Abstract
In this paper, we study the existence of solutions for non-linear fractional differential equations of order involving the p-Laplacian operator with various boundary value conditions including an anti-periodic case. By using the Banach contraction mapping principle, we prove that, under certain conditions, the suggested non-linear fractional boundary value problem involving the p-Laplacian operator has a unique solution for both cases of and . Finally, we illustrate our results with some examples.
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1 Introduction
Recently, boundary value problems for fractional differential equations have gained popularity among researchers since they have many applications in biophysics, blood flow phenomena, aerodynamics, polymer rheology, viscoelasticity, thermodynamics, electro-dynamics of complex medium, capacitor theory, electrical circuits, electro-analytical chemistry, control theory (see [1–8] and [9]). Various results on boundary value problems for fractional differential equations have appeared in the literature (see [10–13] and [14]). Also, solvability of fractional differential equations with anti-periodic boundary value problems have been considered by different authors ([15, 16] and [17]). On the other hand, in studying turbulent flow in a porous medium, Leibenson introduced the concept of p-Laplacian operator [18] and it was used for fractional boundary value problems in [19] and [20]. In this paper, we consider boundary value problems for fractional differential equations of order involving the p-Laplacian operator with various boundary conditions including an anti-periodic case.
Now, we present basic definitions and results that will be needed in the rest of the paper. More detailed information about the theory of fractional calculus and fractional differential equations can be found in [1, 2, 4] and [21].
It is well known that the beta function has the following integral representation:
Moreover, can be expressed in terms of , the gamma function, as follows:
Definition 1 Let α be a positive real number. Then the fractional integral of is defined by
Definition 2 Let α be a positive real number. Then the Caputo fractional derivative of is defined by
where
and denotes the greatest integer less than or equal to α.
Recall that is the space of all real-valued functions which have continuous derivatives up to order on .
In the following lemmas, we give some auxiliary results which will be used in the sequel.
Lemma 1 [1]
Let and . Then
where n is given in (2).
On the other hand, the operator , where , is called the p-Laplacian operator. It is easy to see that , where . The following properties of the p-Laplacian operator will play an important role in the rest of the paper.
Lemma 2 Let be a p-Laplacian operator.
-
(i)
If , , and , then
(3) -
(ii)
If and , then
(4)
In this paper, we focus on the solvability of the following non-linear fractional differential equations of order involving the p-Laplacian operator with boundary conditions:
where , , , and .
Remark 1 For , , the boundary value problem given in (5) becomes anti-periodic.
Lemma 3 Assume that , , and . Then the solution of the boundary value problem
can be represented by the following integral equation:
where , and .
Proof Using (6) and the fact that , we have
or equivalently,
where . Applying the fractional integral operator to both sides of (8), we get
or equivalently,
and
Taking on both sides of (9), (10) and (11), we have
Using equations (12), (13), (14) and the boundary value conditions
we can get that
and
Substituting (15), (16) and (17) into (9) gives (7) and this completes the proof. □
2 Solvability of the fractional boundary value problem
This section is devoted to the solvability of the fractional boundary value problem given in (5). First, we obtain conditions for existence and uniqueness of the solution of the fractional boundary value problem given in (5). Then, each result obtained here is illustrated by examples.
Recall that , the space of continuous functions on is a Banach space with the norm . Now consider , , with
and
Then the operator , defined by , is continuous and compact.
Theorem 1 Suppose , , and the following conditions hold: , and d with
such that
and
Then boundary value problem (5) has a unique solution.
Proof Using inequality (19), we get
By (3) and (20), we have
Moreover,
Finally, substituting (21) in (22), we get
Using the equality
in (23) one can write that
Using (1), we get that
where
Combining (24) with (18) implies that , therefore T is a contraction. As a consequence of the Banach contraction mapping theorem [22] the boundary value problem given in (5) has a unique solution. □
Theorem 2 Suppose , and the following conditions hold for a fixed, , and d with
such that
and
Then boundary value problem (5) has a unique solution.
Proof The inequality implies that . Therefore replace by in the proof of Theorem 1. □
Theorem 3 Suppose , , there exists a non-negative function with such that
and there exists a constant d with
and
Then boundary value problem (5) has a unique solution.
Proof Using (25), we get that
for all . By the definition of the operator , one can write that
As a consequence of (4), (27) and (28), we have
Moreover,
where
By (26), we get , which implies that T is a contraction, therefore the boundary value problem given in (5) has a unique solution. □
In the present part, we illustrate our results by examples.
Example 1 Consider the following anti-periodic boundary value problem:
where
Then , and take , and . Obviously,
On the other hand,
Finally,
Therefore, as a consequence of Theorem 1, the boundary value problem given in (29) has a unique solution.
Example 2 Consider the following anti-periodic boundary value problem:
where
Then, obviously, , . Taking , and , we have
On the other hand,
Finally,
Therefore, as a consequence of Theorem 2, the boundary value problem given in (30) has a unique solution.
Example 3 Now consider the following boundary value problem:
where
Then , and . Also, taking and , we have
and
On the other hand,
therefore, by Theorem 3, the boundary value problem given in (31) has a unique solution.
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We would like to thank the anonymous referees for their valuable suggestions.
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Aktuğlu, H., Özarslan, M.A. Solvability of differential equations of order involving the p-Laplacian operator with boundary conditions. Adv Differ Equ 2013, 358 (2013). https://doi.org/10.1186/1687-1847-2013-358
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DOI: https://doi.org/10.1186/1687-1847-2013-358