Multiple solutions of a p-Laplacian model involving a fractional derivative
Advances in Difference Equations volume 2013, Article number: 126 (2013)
In this paper, we study the p-Laplacian model involving the Caputo fractional derivative with Dirichlet-Neumann boundary conditions. Using a fixed point theorem, we prove the existence of at least three solutions of the model. As an application, an example is included to illustrate the main results.
In this paper, we are concerned with the multiple positive solutions of Dirichlet-Neumann boundary value problems for a type of fractional differential equation involving a p-Laplacian operator as the following form:
where is the p-Laplacian operator, i.e., , , and , ; is the standard Caputo derivative; , and n is an integer; , , are constants, ; f is a given function.
It is well known that both the fractional differential equations and the p-Laplacian operator equations are widely used in the fields of different physical and natural phenomena, non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology, complex geometry and patterns. Many researchers have extensively studied either the fractional differential equations or the p-Laplacian operator equations, respectively. For details of the theory and applications of the fractional differential equations or the p-Laplacian operator equations, see [1–15] and the references therein.
The authors of  studied the boundary value problem of fractional order
By means of the Schauder fixed point theorem and an extension of the Krasnosel’skii fixed point theorem in a cone, the existence of positive solutions is obtained.
In , the authors investigated the nonlinear boundary value problem of fractional differential equation
By means of the Amann theorem and the method of upper and lower solutions, some results on the multiple solutions are obtained.
Liu  was concerned with the mixed type multi-point boundary value problem
the existence of at least three positive solutions of the above mentioned boundary value problem is established.
Recently, a few researchers were devoted to the study of boundary value problems for the fractional differential equations with the p-Laplacian operator equations (see [16, 17]). In , some results on the existence and uniqueness of a solution for the following boundary value problem of a fractional differential equation are obtained:
Motivated by the above, the purpose of this paper is to establish the existence of multiple positive solutions to boundary value problems for a fractional differential equation involving a p-Laplacian operator (1). If p is an integer, the equation in (1) reduces to a standard nonlinear fractional differential equation. And it will become a standard p-Laplacian operator equation when α is an integer. Therefore, our results in this paper are the promotion and more general case of these two types of problems. By means of the fixed point theorem due to Avery and Peterson, we prove the results that there exist at least three positive solutions of the boundary value problem (1). As an application, an example is included to illustrate the main results.
In this section, we give the definition of a fractional derivative and some lemmas, which will be used later.
Definition 2.1 
Let for a function . The fractional integral of order α of y is defined by
provided the integral exists. The Caputo derivative of a function is given by
provided the right-hand side is pointwise defined on , where n is an integer, with .
Γ denotes the gamma function, that is,
From Definition 2.1, we can obtain the following lemma.
Lemma 2.1 Let . If we assume , the fractional differential equation
has a unique solution
Throughout this paper, we always suppose the following condition holds.
(H0) The parameters in the boundary value problem (1) satisfy the following conditions:
Lemma 2.2 Suppose that (H0) holds and . Then the boundary value problem
has a unique solution
That is, every solution of (2) is also a solution of (3) and vice versa.
Proof The definition of the Caputo derivative implies that , and from (2) we have
By using the property of the fractional derivatives and integrals, we can get
From the boundary condition , and , we can obtain that
Substituting (5) and (6) into (4), we can obtain that
The proof is completed. □
Let , , then is a Banach space. Set
then P is a cone on E.
Define the operator by
It is clear that is the solution of the boundary value problem (1) if and only if is the fixed point of the operator T.
Lemma 2.3 Suppose that (H0) holds and the function . Then is completely continuous.
Proof By the definition of the operator T, it is easy to see for any . And using the property of the fractional integrals and derivatives, we can get that
Then Tx is nonnegative, monotone increasing and convex on . They imply that
Thus, , so .
It is easy to prove that T is continuous and compact if the conditions of the lemma hold.
The proof is complete. □
For convenience of the readers, we provide some background material from the theory of cones in Banach spaces and the Avery-Peterson fixed point theorem.
Definition 2.2 Let E be a Banach space and let be a cone. A continuous map γ is called a concave (resp. convex) functional on P if and only if (resp. ) for all and .
Let β and ρ be nonnegative continuous convex functionals on a cone P, let ω be a nonnegative continuous concave functional on a cone P, and let ψ be a nonnegative continuous functional on a cone P. Then, for positive real numbers a, b, c and d, we define the following convex sets:
Lemma 2.4 (Avery-Peterson fixed point theorem )
Let P be a cone in a real Banach space E. Let β and ρ be nonnegative continuous convex functionals on P, let ω be a nonnegative continuous concave functional on P, and let ψ be a nonnegative continuous functional on P satisfying for , such that for some positive numbers M and d, and for all .
Suppose that is completely continuous and there exist positive numbers a, b and c with such that
(A1) , and for all ;
(A2) for all with ;
(A3) and for with .
Then T has at least three fixed points such that , ; ; , with ; and .
3 Multiple positive solutions of the boundary value problems
In this section, we establish the existence of multiple positive solutions of the boundary value problem (1).
Then , , , and if (H0) holds. And , if .
Theorem 3.1 Suppose that (H0) holds, there exist constants a, b, c, d such that , and satisfies the following conditions:
(H1) for any ;
(H2) for any ;
(H3) for any .
Then the boundary value problem (1) has at least three positive solutions , and the solutions are increasing and convex on . Moreover, for any ,
Proof Define the nonnegative continuous convex functionals β, ρ and the nonnegative continuous functional ψ, the nonnegative continuous concave functional ω on the cone P by
Obviously, for any , .
Following from the proof of Lemma 2.3, we can get that , , Tx is increasing and convex on .
For any implies that , , and
By the condition (H1), we have . Hence,
Therefore . By Lemma 2.3, is a completely continuous operator.
Let , then , , and , so
For any , it follows from the condition (H2), . By the proof of Lemma 2.3 and (7), we can get
Thus, the condition (A1) in the Avery-Peterson theorem is satisfied.
For any with , i.e., . Then
Consequently, the condition (A2) in the Avery-Peterson theorem is satisfied.
It is clear that .
For any with , it implies that
It is easy to get that
It follows from the condition (H3) that
then we have
So, the condition (A3) in the Avery-Peterson theorem holds.
Therefore, the conditions in the Avery-Peterson theorem are satisfied, and we can obtain that there exist three positive fixed points for the operator T corresponding to positive solutions to the discrete second-order boundary value problem (1) such that
By Lemma 2.3, we can get that the solutions are increasing and convex on . Hence
In this section, we give an example to illustrate Theorem 3.1.
Example 4.1 Consider the following Dirichlet-Neumann boundary value problem:
We choose , , , . We can easily get that , , , , .
After some calculation, we can check that satisfies following conditions:
for any ;
for any ;
for any .
Then all the conditions of Theorem 3.1 hold.
Hence, by Theorem 3.1, the boundary value problem (8) has at least three positive solutions , , such that
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The authors wish to acknowledge the support by the Innovation Program of Shanghai Municipal Education Commission (No. 10ZZ93), and the National Natural Science Foundation of China (No. 11171220).
The authors declare that they have no competing interests.
All authors jointly worked on the results and they read and approved the final manuscript.
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Liu, X., Jia, M. & Ge, W. Multiple solutions of a p-Laplacian model involving a fractional derivative. Adv Differ Equ 2013, 126 (2013). https://doi.org/10.1186/1687-1847-2013-126