JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 605169 10.1155/2013/605169 605169 Research Article Sequential Derivatives of Nonlinear q-Difference Equations with Three-Point q-Integral Boundary Conditions Pongarm Nittaya Asawasamrit Suphawat Tariboon Jessada Lam Hak-Keung Department of Mathematics Faculty of Applied Science King Mongkut's University of Technology North Bangkok Bangkok 10800 Thailand kmutt.ac.th 2013 31 3 2013 2013 31 10 2012 24 01 2013 2013 Copyright © 2013 Nittaya Pongarm 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.

This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinear q-difference equations with three-point q-integral boundary conditions. Our results are concerned with several quantum numbers of derivatives and integrals. By using Banach's contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.

1. Introduction

The study of q-calculus or quantum calculus was initiated by the pioneer works of Jackson , Carmichael , Mason , Adams , Trjitzinsky , and so forth. Since then, in the last few decades, this subject has evolved into a multidisciplinary research area with many applications; for example, see . For some recent works, we refer the reader to  and references therein. However, the theory of boundary value problems for nonlinear q-difference equations is still in the beginning stages and it needs to be explored further.

In , Ahmad investigated the existence of solutions for a nonlinear boundary value problem of third-order q-difference equation: (1)Dq3u(t)=f(t,u(t)),0t1,u(0)=0,Dqu(0)=0,u(1)=0. Using Leray-Schauder degree theory and standard fixed-point theorems, some existence results were obtained. Moreover, he showed that if q1, then his results corresponded to the classical results. Ahmad et al.  studied a boundary value problem of a nonlinear second-order q-difference equation with nonseparated boundary conditions (2)Dq2u(t)=f(t,u(t)),t[0,T],u(0)=ηu(T),Dqu(0)=ηDqu(T). They proved the existence and uniqueness theorems of the problem (2) using the Leray-Schauder nonlinear alternative and some standard fixed-point theorems. For some very recent results on nonlocal boundary value problems of nonlinear q-difference equations and inclusions, see .

In this paper, we discuss the existence of solutions for the following nonlinear q-difference equation with three-point integral boundary condition: (3)Dq(Dp+λ)x(t)=f(t,x(t)),t[0,T],x(0)=0,β0ηx(s)drs=x(T), where 0<p, q, r<1, fC([0,T]×, ), βT(1+r)/η2, η(0,T) is a fixed point, and λ is a given constant.

The aim of this paper is to prove some existence and uniqueness results for the boundary value problem (3). Our results are based on Banach’s contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory. Since the problem (3) has different values of the quantum numbers of the q-derivative and the q-integral, the existence results of such problem are also new.

2. Preliminaries

Let us recall some basic concepts of quantum calculus .

For 0<q<1, we define the q-derivative of a real-valued function f as (4)Dqf(t)=f(t)-f(qt)(1-q)t,Dqf(0)=limt0Dqf(t). The higher-order q-derivatives are given by (5)Dq0f(t)=f(t),Dqnf(t)=DqDqn-1f(t),n. The q-integral of a function f defined on the interval [0,T] is given by (6)atf(s)dqs:=n=0(1-q)qn×[tf(tqn)-af(qna)],t[0,T], and for a = 0, we denote (7)Iqf(t)=0tf(s)dqs=n=0t(1-q)qnf(tqn), provided the series converges. If a[0,T] and f is defined on the interval [0,T], then (8)abf(s)dqs=0bf(s)dqs-0af(s)dqs. Similarly, we have (9)Iq0f(t)=f(t),Iqnf(t)=IqIqn-1f(t),n. Observe that (10)DqIqf(t)=f(t), and if f is continuous at t=0, then (11)IqDqf(t)=f(t)-f(0). In q-calculus, the product rule and integration by parts formula are (12)Dq(gh)(t)=(Dqg(t))h(t)+g(qt)Dqh(t),0tf(s)Dqg(s)dqs=[f(s)g(s)]0t-0tDqf(s)g(qs)dqs. In the limit q1, the q-calculus corresponds to the classical calculus. The above results are also true for quantum numbers p, r such that 0<p<1 and 0<r<1.

Lemma 1.

Let T(1+r)βη2, 0<p, q, r<1, and let λ be a constant. Then for any hC[0,T], the boundary value problem (13)Dq(Dp+λ)x(t)=h(t),t[0,T],(14)x(0)=0,β0ηx(s)drs=x(T),0<η<T, is equivalent to the integral equation (15)x(t)=0t0sh(u)dqudps-λ0tx(s)dps+β(1+r)tT(1+r)-βη2×0η0v(0sh(u)dqu-λx(s))dpsdrv+λ(1+r)tT(1+r)-βη20Tx(s)dps-(1+r)tT(1+r)-βη20T0sh(u)dqudps.

Proof.

For t[0,T], q-integrating (13) from 0 to t, we obtain (16)(Dp+λ)x(t)=0th(s)dqs+c1. Equation (16) can be written as (17)Dpx(t)=0th(s)dqs-λx(t)+c1. For t[0,T], p-integrating (17) from 0 to t, we have (18)x(t)=0t0sh(u)dqudps-λ0tx(s)dps+c1t+c2. From the first condition of (14), it follows that c2=0. For t[0,T], r-integrating equation (18) from 0 to t, we get (19)0tx(v)drv=0t0v0sh(u)dqudpsdrv-λ0t0vx(s)dpsdrv+c1t21+r. The second boundary condition (14) implies that (20)β0ηx(v)drv=β0η0v0sh(u)dqudpsdrv-βλ0η0vx(s)dpsdrv+c1βη21+r=β0η0v(0sh(u)dqu-λx(s))dpsdrv+c1βη21+r=0T0sh(u)dqudps-λ0Tx(s)dps+c1T. Therefore, (21)c1=β(1+r)T(1+r)-βη20η0v(0sh(u)dqu-λx(s))dpsdrv+λ(1+r)T(1+r)-βη20Tx(s)dps-1+rT(1+r)-βη20T0sh(u)dqudps. Substituting the values of c1 and c2 in (18), we obtain (15). This completes the proof.

For the forthcoming analysis, let 𝒞=C([0,T],) denote the Banach space of all continuous functions from [0,T] to endowed with the norm defined by x=sup{|x(t)|,t[0,T]}.

In the following, for the sake of convenience, we set (22)Ω=11+p×(T2+|β|Tη3(1+r)|T(1+r)-βη2|(1+r+r2)+T3(1+r)|T(1+r)-βη2|),(23)Φ=|λ|T+|β||λ|Tη2|T(1+r)-βη2|+|λ|T2(1+r)|T(1+r)-βη2|.

3. Main Results

Now, we are in the position to establish the main results. We transform the boundary value problem (3) into a fixed-point problem. In view of Lemma 1, for t[0,T], x𝒞, we define the operator A:𝒞𝒞 as (24)(Ax)(t)=0t0sf(u,x(u))dqudps-λ0tx(s)dps+β(1+r)tT(1+r)-βη2×0η0v(0sf(u,x(u))dqu-λx(s))dpsdrv+λ(1+r)tT(1+r)-βη20Tx(s)dps-(1+r)tT(1+r)-βη20T0sf(u,x(u))dqudps. Note that the problem (3) has solutions if and only if the operator equation Ax=x has fixed points.

Our first result is based on Banach’s fixed-point theorem.

Theorem 2.

Assume that f:[0,T]× is a jointly continuous function satisfying the conditions

|f(t,x)-f(t,y)|L|x-y|, for all t[0,T], x,y;

Λ:=(Φ+LΩ)<1,

where L is a Lipschitz constant, Ω and Φ are defined by (22) and (23), respectively.

Then, the boundary value problem (3) has a unique solution.

Proof.

Assume that supt[0,T]|f(t,0)|=M0; we choose a constant (25)RM0Ω1-Λ. Now, we will show that ABRBR, where BR={x𝒞:xR}. For any xBR, we have (26)(Ax)=supt[0,T]|0t0sf(u,x(u))dqudps-λ0tx(s)dps+β(1+r)tT(1+r)-βη2×0η0v(0sf(u,x(u))dqu-λx(s))dpsdrv+λ(1+r)tT(1+r)-βη20Tx(s)dps-(1+r)tT(1+r)-βη20T0sf(u,x(u))dqudps|supt[0,T]{0t0s(|f(u,x(u))-f(u,0)|+|f(u,0)|)dqudps+|λ|0t|x(s)|dps+|β|(1+r)t|T(1+r)-βη2|×(0η0v0s(|f(u,x(u))-f(u,0)|+|f(u,0)|)×dqudpsdrv+|λ|0η0v|x(s)|dpsdrv)+|λ|(1+r)t|T(1+r)-βη2|0T|x(s)|dps+(1+r)t|T(1+r)-βη2|×0T0s(|f(u,x(u))-f(u,0)|+|f(u,0)|)dqudps0t}supt[0,T]{0t0s(L|x(u)|+|f(u,0)|)dqudps+|λ|0t|x(s)|dps+|β|(1+r)t|T(1+r)-βη2|×(0η0v0s(L|x(u)|+|f(u,0)|)dqudpsdrv+|λ|0η0v|x(s)|dpsdrv)+|λ|(1+r)t|T(1+r)-βη2|0T|x(s)|dps+(1+r)t|T(1+r)-βη2|×0T0s(L|x(u)|+|f(u,0)|)dqudps}supt[0,T]{(Lx+M0)0t0sdqudps+|λ|x0tdps+|β|(1+r)t|T(1+r)-βη2|×((Lx+M0)0η0v0sdqudpsdrv+|λ|x0η0vdpsdrv)+|λ|x(1+r)t|T(1+r)-βη2|0Tdps+(1+r)t|T(1+r)-βη2|(Lx+M0)0T0sdqudps}=supt[0,T]{(Lx+M0)t21+p+|λ|xt+|β|(1+r)t|T(1+r)-βη2|×((Lx+M0)(1-r)η3(1+p)(1-r3)+|λ|xη21+r)+|λ|x(1+r)t|T(1+r)-βη2|T+(1+r)t|T(1+r)-βη2|(Lx+M0)T21+p}=supt[0,T]{x(|λ|t+|β||λ|η2t|T(1+r)-βη2|+|λ|T(1+r)t|T(1+r)-βη2|)+(Lx+M0)1+p×(t2+|β|(1+r)η3t|T(1+r)-βη2|(1+r+r2)+T2(1+r)t|T(1+r)-βη2|)}R(|λ|T+|β||λ|Tη2|T(1+r)-βη2|+|λ|T2(1+r)|T(1+r)-βη2|)+(LR+M0)1+p×(T2+|β|Tη3(1+r)|T(1+r)-βη2|(1+r+r2)+T3(1+r)|T(1+r)-βη2|)=RΦ+(LR+M)ΩR.

Next, we will show that A is a contraction. For any x,y𝒞 and for each t[0,T], we have (27)(Ax)-(Ay)=supt[0,T]|(Ax)(t)-(Ay)(t)|=supt[0,T]|0t0s(f(u,x(u))-f(u,y(u)))dqudps-λ0t(x(s)-y(s))dps+β(1+r)tT(1+r)-βη2×0η0v(0s(f(u,x(u))-f(u,y(u)))dqu-λ(x(s)-y(s))0s)dpsdrv+λ(1+r)tT(1+r)-βη20T(x(s)-y(s))dqs-(1+r)t|T(1+r)-βη2|×0T0s(f(u,x(u))-f(u,y(u)))dqudps|supt[0,T]{Lx-y0t0sdqudps+|λ|x-y0tdps+|β|(1+r)t|T(1+r)-βη2|×(Lx-y0η0v0sdqudpsdrv+|λ|x-y0η0vdpsdrv)+|λ|(1+r)tT(1+r)-βη2x-y×0Tdqs+(1+r)t|T(1+r)-βη2|×Lx-y0T0sdqudps}=supt[0,T]{Lx-yt21+p+|λ|x-yt+|β|(1+r)t|T(1+r)-βη2|×(Lx-y(1-r)η3(1+p)(1-r3)+|λ|x-yη21+r)+|λ|x-y(1+r)t|T(1+r)-βη2|T+(1+r)t|T(1+r)-βη2|Lx-yT21+p}x-y(|λ|T+|β||λ|Tη2|T(1+r)-βη2|+|λ|T2(1+r)|T(1+r)-βη2|)+Lx-y1+p(T2+|β|η3T(1+r)|T(1+r)-βη2|(1+r+r2)+T3(1+r)|T(1+r)-βη2|)=(Φ+LΩ)x-yΛx-y. Since Λ<1, A is a contraction. Thus, the conclusion of the theorem follows by Banach's contraction mapping principle. This completes the proof.

Our second result is based on the following Krasnoselskii’s fixed-point theorem .

Theorem 3.

Let K be a bounded closed convex and nonempty subset of a Banach space X. Let A,B be operators such that

Ax+ByK whenever x,yK;

A is compact and continuous;

B is a contraction mapping.

Then, there exists zK such that z=Az+Bz.

Theorem 4.

Assume that (H1) and (H2) hold with

( H 3 ) | f ( t , x ) | μ ( t ) , for all (t,x)[0,T]×, with μL1([0,T],+).

If (28)|β||λ|Tη2|T(1+r)-βη2|+|λ|T2(1+r)|T(1+r)-βη2|+11+p(|β|η3T(1+r)|T(1+r)-βη2|(1+r+r2)+T3(1+r)|T(1+r)-βη2|)<1, then the boundary value problem (3) has at least one solution on [0,T].

Proof.

Setting maxt[0,T]|μ(t)|=||μ|| and choosing a constant (29)RμΩ1-Φ, where Ω and Φ are given by (22) and (23), respectively, we consider that BR={x𝒞:||x||R}.

In view of Lemma 1, we define the operators F1 and F2 on the set BR as (30)(F1x)(t)=0t0sf(u,x(u))dqudps-λ0tx(s)dps,(F2x)(t)=β(1+r)tT(1+r)-βη2×0η0v(0sf(u,x(u))dqu-λx(s))dpsdrv+λ(1+r)tT(1+r)-βη20Tx(s)dps-(1+r)tT(1+r)-βη20T0sf(u,x(u))dqudps, for x,yBR. By computing directly, we have (31)(F1x)+(F2y)μ0t0sdqudps+|λ|x0tdps+|β|(1+r)t|T(1+r)-βη2|×(μ0η0v0sdqudpsdrv+|λ|y0η0vdpsdrv)+|λ|(1+r)t|T(1+r)-βη2|y0Tdps+(1+r)t|T(1+r)-βη2|μ0T0sdqudps,RΦ+μΩR. Therefore, (F1x)+(F2y)BR. The condition (28) implies that F2 is a contraction mapping. Next, we will show that F1 is compact and continuous. Continuity of f coupled with the assumption (H3) implies that the operator F1 is continuous and uniformly bounded on BR. We define sup(t,x)[0,T]×BR|f(t,x)|=fmax<. For t1,t2[0,T] with t2<t1 and xBR, we have (32)|F1x(t1)-F2x(t2)|sup(t,x)[0,T]×BR|0t10sf(u,x(u))dqudps-λ0t1x(s)dps-0t20sf(u,x(u))dqudps+λ0t2x(s)dps|=sup(t,x)[0,T]×BR|t2t10sf(u,x(u))dqudps-λt2t1x(s)dps|fmax|t12-t22|1+p+|λ|(t1-t2)R. Actually, as t1-t20, the right-hand side of the above inequality tends to zero. So F1 is relatively compact on BR. Hence, by the Arzelá-Ascoli theorem, F1 is compact on BR. Therefore, all the assumptions of Theorem 5 are satisfied and the conclusion of Theorem 5 implies that the three-point integral boundary value problem (3) has at least one solution on [0,T]. This completes the proof.

As the third result, we prove the existence of solutions of (3) by using Leray-Schauder degree theory.

Theorem 5.

Let f:[0,T]×. Assume that there exist constants 0κ<(1-Φ)Ω-1, where Ω and Φ are given by (22) and (23), respectively, and M>0 such that |f(t,x)|κ|x|+M for all t[0,T], x𝒞. Then, the boundary value problem (3) has at least one solution.

Proof.

Let us define an operator A:𝒞𝒞 as in (24). We will prove that there exists at least one solution x𝒞 of the fixed-point equation (33)x=Ax. We define a ball BR𝒞, with a constant radius R>0, given by (34)BR={x𝒞:maxt[0,T]|x(t)|<R}. Then, it is sufficient to show that A:B¯R𝒞 satisfies (35)xθAx,xBR,θ[0,1]. Now, we set (36)H(θ,x)=θAx,x𝒞,θ[0,1]. Then, by the Arzelá-Ascoli theorem, we get that hθ(x)=x-H(θ,x)=x-θAx is completely continuous. If (35) holds, then the following Leray-Schauder degrees are well defined. From the homotopy invariance of topological degree, it follows that (37)deg(hθ,BR,0)=deg(I-θA,BR,0)=deg(h,BR,0)=deg(h0,BR,0)=deg(I,BR,0)=10,0BR, where I denotes the unit operator. By the nonzero property of Leray-Schauder degree, h1(x)=x-Ax=0 for at least one xBR. Let us assume that x=θAx for some θ[0,1]. Then, for all t[0,T], we obtain (38)|x(t)|=|θ(Ax)(t)|0t0s|f(u,x(u))|dqudps+|λ|0t|x(s)|dps+|β|(1+r)t|T(1+r)-βη2|×0η0v(0s|f(u,x(u))|dqu+|λx(s)|)dpsdrv+|λ|(1+r)t|T(1+r)-βη2|×0T|x(s)|dps+(1+r)t|T(1+r)-βη2|×0T0s|f(u,x(u))|dqudps(κ|x|+M)0t0sdqudps+|λ||x|0tdps+|β|(1+r)t|T(1+r)-βη2|×((κ|x|+M)0η0v0sdqudpsdrv+|λ||x|0η0vdpsdrv)+|λ||x|(1+r)t|T(1+r)-βη2|×0Tdps+(1+r)t|T(1+r)-βη2|(κ|x|+M)×0T0sdqudps(κ|x|+M)T21+p+|λ||x|T+|β|(1+r)T|T(1+r)-βη2|×((κ|x|+M)(1-r)η3(1+p)(1-r3)+|λ||x|η21+r)+|λ||x|(1+r)T2|T(1+r)-βη2|+(κ|x|+M)(1+r)T3|T(1+r)-βη2|(1+p)=|x|(|λ|T+|β||λ|Tη2|T(1+r)-βη2|+|λ|T2(1+r)|T(1+r)-βη2|)+(κ|x|+M)1+p(T2+|β|Tη3(1+r)|T(1+r)-βη2|(1+r+r2)+T3(1+r)|T(1+r)-βη2|)=|x|Φ+(κ|x|+M)Ω. Taking norm supt[0,T]|x(t)|=x and solving for x, this yields (39)xMΩ1-(Φ+κΩ). Let R=MΩ/(1-(Φ+κΩ))+1, then (35) holds. This completes the proof.

4. Examples

In this section, we give two examples to illustrate our results.

Example 6.

Consider the following nonlinear q-difference equation with boundary value problem: (40)D1/2(D1/3-27)x(t)=1(t+2)2·|x||x|+1,t[0,1],x(0)=0,x(1)+2303/4x(s)d1/4s=0.

Set q=1/2, p=1/3, r=1/4, T=1, λ=-2/7, η=3/4, β=-2/3, and f(t,x)=(1/(t+2)2)(x/(1+x)). Since |f(t,x)-f(t,y)|(1/4)x-y, then, (H1) and (H2) are satisfied with T(1+r)-βη2=13/80, (41)Ω=11+p(T2+|β|Tη3(1+r)|T(1+r)-βη2|(1+r+r2)+T3(1+r)|T(1+r)-βη2|)=13291,Φ=|λ|T+|β||λ|Tη2|T(1+r)-βη2|+|λ|T2(1+r)|T(1+r)-βη2|=47,L=1/4. Hence, Λ=:Φ+LΩ=85/91<1. Therefore, by Theorem 2, the boundary value problem (40) has a unique solution on [0,1].

Example 7.

Consider the following nonlinear q-difference equation with boundary value problem: (42)D2/3(D4/5+19)x(t)=sin(5πx)25π+|x||x|+1,t[0,32],x(0)=0,x(32)=1401x(s)d1/2s.

Set q=2/3, p=4/5, r=1/2, T=3/2, λ=1/9, η=1, and β=1/4. Here, |f(t,x)|=|sin(5πx)/25π+|x|/(1+|x|)|(|x|/5)+1. So, M=1, T(1+r)-βη2=20, and (43)Ω=11+p×(T2+|β|Tη3(1+r)|T(1+r)-βη2|(1+r+r2)+T3(1+r)|T(1+r)-βη2|)=615224,Φ=|λ|T+|β||λ|Tη2|T(1+r)-βη2|+|λ|T2(1+r)|T(1+r)-βη2|=38,κ=15<(1-Φ)Ω-1=28123. Hence, by Theorem 5, the boundary value problem (42) has at least one solution on [0,3/2].

Acknowledgments

The authors would like to thank the reviewers for their valuable comments and suggestions on the paper. This research is supported by King Mongkut’s University of Technology North Bangkok, Thailand.

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