Abstract
In this paper, we discuss the existence and multiplicity of positive solutions to m-point boundary value problems of nonlinear fractional differential equations with p-Laplacian operator
where , and are the standard Riemann-Liouville fractional derivatives with , , , , , , , , , and , , , . Our results are based on the monotone iterative technique and the theory of the fixed point index in a cone. Furthermore, two examples are also given to illustrate the results.
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1 Introduction
Fractional differential equations arise in various areas of science and engineering. Due to their applications, fractional differential equations have gained considerable attention (see, e.g., [1–26] and the references therein).
Recently, there have been some papers dealing with the existence of solutions for nonlinear fractional differential equations with p-Laplacian operator. In [1], Wang et al. investigated the following boundary value problem for fractional differential equations with p-Laplacian operator:
where , are the standard Riemann-Liouville fractional derivatives, , , , .
In [2], Chai studied the existence of positive solutions of the following fractional differential equations with p-Laplacian operator:
where , and are the standard Riemann-Liouville fractional derivatives with , , , , the constant σ is a positive number, .
In [3], Chen and Liu studied the following fractional differential equations with p-Laplacian operator:
where , , , are Caputo fractional derivatives, and is continuous.
In [4], Lu et al. studied the following fractional differential equations with p-Laplacian operator:
where , , , are the standard Riemann-Liouville fractional derivatives, and .
On the other hand, in [5], Bai studied an eigenvalue interval of the following fractional boundary problem:
where , is the standard Caputo fractional derivative, .
In [6], Zhang et al. studied the following singular eigenvalue problem for a higher order fractional differential equation:
where , , for , and , , . is the standard Riemann-Liouville fractional derivative.
Moreover, in recent years, we have done some work on fractional differential equations [7–9]. In [7], we considered the following m-point boundary value problem for fractional differential equations:
where is the standard Riemann-Liouville fractional derivative, , is continuous, , , , , , and .
Combining our work, in this paper, we discuss the existence of positive solutions for the following fractional differential equations with p-Laplacian operator:
where , and are the standard Riemann-Liouville fractional derivatives with , , , , , , , , , and , , , .
Our work presented in this paper has the following features. Firstly, to the best of the author’s knowledge, there are few results on the existence of solutions for nonlinear fractional p-Laplacian differential equations with m-point boundary value problems. Secondly, we transform (1.1) into an equivalent integral equation and discuss the eigenvalue interval for the existence of multiplicity of positive solutions. The paper is organized as follows. In Section 2, we present some background materials and preliminaries. Section 3 deals with some existence results. In Section 4, two examples are given to illustrate the results.
2 Background materials and preliminaries
The fractional integral of order α with the lower limit for a function f is defined as
where Γ is the gamma function.
The Riemann-Liouville derivative of order α with the lower limit for a function f is
Lemma 2.1 ([12])
Assume that with a fractional derivative of order that belongs to . Then
where N is the smallest integer greater than or equal to α.
Lemma 2.2 ([7])
Let . Then the fractional differential equation
has a unique solution which is given by
where
in which
where
Lemma 2.3 ([7])
If , then the function in Lemma 2.2 satisfies the following conditions:
-
(i)
, for ,
-
(ii)
, for ,
where
Lemma 2.4 in [7]has the following property:
Proof For , we conclude that
Thus
It is obvious that
Therefore
□
Lemma 2.5 Let , then BVP (1.1) has a unique solution
Proof Let , . From (1.1), we have
By Lemma 2.1, we have
It follows from that
Thus, from (1.1) we know that
By Lemma 2.2, (1.1) has a unique solution
It follows from that
Thus
□
Lemma 2.6 ([27])
Let E be a real Banach space, be a cone, . Let the operator be completely continuous and satisfy , . Then
3 Main results
We consider the Banach space endowed with the norm defined by . Let , then P is a cone in E. Define an operator as
Then T has a solution if and only if the operator T has a fixed point.
Lemma 3.1 If , then the operator is completely continuous.
Proof From the continuity and non-negativeness of and , we know that is continuous.
Let be bounded. Then, for all and , there exists a positive constant M such that . Thus,
where
This means that is uniformly bounded.
On the other hand, from the continuity of on , we see that it is uniformly continuous on . Thus, for fixed and for any , there exists a constant such that and ,
Hence, for all ,
which implies that is equicontinuous. By the Arzela-Ascoli theorem, we obtain that is completely continuous. The proof is complete. □
Theorem 3.2 If , is nondecreasing in u and , then BVP (1.1) has a minimal positive solution in and a maximal positive solution in . Moreover, , as uniformly on , where
and
Proof Let
where
Step 1: Problem (1.1) has at least one solution.
For , there exists a positive constant such that ,
Thus
By Lemma 3.1, we can see that is completely continuous. Hence, by means of the Schauder fixed point theorem, the operator T has at least one fixed point, and BVP (1.1) has at least one solution in .
Step 2: BVP (1.1) has a positive solution in , which is a minimal positive solution.
From (3.1) and (3.2), one can see that
This, together with being nondecreasing in u, yields that
Since T is compact, we obtain that is a sequentially compact set. Consequently, there exists such that ().
Let be any positive solution of BVP (1.1) in . It is obvious that .
Thus,
Taking limits as in (3.5), we get for .
Step 3: BVP (1.1) has a positive solution in , which is a maximal positive solution.
Let , and . From , we have . Thus
This, together with being nondecreasing in u, yields that
Using a proof similar to that of Step 2, we can show that
and
Let be any positive solution of BVP (1.1) in .
Obviously,
Thus
Taking limits as in (3.6), we obtain for .
The proof is complete. □
Define
Let
Theorem 3.3 Assume that , and the following conditions hold:
(H1) .
(H2) There exists a constant such that for , .
Then BVP (1.1) has at least two positive solutions and such that
for
where
Proof Since
there is such that
Let
Then, for any , it follows from Lemma 2.4 that
Thus
This, together with (3.7), yields that
By Lemma 2.6, we have
In view of
there is , such that
Let
Then, for any , it follows from Lemma 2.4 that
Thus
This, together with (3.7), yields that
By Lemma 2.6, we have
Finally, let . For any , it follows from Lemma 2.3 and (H2) that
Thus
This, together with (3.7), yields that
Using Lemma 2.6, we get
From (3.8)-(3.10) and , we have
Therefore, T has a fixed point and a fixed point . Clearly, , are both positive solutions of BVP (1.1) and . The proof of Theorem 3.3 is completed. □
In a similar way, we can obtain the following result.
Corollary 3.4 Assume that , and the following conditions hold:
(H1) .
(H2) There exists a constant such that for , .
Then BVP (1.1) has at least two positive solutions and such that
for
where
4 Examples
Example 4.1 Consider the following boundary value problem:
where
Thus
By computation, we deduce that
and
On the other hand,
Take
Hence, by Theorem 3.2, BVP (4.1) has a minimal positive solution in and a maximal positive solution in .
Example 4.2 Consider the following boundary value problem:
where
and
It follows from Example 4.1 that
By computation, we deduce that
and
Taking
we have
Thus, condition (H2) is satisfied. It is obvious that condition (H1) holds.
On the other hand, let , , we have , and
Hence, by Theorem 3.3, BVP (4.2) has at least two solutions and such that for
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Acknowledgements
This research is supported by Henan Province College Youth Backbone Teacher Funds (2011GGJS-213) and the National Natural Science Foundation of China (11271336).
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Lv, ZW. Existence results for m-point boundary value problems of nonlinear fractional differential equations with p-Laplacian operator. Adv Differ Equ 2014, 69 (2014). https://doi.org/10.1186/1687-1847-2014-69
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DOI: https://doi.org/10.1186/1687-1847-2014-69