Volume 2008, Article ID 430521,22pages doi:10.1155/2008/430521
Research Article
The Generalized Gronwall Inequality
and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic
System on Banach Space
JinRong Wang,1X. Xiang,1, 2W. Wei,2and Qian Chen2
1College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
2College of Science, Guizhou University, Guiyang, Guizhou 550025, China
Correspondence should be addressed to JinRong Wang,[email protected] Received 27 June 2008; Accepted 29 September 2008
Recommended by Ondˇrej Doˇsl ´y
This paper deals with a class of integrodifferential impulsive periodic systems on Banach space.
Using impulsive periodic evolution operator given by us, theT0-periodic PC-mild solution is introduced and suitable Poincar´e operator is constructed. Showing the compactness of Poincar´e operator and using a new generalized Gronwall’s inequality with impulse, mixed type integral operators andB-norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence ofT0-periodic PC-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration.
Copyrightq2008 JinRong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
It is well known that impulsive periodic motion is a very important and special phenomenon not only in natural science, but also in social science such as climate, food supplement, insecticide population, and sustainable development. Periodic system with applications on finite-dimensional spaces has been extensively studied. Particularly, impulsive periodic systems on finite-dimensional spaces are considered and some important resultssuch as the existence and stability of periodic solution, the relationship between bounded solution and periodic solution, and robustness by perturbationare obtainedsee1–4.
Since the end of last century, many researchers pay great attention to impulsive systems on infinite-dimensional spaces. Particulary, Ahmed et al. investigated optimal control problems of system governed by impulsive system see 5–8. Many authors including us also gave a series of results for semilinear integrodifferential, strongly nonlinearimpulsive systems and optimal control problemssee9–20.
Although, there are some papers on periodic solution for periodic system on infinite- dimensional spacessee12,21–23and some results discussing integrodifferential system on finite Banach space and infinite Banach space see 11, 13. To our knowledge, inte- grodifferential impulsive periodic systems on infinite-dimensional spaceswith unbounded operator have not been extensively investigated. Recently, we discuss the impulsive periodic system and integrodifferential impulsive system on infinite-dimensional spaces.
Linear impulsive evolution operator is constructed and T0-periodic P C-mild solution is introduced. The existence of periodic solutions, alternative theorem criteria of Massera type, asymptotical stability, and robustness by perturbation is established see 24–26.
For semilinear impulsive periodic system, a suitable Poincar´e operator is constructed which verifies its compactness and continuity. By virtue of a generalized Gronwall inequality with mixed integral operator and impulse given by us, the estimate of theP C-mild solutions is derived. Some fixed point theorems such as Banach fixed point theorem and Horn fixed point theorem are applied to obtain the existence of periodicP C-mild solutions, respectivelysee 27,28. For integrodifferential impulsive system, the existence ofP C-mild solutions and optimal controls is presentedsee15.
Herein, we go on studying the following integrodifferential impulsive periodic system
xt ˙ Axt f
t, x, t
0
gt, s, xds
, t /τk, Δxt Bkxt ck, tτk.
1.1
on infinite-dimensional Banach spaceX, where 0τ0 < τ1< τ2<· · ·< τk· · ·; limk→ ∞τk ∞, τkδ τk T0;Δxτk xτk−xτk−,k ∈ Z0;T0 is a fixed positive number; andδ ∈ N denoted the number of impulsive points between 0 andT0. The operatorAis the infinitesimal generator of aC0-semigroup{Tt, t ≥ 0}onX;f is aT0-periodic, with respect tot ∈ 0
∞, Carath´edory function; g is a continuous function from0,∞×0,∞×X toX and is T0-periodic int ands; and Bkδ Bk, ckδ ck. This paper is mainly concerned with the existence of periodic solutions for integrodifferential impulsive periodic system on infinite- dimensional Banach spaceX.
In this paper, we use Leray-Schauder fixed point theorem to obtain the existence of periodic solutions for integrodifferential impulsive periodic system 1.1. First, by virtue of impulsive evolution operator corresponding to linear homogeneous impulsive system, we construct a new Poincar´e operator P for integrodifferential impulsive periodic system 1.1, then we overcome some difficulties to show the compactness of Poincar´e operatorP which is very important. By a new generalized Gronwall inequality with impulse, mixed- type integral operators, andB-norm given by us, the estimate of fixed point set{xλP x, λ∈ 0,1}is established. Therefore, the existence ofT0-periodicP C-mild solutions for impulsive integrodifferential periodic system is shown.
In order to obtain the existence of periodic solutions, many authors use Horn fixed point theorem or Banach fixed point theorem. However, the conditions for Horn fixed point theorem are not easy to be verified sometimes and the conditions for Banach fixed point theorem are too strong. Our method is much different from others’, and we give a new way to show the existence of periodic solutions. In addition, the new generalized Gronwall inequality with impulse, mixed-type integral operator, andB-norm given by us, which can be used in other problems, have played an essential role in the study of nonlinear problems on infinite-dimensional spaces.
This paper is organized as follows. In Section 2, some results of linear impulsive periodic system and properties of impulsive periodic evolution operator corresponding to homogeneous linear impulsive periodic system are recalled. InSection 3, the new generalized Gronwall inequality with impulse, mixed-type integral operator, andB-norm are established.
In Section 4, the T0-periodic P C-mild solution for integrodifferential impulsive periodic system 1.1 is introduced. We construct the suitable Poincar´e operator P and give the relation betweenT0-periodicP C-mild solution and the fixed point ofP. After showing the compactness of the Poincar´e operatorPand obtaining the boundedness of the fixed point set {x λP x, λ ∈ 0,1}by virtue of the generalized Gronwall inequality, we can use Leray- Schauder fixed point theorem to establish the existence ofT0-periodicP C-mild solutions for integrodifferential impulsive periodic system. At last, an example is given to demonstrate the applicability of our result.
2. Linear impulsive periodic system
In order to study the integrodifferential impulse periodic system, we first recall some results about linear impulse periodic system here. LetXbe a Banach space. £Xdenotes the space of linear operators inX; £bXdenotes the space of bounded linear operators inX. £bXis the Banach space with the usual supremum norm. DefineD {τ1, . . . , τδ} ⊂0, T0, whereδ∈N denotes the number of impulsive points between0, T0. We introduceP C0, T0;X≡ {x: 0, T0→X|xto be continuous att∈0, T0\D; xis continuous from left and has right-hand limits att∈D}; and P C10, T0;X≡ {x∈P C0, T0;X|x˙ ∈P C0, T0;X}.Set
xP Cmax
sup
t∈0,T0
xt0, sup
t∈0,T0
xt−0
, xP C1xP Cx˙ P C. 2.1
It can be seen that endowed with the norm·P C·P C1,P C0, T0;XP C10, T0;Xis a Banach space.
Firstly, we consider homogeneous linear impulsive periodic system x. t Axt, t /τk,
Δxt Bkxt, tτk. 2.2
We introduce the following assumptionH1.
H1.1 A is the infinitesimal generator of a C0-semigroup {Tt, t ≥ 0} on X with domainDA.
H1.2There existsδsuch thatτkδτkT0. H1.3For eachk∈Z0, Bk∈£bXandBkδBk.
In order to study system2.2, we need to consider the associated Cauchy problem x. t Axt, t∈0, T0\D,
Δxτk Bkxτk, k1,2, . . . , δ, x0 x.
2.3
Ifx ∈ DAandDAis an invariant subspace ofBk, using Theorem 5.2.2,see29, page 144, step by step, one can verify that the Cauchy problem2.3has a unique classical solutionx∈P C10, T0;Xrepresented byxt St,0x,where
S·,·:Δ t, θ∈0, T0×0, T0|0≤θ≤t≤T0
−→£bX 2.4
given by
St, θ
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
Tt−θ, τk−1 ≤θ≤t≤τk,
T
t−τk IBk
T τk−θ
, τk−1≤θ < τk< t≤τk1, T
t−τk
θ<τj<t
IBj T
τj−τj−1 IBi
T τi−θ
,
τi−1≤θ < τi≤ · · ·< τk< t≤τk1. 2.5 The operator {St, θ,t, θ ∈ Δ} is called impulsive evolution operator associated with {Bk;τk}∞k1.
Now we introduce theP C-mild solution of Cauchy problem2.3andT0-periodicP C- mild solution of the system2.2.
Definition 2.1. For everyx∈X, the functionx∈P C0, T0;Xgiven byxt St,0xis said to be theP C-mild solution of the Cauchy problem2.3.
Definition 2.2. A functionx∈P C0,∞;Xis said to be aT0-periodicP C-mild solution of system2.2if it is aP C-mild solution of Cauchy problem2.3corresponding to somexand xtT0 xtfort≥0.
The following lemma gives the properties of the impulsive evolution operator {St, θ,t, θ∈Δ}associated with{Bk;τk}∞k1which are widely used in sequel.
Lemma 2.3 see 24, Lemma 1. Impulsive evolution operator {St, θ,t, θ ∈ Δ} has the following properties.
1For 0≤θ≤t≤T0,St, θ∈£bX, that is, sup0≤θ≤t≤T0St, θ ≤MT0, whereMT0 >0.
2For 0≤θ < r < t≤T0,r /τk,St, θ St, rSr, θ.
3For 0≤θ≤t≤T0andN∈Z0,StNT0, θNT0 St, θ.
4For 0≤t≤T0andN∈Z0,SNT0t,0 St,0ST0,0N.
5If{Tt, t≥0}is a compact semigroup inX, thenSt, θis a compact operator for 0≤θ <
t≤T0.
Here, we note that system2.2has a T0-periodicP C-mild solution xif and only if ST0,0has a fixed point. The impulsive evolution operator{St, θ,t, θ∈Δ}can be used to reduce the existence ofT0-periodicP C-mild solutions for linear impulsive periodic system to the existence of fixed points for an operator equation. This implies that we can build up
the new framework to study the periodicP C-mild solutions for integrodifferential impulsive periodic system on Banach space.
Consider nonhomogeneous linear impulsive periodic system
xt ˙ Axt ft, t /τk,
Δxt Bkxt ck, tτk, 2.6
and the associated Cauchy problem
xt ˙ Axt ft, t∈0, T0\D, Δxτk Bkxτk ck, k1,2, . . . , δ,
x0 x.
2.7
wheref∈L10, T0;X,ftT0 ftandckδck.
Now we introduce theP C-mild solution of Cauchy problem2.7andT0-periodicP C- mild solution of system2.6.
Definition 2.4. A functionx∈P C0, T0;X, for finite interval0, T0, is said to be aP C-mild solution of the Cauchy problem 2.6 corresponding to the initial value x ∈ X and input f∈L10, T0;Xifxis given by
xt St,0x t
0
St, θfθdθ
0≤τk<t
S t, τk
ck. 2.8
Definition 2.5. A functionx∈P C0,∞;Xis said to be aT0-periodicP C-mild solution of system2.6if it is aP C-mild solution of Cauchy problem2.7corresponding to somexand xtT0 xtfort≥0.
3. The generalized Gronwall inequality
In order to use Leray-Schauder theorem to show the existence of periodic solutions, we need a new generalized Gronwall inequality with impulse, mixed-type integral operator, andB- norm which is much different from classical Gronwall inequality and can be used in other problemssuch as discussion on integrodifferential equation of mixed type, see15. It will play an essential role in the study of nonlinear problems on infinite-dimensional spaces.
We first introduce the following generalized Gronwall inequality with impulse and B-norm.
Lemma 3.1. Letx∈P C0,∞, Xand satisfy the following inequality:
xt ≤ab t
0
xθλ1dθd t
0
xθλB3dθ, 3.1
wherea, b, d≥0, 0≤λ1, λ3≤1 are constants, andxθBsup0≤ξ≤θxξ. Then
xt≤a1ebct. 3.2 Proof. iFor 0≤λ1,λ3<1, letλmax{λ1, λ3} ∈0,1and
yt
⎧⎨
⎩
1, xt ≤1,
xt, xt>1.
3.3
Then
xt≤yt≤a1 b t
0
yθλdθd t
0
yθλ
Bdθ ∀t∈ 0, T0
. 3.4
Using3.4, we obtain ytλ
B≤a1 bd
t
0
yθλ
Bdθ. 3.5
Define
ut a1 bd
t
0
yθλ
Bdθ, 3.6
we get
ut b˙ dytλ
B, t /τk,
u0 a1, u
τk0 u
τk
.
3.7
SinceytλB≤ut, we then have
ut˙ ≤bdut, t /τk,
u0 a1, u
τk0 u
τk
. 3.8
Fort∈τk, τk1, by3.8, we obtain ut≤u
τk0
ebdt−τku τk
ebdt−τk, 3.9
further,
ut≤a1ebdt, 3.10
thus,
xt≤yt
B≤a1ebdt. 3.11
iiForλ1λ3 1, we only need to define
u1t a bd t
0
xθ
Bdθ, 3.12
Similar to the proof ini, one can obtain xt≤xt
B≤aebdt. 3.13
Combiningiandii, one can complete the proof.
Using Gronwall’s inequality with impulse andB-norm, we can obtain the following new generalized Gronwall Lemma.
Lemma 3.2. Letx∈P C0, T0;Xsatisfy the following inequality:
xt
≤ab t
0
xθλ1dθc T0
0
xθλ2dθd t
0
xθλ3
Bdθe T0
0
xθλ4
Bdθ ∀t∈
0, T0
, 3.14 where λ1, λ3 ∈ 0,1,λ2, λ4 ∈ 0,1,a, b, c, d, e ≥ 0 are constants. Then there exists a constant M∗>0 such that
xt≤M∗. 3.15
Proof. ByLemma 3.1, we obtain that
xt≤yt≤yt
B≤ebdt
a1 c T0
0
yθλdθe T0
0
yθλ
Bdθ
, 3.16
where
yt
⎧⎨
⎩
1, xt≤1, xt, xt>1, λ
⎧⎨
⎩
max λ1, λ2, λ3, λ4
∈0,1, if λ1, λ2, λ3, λ4∈0,1, max λ2, λ4
∈0,1, if λ1λ31, λ2, λ4∈0,1.
3.17
Define qt
≡ebdT0
a1 c t
0
yθλdθc T0
0
yθλdθe t
0
yθλ
Bdθe T0
0
yθλ
Bdθ
, 3.18
thenqis a monotone increasing function and qt˙
ebdT0
cytλeytλ
B
≤ceebdT0ytλytλ
B
≤2ceebdT0qλt.
3.19
Consider d
dtq1−λt 1−λq−λtqt˙ ≤2ceebdT01−λ. 3.20 Integrating from 0 tot, we obtain
q1−λt−q1−λ0≤2ceebdT01−λt, 3.21 that is,
qt≤
q1−λ0 2ceebdT01−λt1/1−λ
. 3.22
On the other hand,
2q0 2ebdT0
a1 c T0
0
yθλdθe T0
0
yθλ
Bdθ
;
qT0 ebdT0
a1 2c T0
0
yθλdθ2e T0
0
yθλ
Bdθ
.
3.23
Now, we observe that
2q0−ebdT0a1 q T0
≤
q1−λ0 2ceebdT0T01−λ1/1−λ
. 3.24
As a result, we get
2q0−ebdT0a11−λ
−q1−λ0≤2ceebdT0T01−λ. 3.25
Letting
Υz
2z−ebdT0a11−λ−z1−λ−2ceebdT0T01−λ, 3.26
we haveΥ∈CebdT0a1/2,∞;RandΥebdT0a1/2<0. Moreover,
z→lim∞
Υz
z1−λ21−λ−1>0. 3.27
Hence, there exists enough largez0 > ebdT0a1/2 > 0 such thatΥz > 0 for arbitrary z≥z0. Meanwhile,Υq0≤0. Thus,q0≤z0.
As a result, we obtain
xt≤yt≤q T0
2q0−ebdT0a1
≤2z0−ebdT0a1≡M∗>0 ∀t∈ 0, T0
.
3.28
4. Periodic solutions of integrodifferential impulsive periodic system
In this section, we consider the following integrodifferential impulsive periodic system:
xt ˙ Axt f
t, x, t
0
gt, s, xds
, t /τk, Δxt Bkxt ck, tτk.
4.1
and the associated Cauchy problem
xt ˙ Axt f
t, x, t
0
gt, s, xds
, t∈0, T0\D, Δx
τk Bkx
τk
ck, k1,2, . . . , δ, x0 x.
4.2
By virtue of the expression of theP C-mild solution of the Cauchy problem2.7, we can introduce theP C-mild solution of the Cauchy problem4.2.
Definition 4.1. A functionx ∈ P C0, T0;Xis said to be aP C-mild solution of the Cauchy problem4.2corresponding to the initial valuex ∈ X if xsatisfies the following integral equation:
xt St,0x t
0
St, θf
θ, xθ, θ
0
g
θ, s, xs ds
dθ
0≤τk<t
S t, τk
ck fort∈ 0, T0
.
4.3
Now, we introduce theT0-periodicP C-mild solution of system4.1.
Definition 4.2. A functionx∈P C0,∞;Xis said to be aT0-periodicP C-mild solution of system4.1if it is aP C-mild solution of Cauchy problem4.2corresponding to somexand xtT0 xtfort≥0.
AssumptionH2includes the following.
H2.1f:0,∞×X×X→Xsatisfies the following.
iFor eachx, y∈X×X,t→ft, x, yis measurable.
iiFor eachρ >0,there existsLfρ>0 such that, for almost allt∈0,∞and allx1, x2,y1, y2∈X,x1,x2,y1,y2 ≤ρ, we have
f t, x1, y1
−f
t, x2, y2≤Lfρx1−x2y1−y2. 4.4
H2.2There exists a positive constantMf such that ft, x, y≤Mf
1xy
∀x, y∈X. 4.5
H2.3ft, x, yisT0-periodic int, that is,ftT0, x, y ft, x, y, t≥0.
H2.4LetD {t, s ∈ 0∞×0∞; 0 ≤ s ≤ t}. The functiong : D×X→X is continuous for eachρ > 0, there existsLgρ > 0 such that, for eacht, s ∈ Dand each x, y∈Xwithx,y ≤ρ, we have
gt, s, x−gt, s, y≤Lgρx−y. 4.6
H2.5There exists a positive constantMgsuch that gt, s, x≤Mg
1x
∀x, y∈X. 4.7
H2.6gt, s, xare T0-periodic intands, that is,gtT0, sT0, x gt, s, x, t≥s≥0 and
T0
0
gt, s, xds0, t≥s≥0. 4.8
H2.7For eachk∈Z0 andck∈X, there existsδ∈Nsuch thatckδck. Lemma 4.3. Under assumptions [H2.4] and [H2.5], one has the following properties:
1·
0g·, s, xsds:P C0, T0;X→P C0, T0;X.
2For allx1, x2∈P C0, T0;Xandx1P C0,T0;X,x2P C0,T0;X≤ρ, t
0
g
t, s, x1s ds−
t
0
g
t, s, x2s ds
≤LgρT0x1−x2
B. 4.9
3Forx∈P C0, T0;X, t
0
g
t, s, xs ds
≤MgT0
1xt
B
. 4.10
Proof. 1Sincegis continuous in its variables and satisfies linear growth conditions, one can verify that·
0g·, s, xsdsmapsP C0, T0;XtoP C0, T0;X.
2Letx1, x2∈P C0, T0;X,x1P C0,T0;X,x2P C0,T0;X≤ρ, we have t
0
g
t, s, x1s ds−
t
0
g
t, s, x2s ds
≤ t
0
g
t, s, x1s
−g
t, s, x2sds
≤ t
0
Lgρx1s−x2sds
≤Lgρtx1t−x2t
B
≤LgρT0x1t−x2t
B.
4.11
3Forx∈P C0, T0;X, t
0
g
t, s, xs ds
≤Mg
t
0
1xsds
≤Mgt
1xt
B
≤MgT0
1xt
B
.
4.12
Now we present the existence ofP C-mild solution for system4.2.
Theorem 4.4. Assumptions [H1.1], [H2.1], [H2.4], and [H2.5] hold. Then system4.2has a unique P C-mild solution given by
x t, x
St,0x t
0
St, θf
θ, xθ, θ
0
g
θ, s, xs ds
dθ
0≤τk<t
S t, τk
ck. 4.13
Proof. A similar result is given by Wei et al.15. Thus, we only sketch the proof here. In order to make the process clear, we divide it into three steps.
Step 1. We consider the following general integrodifferential equation without impulse
xt ˙ Axt f
t, x, t
0
gt, s, xds
, t∈s, τ, xs x∈X.
4.14
In order to obtain the local existence of mild solution for system4.14, we only need to set up the framework for use of the contraction mapping theorem. Consider the ball given by
B x∈C s, t1
;X
|xt−x≤1, s≤t≤t1
, 4.15
wheret1would be chosen, andxt ≤1x ρ,s ≤t ≤t1.B ⊆Cs, t1, Xis a closed convex set. Define a mapQ onBgiven by
Qxt Ttx t
s
Tt−θf
θ, xθ, θ
0
g
θ, s, xs ds
dθ. 4.16
Under assumptionsH1.1,H2.1,H2.2,H2.4,H2.5andLemma 3.1, one can verify that mapQ is a contraction map onBwith chosent1 > 0. This means that system 4.14has a unique mild solutionx∈Cs, t1;Xgiven by
xt Ttx t
s
Tt−θf
θ, xθ, θ
0
g
θ, s, xs ds
dθ on s, t1
. 4.17
Again, usingLemma 3.1, we can obtain the a priori estimate of the mild solutions for system 4.14and present the global existence of mild solutions.
Step 2. Fort∈τk, τk1, consider the Cauchy problem
xt ˙ Axt f
t, x, t
0
gt, s, xds
, t∈
τk, τk1 , x
τk
xk≡ IBk
x τk
ck∈X.
4.18
ByStep 1, Cauchy problem4.18also has a uniqueP C-mild solution
xt T t−τk
xk t
τk
Tt−θf
θ, xθ, θ
0
g
θ, s, xs ds
dθ. 4.19
Step 3. Combining all of the solutions onτk, τk1 k1, . . . , δ, one can obtain theP C-mild solution of Cauchy problem4.2given by
x t, x
St,0x t
0
St, θf
θ, xθ, θ
0
g
θ, s, xs ds
dθ
0≤τk<t
S t, τk
ck. 4.20
This completes the proof.
To establish the periodic solutions for system4.1, we define a Poincar´e operator from XtoXas follows:
P x
x T0, x S
T0,0 x
T0
0
S T0, θ
f
θ, x θ, x
, θ
0
g θ, s, x
s, x ds
dθ
0≤τk<T0
S T0, τk
ck, 4.21
where x·, x denote the P C-mild solution of Cauchy problem 4.2 corresponding to the initial valuex0 x, then we examine whetherPhas a fixed point.
We first note that a fixed point ofPgives rise to a periodic solution.
Lemma 4.5. System4.1has aT0-periodicP C-mild solution if and only ifPhas a fixed point.
Proof. Suppose x· x·T0, thenx0 xT0 Px0. This implies that x0 is a fixed point of P. On the other hand, if P x0 x0, x0 ∈ X, then for the P C-mild solution x·, x0of Cauchy problem4.2corresponding to the initial valuex0 x0, we can define
y· x·T0, x0, theny0 xT0, x0 P x0 x0. Now, fort >0, we can use2,3, and 4ofLemma 2.3and assumptionsH1.2,H1.3,H2.3,H2.6, andH2.7to arrive at
yt x
tT0, x0
S
tT0, T0 S
T0,0 x0
T0
0
S
tT0, T0
S T0, θ
f
θ, x θ, x0
, θ
0
g θ, s, x
s, x0
ds
dθ
0≤τk<T0
S
tT0, T0
S T0, τk
ck
tT0
T0
S
tT0, θ f
θ, x
θ, x0
, θ
0
g θ, s, x
s, x0
ds
dθ
T0≤τkδ<tT0
S
tT0, τkδ ckδ
St,0
S T0,0
x0 T0
0
S T0, θ
f
θ, x θ, x0
, θ
0
g θ, s, x
s, x0
ds
dθ
0≤τk<T0
S T0, τk
ckδ
t
0
S
tT0, θT0 f
θT0, x
θT0, x0 ,
θT0
0
g
θT0, s, x s, x0
ds
dθ
T0≤τkδ<tT0
S
tT0, τkδ ckδ
St,0x T0
t
0
S
tT0, θT0 f
θT0, x
θT0, x0 ,
θT0
T0
g
θT0, s, x s, x0
ds
dθ
T0≤τkδ<tT0
S
tT0, τkδ ckδ
St,0x T0
t
0
St, θf
θ, x
θT0, x0 ,
θ
0
g
θT0, sT0, x
sT0, x0 ds
dθ
T0≤τkδ<tT0
S
tT0, τkδ ckδ
St,0y T0
t
0
St, θf
θ, y θ, y0
, θ
0
g θ, s, y
s, y0 ds
dθ
0≤τk<t
S t, τk
ck
St,0y0 t
0
St, θf
θ, y θ, y0
, s
0
g θ, s, y
s, y0 ds
dθ
0≤τk<t
S t, τk
ck. 4.22
This implies thaty·, y0is aP C-mild solution of Cauchy problem4.2with initial value y0 x0. Thus the uniqueness implies thatx·, x0 y·, y0 x·T0, x0, so thatx·, x0 is aT0-periodic.
Next, we show thatPdefined by4.21is a continuous and compact operator.
Lemma 4.6. Suppose that {Tt, t ≥ 0} is a compact semigroup in X. Then the operatorP is a continuous and compact operator.
Proof. 1Show thatPis a continuous operator onX.
Letx, y ∈ Ξ⊂ X, whereΞis a bounded subset ofX. Suppose thatx·, xandx·, y are theP C-mild solutions of Cauchy problem4.2corresponding to the initial values xand y∈X,respectively, given by
x t, x
St,0x t
0
St, θf
θ, x θ, x
, θ
0
g θ, s, x
s, x ds
dθ
0≤τk<t
S T0, τk
ck;
x t, y
St,0y t
0
St, θf
θ, x θ, y
, θ
0
g θ, s, x
s, y ds
dθ
0≤τk<t
S T0, τk
ck. 4.23
Thus, we obtain x
t, x
≤MT0x
1MgT0
MT0MfT0MT0
0≤τk<T0
ckMT0Mf
t
0
x
θ, xdθ MT0MfMgT0
θ
0
x
s, xds
≤a0MT0Mf
t
0
x
θ, xdθMT0MfMgT0
t
0
xs,x
Bds, x
t, y
≤MT0y
1MgT0
MT0MfT0MT0
0≤τk<T0
ckMT0Mf t
0
x
θ, ydθ MT0MfMgT0
t
0
x
s, yds
≤b0MT0Mf
t
0
x
θ, ydθMT0MfMgT0
t
0
xs,y
Bds,
4.24
wherexs,xBsup0≤ξ≤sxξ, xandxs,yBsup0≤ξ≤sxξ, y.
ByLemma 3.1, one can verify that there exist constantsM∗1andM2∗>0 such that x
t, x≤M∗1, x
t, y≤M∗2. 4.25
Let ρ max{M∗1, M∗2} > 0, then x·, x,x·, y ≤ ρ which imply that they are locally bounded.
By assumptionsH2.1,H2.2,H2.4,H2.5, and2ofLemma 4.3, we obtain x
t, x
−x
t, y≤St,0x−y
t
0
St, θ f
θ, x
θ, x ,
θ
0
g θ, s, x
s, x ds
−f
θ, x θ, y
, θ
0
g θ, s, x
s, y
ds
dθ
≤MT0x−yMT0Lfρ t
0
x θ, x
−x
θ, ydθ MT0LfρLgρT0
t
0
xs,x−xs,y
Bds.
4.26
ByLemma 3.1again, one can verify that there exists constantM∗3>0 such that x
t, x
−x
t, y≤M∗3MT0x−y≡Lx−y, ∀t∈ 0, T0
, 4.27
which implies that P
x
−P
yx T0, x
−x
T0, y≤Lx−y. 4.28 Hence,Pis a continuous operator onX.
2Verify thatPtakes a bounded set into a precompact set inX.
LetΓis a bounded subset ofX. DefineKPΓ {Px∈X|x∈Γ}.
For 0< ε≤T0, define
KεPεΓ S
T0, T0−ε x
T0−ε, x
|x∈Γ
. 4.29
Next, we show thatKεis precompact inX. In fact, forx∈Γfixed, we have x
T0−ε, x
≤S
T0−ε,0 x
T0−ε
0
S
T0−ε, θ f
θ, x
θ, x ,
θ
0
g θ, s, x
s, x
ds
dθ
0≤τk<T0−ε
S
T0−ε, τk ck
≤MT0xMT0MfT0
1MgT0
MT0Mf
T0
0
x
θ, xdθ MT0
0≤τk<T0
ckMT0MfMgT0
T0
0
xs,x
Bds
≤MT0xMT0MfT0
1MgT0
1MgT0
MT0MfT0ρMT0
0≤τk<T0
ck. 4.30 This implies that the set{xT0−ε, x|x∈Γ}is totally bounded.
By virtue of {Tt, t ≥ 0} which is a compact semigroup and 5 of Lemma 2.3, ST0, T0−εis a compact operator. Thus,Kεis precompact inX.
On the other hand, for arbitraryx∈Γ,
Pε x
S T0,0
x T0−ε
0
S T0, θ
f
θ, x θ, x
, θ
0
g θ, s, x
s, x ds
dθ
0≤τk<T0−ε
S T0, τk
ck. 4.31
Thus, having this combined with4.21, we have Pε
x
−P x
≤ T0−ε
0
S T0, θ
f
θ, x θ, x
, θ
0
g θ, s, x
s, x ds
dθ
− T0
0
S T0, θ
f
θ, x θ, x
, θ
0
g θ, s, x
s, x ds
dθ
0≤τk<T0−ε
S T0, τk
ck−
0≤τk<T0
S T0, τk
ck
≤ T0
T0−ε
S
T0, θ f
θ, x
θ, x ,
θ
0
g θ, s, x
s, x
ds
dθMT0
T0−ε≤τk<T0
ck
≤MT0Mf
1MgT01ρεMT0
T0−ε≤τk<T0
ck.
4.32
It is shown that the set K can be approximated to an arbitrary degree of accuracy by a precompact setKε. HenceK itself is a precompact set inX. That is,P takes a bounded set into a precompact set inX. As a result,Pis a compact operator.
In order to use Leray-Schauder fixed pointed theorem to examine whether the operator P has a fixed point, we have to make assumptions H2.2 and H2.5 a little stronger as follows.
H2.2There exists constantNf >0 and 0< λ <1 such that ft, x, y≤Nf
1xλyλ
∀x, y∈X. 4.33
H2.5There exists a positive constantNg>0 and 0< λ <1 such that gt, s, x≤Ng
1xλ
∀x∈X. 4.34
Now, we can give the main results in this paper.
Theorem 4.7. Assumptions [H1], [H2.1], [H2.2], [H2.3], [H2.4], [H2.5], [H2.6], and [H2.7] hold.
Suppose that{Tt, t≥0}is a compact semigroup inX. Then system4.1has aT0-periodicP C-mild solution on0,∞.
Proof. By virtue of{Tt, t≥0}which is a compact semigroup and5ofLemma 2.3,ST0,0 is a compact operator on infinite-dimensional spaceX. Thus,ST0,0/αI,α∈R. Then, there existsβ >0 such thatσST0,0−Ix ≥βxforσ∈0,1. In fact, defineΠσI−σST0,0, σ ∈ 0,1, and Πσ : 0,1→£bX and hσ Πσ : 0,1→R. It is obvious thath ∈ C0,1;R. Thus, there existσ∗∈0,1andβ >0 such that
hσ∗ min hσ|σ∈0,1
≥β >0. 4.35
If not, there exists σ ∈ 0,1such thathσ 0. We can assert thatσ /0 unlesshσ 1.
Thus, forσ∈0,1,
S T0,0
1
σI, where 1
σ ≥1, 4.36
which is a contradiction withST0,0/αI,α∈R.
ByTheorem 4.4, for fixed x ∈ X, Cauchy problem4.2corresponding to the initial value x0 xhas theP C-mild solution x·, x. ByLemma 4.6, the operatorP defined by 4.21, is compact.
According to Leray-Schauder fixed point theory, it suffices to show that the set{x ∈ X|xσP x, σ ∈0,1}is a bounded subset ofX. In fact, letx∈ {x∈X|xσP x, σ ∈0,1}, we have
βx≤σS T0,0
−I x σ
T0
0
S
T0, θ f
θ, x
θ, x ,
θ
0
g θ, s, x
s, x
ds
dθ σ
0≤τk<T0
S
T0, τkck.
4.37
