Abstract and Applied Analysis

Volume 2013 (2013), Article ID 658617, 8 pages

http://dx.doi.org/10.1155/2013/658617

Research Article

On the Solvability of Caputo -Fractional Boundary Value Problem Involving -Laplacian Operator

Hüseyin Aktuğlu and Mehmet Ali Özarslan

Eastern Mediterranean University, Gazimagusa, Mersin 10, Turkey

Received 2 April 2013; Accepted 10 June 2013

Academic Editor: Nazim Idrisoglu Mahmudov

Copyright © 2013 Hüseyin Aktuğlu and Mehmet Ali Özarslan. 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 model of a Caputo -fractional boundary value problem involving -Laplacian operator. By using the Banach contraction mapping principle, we prove that, under some conditions, the suggested model of the Caputo -fractional boundary value problem involving -Laplacian operator has a unique solution for both cases of and . It is interesting that in both cases solvability conditions obtained here depend on , , and the order of the Caputo -fractional differential equation. Finally, we illustrate our results with some examples.

1. Introduction

In this section we will give some basic definitions and results that will be needed in the sequel. For more details about the theory of -calculus, fractional calculus, and -fractional calculus, we refer readers to [1–10].

Let be a fixed real number. Then for any ,

The -binomial function is defined for all as

where is not a positive integer. It is easy to see that

The -analog of Euler's gamma function is denoted by and defined as

The following theorem will be used to compare values of , the usual gamma function, with values of for a fixed .

Theorem 1 (see [11]). For , one has

It is known that for ,

The nabla -derivative of the function is defined by

The nabla -integral of is defined by

Jackson in [12] and Thomae in [13] showed that the -beta function, which is defined by

has the following -integral representation:

The fundamental theorem of -calculus states that

and if is continuous at , then

Moreover,

where the derivative is applied with respect to .

The nabla -fractional derivative of with respect to and for all is given by

Moreover, the -fractional integral of order is defined by

The -order Caputo -fractional derivative of a function is defined by

where and denotes the greatest integer less than or equal to .

The following lemma enables us to transfer Caputo -fractional differential equations into an equivalent -fractional integral equation.

Lemma 2 (see [3]). Assume that and is defined on a suitable domain. Then

and if , then

On the other hand the operator , where is called the -Laplacian operator. It is easy to see that , where . The following properties of -Laplacian operator will be used in the rest of the paper.

 (P1) if ,  , and , then ; (P2) if and then, .

2. A Model of Caputo -Fractional Boundary Value Problem Involving -Laplacian Operator

In this paper, our main aim is to prove the existence and uniqueness of the solution for the following Caputo -fractional boundary value problem involving the -Laplacian operator:

where ,  , and .

Note that, the boundary value problem given in (19) is antiperiodic for .

In the following lemma we obtain a -integral equation which is equivalent to the Caputo -fractional boundary value problem given in (19).

Lemma 3. Assume that ,, and . Then

are equivalent to the following -integral equation:

where ,  , and   .

Proof. Using (20) and the fact that, we have

or equivalently,

Applying -fractional integral operator to both sides and using Lemma 2, we get

Using ,   for in (24), we obtain

According to (13) and (14), we have

Taking in both sides of (25) and (26), we get

Solving equations obtained by the given boundary value conditions and , it follows that

Substituting (28) into (25) gives (21) which completes the proof.

3. Solvability of the Caputo -Fractional Boundary Value Problem

This section is devoted to the solvability of the Caputo -fractional boundary value problem given in (19). In the first part we shall prove the existence and uniqueness of the solution, and then we shall illustrate our main results with some examples.

Recall that is a Banach space with the norm . Now consider , with

Then is a continuous and compact operator.

Theorem 4. Suppose that ,   is fixed, and the following conditions hold:

  and with

such that

Then the boundary value problem (19) has a unique solution.

Proof. Inequality given in (31) implies that

On the other hand using and (32), we have

Similarly,

Finally using (34) in (35), we get

Since

we have

In other words,

where  .

By condition (30), we get , which implies that is a contraction. As a consequence of the Banach contraction mapping theorem and Lemma 3, the boundary value problem given in (19) has a unique solution.

Theorem 5. Suppose that ,    , and the following conditions hold for a fixed ,

, , and with

such that

Then the boundary value problem (19) has a unique solution.

Remark 6. When , Theorems 4 and 5 reduce to Theorems 3.1 and 3.2 of [14].

Theorem 7. Suppose that , and the following conditions hold for a fixed . There exists a nonnegative function with such that

and there exists a constant with

Then the boundary value problem (19) has a unique solution.

Proof. By (42), we can get that

for all . By the definition of , we have

Using (P2) and (45) gives

Therefore,

Since

we have

Using -Beta function and the fact that , we get

where

By condition (43), we get which implies that is a contraction; therefore boundary value problem given in (19) has a unique solution.

Next, we give some examples to illustrate our results.

Example 8. Consider the following Boundary value problem

where

Then , and take ,  , and . Using Theorem 1 and the fact that for any fixed , we have

Moreover, it can be easily seen that

Finally,

Therefore as a consequence of Theorem 4, boundary value problem given in (53) has a unique solution.

Example 9. Consider the following boundary value problem:

Then

Therefore boundary value problem given in (58) reduces to the boundary value problem given in (53), and it has a unique solution.

Example 10. Now consider the following antiperiodic boundary value problem:

where

Then , and take . Using Theorem 1 and taking , we get that

On the other hand,

Therefore by Theorem 7, the antiperiodic boundary value problem given in (60) has a unique solution.

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