- Open Access
Multiple solutions of a p-Laplacian model involving a fractional derivative
© Liu et al.; licensee Springer 2013
- Received: 17 February 2013
- Accepted: 17 April 2013
- Published: 2 May 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.
- Banach Space
- Fractional Derivative
- Fixed Point Theorem
- Fractional Differential Equation
- Caputo Fractional Derivative
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.
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.
By means of the Amann theorem and the method of upper and lower solutions, some results on the multiple solutions are obtained.
the existence of at least three positive solutions of the above mentioned boundary value problem is established.
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 
provided the right-hand side is pointwise defined on , where n is an integer, with .
From Definition 2.1, we can obtain the following lemma.
Throughout this paper, we always suppose the following condition holds.
That is, every solution of (2) is also a solution of (3) and vice versa.
The proof is completed. □
then P is a cone on E.
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.
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 .
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 .
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 .
Obviously, for any , .
Following from the proof of Lemma 2.3, we can get that , , Tx is increasing and convex on .
Therefore . By Lemma 2.3, is a completely continuous operator.
Thus, the condition (A1) in the Avery-Peterson theorem is satisfied.
Consequently, the condition (A2) in the Avery-Peterson theorem is satisfied.
It is clear that .
So, the condition (A3) in the Avery-Peterson theorem holds.
In this section, we give an example to illustrate Theorem 3.1.
We choose , , , . We can easily get that , , , , .
for any ;
for any ;
for any .
Then all the conditions of Theorem 3.1 hold.
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).
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