R E S E A R C H
Open Access
Positive solutions of nonlinear boundary
value problems for delayed fractional
q-difference systems
Qiaozhen Yuan and Wengui Yang
**Correspondence: [email protected] Ministry of Public Education, Sanmenxia Polytechnic, Sanmenxia, Henan 472000, China
Abstract
In this paper, we investigate the existence and uniqueness of positive solutions to nonlinear boundary value problems for delayed fractionalq-difference systems by applying the properties of the Green function and some well-known fixed-point theorems. As applications, some examples are presented to illustrate the main results.
MSC: 39A13; 34B18; 34A08
Keywords: fractionalq-difference systems; boundary value problems; positive solutions; fixed-point theorems
1 Introduction
In the past decades, fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in various fields of science and engineering such as physics, mechanics, chemistry, biology, engineering, etc. Therefore, the subject of fractional differential equations has gained considerable attention by many researchers. Some recent results on fractional boundary value problems can be found in [–] and references therein. For example, Ahmad and Nieto [] dealt with some existence results for a boundary value problem involving a nonlinear fractional order integrodifferential equation with integral boundary conditions based on a contraction mapping principle and Krasnoselskiii’s fixed-point theorem. Ahmadet al.[] investigated the existence and uniqueness of solutions for a class of Caputo-type fractional boundary value problems in-volving four-point nonlocal Riemann-Liouville integral boundary conditions of different order by means of standard tools of fixed-point theory and Leray-Schauder nonlinear al-ternative. Ouyanget al.[] considered the following nonlinear system of fractional order differential equations with delays:
Dαiu
i
(t) +fi
t,u
τi(t)
, . . . ,uN
τiN(t)
= , <t< ,
u(ij)() = , j= , , . . . ,ni– ,i= , , . . . ,N,
u(ni–)
i () =ηi, i= , , . . . ,N,
whereDαiis the standard Riemann-Liouville fractional derivative. By using some
fixed-point theorems and some properties of the Green function, the existence of positive solu-tions was obtained.
The q-difference calculus or quantum calculus is an old subject that was initially de-veloped by Jackson [, ]; basic definitions and properties ofq-difference calculus can be found in the book mentioned in [].
The fractionalq-difference calculus had its origin in the works by Al-Salam [] and Agarwal []. Recently, maybe due to the explosion in research within the fractional differ-ential calculus setting, new developments in this theory of fractionalq-difference calculus were made; for example,q-analogues of the integral and differential fractional operators properties such as theq-Laplace transform,q-Taylor’s formula, Mittage-Leffler function [–], just to mention some.
More recently, boundary value problems of nonlinear fractionalq-difference equations have gained popularity and importance. Many researchers pay attention to the existence and multiplicity of solutions or positive solutions for nonlinear boundary value problems of fractionalq-difference equations by means of upper and lower solutions method and some fixed-point theorems such as the Krasnoselskii fixed-point theorem, the Leggett-Williams fixed-point theorem, and the Schauder fixed-point theorem; for examples, see [–] and the references therein. El-Shahed and Al-Askar [] studied the existence of multiple positive solutions to the nonlinearq-fractional boundary value problems by us-ing Guo-Krasnoselskii’s fixed-point theorem in a cone. Graef and Kong [] investigated the uniqueness, existence, and nonexistence of positive solutions for the boundary value problem with fractionalq-derivatives in terms of different ranges ofλ. Ma and Yang [] obtained the existence of solutions for multi-point boundary value problems of nonlinear fractionalq-difference equations by means of the Banach contraction principle and Kras-noselskii’s fixed-point theorem. Zhaoet al.[] showed some existence results of posi-tive solutions to nonlocalq-integral boundary value problems of a nonlinear fractional q-derivative equation using the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s fixed-point theorem. Ferreira [] and [] dealt with the existence of positive solutions to nonlinearq-difference boundary value problems,
Dαqu(t) = –ft,u(t), ≤t≤, <α≤, u() =u() = ,
and
Dαqu(t) = –ft,u(t), ≤t≤, <α≤, u() = (Dqu)() = , (Dqu)() =β≥,
respectively. By applying a fixed-point theorem in cones, sufficient conditions for the ex-istence of nontrivial solutions were enunciated.
In [], Liang and Zhang discussed the following nonlinear q-fractional three-point boundary value problem:
By using a fixed-point theorem in partially ordered sets, the authors obtained sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.
In [], Ahmadet al.studied the following nonlocal boundary value problems of non-linear fractionalq-difference equations,
c
Dαqu(t) =ft,u(t), t∈[, ],α∈(, ],
au() –b(Dqu)() =cu(η), au() +b(Dqu)() =cu(η),
wherecDα
q denotes the Caputo fractionalq-derivative of orderα, andai,bi,ci,ηi∈R(i=
, ). The existence of solutions for the problem was shown by applying some well-known tools of point theory, such as Banach contraction principle, Krasnoselskii’s fixed-point theorem, and Leray-Schauder nonlinear alternative.
In [], Alsaedi et al. were concerned with the following nonlinear fractional q -difference equations with nonlocal integral boundary conditions:
c
Dβqu(t) +λu(t) =ft,u(t), t∈[, ],β∈(, ], u() = , u() = (Iqu)(η) =
η
u(s)dqs, <η< .
The existence results were obtained by applying some well-known fixed-point theorems. Motivated by the above works, in this paper, we consider the following system of non-linear fractionalq-difference equations with delays:
Dαi
qui
(t) +fi
t,u
τi(t)
, . . . ,uN
τiN(t)
= , <t< ,
Djqui
() = , j= , , . . . ,ni– ,i= , , . . . ,N,
Dni–
q ui
() =ηi, i= , , . . . ,N,
(.)
whereDαi
q is the fractionalq-derivative of the Riemann-Liouville type,αi∈(ni– ,ni] for
someni> ,ηi≥ fori= , , . . . ,N, ≤τij(t)≤tfori,j= , , . . . ,N, andfiis a nonlinear
function from [, ]×RN
+ toR+= [,∞). The purpose of this paper is to establish suffi-cient conditions on the existence of positive solutions for fractionalq-difference system (.) by using some properties of the Green function and some fixed-point theorems such as the Banach contraction principle, Krasnoselskii’s fixed-point theorem, and the Leray-Schauder nonlinear alternative. By a positive solution for the fractionalq-difference sys-tem (.) we mean a mapping with positive components on [, ] such that (.) is satis-fied. Obviously, (.) includes the usual system of fractionalq-difference equations when τij(t)≡tfor alliandj. Therefore, the obtained results generalize and include some existing
ones.
2 Preliminaries
Letq∈(, ) and define
[a]q=
qa–
q– , a∈R.
Theq-analogue of the power (a–b)nwithn∈N
is
(a–b)()= , (a–b)(n)=
n–
k=
a–bqk, n∈N,a,b∈R.
More generally, ifα∈R, then
(a–b)(α)=aα
∞
n=
a–bqn
a–bqα+n.
Note that, ifb= , thena(α)=aα. Theq-gamma function is defined by
q(x) =
( –q)(x–)
( –q)x–, x∈R\ {, –, –, . . .},
and satisfiesq(x+ ) = [x]qq(x).
Theq-derivative of a functionf is here defined by
(Dqf)(x) =
f(x) –f(qx)
( –q)x , (Dqf)() =xlim→(Dqf)(x),
andq-derivatives of higher order by
Dqf(x) =f(x) and Dnqf(x) =Dq
Dnq–f(x), n∈N.
Theq-integral of a functionf defined in the interval [,b] is given by
(Iqf)(x) =
x
f(t)dqt=x( –q)
∞
n=
fxqnqn, x∈[,b].
Ifa∈[,b] andf is defined in the interval [,b], its integral fromatobis defined by
b
a
f(t)dqt=
b
f(t)dqt–
a
f(t)dqt.
Similarly to what is done for derivatives, an operatorIn
q can be defined, namely,
Iqf(x) =f(x) and Iqnf(x) =Iq
Iqn–f(x), n∈N.
The fundamental theorem of calculus applies to these operatorsIqandDq,i.e.,
and iff is continuous atx= , then
(IqDqf)(x) =f(x) –f().
Basic properties of the two operators can be found in the book []. We now point out three formulas that will be used later (iDqdenotes the derivative with respect to variablei)
a(t–s)(α)=aα(t–s)(α),
tDq(t–s)(α)= [α]q(t–s)(α–), xDq
x
f(x,t)dqt
(x) =
x
xDqf(x,t)dqt+f(qx,x).
We note that ifα> anda≤b≤t, then (t–a)(α)≥(t–b)(α)[].
Definition . ([]) Letα ≥ and f be function defined on [, ]. The fractional q -integral of the Riemann-Liouville type isI
qf(x) =f(x) and
Iqαf(x) = q(α)
x
(x–qt)(α–)f(t)dqt, α> ,x∈[, ].
Definition .([]) The fractionalq-derivative of the Riemann-Liouville type of order
α≥ is defined byD
qf(x) =f(x) and
Dαqf(x) =DmqIqm–αf(x), α> ,
wheremis the smallest integer greater than or equal toα.
Definition .([]) The fractional q-derivative of the Caputo type of orderα≥ is
defined by
c
Dα
qf
(x) =Im–α
q Dmqf
(x), α> ,
wheremis the smallest integer greater than or equal toα.
Lemma .([, ]) Letα,β≥and f be a function defined on[, ].Then the following
formulas hold:
() (IqβIqαf)(x) =I
α+β
q f(x), () (Dα
qIqαf)(x) =f(x).
Theorem .([]) Letα> and p be a positive integer.Then the following equality holds:
IqαDαqf(x) =DαqIqαf(x) –
p–
k=
xα–p+k
q(α+k–p+ )
Dkqf().
Theorem .(Banach contraction mapping theorem []) Let M be a complete metric
Theorem .([, ]) Let C be a closed and convex subset of a Banach space X.Assume
that U is a relatively open subset of C with∈U and T:U→C is completely continuous.
Then at least one of the following two properties holds:
(i) T has a fixed point inU,
(ii) there existu∈∂Uandλ∈(, )withu=λTu.
Theorem .(Krasnoselskii fixed-point theorem [, ]) Let P be a cone in a Banach
space X.Assume that and are open subsets of X with∈ and ⊆ .Suppose
that T:P∩( \ )is a completely continuous operator such that either
(i) Tu ≤ uforu∈P∩∂ andTu ≥ uforu∈P∩∂ ,or
(ii) Tu ≥ uforu∈P∩∂ andTu ≤ uforu∈P∩∂ .
Then T has a fixed point in \ .
3 Existence of positive solutions
Throughout this paper, we letE=C([, ],RN). Then (E, ·
E) is a Banach space, where
uE= max
≤i≤N≤maxt≤
ui(t) foru= (u, . . . ,uN)T∈E.
In this section, we always assume thatf= (f, . . . ,fN)T∈C([, ]×RN+,RN+).
Lemma . Fractional q-difference systems(.)is equivalent to the following system of
q-integral equations:
ui(t) =
Gi(t,qs)fi
s,u
τi(s)
, . . . ,uN
τiN(s)
dqs+
ηitαi–
[αi– ]q· · ·[αi–ni+ ]q
(.)
for i= , , . . . ,N,where
Gi(t,s) =
q(αi)
tαi–( –s)(αi–ni)– (t–s)(αi–), ≤s≤t≤,
tαi–( –s)(αi–ni), ≤t≤s≤. (.)
Proof It is easy to see that if (u,u, . . . ,uN)Tsatisfies (.), then it also satisfies (.). So,
assume that (u,u, . . . ,uN)Tis a solution to (.). In view of Lemma . and Theorem .,
integrating both sides of the first equation of (.) of orderαiwith respect tot, we can see
that
ui(t) = –
q(αi)
t
(t–qs)(αi–)f i
s,u
τi(s)
, . . . ,uN
τiN(s)
dqs
+citαi–+citαi–+· · ·+cniit
αi–ni
for ≤t≤,i= , , . . . ,N. It follows that
(Dqui)(t) = –
[αi– ]q
q(αi)
t
(t–qs)(αi–)f i
s,u
τi(s)
, . . . ,uN
τiN(s)
dqs
+ [αi– ]qcitαi–+ [αi– ]qcitαi–+· · ·+ [αi–ni+ ]qcni–,it
for ≤t≤,i= , , . . . ,N. Combining with the boundary conditions in (.), this yields
Then it follows from (.) and the boundary condition (Dni–
q u)() =ηithat
Some properties of the Green functionsGi(t,s) needed in the sequel are now stated and
proved.
Lemma . Function Gi(t,s)defined above satisfies the following conditions:
Proof We start by defining the two functions
ϕi(t,s) =tαi–( –s)(αi–ni)– (t–s)(αi–), ≤s≤t≤,
and
ψi(t,s) =tαi–( –s)(αi–ni), ≤t≤s≤.
It is clear thatψi(t,qs)≥ andψi(,qs) = . On the other hand, fort=
ϕi(t,qs) =tαi–( –qs)(αi–ni)–tαi– –q
s t
(αi–)
≥tαi–( –qs)(αi–ni)– ( –qs)(αi–)≥.
Therefore,Gi(t,qs)≥. Moreover, for fixeds∈[, ],
tDqϕi(t,qs) = [αi– ]qtαi–( –qs)(αi–ni)– [αi– ]q(t–qs)(αi–)
≥[αi– ]qtαi–( –qs)(αi–ni)– [αi– ]qt(αi–) –q
s t
(αi–)
≥[αi– ]qtαi–
( –qs)(αi–ni)– ( –qs)(αi–)≥,
i.e.,ϕi(t,qs) is an increasing function oft. Obviously,ψi(t,qs) is increasing int, therefore
Gi(t,qs) is an increasing function oftfor fixeds∈[, ]. This concludes the proof of (a).
Suppose now thatt≥qs. Then we have
Gi(t,qs)
Gi(,qs)
= t
αi–( –qs)(αi–ni)– (t–qs)(αi–)
( –qs)(αi–ni)– ( –qs)(αi–)
= t
αi–(( –qs)(αi–ni)– ( –qs t)
(αi–))
( –qs)(αi–ni)– ( –qs)(αi–)
≥tαi–(( –qs)(αi–ni)– ( –qs)(αi–)) ( –qs)(αi–ni)– ( –qs)(αi–) =t
αi–.
On the other hand, ift≤qs, then we have
Gi(t,qs)
Gi(,qs)
=tαi–,
and this finishes the proof of (b).
Now, we are ready to present the main results.
Theorem . Suppose that there exist functionsλij(t)∈C([, ],R+),i,j= , , . . . ,N,such
that
fi(t,u, . . . ,uN) –fi(t,v, . . . ,vN)≤ N
j=
for t∈[, ], (u,u, . . . ,uN)T, (v,v, . . . ,vN)T∈RN+.If
then(.)has a unique positive solution.
Proof Let
=u∈E|ui(t)≥ fort∈[, ],i= , , . . . ,N
. (.)
It is easy to see that is a complete metric space. Define an operatorTon by
Tu(t) =
Because of the continuity ofGandf, it follows easily from Lemma . thatT maps into itself. To finish the proof, we only need to show thatT is a contraction. Indeed, for u,v∈ , by (.), we have
This, combined with Theorem . and (.), immediately implies that T : → is a contraction. Therefore, the proof is complete with the help of Lemma . and Theorem ..
Theorem . Suppose that there exist functionsλi(t)∈C([, ],R+),i,j= , , . . . ,N,and
then(.)has a unique positive solution.
Theorem . Suppose that there exist nonnegative real-valued functions mi,ni, . . . ,niN ∈
then(.)has at least one positive solution.
Proof Let andT: → be defined by (.) and (.), respectively. We first show that
T is completely continuous through the following three steps.
Step . Show thatT : → is continuous. Let{uk(t)}be a sequence in such that
uk(t)→u(t)∈ . Then
= [, ]× {u(t)|uk(t)∈ ,t∈[, ],k≥}is bounded in [, ]×
RN
+. Sincef is continuous, it is uniformly continuous on any compact set. In particular, for anyε> , there exists a positive integerKsuch that
–fi
, which implies thatTis continuous.
Step . Show thatT maps bounded sets of into bounded sets. LetAbe a bounded subset of . Then [, ]× {u(t)|t∈[, ],u∈A} ⊆[, ]×RN+ is bounded. Sincef is con-tinuous, there exists anM> such that
fi
Immediately, we can easily see thatTAis a bounded subset of .
Step . Show that T maps bounded sets of into equicontinuous sets. Let B be a bounded subset of . Similarly as in Step , there existsL> such that
fi
Now the equicontinuity ofT onBfollows easily from the fact thatGiis continuous and
hence uniformly continuous on [, ]×[, ].
Now we have shown thatT is completely continuous. To apply Theorem ., let
Fixr>μand defineU={u∈ :u<r}. We claim that there is nou∈U such that
Lemma .. Therefore, the proof is complete.
Corollary . If all fi,i,j= , , . . . ,N,are bounded, then(.) has at least one positive
solution.
To state the last result of this section, we introduce
Proof Let be defined by (.) and i={u∈E|u<ri},i= , . Obviously, is a cone
(.). This completes the proof.
4 Some examples
In this section, we demonstrate the feasibility of some of the results obtained in Section .
Example . Consider the following fractionalq-difference system:
One can easily see that (.) is satisfied with
λ(t) =λ(t) =
e–t
+et, λ(t) =λ(t) =
Moreover,
It follows from Theorem . that (.) has a unique positive solution on [, ].
Example . Consider the following fractionalq-difference system:
One can easily see that (.) is satisfied. Moreover,
G(,s) =G(,s) =
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed to each part of this work equally and read and approved the final version of the manuscript.
Acknowledgements
The authors sincerely thank the editor and reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.
Received: 18 October 2013 Accepted: 14 January 2014 Published:31 Jan 2014 References
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