DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2018/2475284 2475284 Research Article Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractional q-Difference Equation Chen Huiqin 1 http://orcid.org/0000-0003-2216-6469 Kang Shugui 1 Kong Lili 1 Gao Ying 1 Anderson Douglas R. School of Mathematics and Computer Sciences Shanxi Datong University Datong Shanxi 037009 China sxdtdx.edu.cn 2018 2622018 2018 28 11 2017 09 01 2018 28 01 2018 2622018 2018 Copyright © 2018 Huiqin Chen 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.

A class of boundary value problems of Caputo fractional q-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.

National Natural Science Foundation of China 11271235 Development Foundation of Higher Education Department of Shanxi Province 20101109 20111117 20111020 Shanxi Datong University 2016K9 2017K4
1. Introduction

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. , many papers on the fractional q-difference equation kept emerging, such as the papers  and their references. Among them, Li and Yang  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  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.  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 fractional q-difference equations; see  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, 1520], the authors focused on the fractional q-difference equation with integral boundary value conditions.

Motivated by the methods of  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=λ01xsdqs,where DqνC denotes the Caputo fractional q-derivative of order ν and  fC([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.

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.

Setting 0<q<1, we define(3)νq=1-qν1-q,νR.

The q-analogue of the power function (t-s)n with nN is(4)t-s0=1,t-sn=k=0n-1t-sqk,nN,t,sR.

If αR, then (5)t-sα=tαn=0t-sqnt-sqn+α.

We define the q-gamma function as follows: (6)Γqs=1-qs-11-q1-s,sR0,-1,-2,, which satisfies Γq(s+1)=[s]qΓq(s).

For 0<q<1, the q-derivative of a function f is defined by(7)Dqft=dqdqtft=ft-fqt1-qt,Dqf0=limt0Dqft,t0.

The higher order q-derivatives are defined by(8)Dq0ft=ft,Dqnft=DqDqn-1ft,nN.

The q-integral of a function f defined on the interval [0,b] is given by(9)Iqft=0tfsdqs=t1-qn=0ftqnqn,t0,b provided that the series converges.

If f 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 order q-integrals are defined by(11)Iq0ft=ft,Iqnft=IqIqn-1ft,nN.

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 .

Now let us give definitions of fractional q-integral and q-derivative.

Definition 1 (see [<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr" rid="B6">6</xref>]).

Let ν0 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=1Γqν0tt-qsν-1fsdqs,ν>0,t0,1, where L1([0,1]) denotes the classical Banach space consisting of measurable functions on [0,1] that are integrable.

Definition 2 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

The fractional q-derivative of the Riemann-Liouville type of order ν0 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 [<xref ref-type="bibr" rid="B13">13</xref>]).

The fractional q-derivative of Caputo type of order ν0 for a function f is defined by (14)DqνfCt=Iqn-νDqnft.

Lemma 4 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

Let ν>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 [<xref ref-type="bibr" rid="B13">13</xref>]).

Let αR+,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.

Given g(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=λ01xsdqsis (19)xt=01Gt,qsgsdqs, where (20)Gt,qs=1τΓqν2qt1-qsν-1νq-λ+λsqν-τt-qsν-1,0qst1,2qt1-qsν-1νq-λ+λsqν,0tqs1,and τ=(q-λ)[ν]q. Here G(t,qs) is called Green’s function of boundary value problem (17)-(18).

Proof.

By Lemma 4, it is clear that (17) is equivalent to(21)xt=-1Γqν0tt-qsν-1gsdqs+c1+c2t+c3t2 for some ciR,i=1,2,3. Applying the boundary condition x(0)=Dq2x(0)=0, there is c1=c3=0, and then (22)xt=-1Γqν0tt-qsν-1gsdqs+c2t.Using the condition x(1)=λ01x(s)dqs, there is (23)c2=1Γqν011-qsν-1gsdqs+λ01xsdqs. Substituting c2 into (22), we get (24)xt=-1Γqν0tt-qsν-1gsdqs+tΓqν011-qsν-1gsdqs+λt01xsdqs. 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=-1Γqν01dqt0tt-qsν-1gsdqs+1Γqν01tdqt011-qsν-1gsdqs+λξ01sdqs=-1νqΓqν011-qsνgsdqs+12qΓqν011-qsν-1gsdqs+λξ2q. Solving the above equation, then (26)ξ=-2qτΓqν011-qsνgsdqs+νqτΓqν011-qsν-1gsdqs. Substituting ξ into (24), we get (27)xt=-1Γqν0tt-qsν-1gsdqs+1τΓqν012qt1-qsν-1νq-λ+λsqνgsdqs=1τΓqν0t2qt1-qsν-1νq-λ+λsqν-τt-qsν-1gsdqs+1τΓqνt12qt1-qsν-1νq-λ+λsqνgsdqs=01Gt,qsgsdqs.

The proof is complete.

Remark 7.

It is obvious that G(0,qs)=G(t,1)=0 for all t,qs[0,1] and λq.

Lemma 8.

Suppose 2<ν<3 and 0<λ<q. Then the function G(t,qs) defined by (20) satisfies the following inequalities: (28)tG1,qsGt,qs2qνqλνq-2qG1,qs,(t,qs)[0,1]×[0,1].

Proof.

According to the expression of G(t,qs), we get (29)Gt,qsG1,qs=2qt1-qsν-1νq-λ+λsqν-τt-qsν-11-qsν-1λνq-2q1-sqν,0qst1,2qtνq-λ+λsqνλνq-2q1-sqν,0tqs1.

For the case 0tqs1, it is clear that (30)Gt,qsG1,qs2qνqλνq-2q,Gt,qsG1,qs2qt2qsqν-1+νqλ2qsqν-1+νq=2qtλt.

For the case 0qst1, it is easy to see that (31)Gt,qsG1,qs2qtνq-λ+λsqνλνq-2q1-sqν2qνqλνq-2q. On the other hand, (32)Gt,qsG1,qs2qt1-qsν-1νq-λ+λsqν-τt-tqsν-11-qsν-1λνq-2q1-sqν=2qtνq-λ+λsqν-2q-λνqtν-1λνq-2q1-sqν2qtνq-λ+λsqν-2q-λνqtλνq-2q1-sqν=t-2qλ+2qλsqν+λνqλνq-2q1-sqν=t. Therefore (33)tG1,qsGt,qs2qνqλνq-2qG1,qs,0qs<t1. When t=1, inequalities (28) are obvious. In conclusion, inequalities (28) are fulfilled. The proof is complete.

Corollary 9.

If [ν]q-q>0, then G(t,qs)>0, for t,qs(0,1).

Definition 10 (see [<xref ref-type="bibr" rid="B25">24</xref>]).

If P is a cone of the real Banach space E, a mapping θ:P[0,) is continuous and with (34)θtx+1-tytθx+1-tθy,x,yP,t0,1, it is called a nonnegative concave continuous functional θ on P.

Assuming that r, a, b are positive constants, we will employ the following notations: (35)Pr=xP:x<r,P¯r=xP:xr,Pθ,a,b=xP:θxa,xb.

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

Lemma 11 (see [<xref ref-type="bibr" rid="B25">24</xref>]).

Let E=(E,·) be a Banach space, PE 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 xP¯c. Let T:P¯cP¯c be a completely continuous operator. Assume there are numbers , a, and b with 0<d<a<bc such that

the set {xP(θ,a,b):θ(x)>a} is nonempty and θ(Tx)>a for all xP(θ,a,b);

Tx<d for xP¯d;

θ(Tx)>a for all xP(θ,a,c) with Tx>b.

Then T has at least three fixed points x1,  x2, and x3P¯c. Furthermore, we have (36)maxt0,1x1t<d,a<mint0,1x2t<maxt0,1x2t<c,d<maxt0,1x3tc,mint0,1x3t<a.

3. Existence of Three Positive Solutions

In this section, the above lemmas will be applied to obtain the main results of this paper.

Let C[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 PC[0,1] as follows: (37)P=xC0,1:xt0,t0,1.

From Lemma 6, 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,t0,1. 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,t0,1. Then T(P)P and using the Ascoli-Arzelà theorem, we are able to confirm that T is completely continuous.

We shall use Lemma 11 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=mintμ,1xt. Denote (41)f0=limsupx0maxt0,1ft,xx,f=limsupx+maxt0,1ft,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:

f(t,x) is a nonnegative continuous function on [0,1]×[0,+) and there exists tn0 such that f(tn,x(tn))>0,n=1,2,.

Theorem 12.

Assume that condition (C) holds and there exist constants 0<d<a such that

f(t,x)B/μa for (t,x)[μ,1]×[a,c], where c>ρa;

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.

Set c>max{β/A-κ,ρa}, and then, for xP¯c, we have from (28)(42)Tx=maxt0,101Gt,qsfs,xsdqs012qνqλνq-2qG1,qsκxs+βdqsκx+β012qνqλνq-2qG1,qsdqs=κx+βA<c. That is, TxPc. Therefore T:P¯cP¯c is a completely continuous operator. By (C1), we can get(43)Tx012qνqλνq-2qG1,qsfs,xsdqs<Ad012qνqλνq-2qG1,qsdqs=d. Hence condition (ii) of Lemma 11 is satisfied.

We choose x0=(ρ+1)a/2 for t[μ,1]; then x0{xP(θ,a,ρa):θ(x)>a}, which implies {xP(θ,a,ρa):θ(x)>a}. Hence, if xP(θ,a,ρa), then ax(t)ρa for μt1. Thus(44)θTx=minμt101Gt,qsfs,xsdqs>μ1minμt1Gt,qsfs,xsdqs>μ1μG1,qsfs,xsdqsBμaμ1μG1,qsdqs=a. From the above inequality, we see that θ(Tx)>a for all xP(θ,a,ρa). This affirms that condition (i) of Lemma 11 is satisfied.

Finally, for xP(θ,a,c) with Tx>ρa, we get(45)θTx=minμt101Gt,qsfs,xsdqsμ01G1,qsfs,xsdqs=1ρ01νq2qλνq-2qG1,qsfs,xsdqs1ρmax0t101Gt,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 constants 0<d<a<c(c>ρa) such that (C1), (C2), and (C4) are satisfied, where

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)maxt0,1x1t<d,a<mintμ,1x2t<maxt0,1x2t<c,d<maxt0,1x3tc,mintμ,1x3t<a.

Proof.

From (C4), we get(47)Tx=maxt0,101Gt,qsfs,xsdqs012qνqλνq-2qG1,qsfs,xsdqs<Ac012qνqλνq-2qG1,qsdqs=c. Therefore, T:P¯cP¯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)maxt0,1x1t<d,a<mintμ,1x2t<maxt0,1x2t<c,d<maxt0,1x3tc,mintμ,1x3t<a. The proof is complete.

Theorem 14.

Let condition (C) hold. Assume that there exist constants 0<d<a such that (C1) and (C2) are satisfied, and function f(t,s) satisfies

f<A.

Then the boundary vale problem (1)-(2) has at least three positive solutions.

Proof.

From hypothesis (C5), there exist 0<σ<A and R>0; when xR, we have (49)ft,xσu. Set M=max(t,x)[0,1]×[0,R]f(t,x); consequently we get (50)0ft,xσx+M,0x<+. 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 constants a,c(c>ρa) such that conditions (C), (C2), and (C4) hold. And function f(t,x) satisfies

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 constant d<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 Theorems 14 and 15, we obtain one theorem and four corollaries as follows.

Theorem 16.

Assume that the function f(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 function f(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 function f(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 function f(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 function f(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 take ν=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 fractional q-difference (53)DqνxCt+ft,x=0,0<t<1,with the boundary conditions (54)x0=Dq2x0=0,x1=1201xsdqs.

Example 1.

We take(55)ft,x=2t125+36x3,t,x0,1×0,1,2t125+35+x,t,x0,1×1,+. There exist constants d=1/36 and a=253/250 such that(56)ft,x=2t125+36x30.0167716<0.65×136d<Adfor  t,x0,1×0,136;ft,x=2t125+35+x36.02>17.785·1μa>Bμafor  t,xμ,1×a,ρa+1;ft,x2/125+36xfor  t,x0,1×0,+.

All the conditions of Theorem 12 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,x0,1×0,1,e-t99.35x1/2+0.65x,t,x0,1×1,+. There exist constants d=0.25 and a=1.06 such that(58)f=0.65<Afor  t,x0,1×0,+;ft,x37.8822>17.785·1μa>Bμafor  t,xμ,1×1.06,ρa+1;ft,x100d5100dfor  t,x0,1×0,0.25.

All the conditions of Theorem 14 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,x0,1×0,1,45x1/2+0.6x,t,x0,1×1,+. 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+0.6x56.0135>17.785·1μa>Bμafor  t,xμ,1×a,ρa+1.

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.

5. Conclusions

The main innovation of this paper was that existence criteria of three positive solutions for a Caputo fractional q-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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (Grant no. 11271235), the Development Foundation of Higher Education Department of Shanxi Province (20101109, 20111117, and 20111020), and Shanxi Datong University Institute (2016K9 and 2017K4).