On the Existence of Three Positive Solutions for a Caputo Fractional Difference Equation
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.
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  considered the nonlinear discrete fractional boundary value problem of the formwhere 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  studied the three-point fractional sum boundary value problem of the formwhere 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 problemwhere , 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.  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 , 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 definefor 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 bywhere .
Definition 2. For and a continuous function defined on , the -th Caputo fractional difference of is given bywhere . If , then .
Lemma 1 (see ). Assume that and is defined on domains , thenwhere and .
Remark 1. Notice that , . could be extended to , so we only discuss .
Definition 3 (see ). If is a cone of the real Banach space , a mapping is continuous and withis called a nonnegative concave continuous functional on .
Assume that , , are positive constants, we will employ the following notationsOur existence criteria will be based on the following Leggett-Williams fixed-point theorem.
Lemma 4 (see ). 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
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
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 numbersThen the boundary value problem (3) has at least three positive solutions , and .
Proof. Set , then for , from (C3), we havenamely, . Therefore is a completely continuous operator. From (C1), we can getTherefore, 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 getthis 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 getTherefore, . 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 satisfyingThe 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 haveSet , consequentlyThis 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 haveNamely,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.
This section, we present two examples to illustrate our results. Set , by estimating, we then have .
Example 1. We takeThere exist constants and such thatAll 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.
No data were used in the study.
Conflicts of Interest
The authors declare that they have no conflicts of interests.
The supporting material of National Natural Science Foundation of China (grant No. 11871314) and Datong Applied Basic Research Project (No. 2019153).
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