Open Access

Existence of solutions for a coupled system of fractional differential equations at resonance

Boundary Value Problems20122012:98

https://doi.org/10.1186/1687-2770-2012-98

Received: 14 May 2012

Accepted: 22 August 2012

Published: 7 September 2012

Abstract

In this paper, by using the coincidence degree theory, we study the existence of solutions for a coupled system of fractional differential equations at resonance. A new result on the existence of solutions for a fractional boundary value problem is obtained.

MSC:34B15.

Keywords

fractional differential equationboundary value problemcoincidence degree theoryresonance

1 Introduction

In recent years, the fractional differential equations have received more and more attention. The fractional derivative has been occurring in many physical applications such as a non-Markovian diffusion process with memory [1], charge transport in amorphous semiconductors [2], propagations of mechanical waves in viscoelastic media [3], etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are also described by differential equations of fractional order (see [49]).

Recently, boundary value problems for fractional differential equations have been studied in many papers (see [1025]). Moreover, the existence of solutions to a coupled systems of fractional differential equations have been studied by many authors (see [2633]). But the existence of solutions for a coupled system of fractional differential equations at resonance are seldom considered. Motivated by all the works above, in this paper, we consider the following boundary value problem (BVP for short) for a coupled system of fractional differential equations given by
{ D 0 + α u ( t ) = f ( t , v ( t ) , v ( t ) ) , t ( 0 , 1 ) , D 0 + β v ( t ) = g ( t , u ( t ) , u ( t ) ) , t ( 0 , 1 ) , u ( 0 ) = v ( 0 ) = 0 , u ( 0 ) = u ( 1 ) , v ( 0 ) = v ( 1 ) ,
(1.1)

where D 0 + α , D 0 + β are the standard Caputo fractional derivatives, 1 < α 2 , 1 < β 2 and f , g : [ 0 , 1 ] × R 2 R is continuous.

The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions and lemmas. In Section 3, we establish a theorem on the existence of solutions for BVP (1.1) under nonlinear growth restriction of f and g, basing on the coincidence degree theory due to Mawhin (see [34]). Finally, in Section 4, an example is given to illustrate the main result.

2 Preliminaries

In this section, we will introduce some notations, definitions and preliminary facts which are used throughout this paper.

Let X and Y be real Banach spaces, and let L : dom L X Y be a Fredholm operator with index zero, and P : X X , Q : Y Y be projectors such that
Im P = Ker L , Ker Q = Im L , X = Ker L Ker P , Y = Im L Im Q .
It follows that
L | dom L Ker P : dom L Ker P Im L

is invertible. We denote the inverse by K P .

If Ω is an open bounded subset of X, and dom L Ω ¯ , the map N : X Y will be called L-compact on Ω ¯ if Q N ( Ω ¯ ) is bounded and K P ( I Q ) N : Ω ¯ X is compact, where I is an identity operator.

Lemma 2.1 [27]

Let L : dom L X Y be a Fredholm operator of index zero and N : X Y L-compact on Ω ¯ . Assume that the following conditions are satisfied:
  1. (1)

    L x λ N x for every ( x , λ ) [ ( dom L Ker L ) ] Ω × ( 0 , 1 ) ;

     
  2. (2)

    N x Im L for every x Ker L Ω ;

     
  3. (3)

    deg ( Q N | Ker L , Ker L Ω , 0 ) 0 , where Q : Y Y is a projection such that Im L = Ker Q .

     

Then the equation L x = N x has at least one solution in dom L Ω ¯ .

Definition 2.1 The Riemann-Liouville fractional integral operator of order α > 0 of a function x is given by
I 0 + α x ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 x ( s ) d s ,

provided that the right-hand side integral is pointwise defined on ( 0 , + ) .

Definition 2.2 The Riemann-Liouville fractional derivative of order α > 0 of a function x is given by
D 0 + α R x ( t ) = d n d t n I 0 + n α x ( t ) = 1 Γ ( n α ) d n d t n 0 t ( t s ) n α 1 x ( s ) d s ,

where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined on ( 0 , + ) .

Definition 2.3 The Caputo fractional derivative of order α > 0 of a function x is given by
D 0 + α x ( t ) = R D 0 + α [ x ( t ) k = 0 n 1 x ( k ) ( 0 ) k ! t k ] ,

where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined on ( 0 , + ) .

Lemma 2.2 [35]

Assume that x C ( 0 , 1 ) L ( 0 , 1 ) with a Caputo fractional derivative of order α > 0 that belongs to C ( 0 , 1 ) L ( 0 , 1 ) . Then
I 0 + α D 0 + α x ( t ) = x ( t ) + c 0 + c 1 t + c 2 t 2 + + c n 1 t n 1 ,

where c i R , i = 0 , 1 , 2 , , n 1 , here n is the smallest integer greater than or equal to α.

Lemma 2.3 [35]

Assume that α > 0 and x C [ 0 , 1 ] . Then
D 0 + α I 0 + α x ( t ) = x ( t ) .

In this paper, we denote X = C 1 [ 0 , 1 ] with the norm x X = max { x , x } and Y = C [ 0 , 1 ] with the norm y Y = y , where x = max t [ 0 , 1 ] | x ( t ) | . Then we denote X ¯ = X × X with the norm ( u , v ) X ¯ = max { u X , v X } and Y ¯ = Y × Y with the norm ( x , y ) Y ¯ = max { x Y , y Y } . Obviously, both X ¯ and Y ¯ are Banach spaces.

Define the operator L 1 : dom L X Y by
L 1 u = D 0 + α u ,
where
dom L 1 = { u X | D 0 + α u ( t ) Y , u ( 0 ) = 0 , u ( 0 ) = u ( 1 ) } .
Define the operator L 2 : dom L 2 X Y by
L 2 v = D 0 + β v ,
where
dom L 2 = { v X | D 0 + β v ( t ) Y , v ( 0 ) = 0 , v ( 0 ) = v ( 1 ) } .
Define the operator L : dom L X ¯ Y ¯ by
L ( u , v ) = ( L 1 u , L 2 v ) ,
(2.1)
where
dom L = { ( u , v ) X ¯ | u dom L 1 , v dom L 2 } .
Let N : X ¯ Y ¯ be the Nemytski operator
N ( u , v ) = ( N 1 v , N 2 u ) ,
where N 1 : Y X
N 1 v ( t ) = f ( t , v ( t ) , v ( t ) )
and N 2 : Y X
N 2 u ( t ) = g ( t , u ( t ) , u ( t ) ) .
Then BVP (1.1) is equivalent to the operator equation
L ( u , v ) = N ( u , v ) , ( u , v ) dom L .

3 Main result

In this section, a theorem on the existence of solutions for BVP (1.1) will be given.

Theorem 3.1 Let f , g : [ 0 , 1 ] × R 2 R be continuous. Assume that

( H 1 ) there exist nonnegative functions p i , q i , r i C [ 0 , 1 ] ( i = 1 , 2 ) with
Γ ( α ) Γ ( β ) 4 ( Q 1 + R 1 ) ( Q 2 + R 2 ) Γ ( α ) Γ ( β ) > 0
such that for all ( u , v ) R 2 , t [ 0 , 1 ]
| f ( t , u , v ) | p 1 ( t ) + q 1 ( t ) | u | + r 1 ( t ) | v | ,
and
| g ( t , u , v ) | p 2 ( t ) + q 2 ( t ) | u | + r 2 ( t ) | v | ,

where P i = p i , Q i = q i , R i = r i ( i = 1 , 2 );

( H 2 ) there exists a constant B > 0 such that for t [ 0 , 1 ] , | u | > B , v R either
u f ( t , u , v ) > 0 , u g ( t , u , v ) > 0 ,
or
u f ( t , u , v ) < 0 , u g ( t , u , v ) < 0 ;
( H 3 ) there exists a constant D > 0 such that for every c 1 , c 2 R satisfying min { c 1 , c 2 } > D either
c 1 N 1 ( c 2 t ) > 0 , c 2 N 2 ( c 1 t ) > 0 ,
or
c 1 N 1 ( c 2 t ) < 0 , c 2 N 2 ( c 1 t ) < 0 .

Then BVP (1.1) has at least one solution.

Now, we begin with some lemmas below.

Lemma 3.1 Let L be defined by (2.1), then
(3.1)
(3.2)
Proof By Lemma 2.2, L 1 u = D 0 + α u ( t ) = 0 has the solution
u ( t ) = c 0 + c 1 t , c 0 , c 1 R .
Combining it with the boundary value conditions of BVP (1.1), one has
Ker L 1 = { u X | u = c 1 t , c 1 R } .
For x Im L 1 , there exists u dom L 1 such that x = L 1 u Y . By Lemma 2.2, we have
u ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 x ( s ) d s + c 0 + c 1 t .
Then, we have
u ( t ) = 1 Γ ( α 1 ) 0 t ( t s ) α 2 x ( s ) d s + c 1 .
By the conditions of BVP (1.1), we can get that x satisfies
0 1 ( 1 s ) α 2 x ( s ) d s = 0 .
On the other hand, suppose x Y and satisfies 0 1 ( 1 s ) α 2 x ( s ) d s = 0 . Let u ( t ) = I 0 + α x ( t ) , then u dom L 1 . By Lemma 2.3, we have D 0 + α u ( t ) = x ( t ) so that x Im L 1 . Then we have
Im L 1 = { x Y | 0 1 ( 1 s ) α 2 x ( s ) d s = 0 } .
Similarly, we can get
Ker L 2 = { v X | v = c 2 t , c 2 t R } , Im L 2 = { y Y | 0 1 ( 1 s ) β 2 y ( s ) d s = 0 } .

Then, the proof is complete. □

Lemma 3.2 Let L be defined by (2.1), then L is a Fredholm operator of index zero, and the linear continuous projector operators P : X ¯ X ¯ and Q : Y ¯ Y ¯ can be defined as
P ( u , v ) = ( P 1 u , P 2 v ) = ( u ( 0 ) t , v ( 0 ) t ) , Q ( x , y ) = ( Q 1 x , Q 2 y ) = ( ( α 1 ) 0 1 ( 1 s ) α 2 x ( s ) d s , ( β 1 ) 0 1 ( 1 s ) β 2 y ( s ) d s ) .
Furthermore, the operator K P : Im L dom L Ker P can be written by
K P ( x , y ) = ( I 0 + α x ( t ) , I 0 + β y ( t ) ) .
Proof Obviously, Im P = Ker L and P 2 ( u , v ) = P ( u , v ) . It follows from ( u , v ) = ( ( u , v ) P ( u , v ) ) + P ( u , v ) that X ¯ = Ker P + Ker L . By simple calculation, we can get that Ker L Ker P = { ( 0 , 0 ) } . Then we get
X ¯ = Ker L Ker P .
For ( x , y ) Y ¯ , we have
Q 2 ( x , y ) = Q ( Q 1 x , Q 2 y ) = ( Q 1 2 x , Q 2 2 y ) .
By the definition of Q 1 , we can get
Q 1 2 x = Q 1 x ( α 1 ) 0 1 ( 1 s ) α 2 d s = Q 1 x .

Similar proof can show that Q 2 2 y = Q 2 y . Thus, we have Q 2 ( x , y ) = Q ( x , y ) .

Let ( x , y ) = ( ( x , y ) Q ( x , y ) ) + Q ( x , y ) , where ( x , y ) Q ( x , y ) Ker Q = Im L , Q ( x , y ) Im Q . It follows from Ker Q = Im L and Q 2 ( x , y ) = Q ( x , y ) that Im Q Im L = { ( 0 , 0 ) } . Then, we have
Y ¯ = Im L Im Q .
Thus
dim Ker L = dim Im Q = codim Im L .

This means that L is a Fredholm operator of index zero.

Now, we will prove that K P is the inverse of L | dom L Ker P . By Lemma 2.3, for ( x , y ) Im L , we have
L K P ( x , y ) = ( D 0 + α ( I 0 + α x ) , D 0 + β ( I 0 + β y ) ) = ( x , y ) .
(3.3)
Moreover, for ( u , v ) dom L Ker P , we have u ( 0 ) = v ( 0 ) = 0 and
K P L ( u , v ) = ( I 0 + α D 0 + α u ( t ) , I 0 + β D 0 + β v ( t ) ) = ( u ( t ) + c 0 + c 1 t , v ( t ) + d 0 + d 1 t ) , c 0 , c 1 , d 0 , d 1 R ,
which, together with u ( 0 ) = v ( 0 ) = 0 , yields that
K P L ( u , v ) = ( u , v ) .
(3.4)

Combining (3.3) with (3.4), we know that K P = ( L | dom L Ker P ) 1 . The proof is complete. □

Lemma 3.3 Assume Ω X ¯ is an open bounded subset such that dom L Ω ¯ , then N is L-compact on Ω ¯ .

Proof By the continuity of f and g, we can get that Q N ( Ω ¯ ) and K P ( I Q ) N ( Ω ¯ ) are bounded. So, in view of the Arzelá-Ascoli theorem, we need only prove that K P ( I Q ) N ( Ω ¯ ) X ¯ is equicontinuous.

From the continuity of f and g, there exists a constant A i > 0 , i = 1 , 2 , such that ( u , v ) Ω ¯
| ( I Q 1 ) N 1 v | A 1 , | ( I Q 2 ) N 2 u | A 2 .
Furthermore, for 0 t 1 < t 2 1 , ( u , v ) Ω ¯ , we have
| K P ( I Q ) N ( u ( t 2 ) , v ( t 2 ) ) ( K P ( I Q ) N ( u ( t 1 ) , v ( t 1 ) ) ) | = | ( I 0 + α ( I Q 1 ) N 1 v ( t 2 ) , I 0 + β ( I Q 2 ) N 2 u ( t 2 ) ) ( I 0 + α ( I Q 1 ) N 1 v ( t 1 ) , I 0 + β ( I Q 2 ) N 2 u ( t 1 ) ) | = | ( I 0 + α ( I Q 1 ) N 1 v ( t 2 ) I 0 + α ( I Q 1 ) N 1 v ( t 1 ) , I 0 + β ( I Q 2 ) N 2 u ( t 2 ) I 0 + β ( I Q 2 ) N 2 u ( t 1 ) ) | .
By
| I 0 + α ( I Q 1 ) N 1 v ( t 2 ) I 0 + α ( I Q 1 ) N 1 v ( t 1 ) | 1 Γ ( α ) | 0 t 2 ( t 2 s ) α 1 ( I Q 1 ) N 1 v ( s ) d s 0 t 1 ( t 1 s ) α 1 ( I Q 1 ) N 1 v ( s ) d s | A 1 Γ ( α ) [ 0 t 1 ( t 2 s ) α 1 ( t 1 s ) α 1 d s + t 1 t 2 ( t 2 s ) α 1 d s ] = A 1 Γ ( α + 1 ) ( t 2 α t 1 α )
and
| ( I 0 + α ( I Q 1 ) N 1 v ) ( t 2 ) ( I 0 + α ( I Q 1 ) N 1 v ) ( t 1 ) | = α 1 Γ ( α ) | 0 t 2 ( t 2 s ) α 2 ( I Q 1 ) N 1 v ( s ) d s 0 t 1 ( t 1 s ) α 2 ( I Q 1 ) N 1 v ( s ) d s | A 1 Γ ( α 1 ) [ 0 t 1 ( t 1 s ) α 2 ( t 2 s ) α 2 d s + t 1 t 2 ( t 2 s ) α 2 d s ] A 1 Γ ( α ) [ t 2 α 1 t 1 α 1 + 2 ( t 2 t 1 ) α 1 ] .
Similar proof can show that
| I 0 + β ( I Q 2 ) N 2 u ( t 2 ) I 0 + β ( I Q 2 ) N 2 u ( t 1 ) | A 2 Γ ( β + 1 ) ( t 2 β t 1 β ) , | ( I 0 + β ( I Q 2 ) N 2 u ) ( t 2 ) ( I 0 + β ( I Q 2 ) N 2 u ) ( t 1 ) | A 2 Γ ( β ) [ t 2 β 1 t 1 β 1 + 2 ( t 2 t 1 ) β 1 ] .

Since t α , t α 1 , t β and t β 1 are uniformly continuous on [ 0 , 1 ] , we can get that K P ( I Q ) N ( Ω ¯ ) X ¯ is equicontinuous.

Thus, we get that K P ( I Q ) N : Ω ¯ X ¯ is compact. The proof is complete. □

Lemma 3.4 Suppose ( H 1 ), ( H 2 ) hold, then the set
Ω 1 = { ( u , v ) dom L Ker L | L ( u , v ) = λ N ( u , v ) , λ ( 0 , 1 ) }

is bounded.

Proof Take ( u , v ) Ω 1 , then N ( u , v ) Im L . By (3.2), we have
0 1 ( 1 s ) α 2 f ( s , v ( s ) , v ( s ) ) d s = 0 , 0 1 ( 1 s ) β 2 g ( s , u ( s ) , u ( s ) ) d s = 0 .

Then, by the integral mean value theorem, there exist constants ξ , η ( 0 , 1 ) such that f ( ξ , v ( ξ ) , v ( ξ ) ) = 0 and g ( η , u ( η ) , u ( η ) ) = 0 . So, from ( H 2 ), we get | v ( ξ ) | B and | u ( η ) | B .

From ( u , v ) dom L , we get u ( 0 ) = v ( 0 ) = 0 , then
(3.5)
(3.6)
By L ( u , v ) = λ N ( u , v ) and ( u , v ) dom L , we have
u ( t ) = λ Γ ( α ) 0 t ( t s ) α 1 f ( s , v ( s ) , v ( s ) ) d s + u ( 0 ) t
and
v ( t ) = λ Γ ( β ) 0 t ( t s ) β 1 g ( s , u ( s ) , u ( s ) ) d s + v ( 0 ) t .
Then we get
u ( t ) = λ Γ ( α 1 ) 0 t ( t s ) α 2 f ( s , v ( s ) , v ( s ) ) d s + u ( 0 )
and
v ( t ) = λ Γ ( β 1 ) 0 t ( t s ) β 2 g ( s , u ( s ) , u ( s ) ) d s + v ( 0 ) .
Take t = η , we get
u ( η ) = λ Γ ( α 1 ) 0 η ( η s ) α 2 f ( s , v ( s ) , v ( s ) ) d s + u ( 0 ) .
Together with | u ( η ) | B , ( H 1 ) and (3.6), we have
| u ( 0 ) | | u ( η ) | + λ Γ ( α 1 ) 0 η ( η s ) α 2 | f ( s , v ( s ) , v ( s ) ) | d s B + 1 Γ ( α 1 ) 0 η ( η s ) α 2 [ p 1 ( s ) + q 1 ( s ) | v ( s ) | + r 1 ( s ) | v ( s ) | ] d s B + 1 Γ ( α 1 ) 0 η ( η s ) α 2 [ P 1 + Q 1 v + R 1 v ] d s B + 1 Γ ( α 1 ) 0 η ( η s ) α 2 [ P 1 + ( Q 1 + R 1 ) v ] d s B + 1 Γ ( α ) [ P 1 + ( Q 1 + R 1 ) v ] .
So, we have
u 1 Γ ( α 1 ) 0 t ( t s ) α 2 | f ( s , v ( s ) , v ( s ) ) | d s + | u ( 0 ) | 1 Γ ( α 1 ) 0 t ( t s ) α 2 [ p 1 ( s ) + q 1 ( s ) | v ( s ) | + r 1 ( s ) | v ( s ) | ] d s + | u ( 0 ) | 1 Γ ( α 1 ) [ P 1 + ( Q 1 + R 1 ) v ] 0 t ( t s ) α 2 d s + | u ( 0 ) | B + 2 Γ ( α ) [ P 1 + ( Q 1 + R 1 ) v ] .
(3.7)
Similarly, we can get
v B + 2 Γ ( β ) [ P 2 + ( Q 2 + R 2 ) u ] .
(3.8)
Together with (3.7) and (3.8), we have
u B + 2 Γ ( α ) { P 1 + ( Q 1 + R 1 ) [ B + 2 Γ ( β ) ( P 2 + ( Q 2 + R 2 ) u ) ] } .
Thus, from Γ ( α ) Γ ( β ) 4 ( Q 1 + R 1 ) ( Q 2 + R 2 ) Γ ( α ) Γ ( β ) > 0 , we obtain that
u Γ ( α ) Γ ( β ) B + 2 Γ ( β ) [ P 1 + ( Q 1 + R 1 ) B ] + 4 P 2 ( Q 1 + R 1 ) Γ ( α ) Γ ( β ) 4 ( Q 1 + R 1 ) ( Q 2 + R 2 ) : = M 1
and
v 1 Γ ( β ) [ P 2 + Q 2 B + ( Q 2 + R 2 ) M 1 ] : = M 2 .
Together with (3.5) and (3.6), we get
( u , v ) X ¯ max { M 1 , M 2 } : = M .

So Ω 1 is bounded. The proof is complete. □

Lemma 3.5 Suppose ( H 3 ) holds, then the set
Ω 2 = { ( u , v ) | ( u , v ) Ker L , N ( u , v ) Im L }

is bounded.

Proof For ( u , v ) Ω 2 , we have ( u , v ) = ( c 1 t , c 2 t ) , c 1 , c 2 R . Then from N ( u , v ) Im L , we get
0 1 ( 1 s ) α 2 f ( s , c 2 s , c 2 ) d s = 0 , 0 1 ( 1 s ) β 2 g ( s , c 1 s , c 1 ) d s = 0 ,
which, together with ( H 3 ), implies | c 1 | , | c 2 | D . Thus, we have
( u , v ) X ¯ D .

Hence, Ω 2 is bounded. The proof is complete. □

Lemma 3.6 Suppose the first part of ( H 3 ) holds, then the set
Ω 3 = { ( u , v ) Ker L | λ ( u , v ) + ( 1 λ ) Q N ( u , v ) = ( 0 , 0 ) , λ [ 0 , 1 ] }

is bounded.

Proof For ( u , v ) Ω 3 , we have ( u , v ) = ( c 1 t , c 2 t ) , c 1 , c 2 R and
(3.9)
(3.10)
If λ = 0 , then | c 1 | , | c 2 | D because of the first part of ( H 3 ). If λ = 1 , then c 1 = c 2 = 0 . For λ ( 0 , 1 ] , we can obtain | c 1 | , | c 2 | D . Otherwise, if | c 1 | or | c 2 | > D , in view of the first part of ( H 3 ), one has
λ c 1 2 t + ( 1 λ ) ( α 1 ) 0 1 ( 1 s ) α 2 c 1 f ( s , c 2 s , c 2 ) d s > 0 ,
or
λ c 2 2 t + ( 1 λ ) ( β 1 ) 0 1 ( 1 s ) β 2 c 2 g ( s , c 1 s , c 1 ) d s > 0 ,

which contradicts (3.9) or (3.10). Therefore, Ω 3 is bounded. The proof is complete. □

Remark 3.1 If the second part of ( H 3 ) holds, then the set
Ω 3 = { ( u , v ) Ker L | λ ( u , v ) + ( 1 λ ) Q N ( u , v ) = ( 0 , 0 ) , λ [ 0 , 1 ] }

is bounded.

Proof of Theorem 3.1 Set Ω = { ( u , v ) X ¯ | ( u , v ) X ¯ < max { M , D } + 1 } . It follows from Lemma 3.2 and 3.3 that L is a Fredholm operator of index zero and N is L-compact on Ω ¯ . By Lemma 3.4 and 3.5, we get that the following two conditions are satisfied:
  1. (1)

    L ( u , v ) λ N ( u , v ) for every ( ( u , v ) , λ ) [ ( dom L Ker L ) Ω ] × ( 0 , 1 ) ;

     
  2. (2)

    N x Im L for every ( u , v ) Ker L Ω .

     
Take
H ( ( u , v ) , λ ) = ± λ ( u , v ) + ( 1 λ ) Q N ( u , v ) .
According to Lemma 3.6 (or Remark 3.1), we know that H ( ( u , v ) , λ ) 0 for ( u , v ) Ker L Ω . Therefore,
deg ( Q N | Ker L , Ω Ker L , ( 0 , 0 ) ) = deg ( H ( , 0 ) , Ω Ker L , ( 0 , 0 ) ) = deg ( H ( , 1 ) , Ω Ker L , ( 0 , 0 ) ) = deg ( ± I , Ω Ker L , ( 0 , 0 ) ) 0 .

So, the condition (3) of Lemma 2.1 is satisfied. By Lemma 2.1, we can get that L ( u , v ) = N ( u , v ) has at least one solution in dom L Ω ¯ . Therefore, BVP (1.1) has at least one solution. The proof is complete. □

4 Example

Example 4.1 Consider the following BVP:
{ D 0 + 3 2 u ( t ) = 1 16 [ v ( t ) 10 ] + t 2 16 e | v ( t ) | , t [ 0 , 1 ] , D 0 + 5 4 v ( t ) = 1 12 [ u ( t ) 8 ] + t 3 12 sin 2 ( u ( t ) ) , t [ 0 , 1 ] , u ( 0 ) = v ( 0 ) = 0 , u ( 0 ) = u ( 1 ) , v ( 0 ) = v ( 1 ) .
(4.1)

Choose p 1 ( t ) = 11 16 , p 2 ( t ) = 3 4 , q 1 ( t ) = 1 16 , q 2 ( t ) = 1 12 , r 1 ( t ) = r 2 ( t ) = 0 , B = D = 10 .

By simple calculation, we can get that ( H 1 ), ( H 2 ) and the first part of ( H 3 ) hold.

By Theorem 3.1, we obtain that BVP (4.1) has at least one solution.

Declarations

Acknowledgements

The authors would like to thank the referees very much for their helpful comments and suggestions. This research was supported by the Fundamental Research Funds for the Central Universities (2010LKSX09) and the National Natural Science Foundation of China (11271364).

Authors’ Affiliations

(1)
Department of Mathematics, China University of Mining and Technology

References

  1. Metzler R, Klafter J: Boundary value problems for fractional diffusion equations. Physica A 2000, 278: 107-125. 10.1016/S0378-4371(99)00503-8MathSciNetView ArticleGoogle Scholar
  2. Scher H, Montroll E: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 1975, 12: 2455-2477. 10.1103/PhysRevB.12.2455View ArticleGoogle Scholar
  3. Mainardi F: Fractional diffusive waves in viscoelastic solids. In Nonlinear Waves in Solids. Edited by: Wegner JL, Norwood FR. ASME/AMR, Fairfield; 1995:93-97.Google Scholar
  4. Diethelm K, Freed AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. Edited by: Keil F, Mackens W, Voss H, Werther J. Springer, Heidelberg; 1999:217-224.Google Scholar
  5. Gaul L, Klein P, Kempfle S: Damping description involving fractional operators. Mech. Syst. Signal Process. 1991, 5: 81-88. 10.1016/0888-3270(91)90016-XView ArticleGoogle Scholar
  6. Glockle WG, Nonnenmacher TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 1995, 68: 46-53. 10.1016/S0006-3495(95)80157-8View ArticleGoogle Scholar
  7. Mainardi F: Fractional calculus: some basic problems in continuum and statistical mechanics. In Fractals and Fractional Calculus in Continuum Mechanics. Edited by: Carpinteri A, Mainardi F. Springer, Vienna; 1997:291-348.View ArticleGoogle Scholar
  8. Metzler F, Schick W, Kilian HG, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 1995, 103: 7180-7186. 10.1063/1.470346View ArticleGoogle Scholar
  9. Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.MATHGoogle Scholar
  10. Agarwal RP, O’Regan D, Stanek S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 2010, 371: 57-68. 10.1016/j.jmaa.2010.04.034MATHMathSciNetView ArticleGoogle Scholar
  11. Bai Z, Hu L: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052MATHMathSciNetView ArticleGoogle Scholar
  12. Kaufmann ER, Mboumi E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 2008, 3: 1-11.MathSciNetView ArticleGoogle Scholar
  13. Jafari H, Gejji VD: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 2006, 180: 700-706. 10.1016/j.amc.2006.01.007MATHMathSciNetView ArticleGoogle Scholar
  14. Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 2009, 71: 2391-2396. 10.1016/j.na.2009.01.073MATHMathSciNetView ArticleGoogle Scholar
  15. Liang S, Zhang J: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 2009, 71: 5545-5550. 10.1016/j.na.2009.04.045MATHMathSciNetView ArticleGoogle Scholar
  16. Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006, 36: 1-12.View ArticleGoogle Scholar
  17. Kosmatov N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 2010, 135: 1-10.MathSciNetGoogle Scholar
  18. Wei Z, Dong W, Che J: Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. Nonlinear Anal. 2010, 73: 3232-3238. 10.1016/j.na.2010.07.003MATHMathSciNetView ArticleGoogle Scholar
  19. Bai Z, Zhang Y: Solvability of fractional three-point boundary value problems with nonlinear growth. Appl. Math. Comput. 2011, 218(5):1719-1725. 10.1016/j.amc.2011.06.051MATHMathSciNetView ArticleGoogle Scholar
  20. Bai Z: Solvability for a class of fractional m-point boundary value problem at resonance. Comput. Math. Appl. 2011, 62(3):1292-1302. 10.1016/j.camwa.2011.03.003MATHMathSciNetView ArticleGoogle Scholar
  21. Ahmad B, Sivasundaram S: On four-point nonlocal boundary value problems of nonlinear integrodifferential equations of fractional order. Appl. Math. Comput. 2010, 217: 480-487. 10.1016/j.amc.2010.05.080MATHMathSciNetView ArticleGoogle Scholar
  22. Wang G, Ahmad B, Zhang L: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 2011, 74: 792-804. 10.1016/j.na.2010.09.030MATHMathSciNetView ArticleGoogle Scholar
  23. Yang L, Chen H: Unique positive solutions for fractional differential equation boundary value problems. Appl. Math. Lett. 2010, 23: 1095-1098. 10.1016/j.aml.2010.04.042MATHMathSciNetView ArticleGoogle Scholar
  24. Yang L, Chen H: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. 2011, 2011: 1-16.View ArticleGoogle Scholar
  25. Jiang W: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal. 2011, 74: 1987-1994. 10.1016/j.na.2010.11.005MATHMathSciNetView ArticleGoogle Scholar
  26. Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 2009, 22: 64-69. 10.1016/j.aml.2008.03.001MATHMathSciNetView ArticleGoogle Scholar
  27. Bai C, Fang J: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 2004, 150: 611-621. 10.1016/S0096-3003(03)00294-7MATHMathSciNetView ArticleGoogle Scholar
  28. Ahmad B, Alsaedi A: Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. Fixed Point Theory Appl. 2010, 2010: 1-17.MathSciNetView ArticleGoogle Scholar
  29. Ahmad B, Nieto J: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 2009, 58: 1838-1843. 10.1016/j.camwa.2009.07.091MATHMathSciNetView ArticleGoogle Scholar
  30. Rehman M, Khan R: A note on boundary value problems for a coupled system of fractional differential equations. Comput. Math. Appl. 2011, 61: 2630-2637. 10.1016/j.camwa.2011.03.009MATHMathSciNetView ArticleGoogle Scholar
  31. Su X: Existence of solution of boundary value problem for coupled system of fractional differential equations. Eng. Math. 2009, 26: 134-137.Google Scholar
  32. Yang W: Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Comput. Math. Appl. 2012, 63: 288-297. 10.1016/j.camwa.2011.11.021MATHMathSciNetView ArticleGoogle Scholar
  33. Zhang Y, Bai Z, Feng T: Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance. Comput. Math. Appl. 2011, 61(4):1032-1047. 10.1016/j.camwa.2010.12.053MATHMathSciNetView ArticleGoogle Scholar
  34. Mawhin J: Topological degree and boundary value problems for nonlinear differential equations in topological methods for ordinary differential equations. Lect. Notes Math. 1993, 1537: 74-142. 10.1007/BFb0085076MathSciNetView ArticleGoogle Scholar
  35. Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.MATHGoogle Scholar

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