Open Access

Nontrivial solutions for a fractional boundary value problem

Advances in Difference Equations20132013:171

https://doi.org/10.1186/1687-1847-2013-171

Received: 24 March 2013

Accepted: 28 May 2013

Published: 17 June 2013

Abstract

In this work, we discuss the existence of nontrivial solutions for the fractional boundary value problem

{ D 0 + α u = f ( t , u ) , t [ 0 , 1 ] , u ( 0 ) = u ( 0 ) = u ( 1 ) = 0 .

Here α ( 2 , 3 ] is a real number, D 0 + α is the standard Riemann-Liouville fractional derivative of order α. By virtue of some inequalities associated with Green’s function, without the assumption of the nonnegativity of f, we utilize topological degree theory to establish our main results.

MSC:26A33, 34B15, 34B18, 34B27.

Keywords

fractional boundary value problemnontrivial solutiontopological degreeRiemann-Liouville derivative

1 Introduction

In this paper, we investigate nontrivial solutions for the boundary value problem of fractional order involving Riemann-Liouville’s derivative
{ D 0 + α u = f ( t , u ) , t [ 0 , 1 ] , u ( 0 ) = u ( 0 ) = u ( 1 ) = 0 ,
(1.1)

where α ( 2 , 3 ] , f : [ 0 , 1 ] × R R ( R : = ( , + ) ) is continuous.

In view of a fractional differential equation’s modeling capabilities in engineering, science, economy and other fields, the last few decades have resulted in a rapid development of the theory of fractional differential equation; see the books [13]. This may explain the reason why the last few decades have witnessed an overgrowing interest in the research of such problems, with many papers in this direction published. We refer the interested reader to [421] and the references therein.

In [4], Bai and Lü studied the existence and multiplicity of positive solutions for the nonlinear fractional differential equation
{ D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = 0 ,
(1.2)

where 1 < α 2 is a real number and f : [ 0 , 1 ] × R + R + is continuous. They obtained the existence of positive solutions by means of Guo-Krasnosel’skii fixed point theorem and Leggett-Williams fixed point theorem.

In [5], Jiang et al. discussed some positive properties of the Green function for boundary value problem (1.2), and as an application, they utilized the Guo-Krasnosel’skii fixed point theorem to obtain the existence of positive solutions for (1.2).

In [6], El-Shahed and Nieto investigated the existence of nontrivial solutions for a multi-point boundary value problem for fractional differential equations. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of a nontrivial solution were obtained by using the Leray-Schauder nonlinear alternative.

In [7], Wang and Liu adopted the same methods in [6] to discuss the existence of solutions for nonlinear fractional differential equations with fractional anti-periodic boundary conditions
{ D α c x ( t ) = f ( t , x ( t ) , c D q x ( t ) ) , t [ 0 , T ] , x ( 0 ) = x ( T ) , c D p x ( 0 ) = c D p x ( T ) .
(1.3)
In [810], Ahmad et al. utilized fixed point theory to consider some fractional differential equations with fractional boundary conditions and obtained some new existence results. In particular, He and his coauthors [10] investigated the existence of solutions for the fractional nonlinear integro-differential equation of mixed type on a semi-infinite interval in a Banach space
{ D α u ( t ) + f ( t , u ( t ) , T u ( t ) , S u ( t ) ) = θ , n 1 < α n , n 2 , u ( 0 ) = u ( 0 ) = = u ( n 2 ) ( 0 ) = θ , D α 1 u ( ) = u .
(1.4)

Meanwhile, we also note that they developed an explicit iterative sequence for approximating the solution together with an error estimate for the approximation.

In [22, 23], Sun and Zhang discussed a class of singular superlinear and sublinear Sturm-Liouville problems, respectively. In the two papers, the Sturm-Liouville problems are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, and the nonnegativity is not necessary to be nonnegative. The existence results of nontrivial solutions and positive solutions are given by means of topological degree theory.

Motivated by the works mentioned above, in our paper, we adopt the methods of [22, 23] to investigate the fractional problem (1.1). As we know, the eigenvalue and eigenfunction of an integer-order differential equation have been a very perfect theory; however, this work on fractional order differential equation has not appeared in the literature. In order to overcome the difficulty arising from it, we establish some inequalities associated with Green’s function; see Lemma 2.3 in Section 2. With the aid of these inequalities, the nonlinear term f can grow superlinearly and sublinearly, and we obtain that problem (1.1) has at least one nontrivial solution by topological degree theory. This means that both our methodology and results in this paper are different from those in [47, 1116].

2 Preliminaries

The Riemann-Liouville fractional derivative of order α > 0 of a continuous function f : ( 0 , + ) ( , + ) is given by
D 0 + α f ( t ) = 1 Γ ( n α ) ( d d t ) n 0 t f ( s ) ( t s ) α n + 1 d s ,

where n = [ α ] + 1 , [ α ] denotes the integer part of number α, provided that the right-hand side is pointwise defined on ( 0 , + ) . For more details on fractional calculus, we refer the reader to the recent books; see [13]. Next, we present Green’s function of fractional differential equation boundary value problem (1.1).

Lemma 2.1 (See [[16], Lemma 2.7])

Let v C [ 0 , 1 ] and α ( 2 , 3 ] . Then D 0 + α u : = v , together with the boundary conditions u ( 0 ) = u ( 0 ) = u ( 1 ) = 0 , is equivalent to u ( t ) = 0 1 G ( t , s ) v ( s ) d s , where
G ( t , s ) : = 1 Γ ( α ) { t α 1 ( 1 s ) α 2 ( t s ) α 1 , 0 s t 1 , t α 1 ( 1 s ) α 2 , 0 t s 1 .
(2.1)

Lemma 2.2 (See [[16], Lemma 2.8])

The functions G ( t , s ) C ( [ 0 , 1 ] × [ 0 , 1 ] , [ 0 , + ) ) . Moreover, G ( t , s ) satisfies the following inequalities:
t α 1 s ( 1 s ) α 2 Γ ( α ) G ( t , s ) s ( 1 s ) α 2 , t , s [ 0 , 1 ] .
(2.2)
Lemma 2.3 Let φ ( t ) = t ( 1 t ) α 2 , t [ 0 , 1 ] , and α Γ ( α 1 ) Γ ( 2 α ) ( : = K 1 ) 1 α ( α 1 ) Γ ( α ) ( : = K 2 ) . Then
K 1 φ ( s ) 0 1 G ( t , s ) φ ( t ) d t K 2 φ ( s ) , s [ 0 , 1 ] .
(2.3)

Proof By (2.2), we arrive at the inequality (2.3) immediately. The proof is completed. □

By simple computation, we have max t [ 0 , 1 ] 0 1 G ( t , s ) d s = 0 1 φ ( t ) d t = 1 α ( α 1 ) = K 2 Γ ( α ) . Let
E : = C [ 0 , 1 ] , u : = max t [ 0 , 1 ] | u ( t ) | , P : = { u E : u ( t ) 0 , t [ 0 , 1 ] } .
Then ( E , ) becomes a real Banach space and P is a cone on E. Now, note that u solves (1.1) if and only if u is a fixed point of the operator
( A u ) ( t ) : = 0 1 G ( t , s ) f ( s , u ( s ) ) d s , u E .
(2.4)
Clearly, f C ( [ 0 , 1 ] × R , R ) implies A : E E is a completely continuous operator. Denote
( L u ) ( t ) : = 0 1 G ( t , s ) u ( s ) d s , u E .
Then L : E E is a completely continuous linear operator, satisfying L ( P ) P . That is, L is a positive, completely continuous, linear operator. Let
P 0 : = { u E : 0 1 u ( t ) φ ( t ) d t ω u , t [ 0 , 1 ] } ,

where φ ( t ) is defined by Lemma 2.3 and ω : = K 1 Γ ( α ) > 0 . By (2.2) and (2.3), we easily have the following result.

Lemma 2.4 L ( P ) P 0 .

Proof From (2.2), for u P , we have
( L u ) ( t ) = 0 1 G ( t , s ) u ( s ) d s 1 Γ ( α ) 0 1 φ ( s ) u ( s ) d s .
On the other hand, from (2.2) and (2.3), we find
0 1 ( L u ) ( t ) φ ( t ) d t = 0 1 ( 0 1 G ( t , s ) u ( s ) d s ) φ ( t ) d t K 1 0 1 φ ( s ) u ( s ) d s K 1 Γ ( α ) L u .

Therefore, L ( P ) P 0 . This completes the proof. □

Lemma 2.5 (See [24])

Let E be a Banach space and let Ω E be a bounded open set. Suppose that A : Ω ¯ E is a completely continuous operator. If there is u 0 0 such that u A u + μ u 0 , u Ω and μ 0 , then the topological degree deg ( I A , Ω , 0 ) = 0 .

Lemma 2.6 (See [24])

Let E be a Banach space and let Ω E be a bounded open set with 0 Ω . Suppose that A : Ω ¯ E is a completely continuous operator. If A u μ u , u Ω and μ 1 , then the topological degree deg ( I A , Ω , 0 ) = 1 .

3 Main results

We denote λ 1 : = K 1 1 > 0 , λ 2 : = K 2 1 > 0 and B ρ : = { u E : u < ρ } for ρ > 0 .

Theorem 3.1 If there exists a constant b 0 such that
f ( t , u ) b , u R , lim inf u + f ( t , u ) u > λ 1 , lim sup u 0 | f ( t , u ) u | < λ 2 ,
(3.1)

then (1.1) has at least one nontrivial solution.

Proof The first two inequalities of (3.1) imply that there are ε > 0 and b 1 > 0 such that
f ( t , u ) ( λ 1 + ε ) u b 1 , u R , t [ 0 , 1 ] .
(3.2)
Take R > b K 2 Γ ( α ) + ε 1 b ( K 1 3 K 2 3 K 1 1 K 2 ) + b K 1 2 K 2 3 + ε 1 b 1 K 1 2 K 2 2 . In what follows, we shall prove that
u A u + μ u , u E , u = R , μ 0 ,
(3.3)
where u P 0 . Indeed, if u 0 E , u 0 = R , and μ 0 0 such that
u 0 ( t ) = ( A u 0 ) ( t ) + μ 0 u ( t ) = 0 1 G ( t , s ) f ( s , u 0 ( s ) ) d s + μ 0 u ( t ) .
(3.4)
Let u ˜ ( t ) = b 0 1 G ( t , s ) d s , then we have
u 0 ( t ) + u ˜ ( t ) = 0 1 G ( t , s ) ( f ( s , u 0 ( s ) ) + b ) d s + μ 0 u ( t ) ,
which leads to u 0 + u ˜ P 0 by Lemma 2.4. Combining this with (3.2), we find
0 1 ( A u 0 ) ( t ) φ ( t ) d t 0 1 u 0 ( t ) φ ( t ) d t = 0 1 φ ( t ) 0 1 G ( t , s ) f ( s , u 0 ( s ) ) d s d t 0 1 u 0 ( t ) φ ( t ) d t ( λ 1 + ε ) 0 1 φ ( t ) 0 1 G ( t , s ) u 0 ( s ) d s d t b 1 0 1 φ ( t ) 0 1 G ( t , s ) d s d t 0 1 u 0 ( t ) φ ( t ) d t = ( λ 1 + ε ) 0 1 φ ( t ) 0 1 G ( t , s ) ( u 0 ( s ) + u ˜ ( s ) ) d s d t ( λ 1 + ε ) 0 1 φ ( t ) 0 1 G ( t , s ) u ˜ ( s ) d s d t b 1 0 1 φ ( t ) 0 1 G ( t , s ) d s d t 0 1 u 0 ( t ) φ ( t ) d t ε λ 1 1 0 1 φ ( t ) ( u 0 ( t ) + u ˜ ( t ) ) d t + 0 1 u ˜ ( t ) φ ( t ) d t ( λ 1 + ε ) 0 1 φ ( t ) 0 1 G ( t , s ) u ˜ ( s ) d s d t b 1 0 1 φ ( t ) 0 1 G ( t , s ) d s d t ε ω λ 1 1 u 0 + u ˜ + b K 1 K 2 Γ ( α ) b ( λ 1 + ε ) K 2 3 Γ ( α ) b 1 K 2 2 Γ ( α ) ε ω λ 1 1 u 0 ε ω λ 1 1 u ˜ + b K 1 K 2 Γ ( α ) b ( λ 1 + ε ) K 2 3 Γ ( α ) b 1 K 2 2 Γ ( α ) ε ω λ 1 1 R ε ω λ 1 1 b K 2 Γ ( α ) + b K 1 K 2 Γ ( α ) b ( λ 1 + ε ) K 2 3 Γ ( α ) b 1 K 2 2 Γ ( α ) > 0 .
On the other hand, we have by (3.4)
0 1 u 0 ( t ) φ ( t ) d t 0 1 ( A u 0 ) ( t ) φ ( t ) d t = μ 0 0 1 u ( t ) φ ( t ) d t 0 .
That is a contradiction. As a result of this, (3.3) holds. Lemma 2.5 gives
deg ( I A , B R , 0 ) = 0 .
(3.5)
It follows from the third inequality of (3.1) that there exists 0 < r < R such that | f ( t , u ) | λ 2 | u | , | u | r , t [ 0 , 1 ] . In the following, we prove
A u μ u , u B r , μ 1 .
(3.6)
In fact, suppose that there exist u 1 B r , μ 1 1 such that A u 1 = μ 1 u 1 . We may suppose that μ 1 > 1 (otherwise we are done). Thus
μ 1 | u 1 ( t ) | λ 2 0 1 G ( t , s ) | u 1 ( s ) | d s .
(3.7)
Multiply by φ ( t ) both sides of the preceding inequality and integrate over [ 0 , 1 ] , and use (2.3) to obtain
μ 1 0 1 | u 1 ( t ) | φ ( t ) d t λ 2 0 1 φ ( t ) 0 1 G ( t , s ) | u 1 ( s ) | d s d t 0 1 | u 1 ( t ) | φ ( t ) d t .
This, together with 0 1 | u 1 ( t ) | φ ( t ) d t > 0 , leads to μ 1 1 , which is a contradiction. So, (3.6) holds. Lemma 2.6 implies
deg ( I A , B r , 0 ) = 1 .
(3.8)

By (3.5) and (3.8), we have deg ( I A , B R B ¯ r , 0 ) = deg ( I A , B R , 0 ) deg ( I A , B r , 0 ) = 0 1 = 1 . Then A has at least one fixed point on B R B ¯ r . This means that problem (1.1) has at least one nontrivial solution. □

In order to prove Theorem 3.2, we need the following result involving the spectral radius of L, denoted by r ( L ) .

Lemma 3.1 0 < r ( L ) K 2 .

Proof We easily obtain the result by Gelfand’s theorem and (2.2). This completes the proof. □

Theorem 3.2 If there exists a constant b 0 such that
f ( t , u ) b , u R , lim inf u 0 f ( t , u ) | u | > λ 1 , lim sup u + f ( t , u ) u < λ 2 ,
(3.9)

then (1.1) has at least one nontrivial solution.

Proof By the second inequality of (3.9), there exist ε > 0 and r 1 > 0 such that
f ( t , u ) ( λ 1 + ε ) | u | , | u | r 1 , t [ 0 , 1 ] .
(3.10)
For every u B ¯ r 1 , we have from (3.10) that
( A u ) ( t ) ( λ 1 + ε ) 0 1 G ( t , s ) | u ( s ) | d s , t [ 0 , 1 ]
and thus A ( B ¯ r 1 ) P . For all u B r 1 P , from (3.10), we know
( A u ) ( t ) ( λ 1 + ε ) 0 1 G ( t , s ) u ( s ) d s , t [ 0 , 1 ] .
We may suppose that A has no fixed point on B r 1 (otherwise, the proof is completed). Now we show that
u A u + μ u , u B r 1 P , μ 0 ,
(3.11)
where u P . Otherwise, there exist u 0 B r 1 P , μ 0 0 such that u 0 = A u 0 + μ 0 u A u 0 . Consequently,
u 0 ( t ) ( A u 0 ) ( t ) ( λ 1 + ε ) 0 1 G ( t , s ) u 0 ( s ) d s .
Multiply by φ ( t ) both sides of the preceding inequality and integrate over [ 0 , 1 ] , and use (2.3) to obtain
0 1 u 0 ( t ) φ ( t ) d t ( λ 1 + ε ) 0 1 φ ( t ) 0 1 G ( t , s ) u 0 ( s ) d s d t ( λ 1 + ε ) λ 1 1 0 1 u 0 ( t ) φ ( t ) d t ,
which implies 0 1 u 0 ( t ) φ ( t ) d t = 0 , and then u 0 ( t ) 0 , t [ 0 , 1 ] . It contradicts u 0 B r 1 P . Hence (3.11) is true. Since A ( B ¯ r 1 ) P , we have, from the permanence property of fixed point index and Lemma 2.5, that
deg ( I A , B r 1 , 0 ) = i ( A , B r 1 P , P ) = 0 ,
(3.12)
where i denotes fixed point index on P. Recall the definition of u ˜ . Clearly, u ˜ P and A : C [ 0 , 1 ] P u ˜ by (3.9). Define A ˜ u = A ( u u ˜ ) + u ˜ , u C [ 0 , 1 ] . We easily find A ˜ : C [ 0 , 1 ] P . By the third inequality of (3.9), there exist r 2 > r 1 + u ˜ = r 1 + b K 2 Γ ( α ) and 0 < σ < 1 such that
f ( t , u ) σ λ 2 u , u r 2 , t [ 0 , 1 ] .
(3.13)
Let L 1 u = σ λ 2 L u , u C [ 0 , 1 ] . Then L 1 : C [ 0 , 1 ] C [ 0 , 1 ] is a bounded linear operator and L 1 ( P ) P . Let
M = 2 max { sup u B ¯ r 2 0 1 G ( t , s ) | f ( s , u ( s ) ) | d s , 2 b K 2 Γ ( α ) } < +
and W : = { u P : u = μ A ˜ u , 0 μ 1 } . In what follows, we will show that W is bounded. For all u W , let ψ ˜ ( t ) = min { u ( t ) u ˜ ( t ) , r 2 } and e ( u ) = { t [ 0 , 1 ] : u ( t ) u ˜ ( t ) > r 2 } . When u ( t ) u ˜ ( t ) < 0 , ψ ˜ ( t ) = u ( t ) u ˜ ( t ) u ( t ) r 2 r 2 , and so ψ ˜ r 2 . Consequently, for u W , we have from (3.13)
u ( t ) = μ ( A ˜ u ) ( t ) 0 1 G ( t , s ) f ( s , u ( s ) u ˜ ( s ) ) d s + u ˜ ( t ) = e ( u ) G ( t , s ) f ( s , u ( s ) u ˜ ( s ) ) d s + [ 0 , 1 ] e ( u ) G ( t , s ) f ( s , u ( s ) u ˜ ( s ) ) d s + u ˜ ( t ) σ λ 2 0 1 G ( t , s ) u ( s ) d s + 0 1 G ( t , s ) f ( s , ψ ˜ ( s ) ) d s + 2 u ˜ ( t ) σ λ 2 0 1 G ( t , s ) u ( s ) d s + M = ( L 1 u ) ( t ) + M

and then ( ( I L 1 ) u ) ( t ) M , t [ 0 , 1 ] . By Lemma 3.1 and 0 < σ < 1 , r ( L 1 ) = σ λ 2 r ( L ) σ λ 2 K 2 < 1 . Therefore, the inverse operator ( I L 1 ) 1 exists and ( I L 1 ) 1 = I + L 1 + L 1 2 + + L 1 n +  . It follows from L 1 ( P ) P that ( I L 1 ) 1 ( P ) P . So, we have u ( t ) ( I L 1 ) 1 M , t [ 0 , 1 ] and W is bounded.

Select r 3 > max { r 2 , sup W + b K 2 Γ ( α ) } and thus A ˜ has no fixed point on B r 3 . Indeed, if there exists u 1 B r 3 such that A ˜ u 1 = u 1 , then u 1 W and u 1 = r 3 > sup W , which is a contradiction. Then we have from the permanence property and the homotopy invariance property of fixed point index that
deg ( I A ˜ , B r 3 , 0 ) = i ( A ˜ , B r 3 P , P ) = i ( 0 , B r 3 P , P ) = 1 .
(3.14)
Set the completely continuous homotopy H ( t , u ) = A ( u t u ˜ ) + t u ˜ , ( t , u ) [ 0 , 1 ] × B ¯ r 3 . If there exists ( t 0 , u 2 ) [ 0 , 1 ] × B r 3 such that H ( t 0 , u 2 ) = u 2 , and then A ( u 2 t 0 u ˜ ) = u 2 t 0 u ˜ and A ˜ ( u 2 t 0 u ˜ + u ˜ ) = u 2 t 0 u ˜ + u ˜ . Thus u 2 t 0 u ˜ + u ˜ W and u 2 t 0 u ˜ + u ˜ u 2 ( 1 t 0 ) u ˜ r 3 u ˜ > sup W , which is a contradiction. From the homotopy invariance of topological degree and (3.14), we have
deg ( I A , B r 3 , 0 ) = deg ( I A ˜ , B r 3 , 0 ) = 1 .
(3.15)

By (3.12) and (3.15), we get deg ( I A , B r 3 B ¯ r 1 , 0 ) = deg ( I A , B r 3 , 0 ) deg ( I A , B r 1 , 0 ) = 1 , which implies that A has at least one fixed point on B r 3 B ¯ r 1 . This means that the problem (1.1) has at least one nontrivial solution. □

Two examples 1. Let
f ( t , u ) = a 1 u + a 2 u 2 + + a n u n , ( t , u ) [ 0 , 1 ] × R ,
where n is a positive even number, a i R ( i = 1 , 2 , , n 1 ), | a 1 | < λ 2 , a n > 0 . It is easy to see that f ( t , u ) is bounded below and usually sign-changing for u 0 . In addition, lim sup u 0 | f ( t , u ) u | = | a 1 | < λ 2 and lim inf u + f ( t , u ) u = + . Thus by Theorem 3.1, we can obtain the existence of a nontrivial solution of (1.1).
  1. 2.
    Let
    f ( t , u ) = 1 u 2 1 + u 2 , ( t , u ) [ 0 , 1 ] × R .
     

It is easy to see that f ( t , u ) is bounded below and usually sign-changing for u 0 . In addition, lim sup u + f ( t , u ) u = 0 < λ 2 and lim inf u 0 f ( t , u ) | u | = + . Thus, by Theorem 3.2, we can obtain the existence of a nontrivial solution of (1.1).

Declarations

Acknowledgements

Research is supported by the NNSF-China (10971046), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007, A2012402036), GIIFSDU (yzc12063), IIFSDU (2012TS020).

Authors’ Affiliations

(1)
Department of mathematics, Qilu Normal University
(2)
School of Mathematics, Shandong University

References

  1. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
  2. Kilbas A, Srivastava H, Trujillo J: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATHGoogle Scholar
  3. Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.MATHGoogle Scholar
  4. Bai Z, Lü H: 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
  5. Jiang D, Yuan C: The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. 2010, 72: 710-719. 10.1016/j.na.2009.07.012MathSciNetView ArticleMATHGoogle Scholar
  6. El-Shahed M, Nieto J: Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order. Comput. Math. Appl. 2010, 59: 3438-3443. 10.1016/j.camwa.2010.03.031MathSciNetView ArticleMATHGoogle Scholar
  7. Wang F, Liu ZH: Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. Adv. Differ. Equ. 2012., 2012: Article ID 116Google Scholar
  8. Ahmad B, Nieto J: Riemann-Liouville fractional differential equations with fractional boundary conditions. Fixed Point Theory 2012, 13: 329-336.MathSciNetMATHGoogle Scholar
  9. Zhang L, Ahmad B, Wang G, Agarwal RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 2013, 249: 51-56.MathSciNetView ArticleMATHGoogle Scholar
  10. Ahmad B, Ntouyas S, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multi-strip boundary conditions. Math. Probl. Eng. 2013., 2013: Article ID 320415Google Scholar
  11. Guo Y: Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations. Bull. Korean Math. Soc. 2010, 47: 81-87. 10.4134/BKMS.2010.47.1.081MathSciNetView ArticleMATHGoogle Scholar
  12. Ferreira R: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 70Google Scholar
  13. Jia M, Zhang X, Gu X: Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions. Bound. Value Probl. 2012., 2012: Article ID 70. doi:10.1186/1687-2770-2012-70Google Scholar
  14. Yang L, Chen HB: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. 2011., 2011: Article ID 404917Google Scholar
  15. Sudsutad W, Tariboon J: Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions. Adv. Differ. Equ. 2012., 2012: Article ID 93Google Scholar
  16. El-Shahed M: Positive solutions for boundary value problems of nonlinear fractional differential equation. Abstr. Appl. Anal. 2007., 2007: Article ID 10368Google Scholar
  17. Xu JF, Wei ZL, Dong W: Uniqueness of positive solutions for a class of fractional boundary value problems. Appl. Math. Lett. 2012, 25: 590-593. 10.1016/j.aml.2011.09.065MathSciNetView ArticleMATHGoogle Scholar
  18. Xu JF, Yang ZL: Multiple positive solutions of a singular fractional boundary value problem. Appl. Math. E-Notes 2010, 10: 259-267.MathSciNetMATHGoogle Scholar
  19. Xu JF, Wei ZL, Ding YZ: Positive solutions for a boundary-value problem with Riemann-Liouville’s fractional derivative. Lith. Math. J. 2012, 52: 462-476. 10.1007/s10986-012-9187-zMathSciNetView ArticleMATHGoogle Scholar
  20. Wei ZL, Li Q, Che J: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 2010, 367: 260-272. 10.1016/j.jmaa.2010.01.023MathSciNetView ArticleMATHGoogle Scholar
  21. Wei ZL, 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
  22. Sun J, Zhang G: Nontrivial solutions of singular superlinear Sturm-Liouville problems. J. Math. Anal. Appl. 2006, 313: 518-536. 10.1016/j.jmaa.2005.06.087MathSciNetView ArticleMATHGoogle Scholar
  23. Sun J, Zhang G: Nontrivial solutions of singular sublinear Sturm-Liouville problems. J. Math. Anal. Appl. 2007, 326: 242-251. 10.1016/j.jmaa.2006.03.003MathSciNetView ArticleMATHGoogle Scholar
  24. Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando; 1988.MATHGoogle Scholar

Copyright

© Zhang and Xu; 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.