R E S E A R C H
Open Access
Existence results for nonlocal boundary value
problems of nonlinear fractional
q
-difference
equations
Bashir Ahmad
1, Sotiris K Ntouyas
2*and Ioannis K Purnaras
2*Correspondence: [email protected] 2Department of Mathematics,
University of Ioannina, Ioannina, 451 10, Greece
Full list of author information is available at the end of the article
Abstract
In this paper, we study a nonlinear fractionalq-difference equation with nonlocal boundary conditions. The existence of solutions for the problem is shown by applying some well-known tools of fixed-point theory such as Banach’s contraction principle, Krasnoselskii’s fixed-point theorem, and the Leray-Schauder nonlinear alternative. Some illustrating examples are also discussed.
MSC: 34A08; 39A05; 39A12; 39A13
Keywords: fractionalq-difference equations; nonlocal boundary conditions; existence; Leray-Schauder nonlinear alternative; fixed point
1 Introduction
In recent years, the topic of fractional differential equations has gained considerable atten-tion and has evolved as an interesting and popular field of research. It is mainly due to the fact that several times the tools of fractional calculus are found to be more practical and effective than the corresponding ones of classical calculus in the mathematical modeling of dynamical systems associated with phenomena such as fractals and chaos. In fact, frac-tional calculus has numerous applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electrodynamics of complex medium, viscoelas-ticity and damping, control theory, wave propagation, percolation, identification, fitting of experimental data,etc.[–]. The development of fractional calculus ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. For some recent work on fractional differential equations, we refer to [–] and the references therein.
The pioneer work onq-difference calculus orquantum calculusdates back to Jackson’s papers [, ], while a systematic treatment of the subject can be found in [, ]. For some recent existence results onq-difference equations, see [–] and the references therein.
There has also been a growing interest on the subject of discrete fractional equations on the time scaleZ. Some interesting results on the topic can be found in a series of papers [–].
Fractionalq-difference equations have recently attracted the attention of several re-searchers. For some earlier work on the topic, we refer to [] and [], whereas some recent work on the existence theory of fractionalq-difference equations can be found in [–]. However, the study of boundary value problems of fractionalq-difference equa-tions is at its infancy and much of the work on the topic is yet to be done.
In this paper, we discuss the existence and uniqueness of solutions for the nonlocal boundary value problem of fractionalq-difference equations given by
cDα
qx(t) =f
t,x(t), ≤t≤, <α≤, (.)
αx() –βDqx() =γx(η), αx() +βDqx() =γx(η), (.)
where f ∈C([, ]×R,R), cDα
q is the fractional q-derivative of the Caputo type, and αi,βi,γi,ηi∈R(i= , ).
2 Preliminaries on fractionalq-calculus
In this section, we cite some definitions and fundamental results of theq-calculus as well as of the fractionalq-calculus [, ]. We also give a lemma that will be used in obtaining the main results of the paper.
Letq∈(, ) and define
[a]q=
qa–
q– , a∈R.
Theqanalogue of the power (a–b)nis
(a–b)()= , (a–b)(n)= n–
k=
a–bqk, a,b∈R,n∈N.
Theq-gamma function is defined by
q(x) =
( –q)(x–)
( –q)x–, x∈R\ {, –, –, . . .}, <q<
and satisfiesq(x+ ) = [x]qq(x) (see, []).
For <q< , we define theq-derivative of a real valued functionf as
Dqf(t) =
f(t) –f(qt)
( –q)t , t∈Iq–{}, Dqf() =tlim→Dqf(t).
The higher orderq-derivatives are given by
Dqf(t) =f(t), Dnqf(t) =DqDnq–f(t), n∈N.
Forx≥, we setJx={xqn:n∈N∪ {}} ∪ {}and define the definiteq-integral of a functionf :Jx→Rby
Iqf(x) =
x
f(s)dqs=
∞
n=
x( –q)qnfxqn
Fora,b∈Jx, we set
b
a
f(s)dqs=Iqf(b) –Iqf(a) = ( –q)
∞
n=
qnbfbqn–afaqn,
provided that the series exist. Throughout the paper, we will assume that the series in the
q-integrals converge.
Note that fora,b∈Jx, we havea=xqn,b=xqn for somen,n∈N, thus the definite
integral abf(s)dqsis just a finite sum, so no question about convergence is raised. We note that
DqIqf(x) =f(x),
while iff is continuous atx= , then
IqDqf(x) =f(x) –f().
For more details of the basic material onq-calculus, see the book [].
Definition .([]) Letα≥ andf be a function defined on [, ]. The fractionalq -integral of the Riemann-Liouville type is (I
qf)(t) =f(t) and
Iqαf(t) =
t
(t–qs)(α–)
q(α)
f(s)dq(s), α> ,t∈[, ].
Definition .([]) The fractionalq-derivative of the Riemann-Liouville type of order α≥ is defined by (D
qf)(t) =f(t) and
Dαqf(t) =Dq[α]Iq[α]–αf(t), α> ,
where [α] is the smallest integer greater than or equal toα.
Definition .([]) The fractionalq-derivative of the Caputo type of order α≥ is defined by
c
Dqαf(t) =Iq[α]–αD[qα]f(t), α> ,
where [α] is the smallest integer greater than or equal toα.
Lemma . Letα,β≥and let f be a function defined on[, ]. Then the next formulas hold:
(i) (IβqIαqf)(t) = (I
α+β
q f)(t), (ii) (Dα
qIqαf)(t) =f(t).
Lemma .([]) Letα≥and n∈N. Then the following equality holds:
IqαDnqf(t) =DnqIqαf(t) –
[α]–
k=
tα–n+k q(α–n+k)
Lemma .([]) Letα> . Then the following equality holds:
In order to define the solution for the problem (.)-(.), we need the following lemma.
Lemma . For a given g∈C([, ],R)the unique solution of the boundary value problem
Proof In view of Lemmas . and ., integrating equation in (.), we have
Using the boundary conditions of (.) in (.), we have
3 Main results
LetC:=C([, ],R) denote the Banach space of all continuous functions from [, ]→R endowed with the norm defined by x =sup{|x(t)|:t∈[, ]}.
For the sake of convenience, we set
k:= +|α|δ
IqαL() +|γ|δ
IqαL(η) +|γ|δ
IqαL(η) +|β|δ
Iqα–L(), (.) where
δ:=
|α–γ|+|α+β–γη|
|| , (.)
and
δ:=|
α–γ|+|β+γη|
|| . (.)
Theorem . Assume that f : [, ]×R→Ris continuous and that there exists a q-integrable function L: [, ]→Rsuch that
(A) |f(t,x) –f(t,y)| ≤L(t)|x–y|,t∈[, ],x,y∈R.
Then the boundary value problem (.)-(.) has a unique solution provided
k< , (.)
where k is given by (.).
Proof Let us fixsupt∈[,]|f(t, )|=Mand choose
ρ≥ MA
–k,
where
A= q(α+ )
+|γ|δη(
α–)
+|γ|δη(
α–) +|α|δ
+|β|δ q(α)
. (.)
We defineBρ={x∈C: x ≤ρ}and show thatFBρ⊂Bρ, whereFis defined by (.). For
x∈Bρ, observe that
ft,x(t)≤ft,x(t)–f(t, )+f(t, )≤L(t)x(t)+f(t, )≤L(t)ρ+M.
Thenx∈Bρ,t∈[, ], we have (Fx)(t)≤
t
(t–qs)(α–)
q(α)
L(s)ρ+Mdqs
+|γ|δ
η
(η–qs)(α–)
q(α)
L(s)ρ+Mdqs
+|γ|δ
η
(η–qs)(α–)
q(α)
+|α|δ
which, in view of (.) and (.), implies that
≤ x–y sup t∈[,]
t
(t–qs)(α–)
q(α)
L(s)ds
+|γ|δ
η
(η–qs)(α–)
q(α)
L(s)dqs+|γ|δ
η
(η–qs)(α–)
q(α)
L(s)dqs
+|α|δ
( –qs)(α–)
q(α)
L(s)dqs+|β|δ
( –qs)(α–)
q(α– )
L(s)dqs
,
which, in view of (.), yields
(Fx)(t) – (Fy)(t)≤k x–y .
Sincek∈(, ) by assumption (.), therefore,F is a contraction. Hence, it follows by Banach’s contraction principle that the problem (.)-(.) has a unique solution.
In caseL(t) =L(Lis a constant), the condition (.) becomesLA< and Theorem . takes the form of the following result.
Corollary . Assume that f : [, ]×R→Ris a continuous function and that
there exists a constant L∈(, /A)with|f(t,x) –f(t,y)| ≤L|x–y|, t∈[, ], x,y∈R, where A is given by (.).
Then the boundary value problem (.)-(.) has a unique solution.
Our next existence results is based on Krasnoselskii’s fixed-point theorem [].
Lemma .(Krasnoselskii) Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be two operators such that:
(i) Ax+By∈Mwheneverx,y∈M; (ii) Ais compact and continuous; (iii) Bis a contraction mapping.
Then there exists z∈M such that z=Az+Bz.
Theorem . Let f : [, ]×R→Rbe a continuous function satisfying (A). In addition,
we assume that
(A) there exists a functionμ∈C([, ],R+)and a nondecreasing functionφ∈C([, ],R+)
with
f(t,x)≤μ(t)φ|x|, (t,x)∈[, ]×R;
(A) there exists a constantrwith
r≥
q(α+ )
+|γ|δ
η(α–) q(α+ )
+|γ|δ
η(α–) q(α+ )
+ |α|δ q(α+ )
+|β|δ q(α)
φ(r) μ , (.)
If
then the boundary value problem (.)-(.) has at least one solution on[, ].
Proof Consider the setBr={x∈C: x ≤r}, whereris given in (.) and define the
as
Px ≤ φ(r)
q(α+ ) μ .
Now, for anyx∈Br, andt,t∈[, ] witht<t, we have
(Px)(t) – (Px)(t)
=
t
(t–qs)(α–)
q(α)
fs,x(s)dqs–
t
(t–qs)(α–)
q(α)
fs,x(s)dqs
≤ t
(t–qs)(α–)– (t–qs)(α–)
q(α)
fs,x(s)dqs
+
t
t
(t–qs)(α–)
q(α)
fs,x(s)dqs
≤φ(r) μ
t
(t–qs)(α–)– (t–qs)(α–)
q(α)
dqs+
t
t
(t–qs)(α–)
q(α)
dqs
which is independent ofxand tends to zero ast→t. Thus,Pis equicontinuous. SoPis
relatively compact onBr. Hence, by the Arzelá-Ascoli theorem,Pis compact onBr. Thus, all the assumptions of Lemma . are satisfied. So the conclusion of Lemma . implies that the boundary value problem (.)-(.) has at least one solution on [, ].
In the special case whenφ(u)≡, we see that there always exists a positiverso that (.) holds true, thus we have the following corollary.
Corollary . Let f: [, ]×R→Rbe a continuous function satisfying (A). In addition,
we assume that
f(t,x)≤μ(t), ∀(t,x)∈[, ]×R,andμ∈C[, ],R+.
If (.) holds, then the boundary value problem (.)-(.) has at least one solution on[, ].
The next existence result is based on Leray-Schauder nonlinear alternative.
Lemma .(Nonlinear alternative for single valued maps []) Let E be a Banach space, C a closed, convex subset of E, U an open subset of C with∈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). Theorem . Let f : [, ]×R→Rbe a continuous function. Assume that:
(A) there exist functionsp,p∈L([, ],R+), and a nondecreasing functionψ:R+→R+
such that|f(t,x)| ≤p(t)ψ(|x|) +p(t),for(t,x)∈[, ]×R;
(A) there exists a numberM> such that
M
ψ(M)ω+ω
where
Then the boundary value problem (.)-(.) has at least one solution on[, ].
Thus, for anyx∈Bρit holds
Fx ≤ψ(ρ)ω+ω,
which proves our assertion.
Now we show thatF maps bounded sets into equicontinuous sets ofC([, ],R). Let
t,t∈[, ] witht<tandx∈Bρ, whereBρis a bounded set ofC([, ],R). Then taking
Obviously, the right-hand side of the above inequality tends to zero independently of
x ∈ Bρ as t –t → . Therefore, it follows by the Arzelá-Ascoli theorem that F :
C([, ],R)→C([, ],R) is completely continuous.
+|β|δ
( –qs)(α–)
q(α– )
p(s)dqs
+
( –qs)(α–)
q(α)
p(s)dqs+|γ|δ
η
(η–qs)(α–)
q(α)
p(s)dqs
+|γ|δ
η
(η–qs)(α–)
q(α)
p(s)dqs+|α|δ
( –qs)(α–)
q(α)
p(s)dqs
+|β|δ
( –qs)(α–)
q(α– )
p(s)dqs
≤ψ(M)ω+ω<M.
Suppose there exists ax∈∂Uand aλ∈(, ) such thatx=λFx. Then for such a choice of
xandλ, we have
M= x =λ Fx <ψ x ω+ω=ψ(M)ω+ω<M,
which is a contradiction. Consequently, by the Leray-Schauder alternative (Lemma .), we deduce thatFhas a fixed pointx∈Uwhich is a solution of the problem (.)-(.). This
completes the proof.
Remark . Ifp,pin (A) are continuous, thenωi≤A pi ,i= , , whereAis defined by (.).
Example . Consider the fractionalq-difference nonlocal boundary value problem cD/
q x(t) =
L
x+tan–x+sint, ≤t≤, (.)
x() – /Dqx() =x(/), x() +
Dqx() =x(/). (.)
Here,α= ,β= /,α= /,β= /,γ= =γ,η= /,η= /, andLis a constant
to be fixed later on. Moreover,= /,δ= /,δ= /,|f(t,x) –f(t,y)| ≤L|x–y|, and
k= L /(/)
√( + √ + √) √(√ – ) +
.
Choosing
L<
/(/)
√( + √ + √) √(√ – ) +
–
,
all the assumptions of Theorem . are satisfied. Therefore, by Theorem ., problem (.)-(.) has a unique solution.
Example . Consider the problem
cD/
q x(t) = cost
sin|x|/+e–x
(t+ )
+ (t+ ) +
, ≤t≤, (.)
x() – /Dqx() =x(/), x() +
whereα= ,β= /,α= /,β= /,γ= =γ,η= /,η= /. In a straightforward
manner, it can be found that= /,δ= /, fδ= /, and
f(t,x)= cost
sin|x|/+e–x
(t+ )
+ (t+ ) +
≤
|x|+ .
Clearly,p= /,p= ,ψ(M) =M. Consequently,ω= .,ω= .,
and the condition (.) implies thatM> .. Thus, all the assumptions of The-orem . are satisfied. Therefore, the conclusion of TheThe-orem . applies to the problem (.)-(.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors, BA, SKN, and IKP contributed to each part of this study equally and read and approved the final version of the manuscript.
Author details
1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 2Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece.
Acknowledgements
The research of B. Ahmad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors thank the reviewers for their useful comments.
Received: 7 May 2012 Accepted: 2 August 2012 Published: 8 August 2012 References
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