R E S E A R C H
Open Access
Existence solutions for boundary value
problem of nonlinear fractional
q-difference
equations
Wen-Xue Zhou
1,2*and Hai-Zhong Liu
1*Correspondence: [email protected] 1Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, 730070, P.R. China 2School of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R. China
Abstract
In this paper, we discuss the existence of weak solutions for a nonlinear boundary value problem of fractionalq-difference equations in Banach space. Our analysis relies on the Mönch’s fixed-point theorem combined with the technique of measures of weak noncompactness.
MSC: 26A33; 34B15
Keywords: boundary value problem; fractionalq-difference equations; Caputo fractional derivative; weak solutions
1 Introduction
Fractional differential calculus is a discipline to which many researchers are dedicating their time, perhaps because of its demonstrated applications in various fields of science and engineering []. Many researchers studied the existence of solutions to fractional boundary value problems, for example, [–].
Theq-difference calculus or quantum calculus is an old subject that was initially devel-oped by Jackson [, ]; basic definitions and properties ofq-difference calculus can be found in [, ].
The fractionalq-difference calculus had its origin in the works by Al-Salam [] and Agarwal []. More recently, maybe due to the explosion in research within the fractional differential 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 Mittage-Leffler function [], just to mention some.
El-Shahed and Hassan [] studied the existence of positive solutions of theq-difference boundary value problem:
⎧ ⎨ ⎩
–Dqu(t) =a(t)f(u(t)), t∈J:= [, ],
αu() –βDqu() = , γu() +δDqu() = .
Ferreira [] considered the existence of positive solutions to nonlinear q-difference boundary value problem:
⎧ ⎨ ⎩
(RLDαqu)(t) = –f(t,u(t)), <t< , <α≤, u() =u() = .
Ferreira [] studied the existence of positive solutions to nonlinearq-difference bound-ary value problem:
⎧ ⎨ ⎩
(RLDα
qu)(t) = –f(t,u(t)), <t< , <α≤, u() = (Dqu)() = , (Dqu)() =β≥.
El-Shahed and Al-Askar [] studied the existence of positive solutions to nonlinear
q-difference equation:
⎧ ⎨ ⎩
CDαqu+a(t)f(u(t)) = , <t< , <α≤, u() = (D
qu)() = , γ(Dqu)() +βDqu() = ,
whereγ,β≥ andCDqis the fractionalq-derivative of the Caputo type.
Ahmad, Alsaedi and Ntouyas [] discussed the existence of solutions for the second-orderq-difference equation with nonseparated boundary conditions
⎧ ⎨ ⎩
D
qu(t) =f(t,u(t)), t∈I,
u() =ηu(T), Dqu() =ηDqu(T),
wheref ∈C(I×R,R),I= [,T]∩qN,qN:={qn:n∈N}∪{}, andT∈qNis a fixed constant, andη= is a fixed real number.
Ahmad and Nieto [] discussed a nonlocal nonlinear boundary value problem (BVP) of third-orderq-difference equations given by
⎧ ⎨ ⎩
D
qu(t) =f(t,u(t)), t∈Iq,
u() = , Dqu() = , u() =αu(η),
wheref ∈C(Iq×R,R),Iq={, } ∪ {qn:n∈N}, andq∈(, ) is a fixed constant,η∈ {qn: n∈N}andα= /ηis a real number.
This paper is mainly concerned with the existence results for the following fractional
q-difference equations:
⎧ ⎨ ⎩
CDαqu+f(t,u) = , t∈J= [, ], <α≤, u() = (D
qu)() = , γ(Dqu)() +βDqu() = ,
(.)
whereγ,β≥ andCDqis the fractionalq-derivative of the Caputo type.f:J×E→Eis a given function satisfying some assumptions that will be specified later, andEis a Banach space with norm u .
[, ] and the references therein. As far as we know, there are very few results devoted to weak solutions of nonlinear fractional differential equations [–]. Motivated by the above mentioned papers, the purpose of this paper is to establish the existence results for the boundary value problem (.) by virtue of the Mönch’s fixed-point theorem combined with the technique of measures of weak noncompactness.
The remainder of this article is organized as follows. In Section , we provide some basic definitions, preliminaries facts and various lemmas, which are needed later. In Section , we give main results of the problem (.). In the end, we also give an example for the illustration of the theories established in this paper.
2 Preliminaries and lemmas
In this section, we present some basic notations, definitions and preliminary results, which will be used throughout this paper.
Letq∈(, ) and define []
[a]q= qa–
q– =a
a–+· · ·+ , a∈R.
Theq-analogue of the power (a–b)nis
(a–b)()= , (a–b)(n)=
n–
k=
a–bqk, a,b∈R,n∈N.
Ifαis not a positive integer, then
(a–b)(α)=aα ∞
i=
( – (b/a)qi) ( – (b/a)qα+i).
Note that ifb= , thena(α)=aα. Theq-gamma function is defined by
q(x) =
( –q)(x–)
( –q)x–, x∈R\{, –, –, . . .}, <q< ,
and satisfiesq(x+ ) = [x]qq(x).
Theq-derivative of a functionf is here defined by
Dqf(x) = dqf(x)
dqx =
f(qx) –f(x) (q– )x
andq-derivatives of higher order by
Dnqf(x) =
⎧ ⎨ ⎩
f(x), ifn= ,
DqDnq–f(x), ifn∈N.
Theq-integral of a functionf defined in the interval [,b] is given by
x
f(t)dqt=x( –q) ∞
n=
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, as done for derivatives, an operatorIn
qcan be defined, namely,
Iqf(x) =f(x), Iqnf(x) =IqIqn–f(x), n∈N.
The fundamental theorem of calculus applies to these operatorsIqandDq, that is, (DqIqf)(x) =f(x),
and iff is continuous atx= , then (IqDqf)(x) =f(x) –f().
Basic properties of the two operators can be found in the book mentioned in []. We now point out three formulas that will be used later (iDqdenotes the derivative with re-spect 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).
Remark . We note that ifα> anda≤b≤t, then (t–a)(α)≥(t–b)(α)[].
LetJ:= [, ] andL(J,E) denote the Banach space of real-valued Lebesgue integrable functions on the interval J,L∞(J,E) denote the Banach space of real-valued essentially bounded and measurable functions defined overJwith the norm · L∞.
Let E be a real reflexive Banach space with norm · and dual E*, and let (E,ω) =
(E,σ(E,E*)) denote the spaceEwith its weak topology. Here,C(J,E) is the Banach space of
continuous functionsx:J→Ewith the usual supremum norm x ∞:=sup{ x(t) :t∈J}.
Moreover, for a given setV of functionsv:J→R, let us denote byV(t) ={v(t) :v∈ V},t∈J, andV(J) ={v(t) :v∈V,t∈J}.
Definition . A functionh:E→Eis said to be weakly sequentially continuous ifhtakes each weakly convergent sequence inEto a weakly convergent sequence inE(i.e.for any (xn)ninEwithxn(t)→x(t) in (E,ω) thenh(xn(t))→h(x(t)) in (E,ω) for eacht→J).
Definition .[] The functionx:J→Eis said to be Pettis integrable onJif and only if there is an elementxJ∈Ecorresponding to eachI⊂Jsuch thatϕ(xI) =Iϕ(x(s))dsfor allϕ∈E*, where the integral on the right is supposed to exist in the sense of Lebesgue. By
definition,xI=Ix(s)ds.
Lemma . [] If x(·)is Pettis integrable and h(·)is a measurable and an essentially bounded real-valued function,then x(·)h(·)is Pettis integrable.
Definition .[] LetEbe a Banach space,Ethe set of all bounded subsets ofE, and
Bthe unit ball inE. TheDe Blasimeasure of weak noncompactness is the mapβ:E→
[,∞) defined by
β(X) =inf{> : there exists a weakly compact subsetofE
such thatX⊂B+}.
Lemma .[] The De Blasi measure of noncompactness satisfies the following
proper-ties:
(a) S⊂T⇒β(S)≤β(T);
(b) β(S) = ⇔Sis relatively weakly compact; (c) β(S∪T) =max{β(S),β(T)};
(d) β(Sω) =β(S),whereSωdenotes the weak closure ofS; (e) β(S+T)≤β(S) +β(T);
(f ) β(aS) =|a|α(S); (g) β(conv(S)) =β(S); (h) β(|λ|≤hλS) =hβ(S).
The following result follows directly from the Hahn-Banach theorem.
Lemma . Let E be a normed space with x= .Then there existsϕ∈E*with ϕ =
andϕ(x) = x .
Definition . [] Let α ≥ and f be a function defined on [, ]. The fractional
q-integral of the Riemann-Liouville type is (RLI
qf)(x) =f(x) and
RLIqαf
(x) =
x
a
(x–qt)(α–)
q(α)
f(t)dqt, α∈R+,x∈[, ].
Definition . [] The fractionalq-derivative of the Riemann-Liouville type of order
α≥ is defined by (RLD
qf)(x) =f(x) and
RLDαqf(x) =Dq[α]Iq[α]–αf(x), α> ,
where [α] is the smallest integer greater than or equal toα.
Definition .[] The fractionalq-derivative of the Caputo type of orderα≥ is
de-fined by
CDqαf(x) =Iq[α]–αD[qα]f(x), α> ,
where [α] is the smallest integer greater than or equal toα.
() (IqβIqαf)(x) = (I α+β q f)(x), () (DβqIqαf)(x) =f(x).
Lemma .[] Let D be a closed convex and equicontinuous subset of a metrizable locally convex vector space C(J,E)such that∈D.Assume that A:D→D is weakly sequentially continuous.If the implication
V=conv{} ∪A(V) ⇒ V is relatively weakly compact, (.)
holds for every subset V of D,then A has a fixed point.
3 Main results
Let us start by defining what we mean by a solution of the problem (.).
Definition . A functionu∈C(J,Eω) is said to be a solution of the problem (.) ifu
satisfies the equationCDα
qu+f(t,u) = onJ, and satisfy the conditionsu() = (Dqu)() = , γ(Dqu)() +βD
qu() = .
For the existence results on the problem (.), we need the following auxiliary lemmas.
Lemma .[] Letα> and n∈N.Then,the following equality holds:
We derive the corresponding Green’s function for boundary value problem (.), which will play major role in our next analysis.
Lemma . Letρ∈C[, ]be a given function,then the boundary-value problem
⎧
Proof By Lemma . and Lemma ., we can reduce the equation of problem (.) to an
Applying the boundary conditionsu() = (D
qu)() = , we have
Therefore, the unique solution of problem (.) is
u(t) =At–
To prove the main results, we need the following assumptions:
(H) For eacht∈J, the functionf(t,·)is weakly sequentially continuous; (H) For eachx∈C(J,E), the functionf(·,x(·))is Pettis integrable onJ; (H) There existspf∈L∞(J,R+)such that
(H) There existspf∈L∞(J,R+)and a continuous nondecreasing functionψ: [,∞)→
(,∞)such that
f(t,u)≤pf(t)ψ
u for a.e.t∈Jand eachu∈E;
(H) For each bounded setD⊂E, and eacht∈J, the following inequality holds:
βf(t,D)≤pf(t)·β(D);
(H) There exists a constantR> such that R
pf L∞ψ(R)G*
> ,
where pf L∞=sup{pf(t) :t∈J}.
Theorem . Let E be a reflexive Banach space and assume that(H)-(H)are satisfied.If pf L∞G*< , (.) then the problem(.)has at least one solution on J.
Proof Let the operatorA:C(J,E)→C(J,E) defined by the formula
(Au)(t) :=
G(t,qs)fs,u(s)dqs, (.)
whereG(·,·) is the Green’s function defined by (.). It is well known the fixed points of the operatorAare solutions of the problem (.).
First notice that, forx∈C(J,E), we havef(·,x(·))∈P(J,E) (assumption (H)). Since,s→ G(t,s)∈L∞(J), thenG(t,·)f(·,x(·)) is Pettis integrable for allt∈Jby Lemma ., and so the operatorAis well defined.
LetR> , and consider the set
D=
x∈C(J,E) : x ∞≤Rand
x(t) –x(t)≤R pf L∞
G(t,qs) –G(t,qs)dqsfort,t∈J
.
Clearly, the subsetDis closed, convex and equicontinuous. We shall show thatAsatisfies the assumptions of Lemma .. The proof will be given in three steps.
Step : We will show that the operatorAmapsDinto itself.
Takex∈D,t∈Jand assume thatAx(t)= . Then there existsψ∈E*such that Ax(t) =
ψ(Ax(t)). Thus,
(Ax)(t)=ψ(Ax)(t)=
G(t,qs)fs,y(s)dqs ≤
≤
G(t,qs)·pf(s)·x(s)dqs
≤ pf L∞RG* ≤R.
Letτ,τ∈J,τ<τand∀x∈D, soAx(τ) –Ax(τ)= . Then there existsψ∈E*, such
that Ax(τ) –Ax(τ) =ψ(Ax(τ) –Ax(τ)). Hence, Ax(τ) –Ax(τ)=
G(τ,qs) –G(τ,qs)
·fs,x(s)dqs
≤
G(τ,qs) –G(τ,qs)·f
s,x(s)dqs ≤R pf L∞
G(τ,qs) –G(τ,qs)dqs,
this means thatA(D)⊂D.
Step : We will show that the operatorAis weakly sequentially continuous.
Let (xn) be a sequence inDand let (xn(t))→x(t) in (E,w) for eacht∈J. Fixt∈J. Since
f satisfies assumptions (H), we havef(t,xn(t)) converge weakly uniformly tof(t,x(t)). Hence, the Lebesgue dominated convergence theorem for Pettis integrals impliesAxn(t) converges weakly uniformly toAx(t) inEω. Repeating this for eacht∈JshowsAxn→Ax.
ThenA:D→Dis weakly sequentially continuous.
Step : The implication (.) holds. Now let V be a subset of D such that V ⊂
conv(A(V)∪ {}). Clearly,V(t)⊂conv(A(V)∪ {}) for allt∈J. Hence,AV(t)⊂AD(t),
t∈J, is bounded inE. Thus,AV(t) is weakly relatively compact since a subset of a re-flexive Banach space is weakly relatively compact if and only if it is bounded in the norm topology. Therefore,
v(t)≤βA(V)(t)∪ {}
≤βA(V)(t) = ,
thus,V is relatively weakly compact inE. In view of Lemma ., we deduce thatAhas a fixed point, which is obviously a solution of the problem (.). This completes the proof.
Remark . In Theorem ., we presented an existence result for weak solutions of the
problem (.) in the case where the Banach spaceEis reflexive. However, in the nonre-flexive case, conditions (H)-(H) are not sufficient for the application of Lemma .; the difficulty is with condition (.).
Theorem . Let E be a Banach space,and assume assumptions(H), (H), (H), (H)
are satisfied.If(.)holds,then the problem(.)has at least one solution on J.
Theorem . Let E be a Banach space,and assume assumptions(H), (H), (H), (H),
Proof Assume that the operatorA:C(J,E)→C(J,E) is defined by the formula (.). It is well known the fixed points of the operatorAare solutions of the problem (.).
First notice that, forx∈C(J,E), we havef(·,x(·))∈P(J,E) (assumption (H)). Since,s→ G(t,s)∈L∞(J), thenG(t,·)f(·,x(·)) for allt∈Jis Pettis integrable (Lemma .), and thus, the operatorAmakes sense.
LetR> , and consider the set
D=
x∈C(J,E) : x ∞≤Rand
x(t) –x(t)≤ pf L∞ψ(R)
G(t,qs) –G(t,qs)dqsfort,t∈J
, (.)
clearly, the subsetDis closed, convex and equicontinuous. We shall show thatAsatisfies the assumptions of Lemma .. The proof will be given in three steps.
Step : We will show that the operatorAmapsDinto itself.
Takex∈D,t∈Jand assume thatAx(t)= . Then there existsψ∈E*such that Ax(t) =
ψ(Ax(t)). Thus,
(Ax)(t)=ψ(Ax)(t)=ψ
G(t,qs)fs,y(s)dqs
≤
T
G(t,qs)·ψfs,x(s)dqs
≤
T
G(t,qs)·pf(s)·ψx(s)dqs
≤ pf L∞ψ(R)G* ≤R.
Letτ,τ∈J,τ<τand∀x∈D, soAx(τ) –Ax(τ)= . Then there existψ∈E*such
that
Ax(τ) –Ax(τ)=ψ
Ax(τ) –Ax(τ)
. Thus,
Ax(τ) –Ax(τ)=ψ
G(τ,qs) –G(τ,qs)
·fs,x(s)dqs
≤
G(τ,qs) –G(τ,qs)·f
s,x(s)dqs
≤ψ(R) pf L∞
G(τ,qs) –G(τ,qs)dqs,
this means thatA(D)⊂D.
Step : We will show that the operatorAis weakly sequentially continuous.
converging weakly uniformly toAx(t) inEω. We do it for eacht∈JsoAxn→Ax. Then
A:D→Dis weakly sequentially continuous.
Step : The implication (.) holds. Now let V be a subset of D such that V ⊂
conv(A(V)∪ {}). Clearly,V(t)⊂conv(A(V)∪ {}) for allt∈J. Hence,AV(t)⊂AD(t),
t∈J, is bounded inE. Using this fact, assumption (H), Lemma . and the properties of the measureβ, we have for eacht∈J
v(t)≤βA(V)(t)∪ {}
≤βA(V)(t)
=β
G(t,qs)fs,V(s)dqs
≤
G(t,qs)·pf(s)·βV(s)dqs
≤ pf L∞·
G(t,qs)·v(s)dqs
≤ pf L∞· v ∞·G*,
which gives
v ∞≤ pf L∞· v ∞·G*.
This means that
v ∞· – pf L∞·G*
≤.
By (.), it follows that v ∞= , that isv(t) = for eacht∈J, and thenV(t) is relatively weakly compact inE. In view of Lemma ., we deduce thatAhas a fixed point which is obviously a solution of the problem (.). This completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
This research was supported by the National Natural Science Foundation of China (11161027, 11262009). The authors are thankful to the referees for their careful reading of the manuscript and insightful comments.
Received: 10 December 2012 Accepted: 4 April 2013 Published: 19 April 2013 References
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Cite this article as:Zhou and Liu:Existence solutions for boundary value problem of nonlinear fractional
