#### Abstract

We consider the following boundary-value problem of nonlinear fractional differential equation with -Laplacian operator , , , , , where , are real numbers, are the standard Caputo fractional derivatives, , , , , , are parameters, and are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters and are obtained. The uniqueness of positive solution on the parameters and is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.

#### 1. Introduction

In 1695, L’Hôpital asked Leibniz: what if the order of the derivative is ? To which Leibniz considered in a useful means, thus it follows that will be equal to , an obvious paradox. In recent years, fractional calculus has been studied by many mathematicians from Leibniz’s time to the present.

Also, fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, fluid flows, electrical networks, viscoelasticity, aerodynamics, and many other branches of science. For details, see [1–9].

In the last few decades, fractional-order models are found to be more adequate than integer-order models for some real world problems. Recently, there have been some papers dealing with the existence and multiplicity of solutions (or positive solutions) of nonlinear initial fractional differential equations by the use of techniques of nonlinear analysis [10–21], upper and lower solutions method [22–24], fixed point index [25, 26], coincidence theory [27], Banach contraction mapping principle [28], and so forth.

Chai [11] investigated the existence and multiplicity of positive solutions for a class of boundary-value problem of fractional differential equation with -Laplacian operator where , , , is a positive constant number, and are the standard Riemann-Liouville derivatives. By means of the fixed point theorem on cones, some existence and multiplicity results of positive solutions are obtained.

Although the fractional differential equation boundary-value problems have been studied by several authors, very little is known in the literature on the existence and nonexistence of positive solutions of fractional differential equation boundary-value problems with -Laplacian operator when a parameter is involved in the boundary conditions. We also mention that, there is very little known about the uniqueness of the solution of fractional differential equation boundary-value problems with -Laplacian operator on the parameter . Han et al. [29] studied the existence and uniqueness of positive solutions for the fractional differential equation with -Laplacian operator where , are real numbers; are the standard Caputo fractional derivatives; , . Therefore, to enrich the theoretical knowledge of the above, in this paper, we investigate the following -Laplacian fractional differential equation boundary-value problem: where , are real numbers, are the standard Caputo fractional derivatives, , , , , , , are parameters, , and are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters and are obtained. The uniqueness of positive solution on the parameters and , is also studied.

#### 2. Preliminaries and Related Lemmas

*Definition 1 (see [4]). *The Riemann-Liouville fractional integral of order of a function is given by
provided the right side is pointwise defined on .

*Definition 2 (see [4]). *The Caputo fractional derivative of order of a continuous function is given by
where is the smallest integer greater than or equal to , provided that the right side is pointwise defined on .

*Remark 3 (see [8]). *By Definition 2, under natural conditions on the function , for , the Caputo derivative becomes a conventional th derivative of the function .

*Remark 4 (see [4]). *As a basic example,
In particular, , , where is the Caputo fractional derivative, is the smallest integer greater than or equal to .

From the definition of the Caputo derivative and Remark 4, we can obtain the following statement.

Lemma 5 (see [4]). *Let . Then, the fractional differential equation
**
has a unique solution
**
where is the smallest integer greater than or equal to .*

Lemma 6 (see [4]). *Let . Assume that . Then
**
for some , , where is the smallest integer greater than or equal to .*

Lemma 7. *Let and . Then, fractional differential equation boundary-value problem
**
has a unique solution
*

*Proof. *We apply Lemma 6 to reduce (10) to an equivalent integral equation,
Consequently, the general solution of (10) is
By (11), we have
and since , we have by (11)
Therefore, the unique solution of problem (10) and (11) is

Lemma 8. *Let and , . Then, fractional differential equation boundary-value problem
**
has a unique solution
**
where
*

*Proof. *From Lemma 6, the boundary-value problems (18) and (19) are equivalent to the integral equation
for some ; that is,
By the boundary conditions , we have
Therefore, the solution of fractional differential equation boundary-value problems (18) and (19) satisfies
Consequently, . Thus, fractional differential equation boundary-value problem (18) and (19) is equivalent to the problem
Lemma 7 implies that the fractional differential equation boundary-value problems (18) and (19) have a unique solution,
The proof is complete.

Lemma 9 (see [15]). *Let . The function is continuous on and satisfies *(1)*, for ;*(2)*, for . *

Lemma 10 (Schauder fixed point theorem [30]). *Let be a complete metric space, be a closed convex subset of , and be a mapping such that the set is relatively compact in . Then, has at least one fixed point.*

To prove our main results, we use the following assumptions.(H1);(H2) there exist and such that where satisfies (H3) there exist such that where satisfies (H4) there exist and such that where satisfies with (H5) is nondecreasing in ;(H6) there exist such that

*Remark 11. *Let
Then, (H2) holds if , (H3) holds if , and (H4) holds if .

#### 3. Existence

Theorem 12. *Assume that (H1), (H2) hold. Then, the fractional differential equation boundary-value problem (3) has at least one positive solution for .*

*Proof. *Let be given in (H2). Define
and an operator by
Then, is a closed convex set. From Lemma 8, is a solution of fractional differential equation boundary-value problem (3) if and only if is a fixed point of . Moreover, a standard argument can be used to show that is compact.

For any , from (28) and (29), we obtain
Let . Then, from Lemma 6 and (38), it follows that
Thus, , by Schauder fixed point theorem, has a fixed point ; that is, the fractional differential equation boundary-value problem (3) has at least one positive solution. The proof is complete.

Corollary 13. *Assume that (H1) holds and . Then, the fractional differential equation boundary-value problem (3) has at least one positive solution for sufficiently small .*

Theorem 14. *Assume that (H1), (H3) hold. Then, the fractional differential equation boundary-value problem (3) has at least one positive solution for all .*

*Proof. *Let be fixed and be given in (H3). Define . Then
From (31), we have
Thus, there exists large enough so that
Let
For , define
Then, , , and in view of (30), we have
Let the compact operator be defined by (38). Then, from Lemma 6, (30), and (41), we have
From (43) and the inequality for any (see, e.g., [31]), we obtain
Thus, . Consequently, by Schauder fixed point theorem, has a fixed point , that is, the fractional differential equation boundary-value problem (3) has at least one positive solution. The proof is complete.

Corollary 15. *Assume that (H1) holds and . Then, the fractional differential equation boundary-value problem (3) has at least one positive solution for all .*

#### 4. Uniqueness

*Definition 16 (see [32]). *A cone in a real Banach space is called solid if its interior is not empty.

*Definition 17 (see [32]). *Let be a solid cone in a real Banach space be an operator, and . Then, is called a -concave operator if

Lemma 18 (see [32, Theorem ]). *Assume that is a normal solid cone in a real Banach space , , and is a -concave increasing operator. Then, has only one fixed point in .*

Theorem 19. *Assume that (H1), (H5), (H6) hold. Then, the fractional differential equation boundary-value problem (3) has a unique positive solution for any .*

*Proof. *Define . Then, is a normal solid cone in with
For any fixed , let be defined by (38). Define by
Then, from (H5), we have increasing in and
Clearly, . Next, we prove that is a -concave increasing operator. In fact, for with on , we obtain
that is, is increasing. Moreover, (H6) implies
that is, is -concave. By Lemma 8, has a unique fixed point in , that is, the fractional differential equation boundary-value problem (3) has a unique positive solution. The proof is complete.

#### 5. Nonexistence

In this section, we let the Banach space be endowed with the norm .

Lemma 20. *Assume (H1) holds and let be given in (H4). Then, the unique solution of fractional differential equation boundary-value problem (18) and (19) satisfies
**
where is defined by (34).*

*Proof. *In view of Lemma 9 and (19), we have
for , and
for . Therefore, for . The proof is complete.

Theorem 21. *Assume that (H1), (H4) hold. Then, the fractional differential equation boundary-value problem (3) has no positive solution for .*

*Proof. *Assume, to the contrary, the fractional differential equation boundary-value problem (3) has a positive solution for . Then, by Lemma 8, we have
Therefore, on . In view of (32) and (33), we obtain
Then, by Lemmas 6 and 9, we obtain
This contradiction completes the proof.

Corollary 22. *Assume that (H1) holds and . Then, the fractional differential equation boundary-value problem (3) has no positive solution for sufficiently large .*

#### 6. Conclusion: Identities on the Special Polynomials whereby Caputo Fractional Derivative

In this final part, we will focus on the new interesting identities related to special polynomials by means of Caputo fractional derivative.

As well known, the Bernoulli polynomials may be defined to be where usual convention about replacing by in is used. Also, we note that the Bernoulli polynomials is analytic on the region (see [33]).

Let be familiar normal derivative, then we can obtain the following identity

Differentiating in both sides of (61), we have (see [33]).

When in (61), we have are called Bernoulli numbers, which can be generated by

By (61) and (64), we have the following functional equation: and this equation yields to (see [33]).

Let us now take in Definition 2 leads to

Therefore, we procure the following theorem.

Theorem 23. *The following identity holds true:
*

In [34], the Bernoulli polynomials of higher order are defined by we note that is analytic on . It follows from (69), we have (see [34]).

Substituting into (69), are called Bernoulli polynomials of higher order.

Owing to (69) and (70), we readily see that

Therefore, we can state the following theorem.

Theorem 24. *The following identity holds true:
*