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

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 +anin Sitthiwirattham [20] considered the nonlinear discrete fractional boundary value problem of the form


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][2][3][4][5][6][7][8][9].
Based on the above research results, this article considers the existence of three positive solutions for the nonlinear fractional difference equation boundary value problem is continuous and f is not identically zero, 2 < ] ≤ 3, and △ ] C u(t) 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. e 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.

Background Materials and Preliminaries
For convenience, we first review some basic results about fractional sums and differences. For any t ∈ N b+1 for which the right-hand side is defined. We appeal to the convention that if t + 1 − ] is a pole of the Gamma function and t + 1 is not a pole, then t ] � 0.
Lemma 2 (see [22]). Let 2 < ] ≤ 3 and g: is given by where function G: Here G(t, s) is called the Green function of boundary value problem (8).
Lemma 3 (see [22]). e Green function G(t, s) defined by (10) satisfies Definition 3 (see [12]). If P is a cone of the real Banach space E, a mapping ψ: P ⟶ [0, ∞) is continuous and with is called a nonnegative concave continuous functional ψ on P.
Assume that r, a, b 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 E � (E, ‖ · ‖) be a Banach space, P ⊂ E be a cone of E, and c > 0 be a constant. Suppose there exists a concave nonnegative continuous functional ψ on P with ψ(u) ≤ ‖u‖ for u ∈ P c . Let A: P c ⟶ P c be a completely continuous operator. Assume there are numbers d, a, and b with 0 < d < a < b ≤ c such that (i) e set u ∈ P(ψ, a, b): ψ(u) > a is nonempty and ψ(Au) > a for all u ∈ P (ψ, a, b).
en A has at least three fixed points u 1 , u 2 , and u 3 ∈ P c . Furthermore, we have

Set
en B is a Banach space with respect to the norm ‖u‖ � max Now consider the operator T defined by It is easy to see that u � u(t) is a solution of the FBVP (3) if and only if u � u(t) is a fixed point of T. We shall obtain conditions for the existence of three fixed points of T. First, we notice that T is a summation operator on a discrete finite set. Hence, T is trivially completely continuous. From (16), hence, TP ⊂ P. We will discuss the existence of three fixed points of T by using Lemma 4. us, the conditions for the existence of the three positive solutions of (3) are obtained. For this purpose, let the nonnegative concave continuous function ψ on P be defined by Denote Suppose that the function f(t, u) satisfies the following condition (C) f(t, u) is a nonnegative continuous function on [0, 1] × [0, +∞) and there exists t n ⟶ 0 such that f(t n , u(t n )) > 0, n � 1, 2, · · ·. Theorem 1. Assume condition (C) holds and there exist constants 0 < d < a such that Proof. Set c > max βl/(1 − κl), 4a , then for u ∈ P c , from (C3), we have ‖Tu‖ � max namely, Tu ∈ P c . erefore T: P c ⟶ P c is a completely continuous operator. From (C1), we can get Discrete Dynamics in Nature and Society ‖Tu‖ � max erefore, assumption (ii) of Lemma 4 is satisfied.
from which we see that ψ(Tu) > a for all u ∈ P(ψ, a, c). is shows that condition (i) of Lemma 4 is satisfied. Finally, for u ∈ P(ψ, a, c) with ‖Tu‖ > 4a, we get ψ(Tu) � min 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 u 1 , u 2 , and u 3 . Take into account that condition (C) holds, we have u i (t) > 0, 0 < t < 1, i � 1, 2, 3. e proof is completed.
Proof. From (C4), we get ‖Tu‖ � max erefore, T: P c ⟶ P c . e remainder of proof is essentially the same as that of eorem 1 and is therefore omitted. By Lemma 4, the boundary value problem (3) has at least three positive solutions u 1 , u 2 , and u 3 satisfying is shows that condition (C3) of eorem 1 is satisfied. By eorem 1, the boundary value problem (3) has at least three positive solutions. e proof is completed.
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