Boundary Value Problems volume 2011, Article number: 720702 (2011)
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.
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 
.
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.
It is obvious that 
is continuous on 
and 
when 
.
For 
, we have
So, by (2.4), we have
Similarly, for 
, we have 
.
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
Graph of functions 
for 
.
Remark 2.4.
From Figure 2, we can see that 
is increasing with respect to 
.
Graph of function
for
.
Remark 2.5.
From Figure 3, we can see that 
for 
, where 
.
Graph of function
for
,
.
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
Graph of function
for
.
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.
From Propositions 2.2 and 2.10, we obtain that 
is continuous for all 
, and 
.
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 
.
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 
.
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.
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 
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.
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 
.
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.
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
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).
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.
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