Research Article | Open Access

# Positive Solutions of Fractional Differential Equation with -Laplacian Operator

**Academic Editor:**Fuding Xie

#### Abstract

The basic assumption of ecological economics is that resource allocation exists social optimal solution, and the social optimal solution and the optimal solution of enterprises can be complementary. The mathematical methods and the ecological model are one of the important means in the study of ecological economics. In this paper, we study an ecological model arising from ecological economics by mathematical method, that is, study the existence of positive solutions for the fractional differential equation with -Laplacian operator , , , , , and , where are the standard Riemann-Liouville derivatives, -Laplacian operator is defined as , and the nonlinearity may be singular at both and By finding more suitable upper and lower solutions, we omit some key conditions of some existing works, and the existence of positive solution is established.

#### 1. Introduction

It is well known that differential equation models can describe many nonlinear phenomena such as applied mathematics, economic mathematics, and physical and biological processes. Undoubtedly, the application of differential equation in the economics, management science, and engineering that is most successful especially plays an important role in the construction of the model for the corresponding phenomenon. In fact, many economic processes such as ecological economics model, risk model, the CIR model, and the Gaussian model in [1] can be described by differential equations. Recently, fractional-order models have proved to be more accurate than integer-order models; that is, there are more degrees of freedom in the fractional-order models. So complicated dynamic phenomenon of fractional-order calculus system has received more and more attention; see [2–15].

In this paper, we study an ecological model arising from ecological economics by mathematical method, that is, study the existence of positive solutions for the following -Laplacian fractional boundary value problem: where are the standard Riemann-Liouville derivatives with , , , , , and , and -Laplacian operator is defined as , , , and .

The upper and lower solutions method is a powerful tool to achieve the existence results for boundary value problem; see [2–6]. Recently, Zhang and Liu [2] considered the existence of positive solutions for the singular fourth-order -Laplacian equation with the four-point boundary conditions where , , , , and may be singular at and/or 1 and . By using the upper and lower solutions method and fixed-point theorems, the existence of positive solutions to the boundary value problem is obtained. In [2], a upper and lower solution condition (H3) is used.

There exist a continuous function and some fixed positive number , such that , , and where , are the associated Green’s functions for the relevant problems. And then, the condition (H3) was also adopted by Wang et al. [3] to deal with the -Laplacian fractional boundary value problem (1). By using similar method as [2], the existence results of at least one positive solution for the above fractional boundary value problem are established. Recently, replaced (H3) with a simple integral condition, Jia et al. [8] studied the existence, uniqueness, and asymptotic behavior of positive solutions for the higher nonlocal fractional differential equation by using upper and lower solutions method.

In this paper, we restart to establish the existence of positive solutions for the BVP (1) when the nonlinearity may be singular at both and . By finding more suitable upper and lower solutions of (1), we completely omit the condition (H3) in [2, 3] and integral condition in [8], thus our work improves essentially the results of [2, 3, 8].

#### 2. Basic Definitions and Preliminaries

In this section, we present some necessary definitions and lemmas from fractional calculus theory, which can be found in the recent literatures [7, 16, 17].

*Definition 1. * The Riemann-Liouville fractional integral of order of a function is given by
provided that the right-hand side is pointwise defined on .

*Definition 2. *The Riemann-Liouville fractional derivative of order of a function is given by
where , denotes the integer part of number , provided that the right-hand side is pointwise defined on .

Lemma 3. *
If , then
**
If , then
*

Lemma 4. * Let , and let be integrable, then
**
where , and is the smallest integer greater than or equal to . *

*Definition 5. * A continuous function is called a lower solution of the BVP (1), if it satisfies

*Definition 6. * A continuous function is called an upper solution of the BVP (1), if it satisfies
For forthcoming analysis, we first consider the following linear fractional differential equation:

Lemma 7. * If and , then the boundary value problem (12) has the unique solution
**
where
*

*Proof. * By applying Lemma 4, we may reduce (12) to an equivalent integral equation
From and (16), we have . Consequently the general solution of (12) is
By (17), one has
And then, we have
So, the unique solution of problem (12) is
The proof is completed.

Lemma 8. * Let , , , , , and , . The fractional boundary value problem
**
has unique solution
**
where
**
and , and is defined by (14).*

*Proof. *At first, by Lemma 4, (21) is equivalent to the integral equation
From , and (24), we have . Consequently the general solution of (21) is
It follows from (25) that
Thus (25) and (26) imply
where . Similar to Lemma 7, we have
Consequently, fractional boundary value problem (21) is equivalent to the following problem:
Lemma 7 implies that fractional boundary value problem (21) has a unique solution
The proof is completed.

Lemma 9. *Let , , , , and , . The functions and defined by (14) and (23), respectively, are continuous on and satisfy* *, for .* * For ,
**where
*

*Proof. * The proof is obvious, so we omit the proof.

Set We present the following assumptions: (S1) is is continuous and decreasing in . (S2) For any ,

From Lemmas 7 and 9, it is easy to obtain the following conclusion.

Lemma 10. * If satisfies
**
and for any , then , for .*

#### 3. Main Results

Let , and Clearly, , so is nonempty. For any , define an operator by

Theorem 11. *Suppose (S1)-(S2) hold. Then the BVP (1) has at least one positive solution , and there exist two positive constants , , such that
*

*Proof. * We firstly assert that is well defined on and , and is decreasing in .

In fact, for any , by the definition of , there exist two positive numbers , such that for any . It follows from Lemma 9 and (S1)–(S3) that
On the other hand, by Lemma 9, we also have
Take
then by (40) and (41),
which implies that is well defined, and . It follows from that the operator is decreasing in . And by direct computations, we have

Next we focus on lower and upper solutions of the fractional boundary value problem (1). Let
then, if , the conclusion of Theorem 11 holds. If , clearly, and

We will prove that the functions , are a couple of lower and upper solutions of the fractional boundary value problem (1), respectively.

From (S1), is nonincreasing relative to . Thus it follows from (45)-(46) that
and . And it follows from (44)–(47) that
That is, and are a couple of lower and upper solutions of fractional boundary value problem (1), respectively.

Now let us define a function
It follows from (S1) and (49) that is continuous.

We will show that the fractional boundary value problem
has a positive solution. Let us consider the operator
Thus , and the fixed point of the operator is a solution of the BVP (50). Noting that , then there exists a constant , such that , . Thus for all , it follows from Lemma 9, (49), and (S2) that
That is, the operator is uniformly bounded.

From the uniform continuity of and Lebesgue dominated convergence theorem, we easily obtain that is equicontinuous. Thus by the means of the Arzela-Ascoli theorem, we have is completely continuous. The Schauder fixed point theorem implies that has at least a fixed point , such that .

At the end, we claim that
In fact, since is fixed point of and (44), we get

Otherwise, suppose that . According to the definition of , we have
On the other hand, it follows from is an upper solution to (1) that
Let ; it follows from (55) and (56) that
It follows from Lemma 10 that
and then
Notice that is monotone increasing; we have
It follows from Lemma 10 and (54) that
Thus we have on , which contradicts . Hence, is impossible.

By the same way, we also have on . So
Consequently, . Then is a positive solution of the problem (1).

Finally, by , we have
where

*Remark 12. * In Theorem 11, we find more suitable lower and upper solutions, then we refine the proved process, and the key condition (H3) in [2, 3] is removed, but the existence of positive solution is still obtained, thus our result is essential improvement of [2, 3].

Theorem 13. *If is continuous, decreasing in and , for any , then the boundary value problem (1) has at least one positive solution , and there exist two positive constants , , such that
*

*Proof. * The proof is similar to Theorem 11, so we omit it here.

*Example 14. *Consider the following boundary value problem:
Let , , and
Obviously, (S1) holds.

For any ,
which implies that (S2) holds.

By Theorem 11, that the boundary value problem (66) has at least one positive solution.

At the end of this work we also remark that the extension of the pervious results to the nonlinearities depending on the time delayed differential system for energy price adjustment or impulsive differential equation in financial field requires some further nontrivial modifications, and the reader can try to obtain results in our direction. We also anticipate that the methods and concepts here can be extended to the systems with economic processes such as risk model, the CIR model, and the Gaussian model as considered by Almeida and Vicente [1].

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#### Copyright

Copyright © 2013 Teng Ren and Xiaochun Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.