Positive Solutions of Fractional Differential Equation with -Laplacian Operator
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.
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  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  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 , 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.  to deal with the -Laplacian fractional boundary value problem (1). By using similar method as , 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.  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 , thus our work improves essentially the results of [2, 3, 8].
2. Basic Definitions and Preliminaries
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 .
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.
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 ,
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 .
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