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

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 [

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

The upper and lower solutions method is a powerful tool to achieve the existence results for boundary value problem; see [

There exist a continuous function

In this paper, we restart to establish the existence of positive solutions for the BVP (

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

The Riemann-Liouville fractional integral of order

The Riemann-Liouville fractional derivative of order

Let

A continuous function

A continuous function

If

By applying Lemma

Let

At first, by Lemma

Let

The proof is obvious, so we omit the proof.

Set

(S1)

(S2) For any

From Lemmas

If

Let

Suppose (S1)-(S2) hold. Then the BVP (

We firstly assert that

In fact, for any

Next we focus on lower and upper solutions of the fractional boundary value problem (

We will prove that the functions

From (S1),

Now let us define a function

We will show that the fractional boundary value problem

From the uniform continuity of

At the end, we claim that

Otherwise, suppose that

By the same way, we also have

Finally, by

In Theorem

If

The proof is similar to Theorem

Consider the following boundary value problem:

For any

By Theorem

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 [