New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions

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.

(1.1)

subject to the boundary conditions

(1.2)

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

(1.3)

where and

(1.4)

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

(1.5)

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.

(P)

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

(1.6)

where , is called Riemann-Liouville fractional integral of order .

Definition 1.3 (see [4]).

For a function given in the interval , the expression

(1.7)

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

(1.8)

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

(2.1)

is given by

(2.2)

where

(2.3)

(2.4)

(2.5)

Proof.

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

(2.6)

By , there is . Thus,

(2.7)

Differentiating (2.7), we have

(2.8)

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

(2.9)

By , we have

(2.10)

Therefore, the unique solution of BVP (2.1) is

(2.11)

where is defined by (2.4).

From (2.11), we have

(2.12)

It follows that

(2.13)

Substituting (2.13) into (2.11), we obtain

(2.14)

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

(2.15)

where

(2.16)

For , we have

(2.17)

So, by (2.4), we have

(2.18)

Similarly, for , we have .

1. (ii)

   Since , it is clear that is increasing with respect to for .

    

On the other hand, from the definition of , for given , we have

(2.19)

Let

(2.20)

Then, we have

(2.21)

and so,

(2.22)

Noticing , from (2.22), we have

(2.23)

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

(2.24)

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

(2.25)

Remark 2.7.

From (2.25), we have

(2.26)

Remark 2.8.

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

(2.27)

Proposition 2.9.

There exists such that

(2.28)

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

(2.29)

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

(2.30)

Case 2.

If , then from (2.4), we have

(2.31)

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

(2.32)

Therefore, . Letting , we have

(2.33)

Case 3.

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

(2.34)

In view of Remarks 2.6–2.8, we have

(2.35)

From (2.35), there exists a constant such that

(2.36)

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

Let

(2.37)

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

(2.38)

where

(2.39)

is defined by (2.16), is defined in Proposition 2.9.

Now, we show that (2.38) holds.

In fact, from Proposition 2.9, we have

(2.40)

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

(3.1)

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

(3.2)

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

(3.3)

Similarly, by (2.38), we obtain

(3.4)

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:

(4.1)

where denotes or and

(4.2)

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

(4.3)

that is, imply that

(4.4)

Next, turning to , there exists such that

(4.5)

where satisfies .

Set

(4.6)

then .

Chose . Then, for , we have

(4.7)

that is, imply that

(4.8)

Case 2.

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

(4.9)

that is, imply that

(4.10)

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

(4.11)

that is, imply that

(4.12)

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

(5.1)

Proof.

We consider condition . Choose with .

If , then by the proof of (4.4), we have

(5.2)

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

(5.3)

On the other hand, by , for we have

(5.4)

By (5.4), we have

(5.5)

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

(5.6)

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

(6.1)

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:

(7.1)

where

(7.2)

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

(7.3)

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.