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R E S E A R C H

Open Access

Existence of solutions for nonlinear

second-order

q

-difference equations with

first-order

q

-derivatives

Changlong Yu

*

and Jufang Wang

*Correspondence:

changlongyu@126.com College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei, 050018, P.R. China

Abstract

In this paper, we establish the existence of solutions for a boundary value problem with the nonlinear second-orderq-difference equation

D2

qu(t) =f(t,u(t),Dqu(t)), tI,

Dqu(0) = 0, Dqu(1) =

α

u(1).

The existence and uniqueness of solutions for the problem are proved by means of the Leray-Schauder nonlinear alternative and some standard fixed point theorems. Finally, we give two examples to demonstrate the use of the main results. The nonlinear teamfcontainsDqu(t) in the equation.

Keywords: q-difference equations; Leray-Schauder nonlinear alternative; boundary value problem; fixed point theorem

1 Introduction

In this paper, we study the existence of solutions for a boundary value problem with non-linear second-orderq-difference equations

D

qu(t) =f(t,u(t),Dqu(t)), tI,

Dqu() = , Dqu() =αu(),

(.)

wherefC(I×R,R),I={qn:nN} ∪ {, },q(, ), andα=  is a fixed real number.

Theq-difference equations initiated at the beginning of the twentieth century [–] is a very interesting field in difference equations. In the last few decades, it has evolved into a multidisciplinary subject and plays an important role in several fields of physics such as cosmic strings and black holes [], conformal quantum mechanics [], nuclear and high energy physics []. However, the theory of boundary value problems (BVPs) for nonlinear q-difference equations is still in the initial stages and many aspects of this theory need to be explored. To the best of our knowledge, for the BVPs of nonlinearq-difference equations, a few works were done; see [–] and the references therein. In particular, the study of BVPs for nonlinearq-difference equation with first-orderq-difference is yet to be initiated. The main aim of this paper is to develop some existence and uniqueness results for BVP (.). Our results are based on a variety of fixed point theorems such as the Banach

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contraction mapping principle, the Schauder nonlinear alternative and the Leray-Schauder continuous theorem. Some examples and special cases are also discussed.

2 Preliminary results

In this section, firstly, let us recall some basic concepts ofq-calculus [, ].

Definition . For  <q< , we define theq-derivative of a real-value functionf as

Dqf(t) =

f(t) –f(qt)

( –q)t , Dqf() =tlim→Dqf(t).

Note thatlimqDqf(t) =f(t).

Definition . The higher-orderq-derivatives are defined inductively as

Dqf(t) =f(t), Dnqf(t) =DqDnq–f(t), nN.

For example,Dq(tk) = [k]qtk–, wherekis a positive integer and the bracket [k]q= (qk

)/(q– ). In particular,Dq(t) = ( +q)t.

Definition . Theq-integral of a functionf defined in the interval [a,b] is given by

x

a

f(t)dqt:= ∞

n=

x( –q)qnfxqnafqna, x∈[a,b],

and fora= , we denote

Iqf(x) = x

f(t)dqt= ∞

n=

x( –q)qnfxqn.

Then

b

a

f(t)dqt= b

f(t)dqta

f(t)dqt.

Similarly, we have

Iqf(t) =f(t), Iqnf(t) =IqIqn–f(t), nN.

Observe that

DqIqf(x) =f(x),

and iff is continuous atx= , thenIqDqf(x) =f(x) –f().

Inq-calculus, the product rule and integration by parts formula are

Dq(gh)(t) =Dqg(t)h(t) +g(qt)Dqh(t), (.) x

f(t)Dqg(t)dqt=

f(t)g(t)x

x

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Remark . In the limitq→–, the above results correspond to their counterparts in standard calculus.

Definition . f:I×RRis called an S-Carathéodory function if and only if (i) for each(u,v)∈R,tf(t,u,v)is measurable onI;

(ii) for a.e.tI,(u,v)→f(t,u,v)is continuous onR;

(iii) for eachr> , there existsϕr(t)∈L(I,R+)withtϕr(t)∈L(I,R+)onIsuch that

max{|u|,|v|} ≤rimplies|f(t,u,v)| ≤ϕr(t), for a.e.I, where

L(I,R+) ={uC

q: 

u(t)dqtexists}, and normed byuL=

|u(t)|dqtfor all uL(I,R+).

Theorem .(Nonlinear alternative for single-valued maps []) Let E be a Banach space, let C be a closed and convex subset of E,and let U be an open subset of C and∈U.Suppose that F:UC is a continuous,compact(that is,F(U)is a relatively compact subset of C) map.Then either

(i) Fhas a fixed point inU,or

(ii) there is au∂U(the boundary ofUinC)andλ∈(, )withu=λF(u).

Lemma . Let yC[, ],then the BVP

D

qu(t) =y(t), tI,

Dqu() = , Dqu() =αu(),

(.)

has a unique solution

u(t) =

t

(tqs)y(s)dqs+ 

α –  +qs

y(s)dqs

=

G(t,s;q)y(s)dqs, (.)

where

G(t,s;q) = 

α

 –α+αt, st,

 –α+αqs, ts. (.)

Proof Integrating theq-difference equation from  tot, we get

Dqu(t) = t

y(s)dqs+a. (.)

Integrating (.) from  totand changing the order of integration, we have

u(t) =

t

(tqs)y(s)dqs+at+a, (.)

wherea,aare arbitrary constants. Using the boundary conditionsDqu() = ,Dqu() =

αu() in (.), we find thata= , and

a=

α –  +qs

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Substituting the values ofaandain (.), we obtain

u(t) =

t

(tqs)y(s)dqs+ 

α –  +qs

y(s)dqs.

This completes the proof.

Remark . Forq→, equation (.) takes the form

u(t) =

t

(tqs)y(s)dqs+at+a,

which is the solution of a classical second-order ordinary differential equationu(t) =y(t) and the associated form of Green’s function for the classical case is

G(t,s) = 

α

 –α+αt, st,  –α+αs, ts.

We consider the Banach space Cq=C(I,R) equipped with the standard normu=

max{u∞,Dqu∞}, and · ∞=sup{ · ,tI},uCq.

Define an integral operatorT:CpCpby

Tu(t) =

G(t,s;q)fs,u(s),Dqu(s)

dqs

=

t

(tqs)fs,u(s),Dqu(s)

dqs

+

α –  +qs

fs,u(s),Dqu(s)

dqs, tI,uCq. (.)

Obviously,Tis well defined anduCqis a solution of BVP (.) if and only ifuis a fixed

point ofT.

3 Existence and uniqueness results

In this section, we apply various fixed point theorems to BVP (.). First, we give the uniqueness result based on Banach’s contraction principle.

Theorem . Let f :I×RR be a continuous function,and there exists L

(t),L(t)∈ C([, ], [, +∞))such that

f(t,u,v) –f(t,u,v)≤L(t)|uu|+L(t)|vv|, tI, (u,v), (u,v)∈R.

In addition,suppose either

(H) <|α|for <|α|< ,or

(H) < for|α| ≥

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+

α–  +qs

L(s)u(s) –v(s)+L(s)Dqu(s) –Dqv(s)dqs

sup

tI

 +tq+ 

α–  + q  +q

uv

|α|uv<uv,

and

DqTu(t) –DqTv(t)≤sup tI

DqTu(t) –DqTv(t)

≤sup

tI

tfs,u(s),Dqu(s)

fs,v(s),Dqv(s)dqs

≤sup

tI

tL(s)u(s) –v(s)+L(s)Dqu(s) –Dqv(s)dqs

uv

|α|uv<uv.

Therefore, we obtain thatTuTv<uv, soTis a contraction. Thus, the conclusion of the theorem follows by Banach’s contraction mapping principle.

Case :|α| ≥. It is similar to the proof of case . This completes the proof of

Theo-rem ..

Corollary . Assume that f :I×RR is a continuous function and there exist two positive constants L,Lsuch that

f(t,u,v) –f(t,u,v)≤L|uu|+L|vv|, tI, (u,v), (u,v)∈R.

In addition,suppose either

(H) L+L<|α|for <|α|< ,or

(H) L+L< for|α| ≥

holds.Then BVP(.)has a unique solution.

Corollary . Assume that f :I×RR is a continuous function and there exist two functions L(t),L(t)∈L(I,R+)such that

f(t,u,v) –f(t,u,v)≤L|u–u|+L|v–v|, tI, (u,v), (u,v)∈R.

In addition,suppose either

(H) A+Bq|α|<|α|for <|α|< ,or

(H) A< for|α| ≥

holds,where

A=

L(s) +L(s) dqs, B= 

sL(s) +L(s) dqs.

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Proof It is similar to the proof of Theorem .. The next existence result is based on the Leray-Schauder nonlinear alternative theorem.

Lemma . Let f :I×R→R be an S-Carathéodory function.Then T:CqCqis

com-pletely continuous.

Proof The proof consists of several steps.

(i)T maps bounded sets into bounded sets inCq.

LetBr={uCq:ur}be a bounded set inCqanduBr. Then we have (ii)T maps bounded sets into equicontinuous sets ofCq.

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As a consequence of the Arzelá-Ascoli theorem, we can conclude thatT :CqCq is

completely continuous. This proof is completed.

Theorem . Let f :I×RR be an S-Carathéodory function.Suppose further that there exists a real number M> such that

|α|M ( +|α|)ϕrL > 

holds,where

ϕrL=

ϕr(s)dqs= .

Then BVP(.)has at least one solution.

Proof In view of Lemma ., we obtain thatT :CqCqis completely continuous. Let

λ∈(, ) andu=λTu. Then, fortI, we have

u(t)=λTu(t)

t

 |

tqs|fs,u(s),Dqu(s)dqs

+

α –  +qsfs,u(s),Dqu(s)dqs

 + 

|α| 

ϕr(s)dqs=

 + 

|α|

ϕrL,

and

Dqu(t)=DqλTu(t)≤ t

f

s,u(s),Dqu(s)dqs≤ 

ϕr(s)dqs=ϕrL.

Hence, consequently,

|α|u ( +|α|)ϕrL ≤

.

Therefore, there existsM>  such thatu =M. Let us setU={uCq:u<M}. Note

that the operatorT:UCqis completely continuous (which is known to be compact

restricted to bounded sets). From the choice ofU, there is noU∂Usuch thatu=λTu for someλ∈(, ). Consequently, by Theorem ., we deduce thatT has a fixed point uUwhich is a solution of problem (.). This completes the proof.

The next existence result is based on the Leray-Schauder continuation theorem.

Theorem . Let f :I×RR be an S-Carathéodory function.Suppose further that there exist functions p(t),q(t),r(t)∈L(I,R+)with tp(t)∈L(I,R+)such that

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Then BVP(.)has at least one solution provided(N+ )P+P+Q< ,where N=max{α,  – 

α}.

Proof We consider the space

P=uCq:Dqu() = ,Dqu() =αu()

and define the operatorT:P×[, ]→Pby

T(u,λ) =λTu=λ

G(t,s;q)fs,u(s),Dqu(s)

dqs, tI. (.)

Obviously, we can see thatPCq. In view of Lemma ., it is easy to know that for each

λ∈[, ],T(u,λ) is completely continuous inP. It is clear thatuPis a solution of BVP (.) if and only ifuis a fixed point ofT(·, ). Clearly,T(u, ) =  for eachuP. If for each

λ∈[, ] the fixed points ofT(·, ) inPbelong to a closed ball ofPindependent ofλ, then the Leray-Schauder continuation theorem completes the proof.

Next we show that the fixed point ofT(·, ) hasa prioriboundM, which is independent ofλ. Assume thatu=T(u,λ), and set

P=

p(s)dqs, P=

sp(s)dqs,

Q=

q(s)dqs, R= 

r(s)dqs.

By (.), it is clear that|G(t,s;q)| ≤Nfor eachα= . For anyuP, we have

u(t)=u() –

t

Dqu(s)dqs ≤

αDqu()

+

t

Dqu(s)dqs ≤(N+  –t)Dqu∞≤(N+  +t)Dqu∞, tI,

and so it holds that

Dqu∞≤λf

s,u(s),Dqu(s)L≤f

s,u(s),Dqu(s)L

p(t)u(s)+q(t)Dqu(s)+r(s)L

≤(N+ )P+P+Q

Dqu∞+R;

therefore,

Dqu∞≤

R

 – ((N+ )P+P+Q) :=M.

At the same time, we have

u(t)≤λ

G(t,s,q)fs,u(s),Dqu(s)

dqs

N

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N

p(t)u(s)+q(t)Dqu(s)+r(s)

dqs

NPu∞+NQDqu∞+NR, tI,

and so

u∞≤N(QM+R)  –NP :=M.

SetM=max{M,M}, which is independent ofλ. So, BVP (.) has at least one solution.

This completes the proof.

4 Example

Example . Consider the following BVP:

D

qu(t) =et+sin(u(t)) + 

tan–(Dqu(t)), tI, Dqu() = , Dqu() =u().

(.)

Here,f(t,u(t),Dqu(t)) =et+sin(u(t)) +tan–(Dqu(t)),q= ,α=. Clearly,|f(t,u,

v) –f(t,u,v)| ≤ |uu|+ |vv|. ThenL=, L= andL+L= <α. By Corollary ., we obtain that BVP (.) has a unique solution.

Example . Consider the following BVP:

Dqu(t) =t sin(u(t)) + 

(+√)√t(Dqu(t))

β+

t, tI, Dqu() = , Dqu() =u().

(.)

Here,q=,  <β< . It is obvious that|f(t,u,v)| ≤t |u|+  (+√)√t|v|+

t, wherep(t) =

t

, q(t) = 

(+√)√t,r(t) =

t. ThenP=  ,P=

 ,Q=

,R= , so (N+ )P+P+Q=   < . By Theorem ., we obtain that BVP (.) has at least one solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, CLY and JFW contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgements

Research supported by the Natural Science Foundation of China (10901045), (11201112), the Natural Science Foundation of Hebei Province (A2009000664), (A2011208012) and the Foundation of Hebei University of Science and Technology (XL201047), (XL200757).

Received: 16 October 2012 Accepted: 15 April 2013 Published: 2 May 2013

References

1. Jackson, FH: Onq-difference equations. Am. J. Math.32, 305-314 (1910)

2. Carmichael, RD: The general theory of linearq-difference equations. Am. J. Math.34, 147-168 (1912)

3. Mason, TE: On properties of the solutions of linearq-difference equations with entire function coefficients. Am. J. Math.37, 439-444 (1915)

4. Adams, CR: On the linear ordinaryq-difference equation. Ann. Math.30, 195-205 (1928) 5. Strominger, A: Information in black hole radiation. Phys. Rev. Lett.71, 3743-3746 (1993) 6. Youm, D:q-deformed conformal quantum mechanics. Phys. Rev. D62, 095009 (2000)

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8. Ahmad, B: Boundary value problems for nonlinear third-orderq-difference equations. Electron. J. Differ. Equ.94, 1-7 (2011)

9. Ahmad, B, Nieto, JJ: On nonlocal boundary value problem of nonlinearq-difference equations. Adv. Differ. Equ.2012, 81 (2012). doi:10.1186/1687-1847-2012-81

10. Ahmad, B, Alsaedi, A, Ntouyas, SK: A study of second-orderq-difference equations with boundary conditions. Adv. Differ. Equ.2012, 35 (2012). doi:10.1186/1687-1847-2012-35

11. El-Shahed, M, Hassan, HA: Positive solutions ofq-difference equation. Proc. Am. Math. Soc.138, 1733-1738 (2010) 12. Ahmad, B, Ntouyas, SK: Boundary value problems forq-difference inclusions. Abstr. Appl. Anal.2011, Article ID

292860 (2011)

13. Ahmad, B, Nieto, JJ: Basic theory of nonlinear third-orderq-difference equations and inclusions. Math. Model. Anal. 18(1), 122-135 (2013)

14. Gasper, G, Rahman, M: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990) 15. Kac, V, Cheung, P: Quantum Calculus. Springer, New York (2002)

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