Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractional q-Difference Equation

Some researchers have paid close attention to the research of q-difference equation since the q-difference calculus and quantum calculus were discovered by Jackson [1, 2]. After the fractional q-difference calculus was developed by Al-Salam et al. [3–6], many papers on the fractional q-difference equation kept emerging, such as the papers [7–21] and their references. Among them, Li and Yang [7] established the existence of positive solutions for a class of nonlinear fractional qdifference equations with integral boundary conditions by applying monotone iterative method. Koca [8] provided an analytical method that can be used to solve analytically the Caputo fractional q-differential equations with initial condition x(0) = x0. The advantage of the method is that it can be applied to the integer order q-difference equations. By applying themonotone iterative technique combined with the method of lower and upper solutions, Wang et al. [9] obtained the existence of extremal solutions for fractional qdifference equation with initial value problem. There are also many papers about boundary value problems of fractional q-difference equations; see [10–19] and the references therein. These experts did researches about the existence of a positive solution andmultiple positive solutions to this problem by applying some well-known fixed-point theories such as Krasnosel’skii and Schauder fixed-point theorems.Thereinto, in [7, 15–20], the authors focused on the fractional q-difference equation with integral boundary value conditions. Motivated by the methods of [21–23] and the above works, we study the criteria of three positive solutions for a Caputo fractional q-difference equation with integral boundary value conditions by employing properties of Green’s function and the Leggett-Williams fixed-point theorem in this paper. We mainly consider the following problem:


Introduction
Some researchers have paid close attention to the research of -difference equation since the -difference calculus and quantum calculus were discovered by Jackson [1,2].After the fractional -difference calculus was developed by Al-Salam et al. [3][4][5][6], many papers on the fractional -difference equation kept emerging, such as the papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and their references.Among them, Li and Yang [7] established the existence of positive solutions for a class of nonlinear fractional difference equations with integral boundary conditions by applying monotone iterative method.Koca [8] provided an analytical method that can be used to solve analytically the Caputo fractional -differential equations with initial condition (0) =  0 .The advantage of the method is that it can be applied to the integer order -difference equations.By applying the monotone iterative technique combined with the method of lower and upper solutions, Wang et al. [9] obtained the existence of extremal solutions for fractional difference equation with initial value problem.
Motivated by the methods of [21][22][23] and the above works, we study the criteria of three positive solutions for a Caputo fractional -difference equation with integral boundary value conditions by employing properties of Green's function and the Leggett-Williams fixed-point theorem in this paper.We mainly consider the following problem: where   ]  denotes the Caputo fractional -derivative of order ] and  ∈ ([0, 1] × [0, +∞)) → [0, +∞) is continuous function.
In Section 2, basic definitions and some lemmas that will be used in the latter part are presented.In Section 3, some results for the existence of three positive solutions to problem (1)-( 2) are established.And some examples to corroborate our results are given in Section 4.

Background Materials and Preliminaries
In this piece, we show some basic definitions and some lemmas that will be used to demonstrate our main results in the latter section.
Setting 0 <  < 1, we define The -analogue of the power function ( − )  with  ∈ N is We define the -gamma function as follows: which satisfies Γ  ( + 1) = []  Γ  ().For 0 <  < 1, the -derivative of a function  is defined by The higher order -derivatives are defined by The -integral of a function  defined on the interval [0, ] is given by provided that the series converges.
Definition 2 (see [6]).The fractional -derivative of the Riemann-Liouville type of order ] ⩾ 0 is given by  0  () = () and where  is the smallest integer greater than or equal to ].
Lemma 8. Suppose 2 < ] < 3 and 0 <  < [2]  .Then the function (, ) defined by (20) satisfies the following inequalities: Proof.According to the expression of (, ), we get For the case 0 ⩽  ⩽  ⩽ 1, it is clear that For the case 0 ⩽  ⩽  ⩽ 1, it is easy to see that On the other hand, Therefore When  = 1, inequalities (28) are obvious.In conclusion, inequalities (28) are fulfilled.The proof is complete.
Definition 10 (see [24]).If  is a cone of the real Banach space , a mapping  :  → [0, ∞) is continuous and with it is called a nonnegative concave continuous functional  on .

(35)
Our existence criteria will be based on the following Leggett-Williams fixed-point theorem.

Existence of Three Positive Solutions
In this section, the above lemmas will be applied to obtain the main results of this paper.
Let [0, 1] be the space of all continuous real functions defined on [0, 1] with the maximum norm ‖‖ = max ∈[0,1] |()|.We can know it is a Banach space.Define the cone  ⊂ [0, 1] as follows: From Lemma 6, we know that () is a solution of boundary value problem ( 1)-( 2) if and only if it satisfies That is to say, the positive solutions of problem ( 1)-( 2) are equivalent to the fixed points of  in [0, 1] defined by Then () ⊂  and using the Ascoli-Arzelà theorem, we are able to confirm that  is completely continuous.
We shall use Lemma 11 to discuss the existence of three fixed points to .We then obtain sufficient conditions for the existence of three positive solutions to problem (1)-(2).To establish our main results, we take a positive number  ∈ (0, 1), letting the nonnegative concave continuous function  on  be defined by Denote (, )  ; And suppose that the function (, ) satisfies the following condition: Then the boundary value problem ( 1)-( 2) has at least three positive solutions  1 ,  2 , and  3 .
Proof.Set  > max{/( − ), }, and then, for  ∈   , we have from ( 28) That is,  ∈   .Therefore  :   →   is a completely continuous operator.By (C1), we can get Then the boundary value problem ( 1)-( 2) has at least three positive solutions  1 ,  2 , and  3 such that max Proof.From (C4), we get Therefore,  :   →   .The remainder of the proof is similar to the proof of Theorem 12 and is therefore omitted.By Lemma 11, the boundary value problem ( 1)-( 2) has at least three positive solutions  1 ,  2 , and  3 satisfying max The proof is complete.
All the conditions of Theorem 16 hold.Thus, in this case, by using Theorem 16 we know that the boundary value problem (53)-(54) has three positive solutions.

Conclusions
The main innovation of this paper was that existence criteria of three positive solutions for a Caputo fractional -difference equation with integral boundary value conditions are discussed.The study in the paper was to provide an analytical method: The Leggett-Williams fixed-point theorem can be used to solve fractional -difference equation.In order to use the Leggett-Williams fixed-point theorem, Green's function and its properties were derived.By applying these properties and the Leggett-Williams fixed-point theorem, we presented the existence of three positive solutions of this class of fractional -difference equations with integral boundary value conditions.An important advantage of this method is that it can be used to study three positive solutions for integer order -differential equations and fractional differential equation, and so forth.