- Research Article
- Open access
- Published:
New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions
Boundary Value Problems volume 2011, Article number: 720702 (2011)
Abstract
This paper investigates the existence and multiplicity of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions. The results are established by converting the problem into an equivalent integral equation and applying Krasnoselskii's fixed-point theorem in cones. The nonexistence of positive solutions is also studied.
1. Introduction
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode's analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth, and involves derivatives of fractional order. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. An excellent account in the study of fractional differential equations can be found in [1–5]. For the basic theory and recent development of the subject, we refer a text by Lakshmikantham [6]. For more details and examples, see [7–23] 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.
In [23], Zhang used a fixed-point theorem for the mixed monotone operator to show the existence of positive solutions to the following singular fractional differential equation.

subject to the boundary conditions

where is the standard Rimann-Liouville fractional derivative of order
, the nonlinearity
may be singular at
, and function
may be singular at
. The author derived the corresponding Green's function named by fractional Green's function and obtained some properties as follows.
Proposition 1.1.
Green's function satisfies the following conditions:
(i) for all
;
(ii)there exists a positive function such that

where and

here .
It is well known that the cone theoretic techniques play a very important role in applying Green's function in the study of solutions to boundary value problems. In [23], the author cannot acquire a positive constant taking instead of the role of positive function with
in (1.3). At the same time, we notice that many authors obtained the similar properties to that of (1.3), for example, see Bai [12], Bai and L
[13], Jiang and Yuan [14], Li et al, [15], Kaufmann and Mboumi [19], and references therein. Naturally, one wishes to find whether there exists a positive constant
such that

for the fractional order cases. In Section 2, we will deduce some new properties of Green's function.
Motivated by the above mentioned work, we study the following higher-order singular boundary value problem of fractional differential equation.

where is the standard Rimann-Liouville fractional derivative of order
and
may be singular at
or/and at
,
is nonnegative, and
.
For the case of , the boundary value problems () reduces to the problem studied by Eloe and Ahmad in [24]. In [24], the authors used the Krasnosel'skii and Guo [25] fixed-point theorem to show the existence of at least one positive solution if
is either superlinear or sublinear to problem (). For the case of
, the boundary value problems () is related to a m-point boundary value problems of integer-order differential equation. Under this case, a great deal of research has been devoted to the existence of solutions for problem (), for example, see Pang et al. [26], Yang and Wei [27], Feng and Ge [28], and references therein. All of these results are based upon the fixed-point index theory, the fixed-point theorems and the fixed-point theory in cone for strict set contraction operator.
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 (). In Section 3, we discuss some characteristics of the integral operator associated with the problem () and state a fixed-point theorem in cones. In Section 4, we discuss the existence of at least one positive solution of boundary value problem (). In Section 5, we will prove the existence of two or positive solutions, where
is an arbitrary natural number. In Section 6, we study the nonexistence of positive solution of boundary value problem (). In Section 7, one example is also included to illustrate the main results. Finally, conclusions in Section 8 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 1.2 (see [4]).
The integral

where , is called Riemann-Liouville fractional integral of order
.
Definition 1.3 (see [4]).
For a function given in the interval
, the expression

where denotes the integer part of number
, is called the Riemann-Liouville fractional derivative of order
.
Lemma 1.4 (see [13]).
Assume that with a fractional derivative of order
that belongs to
. Then

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 ().
Lemma 2.1.
Assume that Then for any
, the unique solution of boundary value problem

is given by

where



Proof.
By Lemma 1.4, we can reduce the equation of problem (2.1) to an equivalent integral equation

By , there is
. Thus,

Differentiating (2.7), we have

By (2.8) and we have
Similarly, we can obtain that
Then

By , we have

Therefore, the unique solution of BVP (2.1) is

where is defined by (2.4).
From (2.11), we have

It follows that

Substituting (2.13) into (2.11), we obtain

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.
The function defined by (2.4) satisfies
(i) is continuous for all
;
(ii)for all , one has

where

Proof.
-
(i)
It is obvious that
is continuous on
and
when
.
For , we have

So, by (2.4), we have

Similarly, for , we have
.
-
(ii)
Since
, it is clear that
is increasing with respect to
for
.
On the other hand, from the definition of , for given
, we have

Let

Then, we have

and so,

Noticing , from (2.22), we have

Then, for given , we have
arrives at maximum at
when
. This together with the fact that
is increasing on
, we obtain that (2.15) holds.
Remark 2.3.
From Figure 1, we can see that for
. If
, then

Remark 2.4.
From Figure 2, we can see that is increasing with respect to
.
Remark 2.5.
From Figure 3, we can see that for
, where
.
Remark 2.6.
Let . From (2.15), for
, we have

Remark 2.7.
From (2.25), we have

Remark 2.8.
From Figure 4, it is easy to obtain that is decreasing with respect to
, and

Proposition 2.9.
There exists such that

Proof.
For , we divide the proof into the following three cases for
.
Case 1.
If , then from (i) of Proposition 2.2 and Remark 2.5, we have

It is obvious that and
are bounded on
. So, there exists a constant
such that

Case 2.
If , then from (2.4), we have

On the other hand, from the definition of , we obtain that
takes its maximum
at
. So

Therefore, . Letting
, we have

Case 3.
If , from (i) of Proposition 2.2, it is clear that

In view of Remarks 2.6–2.8, we have

From (2.35), there exists a constant such that

Letting and using (2.30), (2.33), and (2.36), it follows that (2.28) holds. This completes the proof.
Let

Proposition 2.10.
If , then one has
(i) is continuous for all
;
(ii).
Proof.
Using the properties of , definition of
, it can easily be shown that (i) and (ii) hold.
Theorem 2.11.
If , the function
defined by (2.3) satisfies
(i) is continuous for all
;
(ii) for each
, and

where

is defined by (2.16),
is defined in Proposition 2.9.
Proof.
-
(i)
From Propositions 2.2 and 2.10, we obtain that
is continuous for all
, and
.
-
(ii)
From (ii) of Proposition 2.2 and (ii) of Proposition 2.10, we have that
for each
.
Now, we show that (2.38) holds.
In fact, from Proposition 2.9, we have

Then the proof of Theorem 2.11 is completed.
Remark 2.12.
From the definition of , it is clear that
.
3. Preliminaries
Let and
denote a real Banach space with the norm
defined by
Let

To prove the existence of positive solutions for the boundary value problem (), we need the following assumptions:
() on any subinterval of (0,1) and
, where
is defined in Theorem 2.11;
() and
uniformly with respect to
on
;
(), where
is defined by (2.37).
From condition , it is not difficult to see that
may be singular at
or/and at
, that is,
or/and
.
Define by

where is defined by (2.3).
Lemma 3.1.
Let hold. Then boundary value problems () has a solution
if and only if
is a fixed point of
.
Proof.
From Lemma 2.1, we can prove the result of this lemma.
Lemma 3.2.
Let hold. Then
and
is completely continuous.
Proof.
For any , by (3.2), we can obtain that
. On the other hand, by (ii) of Theorem 2.11, we have

Similarly, by (2.38), we obtain

So, and hence
. Next by similar proof of Lemma
in [13] and Ascoli-Arzela theorem one can prove
is completely continuous. So it is omitted.
To obtain positive solutions of boundary value problem (), the following fixed-point theorem in cones is fundamental which can be found in [25, page 94].
Lemma 3.3 (Fixed-point theorem of cone expansion and compression of norm type).
Let be a cone of real Banach space
, and let
and
be two bounded open sets in
such that
and
. Let operator
be completely continuous. Suppose that one of the two conditions
(i) and
or
(ii) and
is satisfied. Then has at least one fixed point in
.
4. Existence of Positive Solution
In this section, we impose growth conditions on which allow us to apply Lemma 3.3 to establish the existence of one positive solution of boundary value problem (), and we begin by introducing some notations:

where denotes
or
and

Theorem 4.1.
Assume that hold. In addition, one supposes that one of the following conditions is satisfied:
and
(particularly,
and
).
there exist two constants with
such that
is nondecreasing on
for all , and
, and
for all
. Then boundary value problem () has at least one positive solution.
Proof.
Let be cone preserving completely continuous that is defined by (3.2).
Case 1.
The condition holds. Considering
, there exists
such that
, for
, where
satisfies
. Then, for
, we have

that is, imply that

Next, turning to , there exists
such that

where satisfies
.
Set

then .
Chose . Then, for
, we have

that is, imply that

Case 2.
The Condition satisfies. For
, from (3.1) we obtain that
. Therefore, for
, we have
for
, this together with
, we have

that is, imply that

On the other hand, for , we have that
for
, this together with
, we have

that is, imply that

Applying Lemma 3.3 to (4.4) and (4.8), or (4.10) and (4.12), yields that has a fixed point
or
with
. Thus it follows that boundary value problems () has a positive solution
, and the theorem is proved.
Theorem 4.2.
Assume that hold. In addition, one supposes that the following condition is satisfied:
and
(particularly,
and
).
Then boundary value problem () has at least one positive solution.
5. The Existence of Multiple Positive Solutions
Now we discuss the multiplicity of positive solutions for boundary value problem (). We obtain the following existence results.
Theorem 5.1.
Assume , and the following two conditions:
and
(particularly,
);
there exists such that
Then boundary value problem () has at least two positive solutions , which satisfy

Proof.
We consider condition . Choose
with
.
If , then by the proof of (4.4), we have

If , then similar to the proof of (4.4), we have

On the other hand, by , for
we have

By (5.4), we have

Applying Lemma 3.3 to (5.2), (5.3), and (5.5) yields that has a fixed point
, and a fixed point
Thus it follows that boundary value problem () has at least two positive solutions
and
. Noticing (5.5), we have
and
. Therefore (5.1) holds, and the proof is complete.
Theorem 5.2.
Assume , and the following two conditions:
and
;
there exists such that
Then boundary value problem () has at least two positive solutions , which satisfy

Theorem 5.3.
Assume that ,
and
hold. If there exist
positive numbers
with
such that
for
and
for
Then boundary value problem () has at least positive solutions
satisfying
Theorem 5.4.
Assume that ,
and
hold. If there exist
positive numbers
with
such that
is nondecreasing on
for all
;
, and
Then boundary value problem () has at least positive solutions
satisfying
6. The Nonexistence of Positive Solution
Our last results corresponds to the case when boundary value problem () has no positive solution.
Theorem 6.1.
Assume and
, then boundary value problem () has no positive solution.
Proof.
Assume to the contrary that is a positive solution of the boundary value problem (). Then,
, and

which is a contradiction, and complete the proof.
Similarly, we have the following results.
Theorem 6.2.
Assume and
, then boundary value problem () has no positive solution.
7. Example
To illustrate how our main results can be used in practice we present an example.
Example 7.1.
Consider the following boundary value problem of nonlinear fractional differential equations:

where

It is easy to see that hold. By simple computation, we have

thus it follows that problem (7.1) has a positive solution by .
8. Conclusions
In this paper, by using the famous Guo-Krasnoselskii fixed-point theorem, we have investigated the existence and multiplicity of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and obtained some easily verifiable sufficient criteria. The interesting point is that we obtain some new positive properties of Green's function, which significantly extend and improve many known results for fractional order cases, for example, see [12–15, 19]. The methodology which we employed in studying the boundary value problems of integer-order differential equation in [28] can be modified to establish similar sufficient criteria for higher-order nonlinear fractional differential equations. It is worth mentioning that there are still many problems that remain open in this vital field except for the results obtained in this paper: for example, whether or not we can obtain the similar results of fractional differential equations with p-Laplace operator by employing the same technique of this paper, and whether or not our concise criteria can guarantee the existence of positive solutions for higher-order nonlinear fractional differential equations with impulses. More efforts are still needed in the future.
References
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.
Oldham KB, Spanier J: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering. Volume 11. Academic Press, London, UK; 1974:xiii+234.
Podlubny I: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.
Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, Switzerland; 1993:xxxvi+976.
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier, Amsterdam, The Netherlands; 2006.
Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic, Cambridge, UK; 2009.
Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(8):2677-2682. 10.1016/j.na.2007.08.042
Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(10):3337-3343. 10.1016/j.na.2007.09.025
Lakshmikantham V, Vatsala AS: General uniqueness and monotone iterative technique for fractional differential equations. Applied Mathematics Letters 2008, 21(8):828-834. 10.1016/j.aml.2007.09.006
Ahmad B, Nieto JJ: Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations. Abstract and Applied Analysis 2009, 2009:-9.
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.
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.2009.07.033
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.2005.02.052
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 and Applications 2009, 72(2):710-719.
Li CF, Luo XN, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Computers and Mathematics with Applications 2010, 59(3):1363-1375. 10.1016/j.camwa.2009.06.029
Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Analysis: Theory, Methods and Applications 2009, 71(7-8):2391-2396. 10.1016/j.na.2009.01.073
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.2007.12.015
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.2008.05.033
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, 2008(3):1-11.
Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006, 2006: 1-12.
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.2000.7123
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)00583-8
Zhang S: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Computers and Mathematics with Applications 2010, 59(3):1300-1309. 10.1016/j.camwa.2009.06.034
Eloe PW, Ahmad B: Positive solutions of a nonlinear n th order boundary value problem with nonlocal conditions. Applied Mathematics Letters 2005, 18(5):521-527. 10.1016/j.aml.2004.05.009
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.
Pang C, Dong W, Wei Z: Green's function and positive solutions of n th order m -point boundary value problem. Applied Mathematics and Computation 2006, 182(2):1231-1239. 10.1016/j.amc.2006.05.010
Yang J, Wei Z: Positive solutions of n th order m -point boundary value problem. Applied Mathematics and Computation 2008, 202(2):715-720. 10.1016/j.amc.2008.03.009
Feng M, Ge W: Existence results for a class of n th order m -point boundary value problems in Banach spaces. Applied Mathematics Letters 2009, 22(8):1303-1308. 10.1016/j.aml.2009.01.047
Acknowledgments
The authors thank the referee for his/her careful reading of the manuscript and useful suggestions. These have greatly improved this paper. This work 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 2010 level of scientific research of improving project (5028123900), the Graduate Technology Innovation Project (5028211000) and Beijing Municipal Education Commission (71D0911003).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Feng, M., Zhang, X. & Ge, W. New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions. Bound Value Probl 2011, 720702 (2011). https://doi.org/10.1155/2011/720702
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/720702