- Research Article
- Open Access

# The Existence of Positive Solution to a Nonlinear Fractional Differential Equation with Integral Boundary Conditions

- Meiqiang Feng
^{1}Email author, - Xiaofang Liu
^{1}and - Hanying Feng
^{2}

**2011**:546038

https://doi.org/10.1155/2011/546038

© Meiqiang Feng et al. 2011

**Received:**19 December 2010**Accepted:**26 January 2011**Published:**24 February 2011

## Abstract

The expression and properties of Green's function for a class of nonlinear fractional differential equations with integral boundary conditions are studied and employed to obtain some results on the existence of positive solutions by using fixed point theorem in cones. The proofs are based on the reduction of the problem considered to the equivalent Fredholm integral equation of the second kind. The results significantly extend and improve many known results even for integer-order cases.

## Keywords

- Fractional Derivative
- Fixed Point Theorem
- Fractional Differential Equation
- Real Banach Space
- Fixed Point Theory

## 1. Introduction

Fractional calculus is an area having a long history, its infancy dates back to three hundred years, the beginnings of classical calculus. It had attracted the interest of many old famous mathematicians, such as L'Hospital, Leibniz, Liouville, Riemann, Grünward, Letnikov, and so forth [1, 2]. As the old mathematicians expected, in recent several decades fractional differential equations have been found to be a powerful tool in more and more fields, such as materials, physics, mechanics, and engineering [1–5]. For the basic theory and recent development of the subject, we refer the reader to a text by Lakshmikantham et al. [6]. For more details and examples, see [7–24] and the references therein. However, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages, and many aspects of this theory need to be explored.

where , is the standard Riemann-Liouville differentiation, and is a given continuous function.

where is a real number, and is the Caputo's fractional derivative. The function is continuous on .

, , are real numbers, and is a Banach space.

where , and may be singular at or/and at is the standard Riemann-Liouville differentiation, is nonnegative, and .

In the case of , for all , boundary value problem (1.5) reduces to the problem studied by Kaufmann and Mboumi [19]. In [19], the authors used the fixed point theorems to show sufficient conditions for the existence of at least one and at least three positive solutions to problem (1.5). For the case of , boundary value problem (1.5) is related to a boundary value problems of integer-order differential equation. Feng et al. [25] considered the existence and multiplicity of positive solutions to boundary value problem (1.5) by applying the fixed point theory in a cone for strict set contraction operators.

The organization of this paper is as follows. We will introduce some lemmas and notations in the rest of this section. In Section 2, we present the expression and properties of Green's function associated with boundary value problem (1.5). In Section 3, we give some preliminaries about operator. In particular, we state fixed point theory in cones. In Section 4, the main results of boundary value problem (1.5) will be stated and proved. In Section 5, we offer some interesting discussion of the associated boundary value problem (1.5). Finally, conclusions in Section 6 close the paper.

The fractional differential equations-related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to fractional differential equations. The readers who are unfamiliar with this area can consult for example [1–6] for details.

Definition 1.1 (see [4]).

where is called Riemann-Liouville fractional integral of order .

Definition 1.2 (see [4]).

where , denotes the integer part of number , is called the Riemann-Liouville fractional derivative of order .

Lemma 1.3 (see [13]).

for some , , where is the smallest integer greater than or equal to .

## 2. Expression and Properties of Green's Function

In this section, we present the expression and properties of Green's function associated with boundary value problem (1.5).

Lemma 2.1.

Proof.

where , , and are defined by (2.3), (2.4), and (2.5), respectively. The proof is complete.

From (2.3), (2.4), and (2.5), we can prove that , and have the following properties.

Proposition 2.2.

Proposition 2.3.

Proof.

From the properties of , and the definition of , we can prove that the results of Proposition 2.3 hold.

Theorem 2.4.

- (i)
- (ii)

Remark 2.5.

## 3. Preliminaries

In this section, we give some preliminaries for discussing the existence of positive solutions of boundary value problem (1.5).

On the basis of Lemma 3.3 below we will establish in Section 4 the existence of positive solution to the problem (1.5). Here we make the following hypotheses:

(*H* _{
1
})
,
on any subinterval of (0,1) and
;

(*H* _{
2
})
and
uniformly with respect to
on
;

(*H* _{
3
})
, where
is defined by (2.21).

Lemma 3.1.

Let (H_{1})–(H_{3}) hold. Then boundary value problems (1.5) has a solution
if and only if
is a fixed point of
.

Proof.

From Lemma 2.1, we can prove the results of this Lemma.

Lemma 3.2.

Let (H_{1})–(H_{3}) hold. Then
and
is completely continuous.

Proof.

For any , by (3.2), we can obtain . Next by similar proof of Lemma 3.1 in [12] and Ascoli-Arzela theorem one can prove is completely continuous. So it is omitted.

Lemma 3.3 (see [26]).

## 4. Existence of Positive Solutions

In this section, we apply Lemma 3.3 to establish the existence of positive solutions for boundary value problems (1.5).

Theorem 4.1.

Suppose (H_{1})– (H_{3}) and
satisfies the following conditions.

(H_{4})There exists
such that
;

(H_{5})There exists
such that
.

Then boundary value problems (1.5) has at least one positive solution.

Proof.

Obviously, is a nonnegative continuous function, that is, , and .

which is a contradiction to the definition of . Hence, (4.4) holds.

which is a contradiction to (4.9). So, (4.10) holds.

By (ii) of Lemma 3.3, (4.4) and (4.10) yield that has a fixed point , . Thus it follows that boundary value problems (1.5) has at least one positive solution with . The proof is complete.

## 5. Discussion

In this section, we offer some interesting discussion associated with boundary value problems (1.5).

Obviously is continuous on , and it is easy to see that by , where is defined in (5.4).

## 6. Conclusions

In this paper, by using the fixed point theorem of cone, we have investigated the existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and have obtained some easily verifiable sufficient criteria which extend previous results. It is worth mentioning that there are still many problems that remain open in this vital field other than the results obtained in this paper: for example, whether or not we can study the fractional differential equations with integral boundary conditions at resonance (see, e.g., [27]), and whether or not we can give a unified approach applicable to many BVPs (see, e.g., [28–31]). More efforts are still needed in the future.

## Declarations

### Acknowledgments

The authors thank the referee for his/her careful reading of the paper and useful suggestions. This paper is sponsored by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the jurisdiction of Beijing Municipality (PHR201008430), the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201010772018), the Natural Sciences Foundation of Heibei Province (A2009001426), and the Beijing Excellent Training Grant (2010D005007000002).

## Authors’ Affiliations

## References

- Oldham KB, Spanier J:
*The Fractional Calculus*. Academic Press, London, UK; 1974:xiii+234.MATHGoogle Scholar - Podlubny I:
*Fractional Differential Equations, Mathematics in Science and Engineering*.*Volume 198*. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar - Miller KS, Ross B:
*An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication*. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.Google Scholar - Samko SG, Kilbas AA, Marichev OI:
*Fractional Integrals and Derivatives*. Gordon and Breach Science Publishers, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar - Kilbas AA, Srivastava HM, Trujillo JJ:
*Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies*.*Volume 204*. Elsevier Science, Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar - Lakshmikantham V, Leela S, Vasundhara Devi J:
*Theory of Fractional Dynamic Systems*. Cambridge Academic, Cambridge, UK; 2009.MATHGoogle Scholar - Lakshmikantham V, Vatsala AS:
**Basic theory of fractional differential equations.***Nonlinear Analysis. Theory, Methods & Applications*2008,**69**(8):2677-2682. 10.1016/j.na.2007.08.042MathSciNetView ArticleMATHGoogle Scholar - Daftardar-Gejji V:
**Positive solutions of a system of non-autonomous fractional differential equations.***Journal of Mathematical Analysis and Applications*2005,**302**(1):56-64. 10.1016/j.jmaa.2004.08.007MathSciNetView ArticleMATHGoogle Scholar - Daftardar-Gejji V, Bhalekar S:
**Boundary value problems for multi-term fractional differential equations.***Journal of Mathematical Analysis and Applications*2008,**345**(2):754-765. 10.1016/j.jmaa.2008.04.065MathSciNetView ArticleMATHGoogle Scholar - Ahmad B, Nieto JJ:
**Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions.***Nonlinear Analysis. Theory, Methods & Applications*2008,**69**(10):3291-3298. 10.1016/j.na.2007.09.018MathSciNetView ArticleMATHGoogle Scholar - Ahmad B, Nieto JJ:
**Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions.***Boundary Value Problems*2009,**2009:**-11.Google Scholar - Bai Z:
**On positive solutions of a nonlocal fractional boundary value problem.***Nonlinear Analysis. Theory, Methods & Applications*2010,**72**(2):916-924. 10.1016/j.na.2009.07.033MathSciNetView ArticleMATHGoogle Scholar - Bai Z, Lü H:
**Positive solutions for boundary value problem of nonlinear fractional differential equation.***Journal of Mathematical Analysis and Applications*2005,**311**(2):495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar - 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 Analysis. Theory, Methods & Applications*2010,**72**(2):710-719. 10.1016/j.na.2009.07.012MathSciNetView ArticleMATHGoogle Scholar - Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006, (36):1-12.Google Scholar
- Benchohra M, Hamani S, Ntouyas SK:
**Boundary value problems for differential equations with fractional order and nonlocal conditions.***Nonlinear Analysis. Theory, Methods & Applications*2009,**71**(7-8):2391-2396. 10.1016/j.na.2009.01.073MathSciNetView ArticleMATHGoogle Scholar - Salem HAH:
**On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order.***Mathematical and Computer Modelling*2008,**48**(7-8):1178-1190. 10.1016/j.mcm.2007.12.015MathSciNetView ArticleMATHGoogle Scholar - Salem HAH:
**On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies.***Journal of Computational and Applied Mathematics*2009,**224**(2):565-572. 10.1016/j.cam.2008.05.033MathSciNetView ArticleMATHGoogle Scholar - Kaufmann ER, Mboumi E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electronic Journal of Qualitative Theory of Differential Equations 2008, (3):1-11.Google Scholar
- Li CF, Luo XN, Zhou Y:
**Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations.***Computers & Mathematics with Applications*2010,**59**(3):1363-1375.MathSciNetView ArticleMATHGoogle Scholar - Zhang S:
**The existence of a positive solution for a nonlinear fractional differential equation.***Journal of Mathematical Analysis and Applications*2000,**252**(2):804-812. 10.1006/jmaa.2000.7123MathSciNetView ArticleMATHGoogle Scholar - Zhang S:
**Existence of positive solution for some class of nonlinear fractional differential equations.***Journal of Mathematical Analysis and Applications*2003,**278**(1):136-148. 10.1016/S0022-247X(02)00583-8MathSciNetView ArticleMATHGoogle Scholar - Zhong W, Lin W:
**Nonlocal and multiple-point boundary value problem for fractional differential equations.***Computers & Mathematics with Applications*2010,**59**(3):1345-1351.MathSciNetView ArticleMATHGoogle Scholar - Feng M, Zhang X, Ge W:
**New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions.***Boundary Value Problems*2011,**2011:**-20.Google Scholar - Feng M, Ji D, Ge W:
**Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces.***Journal of Computational and Applied Mathematics*2008,**222**(2):351-363. 10.1016/j.cam.2007.11.003MathSciNetView ArticleMATHGoogle Scholar - Guo D, Lakshmikantham V, Liu X:
*Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications*.*Volume 373*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:viii+341.View ArticleMATHGoogle Scholar - Zhang X, Feng M, Ge W:
**Existence result of second-order differential equations with integral boundary conditions at resonance.***Journal of Mathematical Analysis and Applications*2009,**353**(1):311-319. 10.1016/j.jmaa.2008.11.082MathSciNetView ArticleMATHGoogle Scholar - Infante G, Webb JRL:
**Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations.***Proceedings of the Edinburgh Mathematical Society. Series II*2006,**49**(3):637-656. 10.1017/S0013091505000532MathSciNetView ArticleMATHGoogle Scholar - Webb JRL:
**Positive solutions of some three point boundary value problems via fixed point index theory.***Nonlinear Analysis. Theory, Methods & Applications*2001,**47**(7):4319-4332. 10.1016/S0362-546X(01)00547-8MathSciNetView ArticleMATHGoogle Scholar - Webb JRL, Infante G:
**Positive solutions of nonlocal boundary value problems: a unified approach.***Journal of the London Mathematical Society. Second Series*2006,**74**(3):673-693. 10.1112/S0024610706023179MathSciNetView ArticleMATHGoogle Scholar - Webb JRL, Infante G, Franco D:
**Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions.***Proceedings of the Royal Society of Edinburgh. Section A*2008,**138**(2):427-446.MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.