Journal of Function Spaces
Volume 2018 (2018), Article ID 6835028, 11 pages
https://doi.org/10.1155/2018/6835028

Research Article

Positive Solutions for a System of Semipositone Fractional Difference Boundary Value Problems

Cheng Chen,¹ Jiafa Xu,¹ Donal O’Regan,² and Zhengqing Fu³

¹School of Mathematical Sciences, Chongqing Normal University, Chongqing 40133, China
²School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
³College of Mathematics and System Sciences, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Jiafa Xu; xujiafa292@sina.com

Received 16 April 2018; Accepted 8 July 2018; Published 18 July 2018

Academic Editor: Lishan Liu

Copyright © 2018 Cheng Chen et al. 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.

Abstract.

Using the fixed point index, we establish two existence theorems for positive solutions to a system of semipositone fractional difference boundary value problems. We adopt nonnegative concave functions and nonnegative matrices to characterize the coupling behavior of our nonlinear terms.

1. Introduction

In this paper we study the existence of positive solutions for the system of fractional difference boundary value problems involving semipositone nonlinearities:

where , , , , , and is a discrete fractional operator. For the nonlinear terms , we assume the following.

(H0) are two continuous functions; moreover, there exists a positive constant such that

Note that, in this paper, we use to stand for with , where .

Fractional calculus has been applied in physics, chemistry, aerodynamics, biophysics, and blood flow phenomena. For example, T cells’ infections can be depicted by a fractional order model

where are fractional derivatives (see [1, 2]); we also refer the reader to [1–45] and the references therein. In [3], the authors considered the existence of positive solutions for the semipositone discrete fractional system

where . Using the Guo-Krasnosel’skiĭ fixed point theorem, the authors showed that the problem has positive solutions for sufficiently small values of . The growth conditions on are superlinear; i.e.,

uniformly for . Using conditions of (5) type the existence of solutions for various fractional boundary value problems was considered in [1, 4–6, 9, 11–13].

In this paper, we use the fixed point index to obtain two existence theorems for positive solutions to (1) with semipositone nonlinearities. We adopt some appropriate nonnegative concave functions and nonnegative matrices to characterize the coupling behavior of our nonlinear terms. Moreover, the growth conditions on of our nonlinearities are an improvement of (5); see conditions (H1) and (H3) in Section 3.

2. Preliminaries

We first recall some background materials from discrete fractional calculus; for more details we refer the reader to [10].

Definition 1. We define for any for which the right-hand side is well-defined. We use the convention that if is a pole of the Gamma function and is not a pole, then .

Definition 2. For , the th fractional sum of a function is

We also define the th fractional difference for by

where with .
Let be a continuous function. Then we consider the fractional difference boundary value problems

where are as in (1). The following two lemmas are in [9], so we omit their proofs.

Lemma 3 (see [9], Lemma 4). Problem (8) has a unique solution

where

Lemma 4 (see [9], Lemma 5). Green’s function (10) has the following properties.
(i) , × ,
(ii) ≤ ≤ , ∈ × , where = .
Let for . Then for . From Lemma 4, the following inequalities are satisfied:

For convenience, we let

Let be the collection of all maps from to equipped with the max norm, . Then is a Banach space. Define a set by . Then is a cone in . Note that is a Banach space with the norm , and is a cone in .
From Lemma 3, for all , we have that (1) is equivalent to

where is defined in (10).

Lemma 5 (see [46]). Let be a real Banach space and a cone on . Suppose that is a bounded open set and that is a continuous compact operator. If there exists such that

then , where denotes the fixed point index on .

Lemma 6 (see [46]). Let be a real Banach space and a cone on . Suppose that is a bounded open set with and that is a continuous compact operator. If

then .

3. Main Results

Let be a solution of

where are as in (1). Define , and then, from Lemmas 3 and 4, we have

We note that (1) has a positive solution if and only if is a solution of the fractional difference boundary value problems

and for , where are as in (1) and

and

Note that for , we mean , for all .

For , and , we define the operators

and

Then (H0) and using the Arzelà-Ascoli theorem in a standard way establish that is a completely continuous operator. It is clear that is a positive solution for (18) if and only if is a fixed point of .

Let . Then from Lemma 4 we have

If we seek a fixed point of then and

where , and , so as a result if then for (i.e., is a positive solution for (1)).

For convenience, we use to stand for different positive constants. Let for . Now, we list our assumptions on (the first two are needed for Theorem 10 and the last two are needed for Theorem 11):

(H1) There exist such that

(i) is concave and strictly increasing on ,

(ii) there exists such that

(iii) there is a such that and

for and .

(H2) For any ∈ × × , assume

(H3) There exist with such that

(i) , ,

(ii) there exist such that

(H4) For any ∈ × × , assume

Example 7. Let and for . Then, for any , we have

Let and , where , , for . Then, for any , we have

and

Also,

and

Thus, (H1)-(H2) are satisfied.

Example 8. Let and , for . Then, for any , we have

Also,

and

for . Thus, (H3)-(H4) hold.

Remark 9. (i) In (H1), the growth condition for nonlinear term depends on two variables ; however, in [7], this corresponding condition only involves one variable.
(ii) When nonlinear terms grow sublinearly at , nonnegative matrices are used to depict the coupling behavior of our nonlinearities. This is different from condition (H4) in [7].

Theorem 10. Suppose that (H0)-(H2) hold. Then (1) has at least one positive solution.

Proof. We first claim that there exists a sufficiently large positive number such that

where are two given functions. Suppose not. Then there exist and such that , and so

This implies , and for . From (H1) we have

and

As a result, for , we have

Thus

and, therefore,

Multiply both sides of the above inequality by and sum from to and together with (11) we obtain

From (23), (39), and we have . This implies

and

Note that, from (23), (39), and , we find . Moreover, we may assume , for . Then and . Thus, from the concavity of , we have

This implies that

From (40) and Lemma 4 we obtain

Combining the above two inequalities, we get

From (H1), , and thus there exists such that .
Hence, we have and . As a result, choosing we have a contradiction (recall in general ). Thus (38) is true. Consequently Lemma 5 (with chosen above) implies

Now we show that

Suppose not. Then there exist such that . This implies that

Hence, and . However, from (H2) we have

for all . This implies . Similarly, . Thus, note that , and we have

Clearly, this is a contradiction. Thus (53) is true. It follows from Lemma 6 that

From (52) and (57) we have

Therefore the operator has at least one fixed point in with , and then is a positive solution for (1). This completes the proof.

Theorem 11. Suppose that (H0), (H3), and (H4) hold. Then (1) has at least one positive solution.

Proof. We show there exists a positive constant such that

Suppose not. Then there exist such that . This implies that

From (H3) we have

Similarly, we have

Consequently, for all , multiply both sides of the above two inequalities by and sum from to and together with (11) we obtain

Similarly, we have

Consequently, we obtain

From (H3)(i) we have

Note that from the fact that . This implies

Hence

Thus if we choose , , and we have a contradiction. Thus (59) is true. Lemma 6 (with chosen above) implies

We next prove that

where are two fixed functions. Indeed, if not, there exist such that . This implies that

Hence, and . However, from (H4) we have

for all . This implies . Similarly, . Thus, note that , and we have

This is a contradiction. So (70) is true. It follows from Lemma 5 that

From (69) and (74) we have

Therefore the operator has at least one fixed point in with , and then is a positive solution for (1). This completes the proof.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Research supported by the National Natural Science Foundation of China (Grant no. 11601048), Natural Science Foundation of Chongqing (Grant no. cstc2016jcyjA0181), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant nos. KJ1703050, KJ1703043), Natural Science Foundation of Chongqing Normal University (Grant nos. 16XYY24, 15XLB011), and Graduate Student Scientific Research Innovation Projects of Chongqing (Grant no. CYS18296).

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