# Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractionalq-Difference Equation

Authors: Huiqin Chen; Shugui Kang; Lili Kong; Ying Gao
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

A class of boundary value problems of Caputo fractionalq-difference equation is introduced. Green’s function and its properties for this problem are deduced. By applying these properties and the Leggett-Williams fixed-point theorem, existence criteria of three positive solutions are obtained. At last, some examples are given to illustrate the validity of our main results.


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## Body

## 1. Introduction

Some researchers have paid close attention to the research ofq-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 q-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 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 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 q-difference equation with initial value problem.There are also many papers about boundary value problems of fractionalq-difference equations; see [10–19] and the references therein. These experts did researches about the existence of a positive solution and multiple 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: (1)DqνxCt+ft,xt=0,0<q<1,0<t<1,2<ν<3,(2)x0=Dq2x0=0,x1=λ∫01xsdqs,where DqνC denotes the Caputo fractional q-derivative of order ν and  f∈C([0,1]×[0,+∞))→[0,+∞) is continuous function.In Section2, 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.


## 2. 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.Setting0<q<1, we define(3)νq=1-qν1-q,ν∈R.Theq-analogue of the power function (t-s)n with n∈N is(4)t-s0=1,t-sn=∏k=0n-1t-sqk,n∈N,t,s∈R.Ifα∈R, then (5)t-sα=tα∏n=0∞t-sqnt-sqn+α.We define theq-gamma function as follows: (6)Γqs=1-qs-11-q1-s,s∈R∖0,-1,-2,…, which satisfies Γq(s+1)=[s]qΓq(s).For0<q<1, the q-derivative of a function f is defined by(7)Dqft=dqdqtft=ft-fqt1-qt,Dqf0=limt→0⁡Dqft,t≠0.The higher orderq-derivatives are defined by(8)Dq0ft=ft,Dqnft=DqDqn-1ft,n∈N.Theq-integral of a function f defined on the interval [0,b] is given by(9)Iqft=∫0tfsdqs=t1-q∑n=0∞ftqnqn,t∈0,b provided that the series converges.Iff is defined on the interval [0,b] and a∈[0,b], its q-integral from a to b is defined by(10)∫abfsdqs=∫0bfsdqs-∫0afsdqs.The higher orderq-integrals are defined by(11)Iq0ft=ft,Iqnft=IqIqn-1ft,n∈N.We note that(DqIqf)(t)=f(t) and if f is continuous at x=0, we get (IqDqf)(t)=f(t)-f(0). For more details on the basic material of q-calculus, the readers can refer to [1–6].Now let us give definitions of fractionalq-integral and q-derivative.Definition 1 (see [4, 6]).
Letν⩾0 and let f be a function defined on L1([0,1]). The fractional q-integral of the Riemann-Liouville type is (Iq0f)(t)=f(t) and (12)Iqνft=1Γqν∫0tt-qsν-1fsdqs,ν>0,t∈0,1, where L1([0,1]) denotes the classical Banach space consisting of measurable functions on [0,1] that are integrable.Definition 2 (see [6]).
The fractionalq-derivative of the Riemann-Liouville type of order ν⩾0 is given by Dq0f(t)=f(t) and (13)Dqνft=DqnIqn-νft, where n is the smallest integer greater than or equal to ν.Definition 3 (see [13]).
The fractionalq-derivative of Caputo type of order ν⩾0 for a function f is defined by (14)DqνfCt=Iqn-νDqnft.Lemma 4 (see [13]).
Letν>0 and let n be the smallest integer greater than or equal to ν. Then, for t∈[0,1], the following equality holds: (15)IqνDqνfCt=ft+∑k=0n-1tkΓqk+1Dqkf0.Lemma 5 (see [13]).
Letα∈R+,β∈(-1,+∞), and the following is valid:(16)Iqαt-sβ=Γqβ+1Γqβ+α+1t-sβ+α.Next, Green’s function for integral boundary value problem (1)-(2) is derived and the properties of Green’s function are concluded. These properties will be used to demonstrate the main results in Section 3.Lemma 6.
Giveng(t)∈C[0,1], the unique solution of the following problem (17)DqνxCt+gt=0,0<t<1,2<ν<3,(18)x0=Dq2x0=0,x1=λ∫01xsdqsis (19)xt=∫01Gt,qsgsdqs, where (20)Gt,qs=1τΓqν2qt1-qsν-1νq-λ+λsqν-τt-qsν-1,0⩽qs⩽t⩽1,2qt1-qsν-1νq-λ+λsqν,0⩽t⩽qs⩽1,and τ=([2]q-λ)[ν]q. Here G(t,qs) is called Green’s function of boundary value problem (17)-(18).Proof.
By Lemma4, it is clear that (17) is equivalent to(21)xt=-1Γqν∫0tt-qsν-1gsdqs+c1+c2t+c3t2 for some ci∈R,i=1,2,3. Applying the boundary condition x(0)=Dq2x(0)=0, there is c1=c3=0, and then (22)xt=-1Γqν∫0tt-qsν-1gsdqs+c2t.Using the condition x(1)=λ∫01x(s)dqs, there is (23)c2=1Γqν∫011-qsν-1gsdqs+λ∫01xsdqs. Substituting c2 into (22), we get (24)xt=-1Γqν∫0tt-qsν-1gsdqs+tΓqν∫011-qsν-1gsdqs+λt∫01xsdqs. Let ξ=∫01x(s)dqs; integrating equality (24) with respect to t from t=0 to t=1 and then exchanging integral order, we obtain that (25)ξ=-1Γqν∫01dqt∫0tt-qsν-1gsdqs+1Γqν∫01tdqt∫011-qsν-1gsdqs+λξ∫01sdqs=-1νqΓqν∫011-qsνgsdqs+12qΓqν∫011-qsν-1gsdqs+λξ2q. Solving the above equation, then (26)ξ=-2qτΓqν∫011-qsνgsdqs+νqτΓqν∫011-qsν-1gsdqs. Substituting ξ into (24), we get (27)xt=-1Γqν∫0tt-qsν-1gsdqs+1τΓqν∫012qt1-qsν-1νq-λ+λsqνgsdqs=1τΓqν∫0t2qt1-qsν-1νq-λ+λsqν-τt-qsν-1gsdqs+1τΓqν∫t12qt1-qsν-1νq-λ+λsqνgsdqs=∫01Gt,qsgsdqs.
The proof is complete.Remark 7.
It is obvious thatG(0,qs)=G(t,1)=0 for all t,qs∈[0,1] and λ≠[2]q.Lemma 8.
Suppose2<ν<3 and 0<λ<[2]q. Then the function G(t,qs) defined by (20) satisfies the following inequalities: (28)tG1,qs⩽Gt,qs⩽2qνqλνq-2qG1,qs,(t,qs)∈[0,1]×[0,1].Proof.
According to the expression ofG(t,qs), we get (29)Gt,qsG1,qs=2qt1-qsν-1νq-λ+λsqν-τt-qsν-11-qsν-1λνq-2q1-sqν,0⩽qs⩽t⩽1,2qtνq-λ+λsqνλνq-2q1-sqν,0⩽t⩽qs⩽1.
For the case0⩽t⩽qs⩽1, it is clear that (30)Gt,qsG1,qs⩽2qνqλνq-2q,Gt,qsG1,qs⩾2qt2qsqν-1+νqλ2qsqν-1+νq=2qtλ⩾t.
For the case0⩽qs⩽t⩽1, it is easy to see that (31)Gt,qsG1,qs⩽2qtνq-λ+λsqνλνq-2q1-sqν⩽2qνqλνq-2q. On the other hand, (32)Gt,qsG1,qs⩾2qt1-qsν-1νq-λ+λsqν-τt-tqsν-11-qsν-1λνq-2q1-sqν=2qtνq-λ+λsqν-2q-λνqtν-1λνq-2q1-sqν⩾2qtνq-λ+λsqν-2q-λνqtλνq-2q1-sqν=t-2qλ+2qλsqν+λνqλνq-2q1-sqν=t. Therefore (33)tG1,qs⩽Gt,qs⩽2qνqλνq-2qG1,qs,0⩽qs<t⩽1. When t=1, inequalities (28) are obvious. In conclusion, inequalities (28) are fulfilled. The proof is complete.Corollary 9.
If[ν]q-[2]q>0, then G(t,qs)>0, for t,qs∈(0,1).Definition 10 (see [24]).
IfP is a cone of the real Banach space E, a mapping θ:P→[0,∞) is continuous and with (34)θtx+1-ty⩾tθx+1-tθy,x,y∈P,t∈0,1, it is called a nonnegative concave continuous functional θ on P.
Assuming thatr, a, b are positive constants, we will employ the following notations: (35)Pr=x∈P:x<r,P¯r=x∈P:x⩽r,Pθ,a,b=x∈P:θx⩾a,x⩽b.
Our existence criteria will be based on the following Leggett-Williams fixed-point theorem.Lemma 11 (see [24]).
LetE=(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 θ on P with θ(x)⩽x for x∈P¯c. Let T:P¯c→P¯c be a completely continuous operator. Assume there are numbers , a, and b with 0<d<a<b≤c such that(i)
the set{x∈P(θ,a,b):θ(x)>a} is nonempty and θ(Tx)>a for all x∈P(θ,a,b);(ii)
Tx<d for x∈P¯d;(iii)
θ(Tx)>a for all x∈P(θ,a,c) with Tx>b.
ThenT has at least three fixed points x1,  x2, and x3∈P¯c. Furthermore, we have (36)maxt∈0,1⁡x1t<d,a<mint∈0,1⁡x2t<maxt∈0,1⁡x2t<c,d<maxt∈0,1⁡x3t≤c,mint∈0,1⁡x3t<a.


## 3. Existence of Three Positive Solutions

In this section, the above lemmas will be applied to obtain the main results of this paper.LetC[0,1] be the space of all continuous real functions defined on [0,1] with the maximum norm x=maxt∈[0,1]⁡|x(t)|. We can know it is a Banach space. Define the cone P⊂C[0,1] as follows: (37)P=x∈C0,1:xt⩾0,t∈0,1.From Lemma6, we know that x(t) is a solution of boundary value problem (1)-(2) if and only if it satisfies (38)xt=∫01Gt,qsfs,xsdqs,t∈0,1. That is to say, the positive solutions of problem (1)-(2) are equivalent to the fixed points of T in C[0,1] defined by(39)Txt=∫01Gt,qsfs,xsdqs,t∈0,1. Then T(P)⊂P and using the Ascoli-Arzelà theorem, we are able to confirm that T is completely continuous.We shall use Lemma11 to discuss the existence of three fixed points to T. 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 μ∈(0,1), letting the nonnegative concave continuous function θ on P be defined by (40)θx=mint∈μ,1⁡xt. Denote (41)f0=limsupx→0⁡maxt∈0,1⁡ft,xx,f∞=limsupx→+∞⁡maxt∈0,1⁡ft,xx;A-1=∫012qνqλνq-2qG1,qsdqs,B-1=∫μ1G1,qsdqs,ρ=νq2qλμνq-2q. And suppose that the function f(t,x) satisfies the following condition:(C)
f(t,x) is a nonnegative continuous function on [0,1]×[0,+∞) and there exists tn→0 such that f(tn,x(tn))>0,n=1,2,….Theorem 12.
Assume that condition (C) holds and there exist constants0<d<a such that(C1)
f(t,x)<Ad for (t,x)∈[0,1]×[0,d];(C2)
f(t,x)⩾B/μa for (t,x)∈[μ,1]×[a,c], where c>ρa;(C3)
f(t,x)⩽κx+β for (t,x)∈[0,1]×[0,+∞), where κ,β are positive numbers.
Then the boundary value problem (1)-(2) has at least three positive solutions x1,x2, and x3.Proof.
Setc>max⁡{β/A-κ,ρa}, and then, for x∈P¯c, we have from (28)(42)Tx=maxt∈0,1∫01Gt,qsfs,xsdqs⩽∫012qνqλνq-2qG1,qsκxs+βdqs⩽κx+β∫012qνqλνq-2qG1,qsdqs=κx+βA<c. That is, Tx∈Pc. Therefore T:P¯c→P¯c is a completely continuous operator. By (C1), we can get(43)Tx⩽∫012qνqλνq-2qG1,qsfs,xsdqs<Ad∫012qνqλνq-2qG1,qsdqs=d. Hence condition (ii) of Lemma 11 is satisfied.
We choosex0=(ρ+1)a/2 for t∈[μ,1]; then x0∈{x∈P(θ,a,ρa):θ(x)>a}, which implies {x∈P(θ,a,ρa):θ(x)>a}≠∅. Hence, if x∈P(θ,a,ρa), then a⩽x(t)⩽ρa for μ⩽t⩽1. Thus(44)θTx=minμ⩽t⩽1⁡∫01Gt,qsfs,xsdqs>∫μ1minμ⩽t⩽1⁡Gt,qsfs,xsdqs>∫μ1μG1,qsfs,xsdqs⩾Bμa∫μ1μG1,qsdqs=a. From the above inequality, we see that θ(Tx)>a for all x∈P(θ,a,ρa). This affirms that condition (i) of Lemma 11 is satisfied.
Finally, forx∈P(θ,a,c) with Tx>ρa, we get(45)θTx=minμ⩽t⩽1⁡∫01Gt,qsfs,xsdqs⩾μ∫01G1,qsfs,xsdqs=1ρ∫01νq2qλνq-2qG1,qsfs,xsdqs⩾1ρmax0⩽t⩽1⁡∫01Gt,qsfs,xsdqs>1ρ·ρa=a. This confirms that condition (iii) of Lemma 11 is fulfilled. By virtue of Lemma 11, the boundary value problem (1)-(2) has at least three solutions x1,x2, and x3. Taking into account the fact that condition (C) holds, we have xi(t)>0,0<t<1,i=1,2,3. The proof is complete.Theorem 13.
Let condition (C) hold. Assume that there exist constants0<d<a<c(c>ρa) such that (C1), (C2), and (C4) are satisfied, where (C4)
f(t,x)⩽Ac for (t,x)∈[0,1]×[0,c].
Then the boundary value problem (1)-(2) has at least three positive solutions x1,x2, and x3 such that (46)maxt∈0,1⁡x1t<d,a<mint∈μ,1⁡x2t<maxt∈0,1⁡x2t<c,d<maxt∈0,1⁡x3t⩽c,mint∈μ,1⁡x3t<a.Proof.
From (C4), we get(47)Tx=maxt∈0,1⁡∫01Gt,qsfs,xsdqs⩽∫012qνqλνq-2qG1,qsfs,xsdqs<Ac∫012qνqλνq-2qG1,qsdqs=c. Therefore, T:P¯c→P¯c. 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 x1,x2, and x3 satisfying (48)maxt∈0,1⁡x1t<d,a<mint∈μ,1⁡x2t<maxt∈0,1⁡x2t<c,d<maxt∈0,1⁡x3t⩽c,mint∈μ,1⁡x3t<a. The proof is complete.Theorem 14.
Let condition (C) hold. Assume that there exist constants0<d<a such that (C1) and (C2) are satisfied, and function f(t,s) satisfies(C5)
f∞<A.
Then the boundary vale problem (1)-(2) has at least three positive solutions.Proof.
From hypothesis (C5), there exist0<σ<A and R>0; when x⩾R, we have (49)ft,x⩽σu. Set M=max(t,x)∈[0,1]×[0,R]⁡f(t,x); consequently we get (50)0⩽ft,x⩽σx+M,0⩽x<+∞. This shows that condition (C3) of Theorem 12 is satisfied. By Theorem 12, the boundary value problem (1)-(2) has at least three positive solutions. The proof is complete.Theorem 15.
Assume that there exist two positive constantsa,c(c>ρa) such that conditions (C), (C2), and (C4) hold. And function f(t,x) satisfies(C6)
f0<A.
Then the boundary value problem (1)-(2) has at least three positive solutions.Proof.
In line with (C6), it is easy to see that there exists a positive constantd<a such that, for x<d, we have (51)ft,xt<Ax. That is to say, (52)ft,xt<Ad,x<d. This implies that conditions of Theorem 13 are satisfied. By Theorem 13, the boundary value problem (1)-(2) has at least three positive solutions. The proof is complete.In light of the proof of Theorems14 and 15, we obtain one theorem and four corollaries as follows.Theorem 16.
Assume that the functionf(t,x) satisfies conditions (C), (C2), (C5), and (C6). Then the boundary value problem (1)-(2) has at least three positive solutions.Corollary 17.
Assume that conditions (C), (C2), and (C3) hold. The functionf(t,x) satisfies f0=0. Then the boundary value problem (1)-(2) has at least three positive solutions.Corollary 18.
Assume that conditions (C), (C1), and (C2) hold. The functionf(t,x) satisfies f∞=0. Then the boundary value problem (1)-(2) has at least three positive solutions.Corollary 19.
Assume that conditions (C), (C2), and (C4) hold. The functionf(t,x) satisfies f∞=0. Then the boundary value problem (1)-(2) has at least three positive solutions.Corollary 20.
Assume that conditions (C) and (C2) hold. The functionf(t,x) satisfies f0=0 and f∞=0. Then the boundary value problem (1)-(2) has at least three positive solutions.


## 4. Examples

In this section, we present three examples to illustrate our results. We takeν=5/2,q=1/2,μ=1/2,λ=1/2, and by estimation, we then have A>0.65,B<17.785.Consider the Caputo fractionalq-difference (53)DqνxCt+ft,x=0,0<t<1,with the boundary conditions (54)x0=Dq2x0=0,x1=12∫01xsdqs.Example 1.
We take(55)ft,x=2t125+36x3,t,x∈0,1×0,1,2t125+35+x,t,x∈0,1×1,+∞. There exist constants d=1/36 and a=253/250 such that(56)ft,x=2t125+36x3⩽0.0167716<0.65×136d<Adfort,x∈0,1×0,136;ft,x=2t125+35+x⩾36.02>17.785·1μa>Bμafort,x∈μ,1×a,ρa+1;ft,x⩽2/125+36xfort,x∈0,1×0,+∞.
All the conditions of Theorem12 hold. Thus, at this moment, by virtue of Theorem 12 we know that the boundary value problem (53)-(54) has three positive solutions.Example 2.
We take(57)ft,x=100e-tx5,t,x∈0,1×0,1,e-t99.35x1/2+0.65x,t,x∈0,1×1,+∞. There exist constants d=0.25 and a=1.06 such that(58)f∞=0.65<Afort,x∈0,1×0,+∞;ft,x⩾37.8822>17.785·1μa>Bμafort,x∈μ,1×1.06,ρa+1;ft,x⩽100d5⩽100dfort,x∈0,1×0,0.25.
All the conditions of Theorem14 hold. Thus, in this case, by Theorem 14 we know that the boundary value problem (53)-(54) has three positive solutions.Example 3.
We seek(59)ft,x=0.5x+45.1x3,t,x∈0,1×0,1,45x1/2+0.6x,t,x∈0,1×1,+∞. There exists constant a=1.5 such that(60)f∞=0.6<0.65<A;f0=0.5<0.65<A;ft,x=45x1/2+0.6x⩾56.0135>17.785·1μa>Bμafort,x∈μ,1×a,ρa+1.
All the conditions of Theorem16 hold. Thus, in this case, by using Theorem 16 we know that the boundary value problem (53)-(54) has three positive solutions.


## 5. Conclusions

The main innovation of this paper was that existence criteria of three positive solutions for a Caputo fractionalq-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 q-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 q-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 q-differential equations and fractional differential equation, and so forth.



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