A class of nonlinear fractional order differential equation

is investigated in this paper, where is the standard Riemann-Liouville fractional derivative of order , , . Using intermediate value theorem, we obtain a sufficient condition for the existence of the solutions for the above fractional order differential equations.

Consider the following boundary value problem

where is the standard Riemann-Liouville fractional derivative of order , and .

In the last few decades, many authors have investigated fractional differential equations which have been applied in many fields such as physics, mechanics, chemistry, engineering etc. (see references [1,6,10,21-23]). Especially, many works have been devoted to the study of initial value problems and bounded value problems for fractional order differential equations [12,13,15,24].

Recently, the existence of positive solutions of fractional differential equations has attracted many authors’ attention [2-5,8,9,12,14,17-20,25,26]. Using some fixed point theorems, they obtained some nice existence conditions for positive solutions.

In more recent works, Jiang and Yuan [7] considered the following boundary value problem of fractional differential equations

where is the standard Riemann-Liouville fractional derivative of order and is continuous. Using some properties of the Green function , they obtain some new sufficient conditions for the existence of positive solutions for the above problem.

Further, Li, Luo, and Zhou [4] investigated the following fractional order three-point boundary value problems

where is the standard Riemann-Liouville fractional derivative of order , and is continuous.

In this paper, we discuss the boundary value problem (1.1)-(1.2). Using some properties of the Green function and intermediate value theorem, we establish some sufficient conditions for the existence of the positive solutions of the problem (1.1)-(1.2).

The paper is arranged as follows: In Section 2, we introduce some definitions for fractional order differential equations and give our main results for the boundary value problem (1.1)-(1.2). We give some lemmas for our results in Section 3. In Section 4, we prove our main result; and finally, we give an example to illustrate our results.

In this section, we introduce some definitions and preliminary facts which are used in this paper.

Definition 2.1 ([1,10])

The fractional integral of order α with the lower limit for a function f is defined as

provided that the integral on the right-hand side is point-wise defined on , where Γ is the Gamma function.

Definition 2.2 ([1,10])

Riemann-Liouville derivative of order α with the lower limit for a function can be written as

where n is a positive integer.

We call the function a solution of (1.1)-(1.2) if with a fractional derivative of order α belongs to and satisfies Equation (1.1) and the boundary condition (1.2).

We also need to introduce some lemmas as follows, which will be used in the proof of our main theorems.

Lemma 2.1 ([26])

Assume thatwith a fractional derivative of orderbelongs to. Then, the fractional equation

has solutions

Lemma 2.2 ([26])

Assume thatwith a fractional derivative of orderbelongs to. Then

for some, , .

Lemma 2.3 ([16])

Suppose thatXbe a Banach space, is closed and convex. Assume thatUis a relatively open subset ofCwith, andis a completely continuous operator. Then, either

(i) Thas a fixed point in, or

(ii) there existandwith.

Throughout this paper, we assume that satisfies the following:

(H), and there exist two positive functionsandsuch that

where. Furthermore,

for any.

We have our main results:

Theorem 2.1Suppose that (H) holds. If

then the boundary value problem (1.1)-(1.2) has at least one solution, where

Let , equipped the norm

then Ω is a Banach space.

We first give some lemmas as follows:

Lemma 3.1Problem (1.1)-(1.2) is equivalent to the following integral equation

where

Proof The sufficiency is obvious, we only need to prove the necessity.

Suppose that is a solution of the problem (1.1)-(1.2). Integrating both sides of (1.1) of α order from 0 to t with respect to t, it follows that

According to (1.2) and (3.4), we have

Combining (3.4) and (3.5), we obtain

According to (3.3), it is easy to show that (3.2) holds. The proof is completed. □

Lemma 3.2For any, is continuous, andfor any.

Proof The continuity of for is obvious.

Let

we only need to show that for , the rest of the proof is similar or obvious. From the definition of , we have

for . The proof is completed. □

Let

then

The new Green’s function has the following properties:

Lemma 3.3is continuous for, and

Furthermore, for.

Lemma 3.4For any, is nonincreasing with respect to. Especially, for any, for, andfor. That is, where

and

Let

Then, , we have from Lemma 3.1, (3.6) and (3.9) that the integral Equation (3.2) can be rewritten as follows:

Let

Then, and (3.10) is equivalent to the following

We can divide our proof into the following two steps:

First, we replace by any real number μ, then (3.12) can be rewritten as

It suffices to show that for any given real number μ, (3.13) has a solution , which implies that Equation (1.1) has a solution which satisfies the first boundary value condition .

Second, we show that there exists a μ such that the solution of (3.13) satisfies , which implies that the solution of (1.1) also satisfies the boundary value condition .

In this section, we will prove the first step. For convenience sake, we define an operator T on the set Ω as follows:

Lemma 3.5Suppose that, and (2.4) hold, then the operatorTis completely continuous in Ω.

Proof It is easy to show that the operator T maps Ω into itself. We divide the proof into the following three steps.

Step 1. is continuous with respect to .

In fact, suppose that is a sequence in Ω, and converges to . Since is continuous with respect to , and it is obvious that is uniformly continuous with respect to from Lemma 3.3, then for any , there exists an integer N, when ,

which follows from (3.14)-(3.15) that

Thus, the operator T is continuous in Ω.

Step 2. T maps bounded set in Ω into bounded set.

Suppose that is a bounded set with for any . Then, we have from (2.4) and (3.14) that

This gives that the operator T maps bounded set into bounded set in Ω.

Step 3. T is equicontinuous in Ω.

It suffices to show that for any and any , as . We consider the following three cases:

(i) ;

(ii) ;

(iii) .

We only prove the case (i), the rest two cases are similar. Since B is bounded, then there exists a such that . According to (3.14), we have

According to Step 1-Step 3, the operator T is completely continuous in Ω. The proof is completed. □

Further, we have

Lemma 3.6Suppose that, and (2.4) and (2.6) holds, then, for any real numberμ, the integral Equation (3.13) has at least one solution.

Proof We only need to show that the operator T is priori bounded. Let

Define a set as follows

To show the existence of a fixed point of T by Lemma 2.3, we need to verify that the second possibility in Lemma 2.3 cannot happen.

In fact, assume that there exists with and such that . It follows that

and

Here we have the use of the inequality

It is obvious that (3.18) contradicts our assumption that . Therefore, by Lemma 2.3, it follows that T has a fixed point . Hence, the integral Equation (3.14) has at least a solution . The proof is completed. □

Now, we prove Theorem 2.1 by Lemma 3.4-3.5 and the intermediate value theorem.

Proof of Theorem 2.1 It is obvious that the right-hand side of (3.14) is continuously dependent on the parameter μ, so we need to find a μ such that , which implies that .

For any given real number μ, we rewrite (3.13) as follows:

From (4.1), it suffices to show that there exists a μ such that

It is obvious that is continuously dependent on the parameter μ. In order to prove that there exists a such that , we only need to show that , and .

Now, we show that . On the contrary, we assume that . Then, there exists a sequence , such that , which implies that the sequence is bounded from above. Notice that the function is continuous with respect to and . We first claim that it is impossible to have

as is large enough. Indeed, assume that (4.3) is true. Then, by (4.1), we have

for all . Thus, we get

for all . Since we have assumed in (H) that

by (4.2), (4.5)-(4.6), we have

which contradicts our assumption.

Now, for large , we define

Then, is not empty.

Further, we divide the set into two sets and as follows:

It is easy to know that , and , and we have from (H) that is not empty.

From (H) again, the function is bounded below by a constant for and . Thus, there exists a constant M (<0), independent of t and , such that

Let

From the definitions of and , we have

and it follows that as (since if is bounded below by a constant as , then (4.7) holds). Therefore, we can choose large enough so that

for . From (H), (4.1), (4.8)-(4.9), and the definitions of and , for any , we have

from which it follows that

which implies that

This contradicts (4.9).

Now, we have proved that . By a similar method, we can prove that . The detail is omitted.

Notice that is continuous with respect to . It follows from the intermediate value theorem [11] that there exists a such that , that is , which satisfies the second boundary value condition of (1.2). The proof is completed. □

Example 5.1 Consider the following boundary value problem

where

and

It is easy to show that

and

Thus, the conditions of Theorem 2.1 are satisfied. Therefore, the problem (5.1) has at least a nontrivial solution.

The authors declare that they have no competing interests.

Each of the authors, ZO and GL contributed to each part of this study equally and read and approved the final version of the mnanuscript.

Supported partially by China Postdoctoral Science Foundation under Grant No.20110491280 and the Subject Lead Foundation of University of South China No. 2007XQD13.

1. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amstadam (2006)

2. Bakhani, A, Daftardar-Gejji, V: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl.. 278, 434–442 (2003). Publisher Full Text

3. Bakhani, A, Daftardar-Gejji, V: Positive solutions of a system of non-autonomous fractional differential equations. J. Math. Anal. Appl.. 302, 56–64 (2005). Publisher Full Text

4. Li, CF, Luo, XN, Zhou, Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl.. 59, 1363–1375 (2010). Publisher Full Text

5. Bai, CZ: Positive solutions for nonlinear fractional differential equations with coefficient that changes sign. Nonlinear Anal.. 64, 677–685 (2006). Publisher Full Text

6. Delbosco, D: Fractional calculus and function spaces. J. Fract. Calc.. 6, 45–53 (1994)

7. Jiang, DQ, Yuan, CJ: The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal.. 72, 710–719 (2010). Publisher Full Text

8. Kaufmann, ER, Mboumi, E: Positive solutions of a boundary value problem for a nonlocal fractional differential equations. Electron. J. Qual. Theory Differ. Equ.. 3, 1–11 (2008)

9. Jafari, H, Gejji, VD: Positive solutions fractional nonlinear boundary value problems using Adomian decomposition method. Appl. Math. Comput.. 180, 700–706 (2006). Publisher Full Text

10. Podlubny, I: Fractional Differential Equations, Academic Press, New York (1993)

11. Liao, KR, Li, ZY: Mathematical Analysis, Higher Education Press, Beijing (1986) in Chinese

12. Belmekki, M, Benchohra, M: Existence results for fractional order semilinear functional differential equations with nondense domain. Nonlinear Anal.. 72, 925–932 (2010). Publisher Full Text

13. Benchohra, M, Hamani, S, Ntouyas, SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal.. 71, 2391–2396 (2009). Publisher Full Text

14. EI-Shahed, M: Positive solutions for boundary value problem of nonlinear fractional differential equations. Abstr. Appl. Anal.. 2007, (2007)

15. Kosmatov, N: Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal.. 70, 2521–2529 (2009). Publisher Full Text

16. Agarmal, RP, Meehan, M, O’Regan, D: Fixed Point Theory and Applications, Cambridge University Press, Cambridge (2001)

17. Agarwal, RP, Zhou, Y, He, Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl.. 59, 1095–1110 (2010). Publisher Full Text

18. Liang, SH, Zhang, JH: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal.. 71, 5545–5550 (2009). Publisher Full Text

19. Zhang, SQ: Positive solutions for boundary value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ.. 2006, 1–12 (2006)

20. Gejji, VD: Positive solutions of a system of non-autonomous fractional differential equations. J. Math. Anal. Appl.. 302, 56–64 (2005). Publisher Full Text

21. Lakshmikantham, V: Theory of fractional function differential equations. Nonlinear Anal.. 69, 3337–3343 (2008). Publisher Full Text

22. Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal.. 69, 2677–2682 (2008). Publisher Full Text

23. Lakshmikantham, V, Leela, S, Vasundhara Devi, J: Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge (2009)

24. Zhou, Y, Jiao, F, Li, J: Existence and uniqueness of fractional neutral differential equations with infinite delay. Nonlinear Anal.. 71, 3249–3256 (2009). Publisher Full Text

25. Bai, ZB: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal.. 72, 916–924 (2010). Publisher Full Text

26. Bai, ZB, Lu, HS: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl.. 311, 495–505 (2005). Publisher Full Text