Abstract
In this article, we investigate the SturmLiouville boundary value problems of fractional differential equations with pLaplacian
where
MSC: 34A08, 34B18, 35J05.
Keywords:
SturmLiouville boundary value problem; positive solution of fractional differential equation; LeggettWilliams fixedpoint theorem; fixedpoint index theory; pLaplacian operator1 Introduction
During the past decades, much attention has been focused on the study of equations with pLaplacian differential operator. The motivation for those works stems from the applications in the modeling of different physical and natural phenomena: nonNewtonian mechanics [1], system of MongeKantorovich partial differential equations [2], population biology [3], nonlinear flow laws [4], combustion theory [5]. There exist a very large number of papers devoted to the existence of solutions for the equation with pLaplacian operator.
The ordinary differential equation with pLaplacian operator
subject to various boundary conditions, has been studied by many authors, see [6,7] and the references therein.
The existence of positive solutions of the differential equation with pLaplacian operator
satisfying different boundary conditions have been established by using fixedpoint theorems and monotone iterative technique, see [8,9] and the references therein.
In [10], Hai considered the existence of positive solutions for the boundary value problem
where
However, few papers can be found in the literature on the existence of multiple positive solutions for the thirdorder SturmLiouville boundary value problem with pLaplacian.
In [11], Zhai and Guo studied the thirdorder SturmLiouville boundary value problem with pLaplacian
where
Recently, fractional differential equations have been of great interest. The motivation for those works stems from both the intensive development of the theory of fractional calculus itself and the applications such as economics, engineering and other fields [1317]. Much attention has been focused on the study of the existence and multiplicity of solutions or positive solutions for boundary value problems of fractional differential equations by the use of techniques of nonlinear analysis (fixedpoint theorems [1825], upper and lower solutions method [26], fixedpoint index theory [27,28], coincidence theory [29], etc.).
Although the boundary value problems of fractional differential equation with pLaplacian have been studied in many literature, only few papers can be found in the literature on the existence of multiple positive solutions for the SturmLiouville boundary value problems of fractional differential equations with pLaplacian. As the extension and supplement of some results in [11,12], in this article, we investigate the SturmLiouville boundary value problems of fractional differential equations with pLaplacian subject Robin boundary value conditions
where
The rest of this paper is organized as follows. In Section 2, we shall introduce some definitions and lemmas to prove our main results. In Section 3, we state our main results. We prove our main results by LeggettWilliams fixedpoint theorems and fixedpoint index theory in Section 4. As an application, an example is presented to illustrate our main result in Section 5.
2 Preliminaries and lemmas
For the convenience of the reader, we give some background materials from fractional calculus theory to facilitate analysis of problem (1.1). These materials can be found in the recent literature, see [14,17,18,3133].
Definition 2.1 ([17])
The RiemannLiouville fractional integral of order
provided the right side is pointwise defined on
Definition 2.2 ([17])
The Caputo fractional derivative of order
where n is the smallest integer greater than or equal to α, provided that the right side is pointwise defined on
Remark 2.1 ([14])
By Definition 2.2, under natural conditions on the function
Remark 2.2 ([18])
As a basic example, we have
given in particular that
From the definition of the Caputo derivative and Remark 2.2, we can obtain the following statement.
Lemma 2.1 ([17])
Let
has
as the unique solution, wherenis the smallest integer greater than or equal toα.
Lemma 2.2 ([17])
Let
for some
Lemma 2.3Let
has a unique solution,
where
Proof By the Lemma 2.2, we can reduce the equation of problem (2.1) to an equivalent integral equation
for some constants
From the boundary conditions
So,
Hence, the unique solution of (2.1) is
which completes the proof. □
Lemma 2.4Let
has a unique solution,
where
Proof From Lemma 2.2 and the boundary value problem (2.3), we have
that is
By
Lemma 2.3 implies that boundary value problem (2.3) has a unique solution,
which completes the proof. □
Lemma 2.5The Green’s function
Assume
(1)
(2)
(3) there exists a positive numberλsuch that
The method of proof is similar to Lemma 3.2 in [30], and we omit it here.
Definition 2.3 ([31])
Let E be a real Banach space and P be a nonempty, convex closed set in E. We say that P is a cone if it satisfies the following properties:
(i)
(ii)
If
Definition 2.4 ([31])
The map φ is said to be a nonnegative continuous concave functional on P of a real Banach space E provided that
for all
Definition 2.5 ([31])
Let
Lemma 2.6 (LeggettWilliams [32])
Let
(A_{1})
(A_{2})
(A_{3})
ThenThas at least three fixed points
Lemma 2.7 ([32])
Let
(B_{1})
(B_{2})
(B_{3})
ThenThas at least two fixed points
Lemma 2.8 ([31])
LetPbe a closed convex set in a Banach spaceEand let Ω be a bounded open set such that
(C_{1}) (Existence) If
(C_{2}) (Normalization) If
(C_{3}) (Homotopy) Let
(C_{4}) (Additivity) If
Lemma 2.9 ([33])
LetPbe a cone in a Banach spaceE. For
(D_{1}) if
(D_{2}) if
3 Main theorems
In this section, let
where λ is given as in Lemma 2.5.
For convenience of the reader, we denote
Lemma 3.1Let
Then
Proof By Lemma 2.5, we have
Thus,
Let
Hence,
Hence,
That is to say,
We are now ready to prove our main results.
Theorem 3.1Let
(H_{1})
(H_{2})
Then the boundary value problem (1.1) has at least two positive solutions
Proof Let
Evidently, for each
It’s easy to see that
So
Consequently,
That is,
Therefore, condition (B_{1}) of Lemma 2.7 is satisfied. Now if
which shows that
So condition (B_{3}) of Lemma 2.7 is satisfied. Thus using Lemma 2.7, T has at least two fixed points. Consequently, the boundary value problem (1.1) has
at least two positive solutions
Theorem 3.2Let
(H_{3})
(H_{4})
(H_{5})
Then the boundary value problem (1.1) has at least three positive solutions
Proof If
This shows that
Moreover, for
So all the conditions of Lemma 2.6 are satisfied. Thus using Lemma 2.6, T has at least three fixed points. So, the boundary value problem (1.1) has at least
three positive solutions
Theorem 3.3Let
(H_{6})
(H_{7}) there exists a constant
then the boundary value problem (1.1) has at least two positive solutions
Proof From Lemma 3.1, we obtain
where
Let
which implies
On the other hand, since
where
Let
which implies
Finally, let
which implies
Note that
and
Hence, T has a fixed point
Theorem 3.4Let
(H_{8})
(H_{9}) there exists a constant
Then the boundary value problem (1.1) has at least two positive solutions
Proof From Lemma 3.1, we obtain
where
Let
which implies
Next, since
where
Case 1: Suppose that f is bounded, which implies that there exists
Take
Case 2: Suppose that f is unbounded. In view of
Then, for
So, in either case, if we always choose
Thus, from Lemma 2.9, we have
Finally, Let
which implies
Note that
and
Hence, T has a fixed point
4 Example
In this section, we present an example to illustrate the main result.
Example 4.1 We consider the boundary value problem of the fractional differential equation
where
Let
Choosing
Consequently, all the conditions of Theorem 3.2 are satisfied. With the use of Theorem 3.2,
the boundary value problem (4.1) has at least three positive solutions
5 Conclusion
In this paper, the SturmLiouville boundary value problems of fractional differential equations with pLaplacian are investigated, the existence of at least two or at least three positive solutions for the fractional differential equations with Robin boundary conditions are given by using LeggettWilliams fixedpoint theorems and the fixedpoint index theory, respectively.
It is worth emphasizing that our work presented in this article has the following
features: Firstly, the boundary conditions in (1.1) are important Robin boundary conditions.
Secondly, our results improve and extend the main results of [11,12] for the SturmLiouville boundary value problems of integerorder differential equations
with pLaplacian. For example, if
which is studied in [11,12]. Furthermore, if we take
The method can be applied on the SturmLiouville boundary value problems of higherorder fractional differential equations with pLaplacian and boundary conditions involving fractional derivatives
Based on this paper, one can consider boundary value problems of fractional differential equations with parameters, and also one can do further research on eigenvalue problems of fractional differential equations with pLaplacian.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009).
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