R E S E A R C H
Open Access
Existence and uniqueness of the solution
of fractional damped dynamical systems
Jiale Sheng
*and Wei Jiang
*Correspondence:
School of Mathematical Sciences, Anhui University, Hefei, 230039, China
Abstract
In this paper, we consider the existence and uniqueness of the solution of fractional damped dynamical systems. First, we obtain the spreading form of the Gronwall inequality. Furthermore, several sufficient conditions for the existence of the solutions are derived from the application of fixed point theorems and inequalities such as the Hölder and Gronwall inequalities.
MSC: 34k05; 26A33
Keywords: Caputo fractional derivative; fixed point theorem; Gronwall inequality
1 Introduction
Recently, fractional differential equations [–] have been studied intensively. The moti-vation for this work arises from both the development of the theory of fractional calculus itself and its wide application to various fields of science, such as physics, chemistry, bio-logical, electromagnetic of complex media, robotics, economics, etc.
Much attention has been paid to the existence and uniqueness of the solutions of frac-tional dynamic systems [–] on account of the fact that existence is the fundamental problem and a necessary condition for considering some other properties for fractional dynamic systems, such as controllability, stability, etc. Besides, one has to take into ac-count the peculiarity of the kernel to obtain explicit results in practice not only to reduce such an equation to an integral equation. Moreover, many authors have considered the multi-term fractional order systems [–] due to their successful applications in me-chanical system, the dynamics of certain gases, the dynamics of sphere. And the existence and uniqueness of the solutions of this multi-term fractional order systems becomes more complicated than one-term fractional order systems which have been considered before by many authors. In this paper, motivated by the above, we will consider the existence and uniqueness of the solution of the following fractional damped dynamical systems:
⎧ ⎨ ⎩
cDα
tx(t) –AcD β
tx(t) =f(t,x(t)), t∈J:= [,T],
x() =x, x() =x,
()
where <β≤ <α≤, <T <∞,x∈Rn,Ais anRn×nmatrix andf :J×Rn→Rnis jointly continuous.
The rest of this paper is organized as follows. In Section , we recall some definitions and facts on fractional calculus, the generalized Gronwall inequality, and fixed point theo-rems. In Section , we will give some results as regards the spreading form of the Gronwall inequality. In Section , we present our main results of this paper.
2 Preliminaries
In this section, we introduce notations, definitions and preliminary facts. Throughout this paper, letC(J,Rn) be the Banach space of all continuous function fromJintoRnwith the
normxc=sup{|x(t)|:t∈J}forx∈C(J,Rn). Let|x|(·)be any vector norm (e.g., := , ,∞)
and(·)be the matrix norm induced by this vector. The symbol∼means that the quotient of both sides converges to . Then we recall the following well-known definitions.
Definition . The fractional integral of orderα with the lower zero for a functionf : [,∞)→Ris defined as
Itαf(t) = (α)
t
f(s)
(t–s)–αds, t> ,α> ,
provided the right side is point-wise defined on [,∞), where(·) is a gamma function.
Definition . The Riemann-Liouville derivative of orderαwith the lower limit zero for
a functionf : [,∞)→Rcan be defined as
LDα tf(t) =
(n–α)
dn
dtn t
f(s)
(t–s)α+–nds, t> ,n– <α<n.
Definition . The Caputo derivative of orderα for a functionf : [,∞)→Rcan be
defined as
cDα
tf(t) =LDαt
f(t) –
n–
k=
tk
k!f
(k)()
, t> ,n– <α<n.
Lemma .([]) From the definition of fractional integrals and Caputo derivatives,we
have
Iα
t cDαtx(t)
=x(t) –
n–
k=
tk
k!f
(k)(), t> ,n– <α<n.
Especially,when <α< ,then we have
Itα cDαtx(t)=x(t) –x() –tx().
Lemma .([]) Let <β< <α< ,then we have
Itα cDβtx(t)
=Itα–βx(t) –
x()tα–β
Taking the fractional integral of orderαon both sides on the system(),due to the above two lemmas we can easily obtain the integral equation of the system():
x(t) =x+tx–
Atα–β
(α–β+ )x+ A (α–β)
t
(t–s)α–β–x(s)ds
+ (α)
t
(t–s)α–f s,x(s)ds.
For a measurable functionδ:J→R,define the norm
δLp(J)=
⎧ ⎨ ⎩
(J|δ(t)|pdt)p, ≤p<∞,
infμ(J)={supt∈J–J|δ(t)|}, p=∞,
whereμ(J)is the Lebesgue measure on J.Let Lp(J,R)be the Banach space of all Lebesgue
measure functionsδ:J→R withδLp(J)<∞.
Lemma .(Hölder’s inequality []) Assume that p,q≥,and p+q = .Ifδ∈Lp(J,R),
λ∈Lq(J,R),then,for≤q≤ ∞,δλ∈L(J,R),and
δλL(J)≤ δLp(J)λLq(J).
Lemma .([]) LetCbe a nonempty closed convex subset of a Banach space(X, · ). Suppose that P and Q mapCinto X such that:
(i) Px+Qy∈Cwherex,y∈C; (ii) Pis a compact and continuous; (iii) Qis a contraction mapping.
Then there exists z∈Csuch that z=Pz+Qz.
Lemma .([]) Let X be a Banach spaces and F:X→X be a completely continuous operator,if the set
E(F) =y∈X:y=λFy for someλ∈[, ]
is bounded,then F has at least a fixed point.
Lemma .([]) LetCbe a nonempty convex subset of X.Let U be a nonempty open subset
ofCwith∈U and F:U→Cbe a compact and continuous operators.Then either: (i) Fhas fixed points,or
(ii) there existy∈∂Uandλ∗∈[, ]withy=λ∗F(y).
Lemma . ([]) Suppose β> ,a(t)is a nonnegative function locally integrable on [,T) andg(t) is a nonnegative, nondecreasing, continuous function defined on [,T);
g(t)≤M,where M is constant. Suppose x(t)is a nonnegative and locally integrable on [,T)with
x(t)≤a(t) +g(t)
t
Then
x(t)≤a(t) +
t
∞
n=
(g(t)(β))n
(nβ) (t–s)
nβ–a(t)
ds, t∈[,T).
Corollary .([]) Under the hypothesis of Lemma.,let a(t)be a nondecreasing function on[,T).Then
x(t)≤a(t)Eβ g(t)(β)tβ
,
where Eβis the Mittag-Leffler function defined by
Eβ(z) =
∞
k=
zk
(kβ+ ), z∈C,Re(β) > .
3 Some results of the Gronwall inequality
In this section, we will give some results as regards the extended form of the Gronwall inequality.
Lemma . Let p> ,q> ,then
t
τ
(t–s)p–(s–τ)q–ds=(p)(q) (p+q)(t–τ)
p+q–.
Proof Lets=τ+z(t–τ), then we have
t
τ
(t–s)p–(s–τ)q–ds= (t–τ)p+q–
( –z)p–z(q–)dz = (t–τ)p+q–B(p,q)
=(p)(q) (p+q)(t–τ)
p+q–.
Theorem . Supposeα> ,β> ,a(t)is a nonnegative function locally integrable on [,T),g(t),and g(t)are nonnegative,nondecreasing,continuous functions defined on[,T);
g(t)≤M,g(t)≤M,whereM and M are constants .Suppose x(t)is a nonnegative and locally integrable on[,T)with
x(t)≤a(t) +g(t)
t
(t–s)α–x(s)ds+g(t)
t
(t–s)β–x(s)ds, t∈[,T).
Then
x(t)≤a(t) +
t
∞
n=
g(t)n
× n
k=
Ck
n[(α)]n–k[(β)]k
[(n–k)α+kβ] (t–s)
(n–k)α+kβ–a(s)ds, t∈[,T), ()
where g(t) =g(t) +g(t)and Ck n=
n(n–)(n–)···(n–k+)
Proof Letg(t) =g(t) +g(t), then
Whenn= , the inequality () obviously holds. We assume that it holds forn=m. When n=m+ , we have Lemma ., the inequality () can be rewritten as
Bm+x(t)≤g(t)m+
By interchanging the order of integration, the inequality () can be rewritten
n→ ∞with √x(ξ)
nπ γ → asn→ ∞. Therefore, by equation (), we can easily obtain the
conclusion. The proof is completed.
Corollary . Under the hypothesis of Theorem.,let a(t)be a nondecreasing function on[,T).Then
x(t)≤a(t)Eγ
g(t)(α)tα+(β)tβ,
whereγ =min{α,β}.
Proof Whena(t) is a nondecreasing function on [,T), the inequality () can be rewritten as
The proof is completed.
4 Main results
In this section, we will introduce the existence and uniqueness of the solutions of the sys-tem (). Before stating and proving the main results, we need to give the following hy-potheses.
For brevity, let us denote
a= α–
Proof First, we construct a space
Obviously,Fis well defined due to (H).
Thus,Fis a contraction mapping onCrdue to the condition (). Due to the well-known
Banach contraction mapping principle we know that the operatorFhas a unique fixed point onCr. Therefore, the system () has a unique solution. The proof is completed.
Theorem . Assume that(H)-(H)hold,if
h
L
q(J)T
(+b)(–q)
(α)( +b)(–q) < , ()
then the system()has at least one solution on J.
Proof Consider theCrdefined in Theorem .. We define the operatorsFandFonCr
as follows:
(Fx)(t) =x+tx–
Atα–β
(α–β+ )x+ A (α–β)
t
(t–s)α–β–x(s)ds,
(Fx)(t) =
(α)
t
(t–s)α–f s,x(s)ds.
For anyx,y∈Crandt∈J, using (H) and the Hölder inequality, we have
Fx+Fyc≤ |x|+Tx+
ATα–β
(α–β+ )|x|+
rATα–β
(α–β+ )+
m
L
q(J)T
(+a)(–q)
(α)( +a)–q ≤r.
So we know thatFx+Fy∈Cr. In order to use Lemma ., we will verify thatF is a
contraction mapping, whileFis compact and continuous.
Now we prove the operatorFis a contraction operator. Letx,x∈Crandt∈J, then (Fx)(t) – (Fx)(t)
≤
(α)
t
(t–s)α–f s,x (s)
–f s,x(s)ds
≤
(α)
t
(t–s)α–h(s)x(s) –x(s)ds
≤
(α)
t
(t–s)–α–q ds
–q t
h(s)q ds q
x–xc
≤
h
L
q(J)T
(+b)(–q)
(α)( +b)–q x–xc.
So we see thatFis a contraction operator due to the condition ().
Then we prove the operatorFis compact and continuous. Let{xn}be a sequence such
thatxn→xin theC(J,Rn). Then, for eacht∈J, we have
(Fxn)(t) – (Fx)(t)≤
A (α–β)
t
(t–s)α–β–xn(s) –x(s)ds ≤ ATα–β
Thus,Fxn–Fx → asn→ ∞, which implies that the operatorFis continuous onCr.
For eacht∈J, we obtain
(Fx)(t)
≤ |x|+Tx–
ATα–β
(α–β+ )|x|+
A (α–β)
t
(t–s)α–β–x(s)ds
≤ |x|+Tx–
ATα–β
(α–β+ )|x|+
rATα–β
(α–β+ ) :=r.
It shows that, for anyr> , there exists arsuch that, for eachx∈Cr, we haveFx ≤r.
That is to say,Fmaps bounded sets into bounded sets inCr.
And also fort,t∈J, ≤t<t≤T,x∈Cr, we have (Fx)(t) – (Fx)(t)
≤x(t–t) + |
x|A
(α–β+ ) t
α–β
–t
α–β
+ rA (α–β)
t
t
(t–s)α–β–ds
+ rA (α–β)
t
(t–s)α–β–– (t–s)α–β–
ds
=x(t–t) + |
x|A
(α–β+ ) t
α–β
–t
α–β
+ rA (α–β+ ) t
α–β
–t
α–β
≤x(t–t) +
rA (α–β+ ) t
α–β
–t
α–β
.
Ast→t,|(Fx)(t) – (Fx)(t)| →, which implies thatFis equicontinuous. By
apply-ing the well-known Ascoli theorem, we come to the conclusion that the operatorF is
compact.
Hence from all the above results together with Lemma ., we can see that the operator Fhas at least one fix point. Therefore, the system () has at least one solution onJ. The
proof is completed.
Now, we replace (H) by the following linear growth condition:
(H) For eacht∈Jand allx∈Rn, there exists a constantL> such that
f(t,x)≤L +|x|.
Theorem . Assume that(H)and(H)hold,then the system()has at least one solu-tion on J.
Proof Consider the operatorF:Cr→C(J,Rn) defined as () in Theorem .. And we
de-fine the set
E(F) =x∈Cr:x=λFxfor someλ∈[, ]
.
Now we prove theFis completely continuous. Let{xn}be a sequence such thatxn→x
That is to say,Fmaps bounded sets into bounded sets inCr.
And also due to (H), fort,t∈J, ≤t<t≤T,x∈Cr, we have
≤ |x|+Tx+
ATα–β
(α–β+ )|x|+ LTα
(α+ )
+ A
(α–β)
t
(t–s)α–β–x(s)ds
+ L
(α)
t
(t–s)α–x(s)ds.
For brevity, let us denote
l=|x|+Tx+
ATα–β
(α–β+ )|x|+ LTα
(α+ ),
m= A
(α–β), n=
L (α).
Then by Corollary ., we can obtain
x(t)≤l+m t
(t–s)α–β–x(s)ds+n
t
(t–s)α–x(s)ds
≤lEα–β
(m+n) (α)tα+(α–β)tα–β :=M∗,
which implies that there exists a constantM∗> such that|x(t)| ≤M∗. Then we can say that the setE(F) is bounded.
Hence from all the above results together with Lemma ., we deduce thatFhas a fixed point. Therefore the system () has at least one solution. The proof is completed.
In order to weaken the condition (H), we will use Lemma . to prove the following theorem.
(H) There exists a constantq∈(, ) such that a real valued functionφf(t)∈L
q(J,Rn)
and there exists a L-integrable and nondecreasing ψ : [, +∞)→[, +∞) such that
|f(t,x)| ≤φf(t)ψ(|x|) for eacht∈Jandx∈Rn.
(H) The inequality
η
+ψ(η)T(+γ)(–q)ϑ
(α)(+γ)–q
> ()
has at least a positive solution, where=|x|+T|x|+
xcATα–β
(α–β+) ,ϑ=φfLq(J),γ = α–
–q.
Theorem . Assume that(H), (H),and(H)hold,then the system()has at least one
solution on J.
Proof Proof by contradiction. By the hypothesis (H), there exists aN> such thatxc=
N. LetCN={x∈C(J,Rn) :xc<N}. Consider the operatorFdefined in Theorem ., we
that there existx∈∂CN such thatx=λ∗F(x),λ∗∈[, ], then we have
x(t)≤(Fx)(t)≤ |x|+Tx+
ATα–β
(α–β+ )|x|
+ A
(α–β)
t
(t–s)α–β–x(s)ds
+ψ(xc) (α)
t
(t–s)α–f s,x(s)ds
≤ |x|+Tx+
xcATα–β
(α–β+ )
+ψ(xc) (α)
t
(t–s)–α–q ds
–q t
φf(s)
q ds q
≤+ψ(xc)T
(+γ)(–q)ϑ
(α)( +γ)–q .
Thus
η
+ψ(η)T(+γ)(–q)ϑ
(α)(+γ)–q ≤,
which is a contradiction with the inequality () under the hypothesis (H).
Then we can say that there is nox∈∂CN such thatx=λ∗F(x), λ∗∈[, ], from the
selection ofCN. By the Lemma ., we deduce thatFhas a fixed pointx∈CN. Therefore,
the system () has at least one solution. The proof is completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally to the manuscript. Both of them read and approved the final manuscript.
Acknowledgements
The authors are sincerely thankful to the reviewers for their kind comments and valuable suggestions. This research was jointly supported by National Natural Science Foundation of China (no. 11371027 and no. 11601003).
Received: 28 May 2016 Accepted: 28 November 2016
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