Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 753171, 11 pages

http://dx.doi.org/10.1155/2013/753171

Research Article

Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with -Laplacian Operator and Identities on the Some Special Polynomials

Erdoğan Şen,1,2 Mehmet Acikgoz,³ Jong Jin Seo,4,5 Serkan Araci,⁶ and Kamil Oruçoğlu²

¹Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, 59030 Tekirdağ, Turkey

²Department of Mathematics Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey

³Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey

⁴Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea

⁵Department of Mathematics, College of Natural Sciences, Kwangwoon University, Seoul 139-701, Republic of Korea

⁶Atatürk Street, 31290 Hatay, Turkey

Received 13 May 2013; Accepted 28 September 2013

Academic Editor: Josip E. Pečarić

Copyright © 2013 Erdoğan Şen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:

in which and are Bernoulli numbers and multibinomial coefficients.

In the region , the Euler polynomials and the Euler polynomials of higher order are given, respectively, with the help of the following generating functions:

where are Euler numbers in (see [33–36]). From the last equation, we discover the following:

(see [35]).

Obviously, we have that

Taking in Definition 2, by (73) and (75), we compute

Therefore, we obtain the following theorem.

Theorem 25. The following identity

is true. Obviously, we have that

In the region , Genocchi polynomials, , and Genocchi polynomials of higher order, , are defined as an extension of Genocchi numbers defined in [33, 37, 38], respectively,

In this final section, by the similar method, we arrive at the following theorem.

Theorem 26. The following identity

is true. Obviously, we have that,

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