Open Access

Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance

Advances in Difference Equations20132013:312

https://doi.org/10.1186/1687-1847-2013-312

Received: 24 May 2013

Accepted: 2 October 2013

Published: 8 November 2013

Abstract

In this paper, by using the extension of Mawhin’s continuation theorem due to Ge, we study the existence of solutions for a coupled system of fractional p-Laplacian equations at resonance. A new result on the existence of solutions for a fractional boundary value problem is obtained.

MSC:34B15.

Keywords

fractional p-Laplacian equationcoupled systemboundary value problemdegree theoryresonance

1 Introduction

In the recent years, fractional differential equations have played an important role in many fields such as physics, electrical circuits, biology, control theory, etc. (see [19]). Recently, many scholars have paid more attention to boundary value problems for fractional differential equations (see [1025]).

In [10], by means of a fixed point theorem on a cone, Agarwal et al. considered a two-point boundary value problem at nonresonance given by
{ D 0 + α x ( t ) + f ( t , x ( t ) , D 0 + μ x ( t ) ) = 0 , x ( 0 ) = x ( 1 ) = 0 ,

where 1 < α < 2 , μ > 0 are real numbers, α μ 1 and D 0 + α is the Riemann-Liouville fractional derivative.

By using the coincidence degree theory, Bai (see [20]) considered m-point fractional boundary value problems at resonance in the form
{ D 0 + α u ( t ) = f ( t , u ( t ) , D 0 + α 1 u ( t ) ) + e ( t ) , 0 < t < 1 , I 0 + 2 α u ( t ) | t = 0 = 0 , u ( 1 ) = i = 1 m 2 β i u ( η i ) ,

where 1 < α 2 is a real number, β i R , η i ( 0 , 1 ) are given constants such that i = 1 m 2 β i η i m 1 = 1 , and D 0 + α , I 0 + α are the Riemann-Liouville differentiation and integration.

Moreover, the existence of solutions to a coupled system of fractional differential equations have been studied by many authors (see [2633]).

In [28], relying on Schauder’s fixed point theorem, Ahmad et al. considered a three-point boundary value problem for a coupled system of nonlinear fractional differential equations at nonresonance given by
{ D 0 + α u ( t ) = f ( t , v ( t ) , D 0 + p v ( t ) ) , 0 < t < 1 , D 0 + β v ( t ) = g ( t , u ( t ) , D 0 + q u ( t ) ) , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = γ u ( η ) , v ( 0 ) = 0 , v ( 1 ) = γ v ( η ) ,

where 1 < α , β < 2 , p , q , γ > 0 , 0 < η < 1 , α q 1 , β p 1 , γ η α 1 < 1 , γ η β 1 < 1 , D is the standard Riemann-Liouville differentiation and f , g : [ 0 , 1 ] × R × R R are given continuous functions.

In [33], by using the coincidence degree theory due to Mawhin, Jiang discussed the existence of solutions to a coupled system of fractional differential equations at resonance
{ D 0 + α u ( t ) = f ( t , v ( t ) , D 0 + δ v ( t ) ) , u ( 0 ) = 0 , D 0 + γ u ( 1 ) = i = 1 n a i D 0 + γ u ( ξ i ) , D 0 + β v ( t ) = g ( t , u ( t ) , D 0 + γ u ( t ) ) , v ( 0 ) = 0 , D 0 + δ v ( 1 ) = i = 1 n a i D 0 + δ v ( η i ) ,

where t [ 0 , 1 ] , 1 < α , β 2 , 0 < γ α 1 , 0 < δ β 1 , 0 < ξ 1 < ξ 2 < < ξ n < 1 , 0 < η 1 < η 2 < < η m < 1 .

The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson (see [34]) introduced the p-Laplacian equation as follows:
( ϕ p ( x ( t ) ) ) = f ( t , x ( t ) , x ( t ) ) ,

where ϕ p ( s ) = | s | p 2 s , p > 1 . Obviously, ϕ p is invertible and its inverse operator is ϕ q , where q > 1 is a constant such that 1 p + 1 q = 1 .

In the past few decades, many important results relative to a p-Laplacian equation with certain boundary value conditions have been obtained. We refer the reader to [3538] and the references cited therein. We noticed that ϕ p is a quasi-linear operator. So, Mawhin’s continuation theorem is not suitable for a p-Laplacian operator. In [39], Ge and Ren extended Mawhin’s continuation theorem, which is used to deal with more general abstract operator equations.

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 p-Laplacian equations given by
{ D 0 + β ϕ p ( D 0 + α u ( t ) ) = f ( t , v ( t ) , D 0 + δ v ( t ) ) , t ( 0 , 1 ) , D 0 + γ ϕ p ( D 0 + δ v ( t ) ) = g ( t , u ( t ) , D 0 + α u ( t ) ) , t ( 0 , 1 ) , D 0 + α u ( 0 ) = D 0 + α u ( 1 ) = D 0 + δ v ( 0 ) = D 0 + δ v ( 1 ) = 0 ,
(1.1)

where D 0 + α , D 0 + β , D 0 + γ , D 0 + δ are the standard Caputo fractional derivatives, 0 < α , δ , β , γ 1 , 1 < α + β < 2 , 1 < δ + γ < 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, based on the extension of Mawhin’s continuation theorem due to Ge (see [39]). Finally, in Section 4, an example is given to illustrate the main result.

2 Preliminaries

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

Definition 2.1 Let X and Y be two Banach spaces with norms X and Y , respectively. A continuous operator
M : X dom M Y
is said to be quasi-linear if
  1. (i)

    Im M : = M ( X dom M ) is a closed subset of Y,

     
  2. (ii)

    Ker M : = { X dom M : M u = 0 } is linearly homeomorphic to R n , n < .

     

Definition 2.2 Let X be a real Banach space and X ˆ X . The operator P : X X ˆ is said to be a projector provided P 2 = P , P ( λ 1 x 1 + λ 2 x 2 ) = λ 1 P ( x 1 ) + λ 2 P ( x 2 ) for x 1 , x 2 X , λ 1 , λ 2 R . The operator Q : X X ˆ is said to be a semi-projector provided Q 2 = Q .

Definition 2.3 ([39])

Let X ˆ = Ker M and X ˜ be the complement space of X ˆ in X, then X = X ˆ X ˜ . On the other hand, suppose that Y ˆ is a subspace of Y and Y ˜ is the complement space of Y ˆ in Y so that Y = Y ˆ Y ˜ . Let P : X X ˆ be a projector and Q : Y Y ˆ be a semi-projector, and let Ω X be an open and bounded set with origin θ Ω , where θ is the origin of a linear space.

Suppose that N λ : Ω ¯ Y , λ [ 0 , 1 ] is a continuous operator. Denote N 1 by N. Let Σ λ = { u Ω ¯ : M u = N λ u } . N λ is said to be M-compact in Ω ¯ if there is Y ˆ Y with dim Y ˆ = dim X ˆ and an operator R : Ω ¯ × [ 0 , 1 ] X continuous and compact such that for λ [ 0 , 1 ] ,
( I Q ) N λ ( Ω ¯ ) Im M ( I Q ) Y ,
(2.1)
Q N λ x = θ , λ ( 0 , 1 ) Q N x = θ ,
(2.2)
R ( , 0 )  is the zero operator and  R ( , λ ) | Σ λ = ( I P ) | Σ λ ,
(2.3)
M [ P + R ( , λ ) ] = ( I Q ) N λ .
(2.4)

Lemma 2.1 ([39], Ge-Mawhin’s continuation theorem)

Let X and Y be two Banach spaces with norms X and Y , respectively. Ω X is an open and bounded nonempty set. Suppose that
M : X dom M Y
is a quasi-linear operator and
N λ : Ω ¯ Y , λ [ 0 , 1 ]

is M-compact in Ω ¯ . In addition, if

(C1) M x N λ x , ( x , λ ) ( dom M Ω ) × ( 0 , 1 ) ,

(C2) Q N x 0 , for x dom M Ω ,

(C3) deg ( J Q N , Ker M Ω , 0 ) 0 ,

where N = N 1 and J : Y ˆ X ˆ is a homeomorphism with J ( θ ) = θ , then the equation M u = N u has at least one solution in Ω ¯ .

Definition 2.4 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 2.5 The Caputo fractional derivative of order α > 0 of a continuous function x is given by
D 0 + α x ( t ) = I 0 + n α d n x ( t ) d t n = 1 Γ ( n α ) 0 t ( t s ) n α 1 x ( n ) ( 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 , + ) .

Lemma 2.2 [40]

Assume that D 0 + α x C [ 0 , 1 ] , α > 0 . 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 = x ( i ) ( 0 ) i ! , i = 0 , 1 , 2 , , n 1 , here n is the smallest integer greater than or equal to α.

Lemma 2.3 [40]

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 Y = C [ 0 , 1 ] with the norm y Y = y , X 1 = { x | x , D 0 + α x Y } with the norm x X 1 = max { x , D 0 + α x } and X 2 = { x | x , D 0 + δ x Y } with the norm x X 2 = max { x , D 0 + δ x } , where x = max t [ 0 , 1 ] | x ( t ) | . Then we denote X ¯ = X 1 × X 2 with the norm ( u , v ) X ¯ = max { u X 1 , v X 2 } 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 M 1 : dom M 1 X 1 Y by
M 1 u = D 0 + β ϕ p ( D 0 + α u ) ,
where
dom M 1 = { u X | D 0 + β ϕ p ( D 0 + α u ) Y , D 0 + α u ( 0 ) = D 0 + α u ( 1 ) = 0 } .
Define the operator M 2 : dom M 2 X 2 Y by
M 2 v = D 0 + γ ϕ p ( D 0 + δ v ) ,
where
dom M 2 = { v X 2 | D 0 + γ ϕ p ( D 0 + δ v ) Y , D 0 + δ v ( 0 ) = D 0 + δ v ( 1 ) = 0 } .
Define the operator M : dom M X ¯ Y ¯ by
M ( u , v ) = ( M 1 u , M 2 v ) ,
(2.5)
where
dom M = { ( u , v ) X ¯ | u dom M 1 , v dom M 2 } .
Define the operator N : X ¯ Y ¯ by
N ( u , v ) = ( N 1 v , N 2 u ) ,
where N 1 : X 2 Y
N 1 v ( t ) = f ( t , v ( t ) , D 0 + δ v ( t ) )
and N 2 : X 1 Y
N 2 u ( t ) = g ( t , u ( t ) , D 0 + α u ( t ) ) .
Then BVP (1.1) is equivalent to the operator equation
M ( u , v ) = N ( u , v ) , ( u , v ) dom M .

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

(H1) there exist nonnegative functions p i , q i , r i C [ 0 , 1 ] ( i = 1 , 2 ) with
1 Γ ( β + 1 ) Γ ( γ + 1 ) ( 2 p 1 Q 1 ( Γ ( δ + 1 ) ) p 1 + R 1 ) ( 2 p 1 Q 2 ( Γ ( α + 1 ) ) p 1 + R 2 ) < 1
(3.1)
such that for all ( u , v ) R 2 , t [ 0 , 1 ] ,
| f ( t , u , v ) | p 1 ( t ) + q 1 ( t ) | u | p 1 + r 1 ( t ) | v | p 1
and
| g ( t , u , v ) | p 2 ( t ) + q 2 ( t ) | u | p 1 + r 2 ( t ) | v | p 1 ,

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

(H2) there exists a constant B > 0 such that for all 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 .

Then BVP (1.1) has at least one solution.

In order to prove Theorem 3.1, we need to prove some lemmas below.

Lemma 3.1 Let M be defined by (2.5), then
Ker M = ( Ker M 1 , Ker M 2 ) = { ( u , v ) X ¯ | ( u , v ) = ( a , b ) , a , b R } ,
(3.2)
Im M = ( Im M 1 , Im M 2 ) = { ( x , y ) Y ¯ | 0 1 ( 1 s ) β 1 x ( s ) d s = 0 , 0 1 ( 1 s ) γ 1 y ( s ) d s = 0 } ,
(3.3)

and M is a quasi-linear operator.

Proof By Lemma 2.2, M 1 u = D 0 + β ϕ p ( D 0 + α u ) = 0 has the solution
u ( t ) = u ( 0 ) + I 0 + α ϕ q ( c 0 ) = u ( 0 ) + ϕ q ( c 0 ) Γ ( α + 1 ) t α , c 0 = ϕ p ( D 0 + α u ( 0 ) ) ,
which satisfies
D 0 + α u ( t ) = ϕ q ( c 0 ) .
Combining with the boundary value condition D 0 + α u ( 0 ) = 0 , we have
Ker M 1 = { u X 1 | u = a , a R } .
For x Im M 1 , there exists u dom M 1 such that x = M 1 u Y . By Lemma 2.2, we have
D 0 + α u ( t ) = ϕ q ( I 0 + β x ( t ) + c 0 ) = ϕ q ( 1 Γ ( β ) 0 t ( t s ) β 1 x ( s ) d s + c 0 ) .
From the condition D 0 + α u ( 0 ) = 0 , one has c 0 = 0 . By the condition D 0 + α u ( 1 ) = 0 , we obtain that
0 1 ( 1 s ) β 1 x ( s ) d s = 0 .
(3.4)
On the other hand, suppose that x Y and satisfies 0 1 ( 1 s ) β 1 x ( s ) d s = 0 . Let u ( t ) = I 0 + α ϕ q ( I 0 + β x ( t ) ) , then u dom M 1 . By Lemma 2.3, we have D 0 + α u ( t ) = x ( t ) . So that x Im M 1 . Then we have
Im M 1 = { x Y | 0 1 ( 1 s ) β 1 x ( s ) d s = 0 } .
Then we have dim Ker M 1 = 1 and M 1 ( dom M 1 X 1 ) Y closed. Therefore, M 1 is a quasi-linear operator. Similarly, we can get
Ker M 2 = { v X 2 | v = b , b R } , Im M 2 = { y Y | 0 1 ( 1 s ) γ 1 y ( s ) d s = 0 } ,

and M 2 is a quasi-linear operator. Then the proof is complete. □

Lemma 3.2 Let Ω X ¯ be an open and bounded set, then N λ is M-compact in Ω ¯ .

Proof Define the continuous projector P : X ¯ X ˆ and the semi-projector Q : Y ¯ Y ˆ
P ( u , v ) = ( P 1 u , P 2 v ) = ( u ( 0 ) , v ( 0 ) ) , Q ( x , y ) = ( Q 1 x , Q 2 y ) = ( β 0 1 ( 1 s ) β 1 x ( s ) d s , γ 0 1 ( 1 s ) γ 1 y ( s ) d s ) ,

where X ˆ = Ker M and Y ˆ = Im Q .

Obviously, Im P = Ker M 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 M . By a simple calculation, we can get that Ker M Ker P = { ( 0 , 0 ) } . Then we get
X ¯ = Ker M Ker P = X ˆ X ˜ .
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 β 0 1 ( 1 s ) β 1 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 M , Q ( x , y ) Im Q . It follows from Ker Q = Im M and Q 2 ( x , y ) = Q ( x , y ) that Im Q Im M = { ( 0 , 0 ) } . Then we have
Y ¯ = Im Q Im M = Y ˆ Y ˜ .
Thus
dim X ˆ = dim Ker M = dim Im Q = dim Y ˆ .

Let Ω X ¯ be an open and bounded set with ( θ , θ ) Ω . For each ( u , v ) Ω ¯ , we can get Q [ ( I Q ) N λ ( u , v ) ] = 0 . Thus, ( I Q ) N λ ( u , v ) Im M = Ker Q . Take any ( x , y ) Im M in the type ( x , y ) = ( ( x , y ) Q ( x , y ) ) + Q ( x , y ) . Since Q ( x , y ) = 0 , we can get ( I Q ) ( x , y ) Y ¯ . So (2.1) holds. It is easy to verify (2.2).

Furthermore, define R = ( R 1 , R 2 ) : Ω ¯ × [ 0 , 1 ] X ˜ by
R 1 ( u , λ ) ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 ϕ q ( 1 Γ ( β ) 0 s ( s τ ) β 1 ( ( I Q 1 ) N λ 1 v ( τ ) ) d τ ) d s , R 2 ( v , λ ) ( t ) = 1 Γ ( δ ) 0 t ( t s ) δ 1 ϕ q ( 1 Γ ( γ ) 0 s ( s τ ) γ 1 ( ( I Q 2 ) N λ 2 u ( τ ) ) d τ ) d s .
By the continuity of f and g, it is easy to get that R ( u , v , λ ) is continuous on Ω ¯ × [ 0 , 1 ] . Moreover, for all ( u , v ) Ω ¯ , there exists a constant T > 0 such that max { | I 0 + β ( I Q 1 ) N λ 1 v ( τ ) | , | I 0 + γ ( I Q 2 ) N λ 2 u ( τ ) | } T , so we can easily obtain that R ( Ω ¯ , λ ) is uniformly bounded. By the Arzela-Ascoli theorem, we just need to prove that R : Ω ¯ × [ 0 , 1 ] X ˜ is equicontinuous. Furthermore, for 0 t 1 < t 2 1 , ( u , v , λ ) Ω ¯ × [ 0 , 1 ] = ( Ω ¯ 1 , Ω ¯ 2 ) × [ 0 , 1 ] , we have
| R ( u , v , λ ) ( t 2 ) R ( u , v , λ ) ( t 1 ) | = | ( I 0 + α ϕ q ( I 0 + β ( I Q 1 ) N λ 1 v ( t 2 ) ) , I 0 + δ ϕ q ( I 0 + γ ( I Q 2 ) N λ 2 u ( t 2 ) ) ) ( I 0 + α ϕ q ( I 0 + β ( I Q 1 ) N λ 1 v ( t 1 ) ) , I 0 + δ ϕ q ( I 0 + γ ( I Q 2 ) N λ 2 u ( t 1 ) ) ) | = | ( I 0 + α ϕ q ( I 0 + β ( I Q 1 ) N λ 1 v ( t 2 ) ) I 0 + α ϕ q ( I 0 + β ( I Q 1 ) N λ 1 v ( t 1 ) ) , I 0 + δ ϕ q ( I 0 + γ ( I Q 2 ) N λ 2 u ( t 2 ) ) I 0 + δ ϕ q ( I 0 + γ ( I Q 2 ) N λ 2 u ( t 1 ) ) ) | .
By | I 0 + β ( I Q 1 ) N λ 1 v | T , we have
| I 0 + α ϕ q ( I 0 + β ( I Q 1 ) N λ 1 v ( t 2 ) ) I 0 + α ϕ q ( I 0 + β ( I Q 1 ) N λ 1 v ( t 1 ) ) | 1 Γ ( α ) | 0 t 2 ( t 2 s ) α 1 ϕ q ( I 0 + β ( I Q 1 ) N λ 1 v ( s ) ) d s 0 t 1 ( t 1 s ) α 1 ϕ q ( I 0 + β ( I Q 1 ) N λ 1 v ( s ) ) d s | ϕ q ( T ) 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 ] = ϕ q ( T ) Γ ( α + 1 ) ( t 2 α t 1 α ) .

Since t α is uniformly continuous on [ 0 , 1 ] , so R 1 ( Ω ¯ 1 , λ ) is equicontinuous. Similarly, we can get I 0 + β ( ( I Q 1 ) N λ 1 v ( τ ) ) C [ 0 , 1 ] is equicontinuous. Considering that ϕ q ( s ) is uniformly continuous on [ T , T ] , we have D 0 + α R 1 ( Ω ¯ 1 , λ ) = I 0 + β ( ( I Q 1 ) N λ 1 ( Ω ¯ ) ) is also equicontinuous. So, we can obtain that R 1 ( Ω ¯ 1 , λ ) X ˜ 1 is compact.

Similarly, we can get that R 2 ( Ω ¯ 2 , λ ) X ˜ 2 is compact. So, we can obtain that R : Ω ¯ × [ 0 , 1 ] X ˜ is compact.

For each ( u , v ) Σ λ = { ( u , v ) Ω ¯ : M ( u , v ) = N λ ( u , v ) } , we have ( D 0 + β ϕ p ( D 0 + α u ( t ) ) , D 0 + γ ϕ p ( D 0 + δ v ( t ) ) ) = N λ ( u ( t ) , v ( t ) ) Im M . Thus,
R 1 ( u , λ ) ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 ϕ q ( 1 Γ ( β ) 0 s ( s τ ) β 1 ( ( I Q 1 ) N λ 1 v ( τ ) ) d τ ) d s = 1 Γ ( α ) 0 t ( t s ) α 1 ϕ q ( 1 Γ ( β ) 0 s ( s τ ) β 1 D 0 + β ϕ p ( D 0 + α u ( τ ) ) d τ ) d s ,
which together with D 0 + α u ( 0 ) = 0 yields that
R 1 ( u , λ ) ( t ) = u ( t ) u ( 0 ) = [ ( I P 1 ) u ] ( t ) .

It is easy to verify that R 1 ( u , 0 ) ( t ) is the zero operator. Similarly, we can get R 2 ( v , λ ) ( t ) = [ ( I P 2 ) v ] ( t ) and R 2 ( v , 0 ) ( t ) is the zero operator. So (2.3) holds.

On the other hand,
M 1 [ P 1 u + R 1 ( u , λ ) ] ( t ) = M 1 [ 1 Γ ( α ) 0 t ( t s ) α 1 ϕ q ( 1 Γ ( β ) 0 s ( s τ ) β 1 ( ( I Q 1 ) N λ 1 v ( τ ) ) d τ ) d s + u ( 0 ) ] = [ ( ( I Q 1 ) N λ 1 ) v ] ( t ) .

Similarly, we have M 2 [ P 2 v + R 2 ( v , λ ) ] ( t ) = [ ( ( I Q 2 ) N λ 2 ) u ] ( t ) . So, (2.4) holds. Then we have that N λ is M-compact in Ω ¯ . The proof is complete. □

Lemma 3.3 Suppose that (H1), (H2) hold, then the set
Ω 1 = { ( u , v ) dom M Ker M M ( u , v ) = λ N ( u , v ) , λ ( 0 , 1 ) }

is bounded.

Proof Take ( u , v ) Ω 1 , then N ( u , v ) Im M . By (3.3), we have
0 1 ( 1 s ) β 1 f ( s , v ( s ) , D 0 + δ v ( s ) ) d s = 0 , 0 1 ( 1 s ) γ 1 g ( s , u ( s ) , D 0 + α 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 ) , D 0 + δ v ( ξ ) ) = 0 and g ( η , u ( η ) , D 0 + α u ( η ) ) = 0 . So, from (H2), we get | v ( ξ ) | B and | u ( η ) | B .

By Lemma 2.2,
v ( t ) = v ( 0 ) + I 0 + δ D 0 + δ v ( t ) = v ( 0 ) + 1 Γ ( δ ) 0 t ( t s ) δ 1 D 0 + δ v ( s ) d s .
Take t = ξ , we have
v ( ξ ) = v ( 0 ) + 1 Γ ( δ ) 0 ξ ( ξ s ) δ 1 D 0 + δ v ( s ) d s .
Then we have
| v ( 0 ) | | v ( ξ ) | + 1 Γ ( δ ) 0 ξ ( ξ s ) δ 1 | D 0 + δ v ( s ) | d s | v ( ξ ) | + 1 Γ ( δ ) D 0 + δ v 1 δ ξ δ B + 1 Γ ( δ + 1 ) D 0 + δ v .
So, we get
| v ( t ) | | v ( 0 ) | + 1 Γ ( δ ) 0 t ( t s ) δ 1 | D 0 + δ v ( s ) | d s | v ( 0 ) | + 1 Γ ( δ ) D 0 + δ v 1 δ t δ B + 2 Γ ( δ + 1 ) D 0 + δ v , t [ 0 , 1 ] .
That is,
v B + 2 Γ ( δ + 1 ) D 0 + δ v .
(3.5)
Similarly, we can get
u B + 2 Γ ( α + 1 ) D 0 + α u .
(3.6)
By M ( u , v ) = λ N ( u , v ) and D 0 + α u ( 0 ) = D 0 + δ v ( 0 ) = 0 , we get
ϕ p ( D 0 + α u ( t ) ) = λ I 0 + β N 1 v ( t ) = λ Γ ( β ) 0 t ( t s ) β 1 f ( s , v ( s ) , D 0 + δ v ( s ) ) d s .
So, from (H1), we have
| ϕ p ( D 0 + α u ( t ) ) | 1 Γ ( β ) 0 t ( t s ) β 1 | f ( s , v ( s ) , D 0 + δ v ( s ) ) | d s 1 Γ ( β ) 0 t ( t s ) β 1 ( p 1 ( s ) + q 1 ( s ) | v ( s ) | p 1 + r 1 ( s ) | D 0 + δ v ( s ) | p 1 ) d s 1 Γ ( β ) ( p 1 + q 1 v p 1 + r 1 D 0 + δ v p 1 ) 1 β t β 1 Γ ( β + 1 ) ( P 1 + Q 1 v p 1 + R 1 D 0 + δ v p 1 ) ,
which together with | ϕ p ( D 0 + α u ( t ) ) | = | D 0 + α u ( t ) | p 1 and (3.5) yields that
D 0 + α u p 1 1 Γ ( β + 1 ) [ P 1 + Q 1 ( B + 2 Γ ( δ + 1 ) D 0 + δ v ) p 1 + R 1 D 0 + δ v p 1 ] .
(3.7)
Similarly, we can get
D 0 + δ v p 1 1 Γ ( γ + 1 ) [ P 2 + Q 2 ( B + 2 Γ ( α + 1 ) D 0 + α u ) p 1 + R 2 D 0 + α u p 1 ] .
(3.8)
Then from (3.1), (3.7) and (3.8), we can see that there exists a constant M 1 > 0 such that
D 0 + α u , D 0 + δ v M 1 .
(3.9)
Thus, from (3.5) and (3.6), we get
u , v max { B + 2 M 1 Γ ( α + 1 ) , B + 2 M 1 Γ ( δ + 1 ) } : = M 2 .
(3.10)
Combining (3.9) and (3.10), we have
( u , v ) X ¯ max { M 1 , M 2 } : = M .

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

Lemma 3.4 Suppose that (H3) holds, then the set
Ω 2 = { ( u , v ) | ( u , v ) Ker M , N ( u , v ) Im M }

is bounded.

Proof For ( u , v ) Ω 2 , we have ( u , v ) = ( a , b ) . Then, from N ( u , v ) Im M , we get
0 1 ( 1 s ) β 1 f ( s , b , 0 ) d s = 0 , 0 1 ( 1 s ) γ 1 g ( s , a , 0 ) d s = 0 ,
which together with (H2) implies | a | , | b | B . Thus, we have
( u , v ) X ¯ B .

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

Lemma 3.5 Suppose that the first part of (H2) holds, then the set
Ω 3 = { ( u , v ) Ker M | λ J 1 ( u , v ) + ( 1 λ ) Q N ( u , v ) = ( 0 , 0 ) , λ [ 0 , 1 ] }
is bounded, where J 1 : Ker M Im Q is a homeomorphism defined by
J 1 ( a , b ) = ( b , a ) , a , b R .
Proof For ( u , v ) Ω 3 , we have ( u , v ) = ( a , b ) and
λ b + ( 1 λ ) β 0 1 ( 1 s ) β 1 f ( s , b , 0 ) d s = 0 ,
(3.11)
λ a + ( 1 λ ) γ 0 1 ( 1 s ) γ 1 g ( s , a , 0 ) d s = 0 .
(3.12)
If λ = 1 , then a = b = 0 . For λ [ 0 , 1 ) , we can obtain | a | , | b | B . Otherwise, if | a | or | b | > B , in view of the first part of (H2), one has
λ b 2 + ( 1 λ ) β 0 1 ( 1 s ) β 1 b f ( s , b , 0 ) d s > 0 ,
or
λ a 2 + ( 1 λ ) γ 0 1 ( 1 s ) γ 1 a g ( s , a , 0 ) d s > 0 ,

which contradicts (3.11) or (3.12). Therefore, Ω 3 is bounded. The proof is complete. □

Remark 3.1 If the second part of (H2) holds, then the set
Ω 3 = { ( u , v ) Ker M | λ J 1 ( 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 , B } + 1 } . It follows from Lemmas 3.1 and 3.2 that M is a quasi-linear operator and N λ is M-compact on Ω ¯ . By Lemmas 3.3 and 3.4, we get that the following two conditions are satisfied:

(C1) M x N λ x , ( x , λ ) ( dom M Ω ) × ( 0 , 1 ) ,

(C2) Q N x 0 , for x dom M Ω .

Take
H ( ( u , v ) , λ ) = ± λ ( u , v ) + ( 1 λ ) J Q N ( u , v ) .
According to Lemma 3.5 (or Remark 3.1), we know that H ( ( u , v ) , λ ) 0 for ( u , v ) Ker M Ω . Therefore
deg ( J Q N | Ker M , Ω Ker M , ( 0 , 0 ) ) = deg ( H ( , 0 ) , Ω Ker M , ( 0 , 0 ) ) = deg ( H ( , 1 ) , Ω Ker M , ( 0 , 0 ) ) = deg ( ± I , Ω Ker M , ( 0 , 0 ) ) 0 .

So, condition (C3) of Lemma 2.1 is satisfied. By Lemma 2.1, we can get that M ( u , v ) = N ( u , v ) has at least one solution in dom M 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 4 ϕ 3 ( D 0 + 1 2 u ( t ) ) = 25 16 + 1 16 v 2 ( t ) + t e | D 0 + 4 5 v ( t ) | , t ( 0 , 1 ) , D 0 + 1 4 ϕ 3 ( D 0 + 4 5 v ( t ) ) = 30 17 + 1 17 u 2 ( t ) + sin 2 ( D 0 + 1 2 u ( t ) ) , t ( 0 , 1 ) , D 0 + 1 2 u ( 0 ) = D 0 + 1 2 u ( 1 ) = D 0 + 4 5 v ( 0 ) = D 0 + 4 5 v ( 1 ) = 0 .
(4.1)
Corresponding to BVP (1.1), we have that p = 3 , α = 1 2 , δ = 4 5 , β = 3 4 , γ = 1 4 and
f ( t , u , v ) = 25 16 + 1 16 u 2 + t e | v | , g ( t , u , v ) = 30 17 + 1 17 u 2 + sin 2 v .
Choose p 1 ( t ) = p 2 ( t ) = 10 , q 1 ( t ) = 1 16 , q 2 ( t ) = 1 17 , r 1 ( t ) = r 2 ( t ) = 0 , B = 5 . Then we have P 1 = P 2 = 10 , Q 1 = 1 16 , Q 2 = 1 17 , R 1 ( t ) = R 2 ( t ) = 0 . By a simple calculation, we get
1 Γ ( 3 4 + 1 ) Γ ( 1 4 + 1 ) ( 2 2 1 16 ( Γ ( 4 5 + 1 ) ) 2 ) ( 2 2 1 17 ( Γ ( 1 2 + 1 ) ) 2 ) < 1 .

Then (H1) and the first part of (H2) hold.

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

Declarations

Acknowledgements

The authors are grateful to those who gave useful suggestions about the original manuscript. This research was supported by the Fundamental Research Funds for the Central Universities (2013QNA33).

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 ArticleMATHGoogle Scholar
  2. Scher H, Montroll E: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 1975, 12: 2455-2477.View 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, Wien; 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.2010.04.034MathSciNetView ArticleMATHGoogle 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.2005.02.052MathSciNetView ArticleMATHGoogle 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 ArticleMATHGoogle 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.2006.01.007MathSciNetView ArticleMATHGoogle 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.2009.01.073MathSciNetView ArticleMATHGoogle 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.2009.04.045MathSciNetView ArticleMATHGoogle Scholar
  16. Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006, 36: 1-12.View ArticleMathSciNetGoogle Scholar
  17. Kosmatov N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 2010, 135: 1-10.MathSciNetMATHGoogle 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.2010.07.003MathSciNetView ArticleMATHGoogle 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.2011.06.051MathSciNetView ArticleMATHGoogle 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.2011.03.003MathSciNetView ArticleMATHGoogle 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.2010.05.080MathSciNetView ArticleMATHGoogle 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.2010.09.030MathSciNetView ArticleMATHGoogle 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.2010.04.042MathSciNetView ArticleMATHGoogle Scholar
  24. Hu Z, Liu W: Solvability for fractional order boundary value problems at resonance. Bound. Value Probl. 2011, 20: 1-10.MathSciNetMATHGoogle 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.2010.11.005MathSciNetView ArticleMATHGoogle 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.2008.03.001MathSciNetView ArticleMATHGoogle 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)00294-7MathSciNetView ArticleMATHGoogle 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 ArticleMATHGoogle 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.2009.07.091MathSciNetView ArticleMATHGoogle 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.2011.03.009MathSciNetView ArticleMATHGoogle 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.2011.11.021MathSciNetView ArticleMATHGoogle Scholar
  33. Jiang W: Solvability for a coupled system of fractional differential equations at resonance. Nonlinear Anal. 2012, 13: 2285-2292. 10.1016/j.nonrwa.2012.01.023View ArticleMathSciNetMATHGoogle Scholar
  34. Leibenson LS: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 1945, 9: 7-10.MathSciNetGoogle Scholar
  35. Pang H, Ge W, Tian M: Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a p -Laplacian. Comput. Math. Appl. 2008, 56: 127-142. 10.1016/j.camwa.2007.11.039MathSciNetView ArticleMATHGoogle Scholar
  36. Liu B, Yu J: On the existence of solutions for the periodic boundary value problems with p -Laplacian operator. J. Syst. Sci. Math. Sci. 2003, 23: 76-85.MathSciNetMATHGoogle Scholar
  37. Lian L, Ge W: The existence of solutions of m -point p -Laplacian boundary value problems at resonance. Acta Math. Appl. Sin. 2005, 28: 288-295.MathSciNetGoogle Scholar
  38. Chen T, Liu W, Hu Z: A boundary value problem for fractional differential equation with p -Laplacian operator at resonance. Nonlinear Anal. 2012, 75: 3210-3217. 10.1016/j.na.2011.12.020MathSciNetView ArticleMATHGoogle Scholar
  39. Ge W, Ren J: An extension of Mawhin’s continuation theorem and its application to boundary value problems with a p -Laplacian. Nonlinear Anal. 2004, 58: 477-488. 10.1016/j.na.2004.01.007MathSciNetView ArticleMATHGoogle Scholar
  40. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATHGoogle Scholar

Copyright

© Hu et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.