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)), t∈I,
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)), t∈I,
Dqu() = , Dqu() =αu(),
(.)
wheref ∈C(I×R,R),I={qn:n∈N} ∪ {, },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
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 thatlimq→–Dqf(t) =f(t).
Definition . The higher-orderq-derivatives are defined inductively as
Dqf(t) =f(t), Dnqf(t) =DqDnq–f(t), n∈N.
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)qnfxqn–afqna, 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)dqt– a
f(t)dqt.
Similarly, we have
Iqf(t) =f(t), Iqnf(t) =IqIqn–f(t), n∈N.
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
Remark . In the limitq→–, the above results correspond to their counterparts in standard calculus.
Definition . f:I×R→Ris called an S-Carathéodory function if and only if (i) for each(u,v)∈R,t→f(t,u,v)is measurable onI;
(ii) for a.e.t∈I,(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+) ={u∈C
q:
u(t)dqtexists}, and normed byuL=
|u(t)|dqtfor all u∈L(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:U→C 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 y∈C[, ],then the BVP
D
qu(t) =y(t), t∈I,
Dqu() = , Dqu() =αu(),
(.)
has a unique solution
u(t) =
t
(t–qs)y(s)dqs+
α – +qs
y(s)dqs
=
G(t,s;q)y(s)dqs, (.)
where
G(t,s;q) =
α
–α+αt, s≤t,
–α+αqs, t≤s. (.)
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
(t–qs)y(s)dqs+at+a, (.)
wherea,aare arbitrary constants. Using the boundary conditionsDqu() = ,Dqu() =
αu() in (.), we find thata= , and
a=
α – +qs
Substituting the values ofaandain (.), we obtain
u(t) =
t
(t–qs)y(s)dqs+
α – +qs
y(s)dqs.
This completes the proof.
Remark . Forq→, equation (.) takes the form
u(t) =
t
(t–qs)y(s)dqs+at+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, s≤t, –α+αs, t≤s.
We consider the Banach space Cq=C(I,R) equipped with the standard normu=
max{u∞,Dqu∞}, and · ∞=sup{ · ,t∈I},u∈Cq.
Define an integral operatorT:Cp→Cpby
Tu(t) =
G(t,s;q)fs,u(s),Dqu(s)
dqs
=
t
(t–qs)fs,u(s),Dqu(s)
dqs
+
α – +qs
fs,u(s),Dqu(s)
dqs, t∈I,u∈Cq. (.)
Obviously,Tis well defined andu∈Cqis 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×R→R 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|, t∈I, (u,v), (u,v)∈R.
In addition,suppose either
(H) <|α|for <|α|< ,or
(H) < for|α| ≥
+
α– +qs
L(s)u(s) –v(s)+L(s)Dqu(s) –Dqv(s)dqs
≤sup
t∈I
+tq+
α– + q +q
u–v
≤
|α|u–v<u–v,
and
DqTu(t) –DqTv(t)≤sup t∈I
DqTu(t) –DqTv(t)
≤sup
t∈I
tfs,u(s),Dqu(s)
–fs,v(s),Dqv(s)dqs
≤sup
t∈I
tL(s)u(s) –v(s)+L(s)Dqu(s) –Dqv(s)dqs
≤u–v ≤
|α|u–v<u–v.
Therefore, we obtain thatTu–Tv<u–v, 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×R→R is a continuous function and there exist two positive constants L,Lsuch that
f(t,u,v) –f(t,u,v)≤L|u–u|+L|v–v|, t∈I, (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×R→R 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|, t∈I, (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.
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:Cq→Cqis
com-pletely continuous.
Proof The proof consists of several steps.
(i)T maps bounded sets into bounded sets inCq.
LetBr={u∈Cq:u ≤r}be a bounded set inCqandu∈Br. Then we have (ii)T maps bounded sets into equicontinuous sets ofCq.
As a consequence of the Arzelá-Ascoli theorem, we can conclude thatT :Cq→Cq is
completely continuous. This proof is completed.
Theorem . Let f :I×R→R 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 :Cq→Cqis completely continuous. Let
λ∈(, ) andu=λTu. Then, fort∈I, we have
u(t)=λTu(t)
≤ t
|
t–qs|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={u∈Cq:u<M}. Note
that the operatorT:U→Cqis 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 u∈Uwhich 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×R→R 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
Then BVP(.)has at least one solution provided(N+ )P+P+Q< ,where N=max{α, –
α}.
Proof We consider the space
P=u∈Cq:Dqu() = ,Dqu() =αu()
and define the operatorT:P×[, ]→Pby
T(u,λ) =λTu=λ
G(t,s;q)fs,u(s),Dqu(s)
dqs, t∈I. (.)
Obviously, we can see thatP⊂Cq. In view of Lemma ., it is easy to know that for each
λ∈[, ],T(u,λ) is completely continuous inP. It is clear thatu∈Pis a solution of BVP (.) if and only ifuis a fixed point ofT(·, ). Clearly,T(u, ) = for eachu∈P. 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 anyu∈P, we have
u(t)=u() –
t
Dqu(s)dqs ≤
αDqu()
+
t
Dqu(s)dqs ≤(N+ –t)Dqu∞≤(N+ +t)Dqu∞, t∈I,
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
≤N
p(t)u(s)+q(t)Dqu(s)+r(s)
dqs
≤NPu∞+NQDqu∞+NR, t∈I,
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)), t∈I, 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, t∈I, 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
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