , which is one positive solution of the BVP 1.6 .Next, we suppose that H4 holds.The proof is similar to that of the case in which assumption H3 holds and will only be sketched here.Select two positive constants ε 3 and ε 4 with ε 3 1 0 G 1, s ds < 1 and ε 4 η α−1 1 η G 1, s ds > 1, respectively.Then, there exist two positive numbers r 3 and l 3 , such that f t, u ε 3 u for t ∈ 0, 1 , 0 u r 3 , 3.20 f t, u ε 4 u for t ∈ 0, 1 , u l 3 .

3.21
It follows from Remark 2.6 and 3.20 that for u ∈ P ∩ ∂Ω r 3 ,

Au
Au

3.25
Thus, Au u for u ∈ ∂Ω r 4 ∩ P. Hence, the operator A satisfies condition B1 in Lemma 2.8.As a consequence, the operator A has at least one fixed point u ∈ P ∩ Ω 4 \ Ω 3 .This means that the BVP 1.6 has at least one positive solution u.We complete the whole proof.

3.27
This, with Lemma 3.2, clearly manifests that the operator A : P c → P c is completely continuous.In a similar argument, if u ∈ P a , then assumption H5 yields f t, u t < La, 0 t 1.Therefore, condition C2 of Lemma 2.10 is satisfied.
Moreover, the set {x ∈ P θ, b, c | θ x > b} is not empty, since the constant function u t ≡ b c /2 is contained in the set {x ∈ P θ, b, c | θ x > b}.Now, let u ∈ P θ, b, c , then 0 u t c for t ∈ 0, 1 and b min η t 1 u t .Thus, we obtain b u t c, η t 1.

3.31
These fixed points are positive solutions for the BVP 1.6 .The proof is complete.

13 Remark 2 . 11 .
, and P θ, b, d {x ∈ P | b θ x , x d}.Suppose that A : P c → P c is completely continuous and there exist constants 0 < a < b < d c such that C1 {x ∈ P θ, b, d | θ x > b} / ∅ and θ Ax > b for x ∈ P θ, b, d , C2 Ax < a for x ∈ P a , C3 θ Ax > b for x ∈ P θ, b, c with Ax > d, then A has at least three fixed points x 1 , x 2 , and x 3 such that x 1 < a, b < θ x 2 , a < x 3 with θ x 3 < b. 2.If there holds d c, then condition C1 of Lemma 2.10 implies condition C3 of Lemma 2.10.
Lemma 2.8 see 18 .Let B be a Banach space, P ⊆ B a cone, and Ω 1 , Ω 2 two bounded open balls of B centered at the origin with Ω 1 ⊂ Ω 2 .Suppose that A : P ∩ Ω 2 \ Ω 1 → P is a completely continuous operator such that either B1 Ax x , x ∈ P ∩ ∂Ω 1 and Ax x , x ∈ P ∩ ∂Ω 2 or B2 Ax x , x ∈ P ∩ ∂Ω 1 and Ax x , x ∈ P ∩ ∂Ω 2 Holds, then A has a fixed point in P ∩ Ω 2 \ Ω 1 .Definition 2.9.The map θ is said to be a nonnegative continuous concave functional on a cone P of a real Banach space B provided that θ : P → 0, ∞ is continuous and θ tx 1 − t y tθ x 1 − t θ y for all x, y ∈ P and 0 t 1. Lemma 2.10 see 19 .Let P be a cone in a real Banach space B, P c {x ∈ P | x < c}, θ a nonnegative continuous concave functional on P such that θ x x , for all x ∈ P c Next, we intend to verify that all the conditions in Lemma 2.10 hold with respect to the operator A. Lemma 2.10 involves parameters a, b, c, and d with 0 < a < b < d c.Now, let d c, then by Remark 2.11, it is sufficient to verify that the conditions C1 and C2 in Lemma 2.10 hold.To this end, let u ∈ P c , then u c.This together with assumption H7 implies that for 0 t 1, f t, u t Lc.This relation and Remark 2.6 yield Accordingly, the validity of condition C1 in Lemma 2.10 is verified.Consequently, by virtue of Lemma 2.10 and Remark 2.11, the operator A has at least three fixed points u 1 , u 2 , and u 3 satisfying