#### Abstract

By using the method of upper and lower solutions and fixed point theorems, the existence of solutions for a Riemann-Liouville fractional boundary value problem with the nonlinear term depending on fractional derivative of lower order is obtained under the classical Nagumo conditions. Also, some results concerning Riemann-Liouville fractional derivative at extreme points are established with weaker hypotheses, which improve some works in Al-Refai (2012). As applications, an example is presented to illustrate our main results.

#### 1. Introduction

Fractional calculus, which is a powerful tool for the description of memory and hereditary properties of materials and processes, has attracted the attention of many researchers and has been successfully applied in various fields, such as economics, engineering, and physical sciences. For the development of the theory of fractional calculus, we refer the readers to the monographs [1, 2] and references therein.

Recently, some papers have dealt with the existence of the solutions for fractional boundary value problems mainly by means of fixed point theorems [3–5], Leray-Schauder continuation principle [6], critical point theory [7, 8], and the method of upper and lower solutions [9–12]. Therein, various kinds of boundary value problems for nonlinear fractional differential equations have been studied, and some excellent results have been established. We are particularly interested in the case where the nonlinear term depends explicitly on the fractional derivative of lower order. Specifically, Su and Zhang in [13] deal with a boundary value problem of a fractional differential equation with the nonlinear term dependent on a fractional derivative of lower order on the semi-infinite interval: where , , and and are the standard Riemann-Liouville fractional derivatives. The existence results for solutions are obtained by using Schauder's fixed point theorem on an unbounded domain. And in [14] Agarwal et al. investigate the existence of positive solutions for the singular Riemann-Liouville fractional Dirichlet boundary value problem: where , , and satisfying the Carathéodory conditions and being singular at . The proofs are based on a fixed point theorem on a cone, regularization, and sequential techniques.

It is well known that the method of upper and lower solutions is a powerful tool for proving the existence and multiplicity results of solutions for nonlinear differential equations. Using this method and monotone iterative technique, the authors in [10–12] investigate some nonlinear fractional differential equations with nonlinear boundary conditions and establish some fractional comparison principles and further obtain the existence results of solutions, including extremal solutions, yet, mainly focus on the case of order .

Nagumo conditions play an important role in the boundary value problems with nonlinear term involved in the derivative, since as it is known, for instance, for second order differential equations, the existence of upper and lower solutions, by itself, is not sufficient to ensure the existence of solutions. The studies dealing with the Nagumo conditions are well established by applying the method of upper and lower solutions combined with fixed point theorem or topological degree theory for the case of integer order (see [15–17]). To the best of our knowledge, no work has been done concerning the existence of solutions for fractional boundary value problem with nonlinear terms involving fractional derivative under Nagumo conditions.

Inspirited by the papers mentioned above, in this paper, under Nagumo conditions we aim to apply the method of upper and lower solutions combined with fixed point theorems to discuss the existence of solutions for the following Riemann-Liouville fractional boundary value problem (FBVP for short): where is continuous and is the Riemann-Liouville fractional derivative of order , . Our results extend some classical results for second order differential equations to the case of fractional order .

This paper is organized as follows. In Section 2, some notations, definitions, and lemmas are presented. We establish some results concerning the Riemann-Liouville fractional derivatives at extreme points under weaker conditions than those in [18]. In Section 3, sufficient conditions are given for the existence of at least one solution for FBVP (3). In Section 4, an explicit example is given to illustrate our main results.

#### 2. Preliminaries

In this section, we introduce some definitions and lemmas, which are used throughout this paper.

A function is Hölder continuous, if there exist nonnegative constant and exponent , such that A function is, especially, Lipschitz continuous, if the above inequality holds for .

*Definition 1 (see [1, 2]). *The Riemann-Liouville fractional integral of order of a function is given by
where is the gamma function, provided that the right side is pointwise defined on .

*Definition 2 (see [1, 2]). *The Riemann-Liouville fractional derivative of order of a function is given by
provided that the right side is pointwise defined on .

Lemma 3 (see [2]). *Suppose that exists and is integrable on , ; then
*

*Property 1 (see [1, 2]). *Let . The following properties are well known: (1): (2), , ,(3), , ,(4), where , provided that exists.

Lemma 4. *Suppose that and that , , then .*

*Proof. *For all , , we let
By Lemma 3, (8), and , one gets
Together with Property 1(3) and the continuity of , taking the limit , we obtain
Moreover,
From (11), (12), and the uniform convergence of , it follows that

Using (9) and Property 1(1)-(2) we obtain that exists and belongs to , satisfying

To the end, it suffices to show that , . Obviously, it follows from (9) and (14) that
For the homogeneous Abel integral equation (15), we observe that the integrand belongs to . Then by Lemma 2.5 in [1]
Taking the limit in (16), together with (13) we have
Hence , . The proof is complete.

*Remark 5. *In fact, if it holds that and that , , then by Lemma 3, Property 1(1), (3), and , it directly follows that , . With the above arguments in Lemma 4, we easily get the following result.

Lemma 6. *Suppose that and , , then , .*

Lemma 7. *If , then the following fractional boundary value problem
**
has a unique solution:
**
where
*

The proof is standard; we omit it here.

*Remark 8. *Obviously, and , .

In [18], Al-Refai obtained the following interesting result concerning the Riemann-Liouville fractional derivative at extreme points, where there is a little mistake. Now we state it correctly without proof.

Theorem 9 (see [18]). *Let attain its global minimum at ; then
*

*Remark 10. *In the sense of Riemann-Liouville fractional derivative of order , in general the assumption that is difficult to meet due to the fact that the fundamental solution of certain corresponding homogeneous differential equations possesses a singularity at . Hence, we hope to weaken the conditions of the above theorem.

In the following, we give some lemmas, wherein some ideas in the proofs come from [18, 19], with weaker hypotheses.

Lemma 11. *Assume that satisfies the following conditions: *(i)* exist, , for ;*(ii)*there exists constant , such that is Hölder continuous with exponent ;*(iii)* attains its global minimum at .**Then,
**
Moreover, if , then .*

*Proof. *Let , . Obviously, , satisfying the conditions (i)–(iii). It follows that
Since , we know that . Then for , it follows from proofs of Lemma 2.1 in [1] that

At this point, we choose enough small constant , such that . Since exists, . We consider
It obviously follows from that the second integration in (26) converges, and then the first integration in (26) also converges. For the latter, applying the integration by parts, together with (23), we have
due to the fact , , and .

Hence (26) yields that

From the Hölder continuity of on for some , it follows that there exists constant , such that
For , we have . Thus,
Dividing by on both sides of (30) and taking the limit , one gets
Together with (24), we obtain
The proof is complete.

*Remark 12. *Lemma 11 is an essential improvement of Theorem 9 and crucial for our main theorems. By applying the above results on , analogous results for Riemann-Liouville fractional derivatives at global maximum points are derived. It is worth mentioning that the weaker requirement in (ii) seems to be unsatisfactory as well, since in general the exponent may be guaranteed only up to . To solve the difficulty, the ideas of reducing the order and approach method are employed in our main theorems.

Spontaneously for the case , Lakshmikantham and Vatsala in [19] established the following result concerning Riemann-Liouville fractional derivative.

Theorem 13 (see [19]). *Let be locally Hölder continuous such that for any , we have
**
Then it follows that
*

*Remark 14. *As the literature [20] points out, in general, the function containing term is not locally Hölder continuous of any order. For this reason, in [20, 21] the authors attempt to weaken the locally Hölder continuity to continuity on ; nevertheless, their arguments seem to be flawed as well. Similarl to the proof of Lemma 11, we get the following lemma.

Lemma 15. *Assume that satisfies the following conditions: *(i)* exist, , for ;*(ii)*there exists constant , such that is Hölder continuous with exponent ;*(iii)* attains its global minimum at .**Then,
**
Moreover, if , then .*

*Remark 16. *Note that in the case , we allow to attain the minimum at the endpoint . In fact, analogous result for Lemma 15 at global maximum points is the generalization of Theorem 13.

Recently, some results, for instance, Lemma 2.13 in [22] and Property 4 [23] concerning Hölder continuity for Riemann-Liouville fractional integral operator have been obtained. Therein Bourdin in [23] has proved that fractional integral operator , maps functions to Hölder continuous functions, the exponent of which depends on and . In the following, we give some other results about Hölder continuity.

Lemma 17. *Let , . Then is Hölder continuous with exponent on .*

*Proof. *Similar to the proof of Property 4 [23], Lemma 17 is easily obtained. So we omit the proof.

Lemma 18 (see [24]). *Suppose that , ; then,
*

*Remark 19. *Obviously, Lemma 18 is valid under the assumptions , .

Lemma 20. *Let be Hölder continuous with exponent on . Then , is Hölder continuous with exponent on .*

*Proof. *From the Hölder continuity of on , it obviously follows that there exist some , such that and
Without loss of generality, let . Then,
If , using Lemma 18 we have
If , using the mean value theorem we have
The proof is complete.

Combining Lemmas 17 and 20, we have the following corollary.

Corollary 21. *Let , . Then , is Hölder continuous with exponent on .*

Now we introduce the upper and lower solutions of FBVP (3).

*Definition 22. *A function , satisfying with , , is called a lower solution of FBVP (3), if it satisfies
Analogously, a function , satisfying with , , is called an upper solution of FBVP (3), if it satisfies (41)-(42) with reversed inequalities.

*Definition 23. *Given a pair of functions satisfying . A function is said to satisfy the Nagumo condition with respect to and , if there exists a function such that
for all , and

#### 3. Main Results

In this section, we will apply the method of upper and lower solutions combined with fixed point theorem to consider the existence of solutions of FBVP (3).

Denote by the set consisting of the Hölder continuous functions with exponent on and .

In this paper, we consider the Banach space defined by with the norm . In fact, by Lemmas 3 and 17 it is derived that .

The main results in this paper are the following.

Theorem 24. *Assume that the following conditions hold: * *FBVP (3) has a pair of upper and lower solutions , respectively, with
* * is nonincreasing with respect to the third variable;* * satisfies the Nagumo condition with respect to and .**Then FBVP (3) has at least one solution with , such that
*

*Proof. *From the assumptions and , we know
due to the monotonicity of Riemann-Liouville fractional integral operator . And choose constant , such that

It is easy to obtain that . We let
and consider the following modified fractional boundary value problem:
where
Obviously, is bounded; that is, there exists positive constant , such that

To the end, it is sufficient to show that the modified FBVP (51)-(52) has at least one solution , satisfying

We divide the proof into three steps.

*Step 1. *FBVP (51)-(52) has at least one solution with .

Firstly, we define the operator by
From the continuity of and , it is not difficult to verify that the operator is well defined and continuous.

By Lemma 7 we can see that the fixed points of coincide with the solutions of FBVP (51)-(52). In the following, we prove that has a fixed point in .

Secondly, since , , is bounded by , and , together with (56) we easily obtain that is uniformly bounded (here, is a bounded subset of ).

For , , without loss of generality, let . We have
That is, is equicontinuous. According to the Ascoli-Arzela theorem, we know that is completely continuous.

By the Schauder fixed point theorem, we can easily obtain that has at least one fixed point with .

*Step 2. *The function satisfies , .

Suppose that on is not true; then has a negative minimum at some ; that is, .

If , then . From (42) and (52) we have the contradiction .

If , that is, . Obviously, , , which deduces . Choosing enough small , by Lemma 4 we obtain that exists, and

Now, denote , for brevity, by . From (56) it follows that
Choosing in Lemma 11, then
Obviously, the first term on the right side of (60) is Lipschitz continuous in . For the second term on the right side of (60), by the continuity of and Corollary 21 we obtain . Thus, it is deduced that .

On account of and , we have and is Hölder continuous with exponent . At this point, by Lemma 11 we obtain
Taking the limit in (61), it follows by (58) and that

On the other hand, firstly we claim that it holds that . It obviously follows that . Then by Lemma 6 we know
Analogously with above arguments for (62), by Lemma 15 and (63) it is not hard to obtain that
That is, . The claim is proved. Again together with , , and , for , we have the following two cases.*Case 1*. When ,
which contradicts (62).*Case 2*. When ,
which contradicts (62).

Thus, we know that the minimum point satisfying does not occur on .

If , that is, . By the boundary conditions (42) and (52), we have

However, analogously with above arguments for (64) we obtain
which is a contradiction.

Then it holds that , . Analogously we can also obtain that , . Hence, we have , .

*Step 3. *We prove that on .

We only need to show that on . Similarly we can show that on .

Suppose that on is not true; then there exists , satisfying . Due to , and , , we know that , and by the mean value theorem, there exists , such that
Since , there exists an interval (or ) such that
Thus, by we have for ,
Then,
However, by (49)-(50) and Property 1(4) we have
which is a contradiction with (72). So there holds that , . Hence, we have that , .

Consequently, combining Step 2 and Step 3, we obtain that
That is to say, the solution is a solution of FBVP (3). Then FBVP (3) has at least one solution with , such that , .

*Remark 25. *Observe that the validity of the first inequality in (62) can be guaranteed by means of applying approach method. With this idea, combining Lemma 3, Remark 5, Lemma 11, and Corollary 21, under certain stronger conditions instead of (i)-(ii) in Lemma 11, in the end we have two more concise conclusions as follows.

Theorem 26. *Assume that satisfies the following conditions: *(i)*, for ;*(ii)* attains its global minimum at .**Then,
**
Moreover, if , then .*

*Remark 27. *In fact, with implies . We emphasize that this result may help to establish some fractional comparison principles for the case of order lying in , which play a very important role in studying Riemann-Liouville fractional differential equations by means of monotone iterative method. Some related studies will be given in a future paper.

Theorem 28. *Assume that satisfies the following conditions: *(i)*, for ;*(ii)* attains its global minimum at .**Then,
**
Moreover, if , then .*

#### 4. Examples

*Example 29. *Consider the following fractional boundary value problem:

Let , . Obviously, , and , . It follows that is nonincreasing with respect to . Choose , , then , , and , , which deduces that . And it is not difficult to check out that is a lower solution of FBVP (77). Analogously, , is an upper solution of FBVP (77).

Meanwhile, if we choose , then when , , , it holds that That is, satisfies the Nagumo condition with respect to and .

Hence, by Theorem 24 we have that FBVP (77) has at least one solution , satisfying , .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by the Scientific Research Fund of Hunan Provincial Education Department (13K029) and the Innovation Fund Project for Graduate Student of Hunan Province (CX2013B219) and partially supported by the National Nature Science Foundation of China (61170320) and Nature Science Foundation of Guangdong Medical College (B2012053).