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
Trang 1Volume 2008, Article ID 401947, 15 pages
doi:10.1155/2008/401947
Research Article
Bounded and Periodic Solutions of Semilinear
Impulsive Periodic System on Banach Spaces
JinRong Wang, 1 X Xiang, 1, 2 W Wei, 2 and Qian Chen 3
1 College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
2 College of Science, Guizhou University, Guiyang, Guizhou 550025, China
3 College of Electronic Science and Information Technology, Guizhou University, Guiyang,
Guizhou 550025, China
Correspondence should be addressed to JinRong Wang,wjr9668@126.com
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
the T0-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 of T0-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
Copyrightq 2008 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
1 7
Although, there are some papers on periodic solution of periodic systems on infinite
Trang 2dimensional spaces with unbounded operator have not been extensively investigated.
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
Herein, we go on studying the semilinear impulsive periodic system
˙xt Axt ft, x, t / τ k ,
on infinite dimensional Banach space X, where 0 τ0< τ1 < τ2 < · · · < τ k· · · , limk→∞ τ k ∞,
τ kδ τ k T0,Δxτ k xτ
k − xτ−
k , k ∈ Z
X to X and is T0-periodic in t, andB kδ B k , c kδ c k This paper is mainly concerned with the existence of periodic solution for semilinear impulsive periodic system on infinite
dimensional Banach space X.
In this paper, we use Horn’s fixed-point theorem to obtain the existence of periodic
evolution operator corresponding to homogeneous linear impulsive system, we construct
some difficulties to show the continuity and compactness of Poincar´e operator P which are
very important By virtue of Gronwall lemma with impulse, the estimate of P C-mild solutions
periodic system when P C-mild solutions are ultimate bounded is shown.
periodic system and properties of impulsive evolution operator corresponding to
-periodic P 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
norm Define D {τ1, , τ δ } ⊂ 0, T0 We introduce PC0, T0; X ≡ {x : 0, T0 → X | x is continuous at t ∈ 0, T0 \ D, x is continuous from left and has right-hand limits at t ∈ D}, and
P C10, T0; X ≡ {x ∈ PC0, T0; X | ˙x ∈ PC0, T0; X} Set
sup
t∈0,T0xt 0, sup
t∈0,T0xt − 0
Trang 3
It can be seen that endowed with the norm·P C·P C1, PC0, T0; XPC10, T0; X is 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 / τ k ,
DA.
H1.2: There exists δ such that τ kδ τ k T0
0, B k∈ £b X and B kδ B k
.
x t Axt, t ∈ 0, T0 \ D,
Δxτ k B k xτ k , k 1, 2, , δ,
x0 x.
2.3
x ∈ P C10, T0; X represented by xt St, 0x, where
S·, · : Δ {t, θ ∈ 0, T0 × 0, T0 | 0 ≤ θ ≤ t ≤ T0} −→ £X, 2.4 given by
St, θ
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Tt − θ, τ k−1 ≤ θ ≤ t ≤ τ k , Tt − τ kI B k Tτ k − θ, τ k−1 ≤ θ < τ k < t ≤ τ k1 , Tt − τ k
θ<τ j <t
I B j Tτ j − τ
j−1
I B i Tτ i − θ,
τ i−1 ≤ θ < τ i ≤ · · · < τ k < t ≤ τ k1
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 {B k ; τ k}∞k1
solution of system2.2
Trang 4Definition 2.2 For every x ∈ X, the function x ∈ P C0, T0; X given by xt St, 0x is said
Definition 2.3 A function x ∈ P C0, ∞; X is said to be a T0-periodic P C-mild solution of
xt T0 xt for t ≥ 0.
The following lemma gives the properties of the impulsive evolution operator
{St, θ, t, θ ∈ Δ} associated with {Tt, t ≥ 0} and {B k ; τ k}∞k1are widely used in this paper
Lemma 2.4 see 24, Lemma 1 Impulsive evolution operator {St, θ, t, θ ∈ Δ} has the
follow-ing properties.
1 For 0 ≤ θ ≤ t ≤ T0, St, θ ∈ £ b X, that is, there exists a constant M T0 > 0 such that
sup
0≤θ≤t≤T 0
2 For 0 ≤ θ < r < t ≤ T0, r / τ k , St, θ St, rSr, θ.
3 For 0 ≤ θ ≤ t ≤ T0and N ∈ Z0, St NT0, θ NT0 St, θ.
4 For 0 ≤ t ≤ T0and N ∈ Z0, SNT0 t, 0 St, 0ST0, 0 N
5 If {Tt, t ≥ 0} is a compact semigroup in X, then St, θ is a compact operator for 0 ≤ θ <
t ≤ T0.
Secondly, we recall nonhomogeneous linear impulsive periodic system
˙xt Axt ft, t / τ k ,
where f ∈ L10, T0; X, ft T0 ft for t ≥ 0 and c kδ c k
˙xt Axt ft, t ∈ 0, T0 \ D,
Δxτ k B k xτ k c k , k 1, 2, , δ,
x0 x,
2.8
solution of system2.7
Definition 2.5 A function x ∈ P C0, T0; X, for finite interval 0, T0, is said to be a PC-mild
f ∈ L10, T0; X if x is given by
xt St, 0x
t
0
St, θfθdθ
0≤τk <t
St, τ kc k 2.9
Definition 2.6 A function x ∈ P C0, ∞; X is said to be a T0-periodic P C-mild solution of
xt T0 xt for t ≥ 0.
Trang 5Here, we note that system2.2 has a T0-periodic P C-mild solution x if and only if
ST0, 0 has a fixed point The impulsive periodic evolution operator {St, θ, t, θ ∈ Δ}
existence of fixed points for an operator equation This implies that we can use the uniform
periodic system on Banach space
3 Semilinear impulsive periodic system
In order to derive the estimate of P C-mild solutions, we collect the following Gronwall’s
lemma with impulse which is widely used in sequel
Lemma 3.1 Let x ∈ PC0, T0; X and satisfy the following inequality:
xt ≤ a b
t
0
xθdθ
0<τ k <t
where a, b, ζ k ≥ 0, are constants Then, the following inequality holds:
xt ≤ a
0<τ k <t
Proof Defining
ut a b
t
0
xθdθ
0<τ k <t
we get
˙ut bxt ≤ but, t / τ k ,
u0 a, uτ k uτ k ζ k xτ k ≤ 1 ζ k uτ k .
3.4
For t ∈ τ k , τ k1, by 3.4, we obtain
ut ≤ uτ ke bt−τ k ≤ 1 ζ k uτ k e bt−τ k, 3.5 further,
ut ≤ a
0<τ k <t
thus,
xt ≤ a
0<τ k <t
Trang 6Now, we consider the following semilinear impulsive periodic system
˙xt Axt ft, x, t / τ k ,
system3.8
˙xt Axt ft, x, t ∈ 0, T0 \ D,
Δxτ k B k xτ k c k , k 1, 2, , δ,
x0 x.
3.9
Definition 3.2 A function x ∈ P C0, T0; X is said to be a PC-mild solution of the Cauchy
equation:
xt St, 0x
t
0
St, θfθ, xθdθ
0≤τk <t
St, τ kc k 3.10
Remark 3.3 Since one of the main difference of system 3.9 and other ODEs is the middle
k , θ I
B k Sτ k , θ, for 0 ≤ θ < τ k , k 1, 2, , δ, that
xτ k Sτ
k , 0x
τ
k
0
Sτ k, θfθ, xθdθ
0≤τk <τ k
Sτ k, τ kc k
I B k
Sτ k , 0x
τ k
0
Sτ k , θfθ, xθdθ
0≤τk−1 <τ k Sτ k , τ k−1 c k
c k
I B k xτ k c k
3.11
It shows thatΔxτ k B k xτ k c k , k 1, 2, , δ.
H2.1: f : 0, ∞ × X → X is measurable for t ≥ 0 and for any x, y ∈ X satisfying x, y ≤
ρ, there exists a positive constant L f ρ > 0 such that
H2.2: There exists a positive constant M f > 0 such that
Trang 7H2.3: ft, x is T0-periodic in t, that is, ft T0, x ft, x, t ≥ 0.
0 and c k ∈ X, there exists δ ∈ N such that c kδ c k
Now, we state the following result which asserts the existence of P C-mild solution
proof here
Theorem 3.4 Assumptions [H1.1], [H2.1], and [H2.2] hold, and for each k ∈ Z
0, B k ∈ £ b X,
c k ∈ X be fixed Let x ∈ X be fixed Then Cauchy problem 3.9 has a unique PC-mild solution given
by
xt, x St, 0x
t
0
St, θfθ, xθ, xdθ
0≤τk <t
St, τ kc k 3.14
Further, suppose x ∈ Ξ ⊂ X, Ξ is a bounded subset of X, then there exits a constant M∗> 0 such that
Proof Under the assumptions H1.1, H2.1, and H2.2, using the similar method of
.
x t Axt ft, x, t ∈ s, τ,
has a unique mild solution
xt Ttx
t
s
In general, for t ∈ τ k , τ k1, Cauchy problem
.
x t Axt ft, x, t ∈ τ k , τ k1 ,
has a unique P C-mild solution
xt Tt − τ k x k
t
τ k Tt − θfθ, xθdθ. 3.19
Combining all solutions onτk , τ k1 k 1, , δ, one can obtain the PC-mild solution
xt, x St, 0x
t
0
St, θfθ, xθ, xdθ
0≤τ<t
St, τ kc k 3.20
Trang 8Further, by assumptionH2.2 and 1 ofLemma 2.4, we obtain
xt, x ≤
M T0 x M T0 M f T0 M T0
0≤τk <T0
c k
M T0
t
0
xt, x ≤
M T0 x M T0 M f T0 M T0
0≤τk <T0
c k
e M T0 T0 ≡ M∗, ∀ t ∈ 0, T0. 3.22
Definition 3.5 A function x ∈ P C0, ∞; X is said to be a T0-periodic P C-mild solution of
xt T0 xt for t ≥ 0.
In order to study the periodic solutions of the system3.8, we construct a new Poincar´e operator from X to X as follows:
P x xT0, x ST0, 0x
T0
0
ST0, θfθ, xθ, xdθ
0≤τk <T0
ST0, τ kc k , 3.23
initial value x0 x.
We can note that a fixed point of P gives rise to a periodic solution as follows.
Lemma 3.6 System 3.8 has a T0-periodic P C-mild solution if and only if P has a fixed point Proof Suppose x· x· T0, then x0 xT0 Px0 This implies that x0 is a
x·, x0 of Cauchy problem 3.9 corresponding to the initial value x0 x0, we can define
y· x· T0, x0, then y0 xT0, x0 Px0 x0 Now, for t > 0, we can use 2, 3, and
yt xt T0, x0
St T0, T0ST0, 0x0
T0
0
St T0, T0ST0, θfθ, xθ, x0dθ
0≤τk <T0
St T0, T0ST0, τ kc k
tT0
T0
St T0, θfθ, xθ, x0dθ
T0 ≤τ kδ <tT0
St T0, τ kδ c kδ
St, 0
ST0, 0x0
T0
0
ST0, θfθ, xθ, x0dθ
0≤τk <T0
ST0, τ kc k
t
0
St T0, s T0fs T0, xs T0, x0ds
0≤τk <t
St, τ kc k
St, 0y0
t
0
St, sfs, ys, y0ds
0≤τ<t
St, τ kc k
3.24
Trang 9This implies that y·, y0 is a P C-mild solution of Cauchy problem 3.9 with initial value
y0 x0 Thus, the uniqueness implies that x·, x0 y·, y0 x· T0, x0 so that x·, x0
is a T0-periodic
Next, we show that the operator P is continuous.
Lemma 3.7 Assumptions [H1.1], [H2.1], and [H2.2] hold Then, operator P is a continuous operator
of x on X.
Proof Let x, y ∈ Ξ ⊂ X, where Ξ is a bounded subset of X Suppose x·, x and x·, y are the
P C-mild solutions of Cauchy problem 3.9 corresponding to the initial value x and y ∈ X,
respectively, given by
xt, x St, 0x
t
0
St, θfθ, xθ, xdθ
0≤τk <t
ST0, τ kc k;
xt, y St, 0y
t
0
St, θfθ, xθ, ydθ
0≤τk <t
ST0, τ kc k
3.25
xt, x ≤
M T0 x M T0 M f T0 M T0
0≤τk <T0
c k
M T0
t
0
xθ, xdθ;
xt, y ≤
M T0 y M T0 M f T0 M T0
0≤τk <T0
c k
M T0
t
0
xθ, ydθ.
3.26
ByLemma 3.1, one can verify that there exist constants M∗1and M∗2> 0 such that
xt, x ≤ M∗
Lemma 2.4, we obtain
xt, x − xt, y ≤ St, 0x − y
t
0
St, θfθ, xθ, x − fθ, xθ, ydθ
≤ M T0 x − y M T0 L f ρ
t
0
xθ, x − xθ, ydθ.
3.28
ByLemma 3.1again, one can verify that there exists a constant M > 0 such that
which implies that
Hence, P is a continuous operator of x on X.
Trang 10In the sequel, we need to prove the compactness of operator P , so we assume the
following
Lemma 3.8 Assumptions [H1.1], [H2.1], [H2.2], and [H3] hold Then, the operator P is a compact
operator.
Proof We only need to verify that P takes a bounded set into a precompact set on X Let Γ
K ε P ε Γ ST0, T0− ε{xT0− ε, x | x ∈ Γ}.
xT0− ε, x
ST0− ε, 0x
T0 −ε
0
ST0− ε, θfθ, xθ, xdθ
0≤τk <T0 −ε
ST0− ε, τ
k c k
≤ M T0 x M T0 M f T0
T0
0
0≤τk <T0
c k
≤ M T0 x M T0 M f T0 T0ρ M T0
δ k1
c k .
3.31 This implies that the set{xT0− ε, x | x ∈ Γ} is bounded.
P ε x ST0, 0x
T0 −ε
0
ST0, θfθ, xθ, xdθ
0≤τk <T0 −ε
ST0, τ kc k , 3.32
P ε x − Px ≤
T0 −ε
0
ST0, θfθ, xθdθ −
T0
0
ST0, θfθ, xθdθ
0≤τ
k <T0 −ε
ST0, τ kc k−
0≤τk <T0
ST0, τ kc k
≤ T0
T0 −ε ST0, θfθ, xθdθ M T0
T0 −ε≤τ k <T0
c k
≤ 2M T0 M f 1 ρε M T0
T0 −ε≤τ k <T0
c k .
3.33
It is showing that the set K can be approximated to an arbitrary degree of accuracy by a
a precompact set on X As a result, P is a compact operator.
Trang 11After showing the continuity and compactness of operator P , we can follow and derive periodic P C-mild solutions for system 3.8 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 periodic P C-mild solutions here, so we only define the local boundedness and ultimate
boundedness
Definition 3.9 P C-mild solutions of Cauchy problem 3.9 are said to be bounded if for each
B1> 0, there is a B2 > 0 such that x ≤ B1impliesxt, x ≤ B2for t ≥ 0.
Definition 3.10 P C-mild solutions of Cauchy problem 3.9 are said to be locally bounded if
for each B1 > 0 and k0 > 0, there is a B2 > 0 such that x ≤ B1 impliesxt, x ≤ B2 for
0≤ t ≤ k0
Definition 3.11 P C-mild solutions of Cauchy problem 3.9 are said to be ultimate bounded
if there is a bound B > 0, such for each B3 > 0, there is a k > 0 such that x ≤ B3and t ≥ k
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 3.13 see 10, Lemma 3.1, Horn’s fixed point theorem Let E0 ⊂ E1 ⊂ E2 be convex subsets of Banach space X, with E0and E2compact subsets and E1open relative to E2 Let P : E2→ X
be a continuous map such that for some integer m, one has
P j E1 ⊂ E2, 1≤ j ≤ m − 1,
then P has a fixed point in E0.
With these preparations, we can prove our main result in this paper
Theorem 3.14 Let assumptions [H1], [H2], and [H3] hold If the PC-mild solutions of Cauchy
problem3.9 are ultimate bounded, then system 3.8 has a T0-periodic P C-mild solution.
Proof ByTheorem 3.4andDefinition 3.10, Cauchy problem3.9 corresponding to the initial
value x0 x has a P C-mild solution x·, x which is locally bound From ultimate
impliesxt, x ≤ B1 for t ≥ 0 Furthermore, there is a B2 > B1 such thatx ≤ B1 implies
such thatx ≤ B1impliesxt, x ≤ B for t ≥ m − 2T0
Define y·, y0 x· T0, x, then y0 xT0, x P x From 3.24 inLemma 3.6,
the following:
... class="text_page_counter">Trang 5Here, we note that system 2.2 has a T0 -periodic P C-mild solution x if and only if
ST0,... <t
Trang 6Now, we consider the following semilinear impulsive periodic system< /p>
˙xt Axt... k 3.20
Trang 8Further, by assumptionH2.2 and 1 ofLemma 2.4, we obtain
xt,