Discrete Dynamics in Nature and Society
Volume 2020 (2020), Article ID 2940726, 6 pages
https://doi.org/10.1155/2020/2940726

Research Article

On the Existence of Three Positive Solutions for a Caputo Fractional Difference Equation

Lili Kong, Huiqin Chen, Luping Li, and Shugui Kang

School of Mathematics and Statistics, Shanxi Datong University, Datong, Shanxi 037009, China

Correspondence should be addressed to Huiqin Chen; dtdxchq@126.com

Received 13 July 2020; Revised 1 September 2020; Accepted 4 September 2020; Published 24 September 2020

Academic Editor: Douglas R. Anderson

Copyright © 2020 Lili Kong 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.

In this paper, we introduce the application of three fixed point theorem by discussing the existence of three positive solutions for a class of Caputo fractional difference equation boundary value problem. We establish the condition of the existence of three positive solutions for this problem.

1. Introduction

After being proved to be a valuable tool in science and engineering fields, fractional difference equation has attracted attention of more and more scholars. And the existing results of positive solutions for boundary value problem of nonlinear fractional difference equations is the hot spot which has been discussed in recent years. So, a large number of scholars have devoted themselves to the study of fractional difference equations, such as [1–9].

At the same time, the fixed point theory (see [10–12]) has also been widely applied to study the fractional difference equations. After that, many authors obtained the existence of positive solutions for the fractional difference equations by using the fixed point theorem (see [13–22]). For example, Jiraporn Reunsumrit and Thanin Sitthiwirattham [20] considered the nonlinear discrete fractional boundary value problem of the form

where and is a continuous function. The authors employed some fixed-point theorems to obtain the existence, uniqueness of solutions, and the existence of positive solutions.

Reunsumrit and Sitthiwirattham [21] studied the three-point fractional sum boundary value problem of the form

where and is a continuous function. The authors employed Guo-Krasnoselskii’s fixed-point theorem to obtain the existence of at least one positive solution.

Based on the above research results, this article considers the existence of three positive solutions for the nonlinear fractional difference equation boundary value problem

where , is an integer. is continuous and is not identically zero, , and is the standard Caputo difference. Our analysis relies on Leggett-Williams fixed-point theorem to obtain sufficient conditions of the existence of three positive solutions for Caputo fractional boundary value problem (3). Chen et al. [22] considered the existence of positive solutions for (3). In this article, the authors obtained the existence of one or two positive solutions by means of the cone theoretic fixed-point theorems. Compared with [22], the application of Leggett-Williams fixed-point theorem makes our proving process simpler and the number of solutions increased. The research in this article shows that employing the Leggett-Williams fixed-point theorem to prove the existence of positive solutions for the fractional difference equation can get better results.

In the remainder of this paper, we will present basic definitions and some lemmas in order to prove our main results in Section 2. In Section 3, we establish some results for the existence of three positive solutions to problem (3). And some examples to corroborate our results are given in Section 4.

2. Background Materials and Preliminaries

For convenience, we first review some basic results about fractional sums and differences. For any and , we define

for which the right-hand side is defined. We appeal to the convention that if is a pole of the Gamma function and is not a pole, then .

Definition 1. For and a function defined on , the -th fractional sum of is defined by

where .

Definition 2. For and a continuous function defined on , the -th Caputo fractional difference of is given by

where . If , then .

Lemma 1 (see [13]). Assume that and is defined on domains , then

where and .

Lemma 2 (see [22]). Let and be given. Then the solution of the FBVP

is given by

where function is defined by

Here is called the Green function of boundary value problem (8).

Remark 1. Notice that , . could be extended to , so we only discuss .

Lemma 3 (see [22]). The Green function defined by (10) satisfies

(i), .(ii), .(iii), .

Definition 3 (see [12]). If is a cone of the real Banach space , a mapping is continuous and with

is called a nonnegative concave continuous functional on .
Assume that , , are positive constants, we will employ the following notations

Our existence criteria will be based on the following Leggett-Williams fixed-point theorem.

Lemma 4 (see [12]). Let be a Banach space, be a cone of , and be a constant. Suppose there exists a concave nonnegative continuous functional on with for . Let be a completely continuous operator. Assume there are numbers , , and with such that

(i)The set is nonempty and for all .(ii) for .(iii) for all with .

Then has at least three fixed points , , and . Furthermore, we have

3. Main Results

Set

Then is a Banach space with respect to the norm . We define a cone in by

Now consider the operator defined by

It is easy to see that is a solution of the FBVP (3) if and only if is a fixed point of . We shall obtain conditions for the existence of three fixed points of . First, we notice that is a summation operator on a discrete finite set. Hence, is trivially completely continuous. From (16),

hence, .

We will discuss the existence of three fixed points of by using Lemma 4. Thus, the conditions for the existence of the three positive solutions of (3) are obtained. For this purpose, let the nonnegative concave continuous function on be defined by

Denote

Suppose that the function satisfies the following condition

(C) is a nonnegative continuous function on and there exists such that .

Theorem 1. Assume condition (C) holds and there exist constants such that

(C1) for (C2) for , where (C3) for , where are positive numbers

Then the boundary value problem (3) has at least three positive solutions , and .

Proof. Set , then for , from (C3), we have

namely, . Therefore is a completely continuous operator. From (C1), we can get

Therefore, assumption (ii) of Lemma 4 is satisfied.
We choose for , then which implies . Hence, if , then for . Thus,

from which we see that for all . This shows that condition (i) of Lemma 4 is satisfied.
Finally, for with , we get

this shows that condition (iii) of Lemma 4 is satisfied. By the use of Lemma 4, the boundary value problem (3) has at least three solutions , and . Take into account that condition (C) holds, we have . The proof is completed.

Theorem 2. Assume condition (C) holds. There exist constants such that (C1), (C2), and (C4) are satisfied, where

(C4) for

Then the boundary value problem (3) has at least three positive solutions , and such that

Proof. From (C4), we get

Therefore, . The remainder of proof is essentially the same as that of Theorem 1 and is therefore omitted. By Lemma 4, the boundary value problem (3) has at least three positive solutions , and satisfying

The proof is complete.

Theorem 3. Assume condition (C) holds. There exist constants such that (C1), (C2) are satisfied, and

(C5)

Then the boundary value problem (3) has at least three positive solutions.

Proof. By (C5), there exist and , when , we have

Set , consequently

This shows that condition (C3) of Theorem 1 is satisfied. By Theorem 1, the boundary value problem (3) has at least three positive solutions. The proof is completed.

Theorem 4. Assume there exist two positive constants () such that conditions (C), (C2), and (C4) hold. And function satisfies

(C6)

Then the boundary vale problem (3) has at least three positive solutions.

Proof. In line with (C6), it is easy to see that there exists a positive constant such that for , we have

Namely,

This implies that conditions of Theorem 2 are satisfied. By Theorem 2, the boundary vale problem (3) has at least three positive solutions. The proof is completed.
In the light of the proof of Theorem 3 and Theorem 4, we obtain one theorem as follows.

Theorem 5. Assume conditions (C), (C5), and (C6) hold. Suppose that there exists a positive constants such that for . Then the boundary vale problem (3) has at least three positive solutions.

4. Examples

This section, we present two examples to illustrate our results. Set , by estimating, we then have .

Example 1. We take

There exist constants and such that

All the conditions of Theorem 1 hold. Thus, this moment, by virtue of Theorem 1, we know that the boundary value problem (3) has three positive solutions.

Example 2. We take

There exist constants and such that

All the conditions of Theorem 3 hold. Thus, in this case, by Theorem 3, we know that the boundary value problem (3) has three positive solutions.

Data Availability

No data were used in the study.

Conflicts of Interest

The authors declare that they have no conflicts of interests.

Acknowledgments

The study was supported by the National Natural Science Foundation of China (grant No. 11871314) and Datong Applied Basic Research Project (No. 2019153).

References

R. Ferreira and D. Torres, “Fractional h-difference equations arising from the calculus of variations,” Applicable Analysis and Discrete Mathematics, vol. 5, no. 1, pp. 110–121, 2011.

I. K. Dassios and D. I. Baleanu, “On a singular system of fractional nabla difference equations with boundary conditions,” Boundary Value Problems, vol. 2013, Article ID 148, 24 pages, 2013.

C. Goodrich, “On positive solutions to nonlocal fractional and integer-order difference equations,” Applicable Analysis and Discrete Mathematics, vol. 5, no. 1, pp. 122–132, 2011.

D. Seghar, S. Selvarangam, and E. Thandapani, “Oscillation criteria for even order nonlinear neutral difference equations,” Differential Equations & Applications, vol. 6, no. 3, pp. 441–453, 2014.

R. P. Agarwal, D. Baleanu, S. Rezapour, and S. Saleh, “The existence of solutions for some fractional finite difference equations via sum boundary conditions,” Advances in Difference Equations, vol. 2014, Article ID 282, 16 pages, 2014.

F. Chen and Z. Liu, “Asymptotic Stability Results for nonlinear fractional difference equations,” Journal of Applied Mathematics, vol. 2012, Article ID 879657, 14 pages, 2012.

J. W. He, L. Zhang, Y. Zhou, and B. Ahmad, “Existence of solutions for fractional difference equations via topological degree methods,” Advances in Difference Equations, vol. 2018, Article ID 153, 12 pages, 2018.

J. Soontharanon, S. Chasreechai, and T. Sitthiwirattham, “A coupled system of fractional difference equations with nonlocal fractional sum boundary conditions on the discrete half-line,” Mathematics, vol. 7, Article ID 256, 22 pages, 2019.

M. B. Almatrafi, “Solutions structures for some systems of fractional difference equations,” Open Journal of Mathematical Analysis, vol. 3, no. 1, pp. 52–61, 2019.

S. Carl and S. Heikkil, Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory, Springer, Berlin, Germany, 2011.

E. Zeidler, Nonlinear Functional Analysis and its Applications. I: Fixed-point Theorems, Springer, Berlin, Germany, 1992.

D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Shandong, China, 1985.

T. Abdeljawad, “On riemann and caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011.

F. M. Atici and P. W. Eloe, “Two-point boundary value problems for finite fractional difference equations,” Journal of Difference Equations and Applications, vol. 17, no. 4, pp. 445–456, 2011.

H. Chen, Y. Cui, and X. Zhao, “Multiple solutions to fractional difference boundary value problems,” Abstract and Applied Analysis, vol. 2014, Article ID 879380, 6 pages, 2014.

S. G. Kang, Y. Li, and H. Q. Chen, “Positive solutions to boundary value problems of fractional difference equation with nonlocal conditions,” Advances in Difference Equations, vol. 2014, Article ID 7, 12 pages, 2014.

D. Baleanu, S. Z. Rezapour, and S. Salehi, “The existence of solution for ak-dimensional system of multiterm fractional integrodifferential equations with antiperiodic boundary value problems,” Abstract and Applied Analysis, vol. 2014, Article ID 312578, 8 pages, 2014.

C. Promsakon, S. Chasreechai, and T. Sitthiwirattham, “Existence of positive solution to a coupled system of singular fractional difference equations via fractional sum boundary value conditions,” Advances in Difference Equations, vol. 2019, Article ID 128, 22 pages, 2019.

L. Zhang and Y. Zhou, “Existence and attractivity of solutions for fractional difference equations,” Advances in Difference Equations, vol. 2018, Article ID 191, 15 pages, 2018.

J. Reunsumrit and T. Sitthiwirattham, “On positive solutions to fractional sum boundary value problems for nonlinear fractional difference equations,” Mathematical Methods in the Applied Sciences, vol. 39, no. 10, pp. 2737–2751, 2016.

J. Reunsumrit and T. Sitthiwirattham, “Positive solutions of three-point fractional sum boundary value problem for Caputo fractional difference equations via an argument with a shift,” Positivity, vol. 20, no. 4, pp. 861–876, 2016.

H. Q. Chen, Z. Jin, and S. G. Kang, “Existence of positive solutions for Caputo fractional difference equation,” Advances in Difference Equations, vol. 2015, Article ID 44, 12 pages, 2015.