Skip to main content

Multi-point boundary value problems for a class of Riemann-Liouville fractional differential equations

Abstract

In this paper, we shall study the existence and uniqueness of solutions for the multi-point boundary value problem of fractional differential equations D 0 + α u(t)+f(t,u(t))=0, 0<t<1, 2<α3, with boundary conditions u(0)=0, D 0 + β u(0)=0, D 0 + β u(1)= i = 1 m 2 b i D 0 + β u( ξ i ), 1β2, involving Riemann-Liouville fractional derivatives D 0 + α and D 0 + β . We use the nonlinear alternative of Leray-Schauder and the Banach contraction mapping principle to obtain the existence and uniqueness of solutions. Some examples are given to show the applicability of our main results.

MSC:34A08, 34K10.

1 Introduction

Fractional calculus is the study and application of arbitrary order differential and integral theory; see [15]. It is consistent with integer order calculus and a natural extension of the integer order calculus. Fractional differential equations are developed accompanied by fractional calculus. In recent years, with the wide applications of fractional calculus in the fields of physical, mechanical, biological, ecological, engineering, etc., the theory of fractional calculus has been paid more and more attention. Especially the study of fractional differential equations as abstracted from practical problems attracts much attention of many mathematicians.

Boundary value problems for fractional differential equations belong to the important issues for the theory of fractional differential equations. A lot of papers focused on two-point boundary value problems of fractional ordinary differential equations [616], boundary value problems of fractional difference equations [17, 18], and problems of fractional functional differential equations [1925].

However, the results dealing with multi-point boundary value problems of fractional differential equations are relatively scarce [2631].

In 2010, Li et al. [26] considered the existence and uniqueness for nonlinear fractional differential equation of the type

D 0 + α u(t)+f ( t , u ( t ) ) =0,0<t<1,1<α2,

where D 0 + α is the standard Riemann-Liouville fractional order derivative, subject to the boundary conditions

u(0)=0, D 0 + β u(1)=a D 0 + β u(ξ),0β1.

They obtained the existence and multiplicity results of positive solutions by using some fixed point theorems.

In 2011, Yang et al. [27] discussed the existence and uniqueness for a multi-point boundary value problem of the fractional differential equation

D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , 1 < α 2 , u ( 0 ) = 0 , D 0 + β u ( 1 ) = i = 1 m 2 b i D 0 + β u ( ξ i ) , 0 β 1 ,

where D 0 + α and D 0 + β are the Riemann-Liouville fractional derivatives. By fixed point theorem, they obtained the existence and uniqueness results.

In the previous related studies, scholars mostly used a fixed point theorem in cones and the Schauder fixed point theorem to solve some classes of boundary value problems. On the other hand, the study of these classes of problems has been only limited to the low order.

Motivated by their excellent results and the methods, in this paper, we investigate the existence and uniqueness for the multi-point fractional differential equation

D 0 + α u(t)+f ( t , u ( t ) ) =0,0<t<1,
(1.1)
u(0)=0, D 0 + β u(0)=0, D 0 + β u(1)= i = 1 m 2 b i D 0 + β u( ξ i ),
(1.2)

where D 0 + α and D 0 + β are the Riemann-Liouville fractional derivatives, 2<α3, 1β2 with αβ+1, m N + with m>2, 0< ξ 1 < ξ 2 << ξ m 2 <1.

To the best of our knowledge, no one has studied the existence of positive solutions for boundary value problems (1.1) and (1.2). Our main results of this paper are in extending the results in [27] from low order to high order case. Our problem allows the boundary condition to depend on the lower fractional derivative D 0 + β , which leads to extra difficulties. In particular, the condition D 0 + β u(0)=0 involves not only the properties of the function u(t) at zero but also the slope of tangent which pass through zero if β=1. Key tools in finding our main results are the nonlinear alternative of the Leray-Schauder and the Banach contraction mapping principle.

The plan of this paper is as follows. In Section 2, we shall give some definitions and lemmas to prove our main results. In Section 3, we establish the existence and uniqueness of solutions to multi-point boundary value problems (1.1) and (1.2) by the Banach contraction mapping principle, and we investigate the existence of solutions for (1.1) and (1.2) by the nonlinear alternative of Leray-Schauder. In Section 4, examples are presented to illustrate the main results.

In order to facilitate our study, we make the following assumptions:

(H1) f:[0,1]×[0,)[0,) is a continuous function;

(H2) b i 0 (i=1,2,,m2), i = 1 m 2 b i 2 0, and A= i = 1 m 2 b i ξ i α β 1 <1.

2 Preliminaries

For the convenience of the reader, we present here some necessary definitions and lemmas from the fractional calculus theory.

Definition 2.1 ([4])

The fractional integral of order α (α>0) of a function f:(0,+)R is given by

I 0 + α f(t)= 1 Γ ( α ) 0 t f ( s ) ( t s ) 1 α ds,

where Γ() is the gamma function, provided that the right side is point-wise defined on (0,+).

Definition 2.2 ([4])

The Riemann-Liouville fractional derivative of order α>0 of a continuous function f:(0,+)R is given by

D 0 + α f(t)= 1 Γ ( n α ) ( d d t ) n 0 t ( t s ) n α 1 f(s)ds,

where Γ() is the gamma function, provided that the right side is point-wise defined on (0,+) and n=[α]+1, [α] stands for the largest integer less than α.

Lemma 2.1 ([4])

Let α>1, β>0 and t>0. Then

D 0 + β t α = Γ ( α + 1 ) Γ ( α β + 1 ) t α β .

Lemma 2.2 ([4])

Assume that u(t)J(0,1)L(0,1), D 0 + α uJ(0,1)L(0,1) with the Riemann-Liouville fractional derivative of order α>0, then

I 0 + α D 0 + α u(t)=u(t)+ c 1 t α 1 + c 2 t α 2 ++ c N t α N ,

where c i R, i=1,2,,N, and N is the smallest integer greater than or equal to α.

Lemma 2.3 For Riemann-Liouville fractional derivatives, we have

D 0 + β 0 t ( t s ) α 1 f ( s , u ( s ) ) ds= Γ ( α ) Γ ( α β ) 0 t ( t s ) α β 1 f ( s , u ( s ) ) ds,

where fC[0,1], α, β are two constants with α>β0.

Proof From

D 0 + α I 0 + α f(t)=f(t), I 0 + α I 0 + β f(t)= I 0 + α + β f(t),

we get

D 0 + β 0 t ( t s ) α 1 f ( s , u ( s ) ) d s = D 0 + β Γ ( α ) 1 Γ ( α ) 0 t ( t s ) α 1 f ( s , u ( s ) ) d s = D 0 + β Γ ( α ) I 0 + α f ( t , u ( t ) ) = Γ ( α ) D 0 + β I 0 + α f ( t , u ( t ) ) = Γ ( α ) D 0 + β I 0 + β I 0 + α β f ( t , u ( t ) ) = Γ ( α ) I 0 + α β f ( t , u ( t ) ) = Γ ( α ) 1 Γ ( α β ) 0 t ( t s ) α β 1 f ( s , u ( s ) ) d s .

Then we obtain the result. The proof is complete. □

The following lemma is fundamental in proofs of our main results.

Lemma 2.4 ([32])

Let E be a Banach space with CE closed and convex. Assume U is a relatively open subset of C with 0U and T: U ¯ E is a completely continuous operator, T( U ¯ ) is bounded. Then either

(c1) T has a fixed point in U ¯ ; or

(c2) there exist a uU and λ(0,1) with u=λTu.

3 Main results

For convenience, assume E=C[0,1] is a Banach space with the maximum norm x= max 0 t 1 |x(t)| for xC[0,1]. Let r>0 and define X r ={u:uC[0,1],u<r}, which is the subset of C[0,1]. Let M=max{|f(t,u)|:(t,u)[0,1]×[r,r]}.

Lemma 3.1 Let (H1) and (H2) hold. Then the boundary value problem of the following fractional differential equation:

D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , 2 < α 3 , u ( 0 ) = 0 , D 0 + β u ( 0 ) = 0 , D 0 + β u ( 1 ) = i = 1 m 2 b i D 0 + β u ( ξ i ) , 1 β 2 ,

has a unique solution:

u ( t ) = 0 t 1 Γ ( α ) ( t s ) α 1 f ( s , u ( s ) ) d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 t α 1 f ( s , u ( s ) ) d s 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 t α 1 f ( s , u ( s ) ) d s .

Proof By Definition 2.1 and Lemma 2.2, we get

u(t)= 0 t 1 Γ ( α ) ( t s ) α 1 f ( s , u ( s ) ) ds+ c 1 t α 1 + c 2 t α 2 + c 3 t α 3

is the general solution of equation (1.1). By boundary condition u(0)=0, we find that

c 3 =0.

In view of Lemma 2.3 and D 0 + β u(0)=0, we have

D 0 + β u ( t ) = D 0 + β ( 0 t 1 Γ ( α ) ( t s ) α 1 f ( s , u ( s ) ) d s ) + c 1 Γ ( α ) Γ ( α β ) t α β 1 + c 2 Γ ( α 1 ) Γ ( α β 1 ) t α β 2 .

For α(2,3], β[1,2] and αβ+1, we have αβ2[1,0]. Thus c 2 =0. By

D 0 + β u(1)= i = 1 m 2 b i D 0 + β u( ξ i ),

we get

c 1 = 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 f ( s , u ( s ) ) d s 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 f ( s , u ( s ) ) d s .

Then the boundary value problem has a unique solution

u ( t ) = 0 t 1 Γ ( α ) ( t s ) α 1 f ( s , u ( s ) ) d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 t α 1 f ( s , u ( s ) ) d s 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 t α 1 f ( s , u ( s ) ) d s .

The proof is completed. □

Set f(t,u(t))=g(t) in Lemma 3.1. Since f:[0,1]×[0,+)[0,+) is a continuous function, we deduce that function u is a solution of the boundary value problem (1.1) and (1.2) if and only if it satisfies

u ( t ) = 0 t 1 Γ ( α ) ( t s ) α 1 f ( s , u ( s ) ) d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 t α 1 f ( s , u ( s ) ) d s 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 t α 1 f ( s , u ( s ) ) d s .

Let T: X ¯ r E be the operator defined by

T u ( t ) = 0 t 1 Γ ( α ) ( t s ) α 1 f ( s , u ( s ) ) d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 t α 1 f ( s , u ( s ) ) d s 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 t α 1 f ( s , u ( s ) ) d s .

Lemma 3.2 T: X ¯ r E is a completely continuous operator.

Proof For f(t,u) continuous, it is easy to see that T: X ¯ r E is continuous.

For 2<α3, 1β2 and X ¯ r is bounded, then for any u X ¯ r and t[0,1],

| T u ( t ) | | 0 t 1 Γ ( α ) ( t s ) α 1 f ( s , u ( s ) ) d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 t α 1 f ( s , u ( s ) ) d s 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 t α 1 f ( s , u ( s ) ) d s | M Γ ( α ) 0 t ( t s ) α 1 d s + M t α 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 d s + M t α 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 d s M ( 1 A ) Γ ( α ) ( 1 A α + 1 α β + 1 α β i = 1 m 2 b i ξ i α β ) .

Thus

Tu M ( 1 A ) Γ ( α ) ( 1 A α + 1 α β + 1 α β i = 1 m 2 b i ξ i α β ) for all u X ¯ r .

Hence {Tu,u X ¯ r } is bounded.

On the other hand, we will show that for any given ε>0, there exists

δ=min { 1 , ( ε Γ ( α + 1 ) ( 1 A ) ( α β ) M ( ( 1 A ) ( α β ) + α + α i = 1 m 2 b i ξ i α β ) 2 α ) 1 α 1 } ,

for any u X ¯ r , t 1 , t 2 [0,1], with 0< t 2 t 1 <δ, we get

| T u ( t 2 ) T u ( t 1 ) | <ε.

Thus T: X ¯ r is completely continuous.

In fact, for any u X ¯ r , t 1 , t 2 [0,1] with t 1 < t 2 , we have

| T u ( t 2 ) T u ( t 1 ) | | 0 t 2 1 Γ ( α ) ( t 2 s ) α 1 f ( s , u ( s ) ) d s 0 t 1 1 Γ ( α ) ( t 1 s ) α 1 f ( s , u ( s ) ) d s | + | t 2 α 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 f ( s , u ( s ) ) d s t 1 α 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 f ( s , u ( s ) ) d s | + | t 2 α 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 f ( s , u ( s ) ) d s t 1 α 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 f ( s , u ( s ) ) d s | M | 0 t 1 ( t 2 s ) α 1 ( t 1 s ) α 1 Γ ( α ) d s + t 1 t 2 ( t 2 s ) α 1 Γ ( α ) d s | + M | t 2 α 1 t 1 α 1 | ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 d s + M i = 1 m 2 b i | t 2 α 1 t 1 α 1 | ( 1 A ) Γ ( α ) 0 ξ i ( ξ i s ) α β 1 d s = M ( t 2 α t 1 α ) Γ ( α + 1 ) + M | t 2 α 1 t 1 α 1 | ( 1 A ) Γ ( α ) ( 1 α β + 1 α β i = 1 m 2 b i ξ i α β ) = M ( t 2 α t 1 α ) Γ ( α + 1 ) + M ( 1 + i = 1 m 2 b i ξ i α β ) ( α β ) ( 1 A ) Γ ( α ) | t 2 α 1 t 1 α 1 | .
  1. (1)

    If δ t 1 < t 2 <1, by the mean value theorem, we have

    t 2 α t 1 α α( t 2 t 1 ), t 2 α 1 t 1 α 1 (α1)( t 2 t 1 )α( t 2 t 1 ).
  2. (2)

    If 0 t 1 <δ, t 2 <2δ, then

    t 2 α t 1 α t 2 α < ( 2 δ ) α , t 2 α 1 t 1 α 1 t 2 α 1 < ( 2 δ ) α 1 .

So

max { t 2 α 1 t 1 α 1 , t 2 α t 1 α } 2 α δ α 1 .

Thus

| T u ( t 2 ) T u ( t 1 ) | < M ( ( 1 A ) ( α β ) + α + α i = 1 m 2 b i ξ i α β ) 2 α Γ ( α + 1 ) ( 1 A ) ( α β ) δ α 1 <ε.

By the Arzela-Ascoli theorem, we conclude that T: X ¯ r E is a completely continuous operator. □

Theorem 3.1 Assume that there exists a constant k>0 such that

| f ( t , u ) f ( t , v ) | k|uv|,

where u,vR. Then the boundary value problem (1.1)-(1.2) has a unique solution on [0,1], if

k ( ( 1 A ) ( α β ) + α + α i = 1 m 2 b i ξ i α β ) ( 1 A ) ( α β ) Γ ( α + 1 ) <1

is satisfied.

Proof By the definition of T, we have

| ( T u ) ( t ) ( T v ) ( t ) | 1 Γ ( α ) 0 t ( t s ) α 1 | f ( s , u ( s ) ) f ( s , v ( s ) ) | d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 t α 1 | f ( s , u ( s ) ) f ( s , v ( s ) ) | d s + 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 t α 1 | f ( s , u ( s ) ) f ( s , v ( s ) ) | d s k u v ( 1 Γ ( α ) 0 t ( t s ) α 1 d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 d s + 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 d s ) = k [ ( 1 A ) ( α β ) + α + α i = 1 m 2 b i ξ i α β ] ( 1 A ) ( α β ) Γ ( α + 1 ) u v .

Hence, by the Banach contraction mapping principle, boundary value problems (1.1) and (1.2) have a unique solution on [0,1]. The proof is completed. □

Now we study the existence of solutions for the boundary value problem of (1.1)-(1.2) by the nonlinear alternative of Leray-Schauder.

Theorem 3.2 Suppose that the following condition are satisfied:

(a1) There exist a nonnegative function gC[0,1] such that g>0 on the subset of [0,1], and a nondecreasing function h:[0,)[0,) such that |f(t,u)|g(t)h(u), where (t,u)[0,1]×R.

(a2)

sup r ( 0 , ) r g 0 h ( r ) >1,

where

g 0 = 1 Γ ( α ) 0 1 ( 1 s ) α 1 g ( s ) d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 g ( s ) d s + 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 g ( s ) d s .

Then boundary value problem (1.1)-(1.2) has at least one solution.

Proof In view of (a2) and by the definition of supremum, we can choose a constant r 0 (0,) such that

r 0 g 0 h ( r 0 ) >1.
(3.1)

By Lemma 3.2, we know that T: X ¯ r 0 E is completely continuous and T( X ¯ r 0 ) is bounded. Suppose (c2) in Lemma 2.4 holds, i.e. there exist a λ(0,1), u X r 0 such that

u=λTu.

Then

u ( t ) = λ ( 1 Γ ( α ) 0 t ( t s ) α 1 f ( s , u ( s ) ) d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 t α 1 f ( s , u ( s ) ) d s 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 t α 1 f ( s , u ( s ) ) d s ) .
(3.2)

In view of (a1), (a2), (3.2), and u= r 0 , we obtain

r 0 = u h ( r 0 ) Γ ( α ) 0 1 ( 1 s ) α 1 g ( s ) d s + h ( r 0 ) ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 g ( s ) d s + h ( r 0 ) ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 g ( s ) d s g 0 h ( r 0 ) .

Hence we get r 0 g 0 h ( r 0 ) 1, which is in contradiction with (3.1). Therefore, Lemma 2.4 guarantees that T has at least a fixed point u X ¯ r 0 . Then boundary value problem (1.1)-(1.2) has at least one solution. The proof is completed. □

4 Examples

In this section, we will present some examples to illustrate our main results.

Example 4.1 Consider the following boundary value problem:

D 0 + 5 2 u(t)+ u ( t ) 10 + sin 2 t+1=0,0<t<1,
(4.1)
u(0)=0, D 0 + 3 2 u(0)=0, D 0 + 3 2 u(1)= i = 1 2 b i D 0 + 3 2 u( ξ i ),
(4.2)

where b 1 = 1 10 , b 2 = 1 5 , ξ 1 = 1 10 , ξ 2 = 1 5 .

Here

α= 5 2 ,β= 3 2 ,f(t,u)= u 10 + sin 2 t+1,for (t,u)[0,1]×[0,].

It is clear that |f(t,u)f(t,v)|= 1 10 |uv| and

k ( ( 1 A ) ( α β ) + α + α i = 1 m 2 b i ξ i α β ) ( 1 A ) ( α β ) Γ ( α + 1 ) = 1 10 ( ( 1 3 10 ) ( 5 2 3 2 ) + 5 2 + 5 2 × 1 20 ) ( 1 3 10 ) ( 5 2 3 2 ) Γ ( 5 2 ) × 5 2 = 91 525 π < 1 .

By Theorem 3.1, we see that boundary value problem (4.1)-(4.2) has a unique solution.

Example 4.2 Consider the following boundary value problem:

D 0 + 5 2 u(t)+t u 2 (t)sin u 2 (t)=0,0<t<1,
(4.3)
u(0)=0, D 0 + 3 2 u(0)=0, D 0 + 3 2 u(1)= i = 1 m 2 b i D 0 + 3 2 u( ξ i ),
(4.4)

where b 1 = 1 10 , ξ 1 = 1 10 .

Here α= 5 2 , β= 3 2 . Set g(t)=t and h(u)= u 2 . It is easy to see that

| f ( t , u ) | = | t u 2 sin u 2 | t u 2 ,for (t,u)[0,1]×[0,).

By simply calculating, we get

g 0 = 1 Γ ( α ) 0 1 ( 1 s ) α 1 g ( s ) d s + 1 ( 1 A ) Γ ( α ) 0 1 ( 1 s ) α β 1 g ( s ) d s + 1 ( 1 A ) Γ ( α ) i = 1 m 2 b i 0 ξ i ( ξ i s ) α β 1 g ( s ) d s = 7 , 405 6 , 750 π 0.61892 .

Now

sup r ( 0 , ) r g 0 h ( r ) = sup r ( 0 , ) r 7 , 405 r 2 6 , 750 π =+.

Hence by Theorem 3.2, we obtain the result that boundary value problem (4.3)-(4.4) has at least a solution.

References

  1. 1.

    Oldham K, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.

    Google Scholar 

  2. 2.

    Miller K, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York; 1993.

    Google Scholar 

  3. 3.

    Samko S, Kilbas A, Marichev O: Fractional Integral and Derivative, Theory and Applications. Gordon & Breach, Switzerland; 1993.

    Google Scholar 

  4. 4.

    Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.

    Google Scholar 

  5. 5.

    Kilbas A, Srivastava H, Trujillo J: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.

    Google Scholar 

  6. 6.

    Agarwal R, Benchohra M, Hamani A: Boundary value problems for fractional differential equations. Georgian Math. J. 2009, 16: 401–411.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Agarwal R, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 2010, 59: 1095–1100. 10.1016/j.camwa.2009.05.010

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Zhang M, Sun S, Zhao Y, Yang D: Existence of positive solutions for boundary value problems of fractional differential equations. J. Jinan Univ. 2010, 24: 205–208.

    Google Scholar 

  9. 9.

    Zhao Y, Sun S, Han Z, Li Q: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(4):2086–2097. 10.1016/j.cnsns.2010.08.017

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Zhao Y, Sun S, Han Z, Li Q: Positive solutions to boundary value problems of nonlinear fractional differential equations. Abstr. Appl. Anal. 2011., 2011: Article ID 390543

    Google Scholar 

  11. 11.

    Zhao Y, Sun S, Han Z, Zhang M: Positive solutions for boundary value problems of nonlinear fractional differential equations. Appl. Math. Comput. 2011, 217: 6950–6958. 10.1016/j.amc.2011.01.103

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Feng W, Sun S, Han Z, Zhao Y: Existence of solutions for a singular system of nonlinear fractional differential equations. Comput. Math. Appl. 2011, 62(3):1370–1378. 10.1016/j.camwa.2011.03.076

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Pan Y, Han Z: Existence of solutions for a coupled system of boundary value problem of nonlinear fractional differential equations. In Proceedings of the 5th International Congress on Mathematical Biology. World Academic Press, Nanjing; 2011:109–114.

    Google Scholar 

  14. 14.

    Zhang X, Liu L, Wu Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 2012, 55: 1263–1274. 10.1016/j.mcm.2011.10.006

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006, 36: 1072–6691.

    MathSciNet  Google Scholar 

  16. 16.

    Shi A: Upper and lower solutions method and a fractional differential equation boundary value problem. Electron. J. Qual. Theory Differ. Equ. 2009, 30: 1–13.

    Article  MathSciNet  Google Scholar 

  17. 17.

    Pan Y, Han Z, Sun S, Zhao Y: The existence of solutions to a system of discrete fractional boundary value problems. Abstr. Appl. Anal. 2012., 2012: Article ID 707631

    Google Scholar 

  18. 18.

    Pan Y, Han Z, Sun S, Huang Z: The existence and uniqueness of solutions to boundary value problems of fractional difference equations. Math. Sci. 2012, 6(7):1–10.

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Li Y, Sun S, Han Z, Lu H: The existence of positive solutions for boundary value problem of the fractional Sturm-Liouville functional differential equation. Abstr. Appl. Anal. 2013., 2013: Article ID 301560

    Google Scholar 

  20. 20.

    Zhou Y, Tian Y, He Y: Floquet boundary value problems of fractional functional differential equations. Electron. J. Qual. Theory Differ. Equ. 2010, 50: 1–13.

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Li Y, Sun S, Yang D, Han Z: Three-point boundary value problems of fractional functional differential equations with delay. Bound. Value Probl. 2013., 2013: Article ID 38

    Google Scholar 

  22. 22.

    Li X, Song L, Wei J: Positive solutions for boundary value problem of nonlinear fractional functional differential equations. Appl. Math. Comput. 2011, 217: 9278–9285. 10.1016/j.amc.2011.04.006

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Bai C: Existence of positive solutions for a functional fractional boundary value problem. Abstr. Appl. Anal. 2010., 2010: Article ID 127363

    Google Scholar 

  24. 24.

    Ouyang Z, Chen Y, Zou S: Existence of positive solutions to a boundary value problem for a delayed nonlinear fractional differential system. Bound. Value Probl. 2011., 2011: Article ID 475126

    Google Scholar 

  25. 25.

    Zhao Y, Chen H, Huang L: Existence of positive solutions for nonlinear fractional functional differential equation. Comput. Math. Appl. 2012, 64: 3456–3467. 10.1016/j.camwa.2012.01.081

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Li C, Luo X, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 2010, 59: 1363–1375. 10.1016/j.camwa.2009.06.029

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Yang J, Ma JC, Zhao S, Ge Y: Fractional multi-point boundary value problem of fractional differential equations. Math. Pract. Theory 2011, 41(11):188–194.

    MathSciNet  Google Scholar 

  28. 28.

    Rehman M, Khan R, Asif N: Three point boundary value problems for nonlinear fractional differential equations. Acta Math. Sci. 2011, 31(4):1337–1346. 10.1016/S0252-9602(11)60320-2

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Cernea A: On a multi-point boundary value problem for a fractional order differential inclusion. Arab J. Math. Sci. 2013, 19(1):73–83. 10.1016/j.ajmsc.2012.07.001

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Zhou W, Chu Y: Existence of solutions for fractional differential equations with multi-point boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 1142–1148. 10.1016/j.cnsns.2011.07.019

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Jiang W, Wang B, Wang Z: The existence of positive solutions for multi-point boundary value problems of fractional differential equations. Phys. Proc. 2012, 25: 958–964.

    Article  Google Scholar 

  32. 32.

    Zeidler E: Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems. Springer, Berlin; 1985.

    Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shurong Sun.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, B., Sun, S., Li, Y. et al. Multi-point boundary value problems for a class of Riemann-Liouville fractional differential equations. Adv Differ Equ 2014, 151 (2014). https://doi.org/10.1186/1687-1847-2014-151

Download citation

Keywords

  • fractional differential equations
  • multi-point boundary value problem
  • existence and uniqueness
  • fixed point theorem