Volume 2008, Article ID 401947,15pages doi:10.1155/2008/401947
Research Article
Bounded and Periodic Solutions of Semilinear Impulsive Periodic System on Banach Spaces
JinRong Wang,1X. Xiang,1, 2W. Wei,2and Qian Chen3
1College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
2College of Science, Guizhou University, Guiyang, Guizhou 550025, China
3College of Electronic Science and Information Technology, Guizhou University, Guiyang, Guizhou 550025, China
Correspondence should be addressed to JinRong Wang,[email protected] Received 20 February 2008; Revised 6 April 2008; Accepted 7 July 2008 Recommended by Jean Mawhin
A class of semilinear impulsive periodic system on Banach spaces is considered. First, we introduce theT0-periodic PC-mild solution of semilinear impulsive periodic system. By virtue of Gronwall lemma with impulse, the estimate on the PC-mild solutions is derived. The continuity and compactness of the new constructed Poincar´e operator determined by impulsive evolution operator corresponding to homogenous linear impulsive periodic system are shown. This allows us to apply Horn’s fixed-point theorem to prove the existence ofT0-periodic PC-mild solutions when PC-mild solutions are ultimate bounded. This extends the study on periodic solutions of periodic system without impulse to periodic system with impulse on general Banach spaces. 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. There are many results, such as existence, the relationship between bounded solutions and periodic solutions, stability, food limited, and robustness, about impulsive periodic system on finite dimensional spacessee 1–7.
Although, there are some papers on periodic solution of periodic systems on infinite dimensional spaces see8–13and some results about the impulsive systems on infinite dimensional spaces see 14–18. Particulary, Professor Jean Mawhin investigated the periodic solutions of all kinds of systems oninfinite dimensional spaces extensivelysee 2, 19–23. However, to our knowledge, nonlinear impulsive periodic systems on infinite
dimensional spaces with unbounded operator have not been extensively investigated.
There are only few works done by us about the impulsive periodic systemwith unbounded operatoron infinite dimensional spaces see24–27. We have been established periodic solution theory under the existence of a bounded solution for the linear impulsive periodic system on infinite dimensional spaces. Several criteria were obtained to ensure the existence, uniqueness, global asymptotical stability, alternative theorem, Massera’s theorem, and Robustness of aT0-periodicP C-mild solution for the linear impulsive periodic system.
Herein, we go on studying the semilinear impulsive periodic system xt ˙ Axt ft, x, t /τk,
Δxt Bkxt ck, tτk, 1.1 on infinite dimensional Banach spaceX, where 0τ0< τ1 < τ2 <· · ·< τk· · ·, limk→∞τk∞, τkδ τkT0,Δxτk xτk−xτk−,k ∈ Z0,T0 is a fixed positive number and δ ∈ N denoted the number of impulsive points between 0 andT0. The operatorAis the infinitesimal generator of aC0-semigroup {Tt, t ≥ 0} onX,f is a measurable function from0,∞× X to X and is T0-periodic in t, andBkδ Bk, ckδ ck. This paper is mainly concerned with the existence of periodic solution for semilinear impulsive periodic system on infinite dimensional Banach spaceX.
In this paper, we use Horn’s fixed-point theorem to obtain the existence of periodic solution for semilinear impulsive periodic system 1.1. First, by virtue of impulsive evolution operator corresponding to homogeneous linear impulsive system, we construct a new Poincar´e operatorPfor semilinear impulsive periodic system1.1, then we overcome some difficulties to show the continuity and compactness of Poincar´e operatorP which are very important. By virtue of Gronwall lemma with impulse, the estimate ofP C-mild solutions is given. Therefore, the existence ofT0-periodicP C-mild solutions for semilinear impulsive periodic system whenP C-mild solutions are ultimate bounded is shown.
This paper is organized as follows. In Section 2, some results of linear impulsive periodic system and properties of impulsive evolution operator corresponding to homoge- neous linear impulsive periodic system are recalled. InSection 3, the Gronwall’s lemma with impulse is collected and theT0-periodicP C-mild solution of semilinear impulsive periodic system 1.1 is introduced. The new Poincar´e operator P is constructed and the relation betweenT0-periodicP C-mild solution and the fixed point of Poincar´e operatorP is given.
After the continuity and compactness of Poincar´e operatorP are shown, the existence ofT0- periodicP C-mild solutions for semilinear impulsive periodic system is established by virtue of Horn’s fixed-point theorem when P C-mild solutions are ultimate bounded. At last, an example is given to demonstrate the applicability of our result.
2. Linear impulsive periodic system
LetX be 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. We introduceP C0, T0;X≡ {x:0, T0→ X |xis 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,T0xt0, sup
t∈0,T0xt−0
, xP C1 xP 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.
In order to study the semilinear impulsive periodic system, we first recall linear impulse periodic system here.
Firstly, we recall homogeneous linear impulsive periodic system x. t Axt, t /τk,
Δxt Bkxt, tτk. 2.2 We introduce the following assumptionH1.
H1.1:Ais the infinitesimal generator of aC0-semigroup{Tt, t≥0}onXwith domain DA.
H1.2: There existsδsuch thatτkδτkT0. H1.3: For 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, using28, Theorem 5.2.2, page 144, step by step, one can verify that the Cauchy problem2.3has a unique classical solution x∈P C10, T0;Xrepresented byxt St,0x, where
S·,·:Δ {t, θ∈0, T0×0, T0|0≤θ≤t≤T0} −→£X, 2.4 given by
St, θ
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
Tt−θ, τk−1≤θ≤t≤τk, Tt−τkIBkTτk−θ, τk−1≤θ < τk< t≤τk1, Tt−τk
θ<τj<t
IBjTτj−τj−1
IBiTτi−θ,
τi−1≤θ < τi≤ · · ·< τk< t≤τk1.
2.5
Definition 2.1. The operator {St, θ,t, θ ∈ Δ} given by 2.5 is called the impulsive evolution operator associated with{Tt, t≥0}and{Bk;τk}∞k1.
We introduce theP C-mild solution of Cauchy problem2.3andT0-periodicP C-mild solution of system2.2.
Definition 2.2. 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 2.3. 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{Tt, t≥0}and{Bk;τk}∞k1are widely used in this paper.
Lemma 2.4see24, Lemma 1. Impulsive evolution operator{St, θ,t, θ∈Δ}has the follow- ing properties.
1For 0≤θ≤t≤T0,St, θ∈£bX, that is, there exists a constantMT0 >0 such that sup
0≤θ≤t≤T0
St, θ ≤MT0. 2.6
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.
Secondly, we recall nonhomogeneous linear impulsive periodic system xt ˙ Axt ft, t /τk,
Δxt Bkxt ck, tτk, 2.7 wheref∈L10, T0;X,ftT0 ftfort≥0 andckδck.
In order to study system2.7, we need to consider the associated Cauchy problem xt ˙ Axt ft, t∈0, T0\D,
Δxτk Bkxτk ck, k1,2, . . . , δ, x0 x,
2.8
and introduce the P C-mild solution of Cauchy problem 2.8 and T0-periodic P C-mild solution of system2.7.
Definition 2.5. A functionx∈P C0, T0;X, for finite interval0, T0, is said to be aP C-mild solution of the Cauchy problem 2.8 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
St, τkck. 2.9
Definition 2.6. A functionx∈P C0,∞;Xis said to be aT0-periodicP C-mild solution of system2.7if it is aP C-mild solution of Cauchy problem2.8corresponding to somexand xtT0 xtfort≥0.
Here, we note that system2.2has a T0-periodicP C-mild solution xif and only if ST0,0 has a fixed point. The impulsive periodic evolution operator {St, θ,t, θ ∈ Δ}
can be used to reduce the existence ofT0-periodicP C-mild solutions for system2.7to the existence of fixed points for an operator equation. This implies that we can use the uniform framework in 8, 13 to study the existence of periodic P C-mild solutions for impulsive periodic system on Banach space.
3. Semilinear impulsive periodic system
In order to derive the estimate ofP C-mild solutions, we collect the following Gronwall’s lemma with impulse which is widely used in sequel.
Lemma 3.1. Letx∈P C0, T0;Xand satisfy the following inequality:
xt ≤ab t
0
xθdθ
0<τk<t
ζkxτk, 3.1
wherea, b, ζk≥0, are constants. Then, the following inequality holds:
xt ≤a
0<τk<t
1ζkebt. 3.2
Proof. Defining
ut ab t
0
xθdθ
0<τk<t
ζkxτk, 3.3
we get
ut ˙ bxt ≤but, t /τk, u0 a,
uτk uτk ζkxτk ≤1ζkuτk.
3.4
Fort∈τk, τk1, by3.4, we obtain
ut≤uτkebt−τk ≤1ζkuτkebt−τk, 3.5 further,
ut≤a
0<τk<t
1ζkebt, 3.6
thus,
xt ≤a
0<τk<t
1ζkebt. 3.7
For more details the reader can refer to5, Lemma 1.7.1.
Now, we consider the following semilinear impulsive periodic system xt ˙ Axt ft, x, t /τk,
Δxt Bkxt ck, tτk. 3.8 and introduce a suitable Poincar´e operator and study theT0-periodicP C-mild solutions of system3.8.
In order to study the system3.8, we first consider the associated Cauchy problem xt ˙ Axt ft, x, t∈0, T0\D,
Δxτk Bkxτk ck, k1,2, . . . , δ, x0 x.
3.9
Now, we can introduce theP C-mild solution of the Cauchy problem3.9.
Definition 3.2. A functionx ∈ P C0, T0;Xis said to be aP C-mild solution of the Cauchy problem3.9corresponding to the initial valuex ∈ X if xsatisfies the following integral equation:
xt St,0x t
0
St, θfθ, xθdθ
0≤τk<t
St, τkck. 3.10 Remark 3.3. Since one of the main difference of system3.9and other ODEs is the middle
“jumping condition,” we need verify that the P C-mild solution defined by 3.10 satisfies the middle “jumping condition” in3.9. In fact, it comes from 3.10and Sτk, θ I BkSτk, θ,for 0≤θ < τk,k1,2, . . . , δ,that
xτk Sτk,0x τ
k
0
Sτk, θfθ, xθdθ
0≤τk<τk
Sτk, τkck
IBk
Sτk,0x τk
0
Sτk, θfθ, xθdθ
0≤τk−1<τk
Sτk, τk−1 ck
ck
IBkxτk ck.
3.11
It shows thatΔxτk Bkxτk ck, k1,2, . . . , δ.
In order to show the existence of theP C-mild solution of Cauchy problem3.9and T0-periodicP C-mild solutions for system3.8, we introduce assumptionH2.
H2.1:f :0,∞×X →Xis measurable fort≥0 and for anyx, y∈Xsatisfyingx,y ≤ ρ, there exists a positive constantLfρ>0 such that
ft, x−ft, y ≤Lfρx−y. 3.12 H2.2: There exists a positive constantMf >0 such that
ft, x ≤Mf1x ∀x∈X. 3.13
H2.3:ft, xisT0-periodic int, that is,ftT0, x ft, x, t≥0.
H2.4: For eachk∈Z0 andck∈X, there existsδ∈Nsuch thatckδck.
Now, we state the following result which asserts the existence of P C-mild solution for Cauchy problem3.9and gives the estimate ofP C-mild solutions for Cauchy problem 3.9by virtue ofLemma 3.1. A similar result for a class of generalized nonlinear impulsive integral differential equations is given by Xiang and Wei in17. Thus, we only sketch the proof here.
Theorem 3.4. Assumptions [H1.1], [H2.1], and [H2.2] hold, and for eachk ∈ Z0,Bk ∈ £bX, ck∈Xbe fixed. Letx∈Xbe fixed. Then Cauchy problem3.9has a uniqueP C-mild solution given by
xt, x St,0x t
0
St, θfθ, xθ, xdθ
0≤τk<t
St, τkck. 3.14
Further, supposex∈Ξ⊂X,Ξis a bounded subset ofX, then there exits a constantM∗>0 such that xt, x ≤M∗ ∀t∈0, T0. 3.15 Proof. Under the assumptions H1.1, H2.1, and H2.2, using the similar method of 28, Theorem 5.3.3, page 169, Cauchy problem
x. t Axt ft, x, t∈s, τ,
xs x∈X, 3.16
has a unique mild solution
xt Ttx
t
s
Tt−θfθ, xθdθ. 3.17
In general, fort∈τk, τk1, Cauchy problem
x. t Axt ft, x, t∈τk, τk1,
xτk xk≡IBkxτk ck∈X 3.18
has a uniqueP C-mild solution
xt Tt−τkxk t
τk
Tt−θfθ, xθdθ. 3.19 Combining all solutions onτk, τk1 k1, . . . , δ, one can obtain theP C-mild solution of the Cauchy problem3.9given by
xt, x St,0x t
0
St, θfθ, xθ, xdθ
0≤τk<t
St, τkck. 3.20
Further, by assumptionH2.2and1ofLemma 2.4, we obtain xt, x ≤
MT0xMT0MfT0MT0
0≤τk<T0
ck
MT0
t
0
xθ, xdθ. 3.21 Sincex∈Ξ⊂X,Ξis a bounded subset ofX, usingLemma 3.1, one can obtain
xt, x ≤
MT0xMT0MfT0MT0
0≤τk<T0
ck
eMT0T0≡M∗, ∀t∈0, T0. 3.22
Now, we introduce theT0-periodicP C-mild solution of system3.8.
Definition 3.5. A functionx∈P C0,∞;Xis said to be aT0-periodicP C-mild solution of system3.8if it is aP C-mild solution of Cauchy problem3.9corresponding to somexand xtT0 xtfort≥0.
In order to study the periodic solutions of the system3.8, we construct a new Poincar´e operator fromXtoXas follows:
Px xT0, x ST0,0x T0
0
ST0, θfθ, xθ, xdθ
0≤τk<T0
ST0, τkck, 3.23 wherex·, xdenote theP C-mild solution of the Cauchy problem3.9corresponding to the initial valuex0 x.
We can note that a fixed point ofPgives rise to a periodic solution as follows.
Lemma 3.6. System3.8has 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 problem3.9corresponding 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 2.4and assumptionsH1.2,H1.3,H2.3,H2.4to obtain
yt xtT0, x0
StT0, T0ST0,0x0 T0
0
StT0, T0ST0, θfθ, xθ, x0dθ
0≤τk<T0
StT0, T0ST0, τkck tT0
T0
StT0, θfθ, xθ, x0dθ
T0≤τkδ<tT0
StT0, τkδ ckδ
St,0
ST0,0x0 T0
0
ST0, θfθ, xθ, x0dθ
0≤τk<T0
ST0, τkck
t
0
StT0, sT0fsT0, xsT0, x0ds
0≤τk<t
St, τkck
St,0y0 t
0
St, sfs, ys, y0ds
0≤τk<t
St, τkck.
3.24
This implies thaty·, y0is aP C-mild solution of Cauchy problem3.9with initial value y0 x0. Thus, the uniqueness implies thatx·, x0 y·, y0 x·T0, x0so thatx·, x0 is aT0-periodic.
Next, we show that the operatorPis continuous.
Lemma 3.7. Assumptions [H1.1], [H2.1], and [H2.2] hold. Then, operatorPis a continuous operator ofxonX.
Proof. Letx, y∈Ξ⊂X, whereΞis a bounded subset ofX. Supposex·, xandx·, yare the P C-mild solutions of Cauchy problem3.9corresponding to the initial valuexandy ∈X, respectively, given by
xt, x St,0x t
0
St, θfθ, xθ, xdθ
0≤τk<t
ST0, τkck;
xt, y St,0y t
0
St, θfθ, xθ, ydθ
0≤τk<t
ST0, τkck.
3.25
Thus, by assumptionH2.2and1ofLemma 2.4, we obtain xt, x ≤
MT0xMT0MfT0MT0
0≤τk<T0
ck
MT0
t
0
xθ, xdθ;
xt, y ≤
MT0yMT0MfT0MT0
0≤τk<T0
ck
MT0
t
0
xθ, ydθ.
3.26
ByLemma 3.1, one can verify that there exist constantsM∗1andM∗2>0 such that
xt, x ≤M∗1, xt, y ≤M∗2. 3.27 Let ρ max{M1∗, M2∗} > 0, then x·, x,x·, y ≤ ρ. By assumptionH2.1 and 1 of Lemma 2.4, we obtain
xt, x−xt, y ≤ St,0x−y t
0
St, θfθ, xθ, x−fθ, xθ, ydθ
≤MT0x−yMT0Lfρ t
0
xθ, x−xθ, ydθ.
3.28
ByLemma 3.1again, one can verify that there exists a constantM >0 such that
xt, x−xt, y ≤MMT0x−y ≡Lx−y, ∀t∈0, T0, 3.29 which implies that
Px−PyxT0, x−xT0, y ≤Lx−y. 3.30
Hence,Pis a continuous operator ofxonX.
In the sequel, we need to prove the compactness of operator P, so we assume the following.
AssumptionH3: The semigroup{Tt, t≥0}is compact onX.
Now, we are ready to prove the compactness of operatorPdefined by3.23.
Lemma 3.8. Assumptions [H1.1], [H2.1], [H2.2], and [H3] hold. Then, the operatorPis a compact operator.
Proof. We only need to verify thatP takes a bounded set into a precompact set onX. LetΓ is a bounded subset ofX. DefineK PΓ {Px ∈ X | x ∈Γ}. For 0 < ε < t≤ T0, define KεPεΓ ST0, T0−ε{xT0−ε, x|x∈Γ}.
Next, we show thatKεis precompact onX. In fact, forx∈Γfixed, we have xT0−ε, x
ST0−ε,0x T0−ε
0
ST0−ε, θfθ, xθ, xdθ
0≤τk<T0−ε
ST0−ε, τkck
≤MT0xMT0MfT0 T0
0
xθ, xdθMT0
0≤τk<T0
ck
≤MT0xMT0MfT0T0ρMT0
δ k1
ck.
3.31 This implies that the set{xT0−ε, x|x∈Γ}is bounded.
By assumptionH3and5ofLemma 2.4,ST0, T0−εis a compact operator. Thus, Kεis precompact onX.
On the other hand, for arbitraryx∈Γ, Pεx ST0,0x
T0−ε
0
ST0, θfθ, xθ, xdθ
0≤τk<T0−ε
ST0, τkck, 3.32
thus, combined with3.23, we have Pεx−Px ≤
T0−ε
0
ST0, θfθ, xθdθ− T0
0
ST0, θfθ, xθdθ
0≤τk<T0−ε
ST0, τkck−
0≤τk<T0
ST0, τkck
≤ T0
T0−εST0, θfθ, xθdθMT0
T0−ε≤τk<T0
ck
≤2MT0Mf1ρεMT0
T0−ε≤τk<T0
ck.
3.33
It is showing that the setK can be approximated to an arbitrary degree of accuracy by a precompact setKε. Hence,Kitself is precompact set onX. That is,Ptakes a bounded set into a precompact set onX. As a result,Pis a compact operator.
After showing the continuity and compactness of operatorP, we can follow and derive periodicP C-mild solutions for system3.8. In the sequel, we define the following definitions.
The following definitions are standard, we state them here for convenient references. Note that the uniform boundedness and uniform ultimate boundedness are not required to obtain the periodicP C-mild solutions here, so we only define thelocalboundedness and ultimate boundedness.
Definition 3.9. P C-mild solutions of Cauchy problem3.9are said to be bounded if for each B1>0, there is aB2 >0 such thatx ≤B1impliesxt, x ≤B2fort≥0.
Definition 3.10. P C-mild solutions of Cauchy problem3.9are said to be locally bounded if for eachB1 > 0 andk0 > 0, there is aB2 > 0 such thatx ≤ B1 impliesxt, x ≤ B2 for 0≤t≤k0.
Definition 3.11. P C-mild solutions of Cauchy problem3.9are said to be ultimate bounded if there is a boundB >0, such for eachB3 > 0, there is ak >0 such thatx ≤ B3andt ≥k implyxt, x ≤B.
We also need the following results as a reference.
Lemma 3.12 see 11, Theorem 3.1. Local boundedness and ultimate boundedness implies boundedness and ultimate boundedness.
Lemma 3.13see10, Lemma 3.1, Horn’s fixed point theorem. LetE0 ⊂ E1 ⊂ E2 be convex subsets of Banach spaceX, withE0andE2compact subsets andE1open relative toE2. LetP:E2→X be a continuous map such that for some integerm, one has
PjE1⊂E2, 1≤j ≤m−1,
PjE1⊂E0, m≤j ≤2m−1, 3.34
thenPhas a fixed point inE0.
With these preparations, we can prove our main result in this paper.
Theorem 3.14. Let assumptions [H1], [H2], and [H3] hold. If the P C-mild solutions of Cauchy problem3.9are ultimate bounded, then system3.8has aT0-periodicP C-mild solution.
Proof. ByTheorem 3.4andDefinition 3.10, Cauchy problem3.9corresponding to the initial value x0 x has a P C-mild solution x·, x which is locally bound. From ultimate boundedness andLemma 3.12,x·, xis bound. Next, letB >0 be the bound in the definition of ultimate boundedness. Then, by boundedness, there is a B1 > B such that x ≤ B impliesxt, x ≤ B1 fort ≥ 0. Furthermore, there is aB2 > B1 such thatx ≤ B1 implies xt, x ≤B2fort≥0. Now, using ultimate boundedness again, there is a positive integerm such thatx ≤B1impliesxt, x ≤Bfort≥m−2T0.
Definey·, y0 x·T0, x, theny0 xT0, x Px.From3.24inLemma 3.6, we obtainPy0 yT0, y0 x2T0, x.Thus,P2x PPx Py0 x2T0, x.
Suppose there exists integerm−1 such thatPm−1x xm−1T0, x.By induction, we get the following:
Pmx Pm−1Px Pm−1y0 ym−1T0, y0 xmT0, x. 3.35
Thus, we obtain
Pj−1xxj−1T0, x< B2, j1,2, . . . , m−1, x< B1; Pj−1xxj−1T0, x< B, j≥m, x< B1.
3.36
It comes fromLemma 3.8thatPx xT0, xonXis compact. Now let H{x∈X :x< B2}, E2cl.cov.PH, W{x∈X :x< B1}, E1W∩E2,
G{x∈X :x< B}, E0 cl.cov.PG,
3.37
where cov.Yis the convex hull of the set Y defined by cov.Y {n
i1λiyi | n ≥ 1, yi ∈ Y, λi≥0,n
i1λi1},and cl. denotes the closure. Then, we see thatE0 ⊂E1 ⊂E2are convex subset ofXwithE0,E2compact subsets, andE1open relative toE2, and from3.36, one has
PjE1⊂PjW P Pj−1W⊂PH⊂E2, j1,2, . . . , m−1;
PjE1⊂PjW P Pj−1W⊂PG⊂E0, jm, m1, . . . ,2m−1.
3.38
We see that P : E2 → X is a continuous map continuous fromLemma 3.7. Consequently, from Horn’s fixed-point theorem, we know that the operatorP has a fixed pointx0 ∈ E0 ⊂ X. By Lemma 3.6, we know that the P C-mild solution x·, x0 of Cauchy problem 3.9, corresponding to the initial valuex0 x0, is just T0-periodic. Therefore, x·, x0is a T0- periodicP C-mild solution of system3.8. This proves the theorem.
4. Application
In this section, an example is given to illustrate our theory. Consider the following boundary value problem
∂
∂txt, y Δxt, y
x2t, y 1sint, y, y∈Ω, t /τi, i1,2,3,5,6,7, . . . ,
Δxτi, y
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
0.05Ixτi, y, i1,
−0.05Ixτi, y, i2, 0.05Ixτi, y, i3,
y∈Ω, τi i
2π, i1,2,3,5,6,7, . . . ,
xt, y 0, y∈∂Ω, t >0,
4.1
and the associated initial-boundary value problem
∂
∂txt, y Δxt, y
x2t, y 1sint, y, y∈Ω, t∈0,2π\ 1
2π, π,3 2π
,
Δxτi, y
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
0.05Ixτi, y, i1,
−0.05Ixτi, y, i2, 0.05Ixτi, y, i3,
y∈Ω, τi i
2π, i1,2,3,
xt, y 0, y∈∂Ω, t >0, x0, y x2π, y,
4.2
whereΩ⊂R3is bounded domain and∂Ω∈C3.
DefineXL2Ω,DA H2Ω∩H01Ω, andAx−∂2x/∂y12∂2x/∂y22∂2x/∂y32 forx∈DA. Then,Agenerates a compact semigroup{Tt, t≥0}. Definex·y x·, y, sin·y sin·, y,f·, x·y
x2·, y 1sin·, y, and
Bi
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0.05I, i3m−2,
−0.05I, i3m−1, 0.05I, i3m,
i, m∈N, 4.3
andτi im−1/2π, i, m∈N.
Thus, problem4.1can be rewritten as
xt ˙ Axt ft, x, t /τi, i1,2,3,5,6,7, . . . ,
Δxt Bixt, tτi, i1,2,3,5,6,7, . . . , 4.4 and problem4.2can be rewritten as
xt ˙ Axt ft, x, t∈0,2π\ 1
2π, π,3 2π
, Δx
i 2π
Bix
i 2π
, i1,2,3, x0 x2π.
4.5
If the P C-mild solutions of Cauchy problem 4.5 are ultimate bounded, then all the assumptions inTheorem 3.14are met, our results can be used to system4.4. That is, problem 4.1has a 2π-periodicP C-mild solutionx2π·, y∈P C2π0∞;L2Ω, where
P C2π0,∞;L2Ω≡ {x∈P C0,∞;L2Ω|xt xt2π, t≥0}. 4.6 Acknowledgments
This work is supported by National Natural Science foundation of Chinano. 10661044and Guizhou Province Foundno. 2008008. This work is partially supported by undergraduate carve out project of department of Guiyang City Science and Technology.
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