^{1, 2}

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^{2}

^{3}

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^{2}

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We establish the existence of positive solutions to a class of singular nonlocal fractional order differential system depending on two parameters. Our methods are based on Schauder’s fixed point theorem.

Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Particularly, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of many materials and processes. With this advantage, fractional order models are more realistic and practical than the classical integer-order models in physics, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, fitting of experimental data, and so forth [

However, the research on the systems of fractional differential equations has not received much attention. So motivated by the results mentioned above, in this paper, we study the existence of positive solutions for the following singular nonlocal fractional order differential system depending on two parameters:

The present paper has several interesting features. Firstly, the system depends on two parameters and the nonlinear terms

In this section, we firstly define an appropriate invariant set and then make a change of variables for the system (

Throughout this paper, we mean by

Let

By using semigroup property of the fractional integration operator (see [

On the other hand, if

Now we recall some useful lemmas by [

Let

By Lemma

The Green functions

there exist functions

Clearly, the following maximum principle is direct conclusion of Lemma

If

It is well known that

Let us define a nonlinear operator

In order to find the fixed point of

A continuous function

A continuous function

For the convenience in presentation, we now present some assumptions to be used in the rest of the paper.

For any real numbers

Suppose (A1) and (A2) hold; then for any

We start by showing that (

For any

Take

Let

Let

Define a function

Next we define an operator

Note that

In what follows, we prove

It follows from

Consider the following singular fractional order differential system

It follows from (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors were supported financially by the National Natural Science Foundation of China (no. 11371221) and the Project of Shandong Province Higher Educational Science and Technology Program (no. J13LI12).

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