Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 939062, 19 pages
doi:10.1155/2008/939062
Research Article
Existence of Periodic Solutions for
Integrodifferential Impulsive Periodic
System on Banach Space
JinRong Wang,1 X. Xiang,2 and W. Wei2
1
2
College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
College of Science, Guizhou University, Guiyang, Guizhou 550025, China
Correspondence should be addressed to JinRong Wang, wjr9668@126.com
Received 5 July 2008; Accepted 28 October 2008
Recommended by Agacik Zafer
This paper deals with a class of integrodifferential impulsive periodic systems on Banach space.
Using impulsive periodic evolution operator given by us, the T0 -periodic PC-mild solution
is introduced and suitable Poincaré operator is constructed. By virtue of the generalized new
Gronwall lemma with impulse and B-norm, the estimate on the PC-mild solutions is derived.
Showing the continuity and compactness of the Poincaré operator, we utilize Horn’s fixed point
theorem to prove the existence of T0 -periodic PC-mild solutions when the PC-mild solutions are
bounded and ultimate bounded. This extends the study of periodic solutions of integrodifferential
periodic system without impulse to integrodifferential periodic system with impulse on general
Banach spaces. At last, an example is given for demonstration.
Copyright q 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. 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 results such as the
existence and stability of periodic solution, the relationship between bounded solution and
periodic solution, robustness by perturbation are obtained see 1–4.
Since the end of the last century, many authors including us pay great attention
to impulsive systems on infinite dimensional spaces. Ahmed investigated optimal control
problems of a system governed by an impulsive system see 5–8. We also gave a series
of results for semilinear integrodifferential, strongly nonlinear impulsive systems and
2
Abstract and Applied Analysis
optimal control problems see 9–12. Particulary, Benchohra et al. give many interesting
recent results for various classes of differential equations with impulses on finite and infinite
dimensional Banach spaces see 13.
Although, there are some papers on periodic solution for periodic system on infinite
dimensional spaces see 14–24 and some results discussing integrodifferential system on
finite Banach space and infinite Banach space see 19, 25, to our knowledge, integrodifferential impulsive periodic systems on infinite dimensional spaces with unbounded operator
have not been extensively investigated. Recently, we discuss the impulsive periodic system
and integrodifferential impulsive system on infinite dimensional spaces. For linear impulsive
evolution operator is constructed and T0 -periodic P C-mild solution is introduced. Existence
of periodic solutions and alternative theorem, criteria of Massera type, and asymptotical
stability and robustness by perturbation are established see 26–28. For integrodifferential
impulsive system, existence of P C-mild solutions and optimal controls are presented see
9.
Herein, we go on studying the following integrodifferential impulsive periodic
system:
t
ẋt Axt f t, x, gt, s, xds ,
0
Δxt Bk xt ck ,
t
/ τk ,
1.1
t τk ,
on infinite dimensional Banach space X, where 0 τ0 < τ1 < τ2 < · · · < τk · · · , limk → ∞ τk ∞,
τkδ τk T0 , Δxτk xτk − xτk− , k ∈ Z0 , T0 is a fixed positive number, and δ ∈ N
denotes the number of impulsive points between 0 and T0 . The operator A is the infinitesimal
generator of a C0 -semigroup {T t, t ≥ 0} on X, f is a measurable function from 0, ∞×X ×X
to X and is T0 -periodic in t, g is a continuous function from 0, ∞ × 0, ∞ × X to X and is
T0 -periodic in t and s 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 space X.
In this paper, we use Horn’s 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 Poincaré operator P for integrodifferential impulsive periodic system 1.1, then
overcome some difficulties to show the continuity and compactness of Poincaré operator P
which are very important. By a new generalized Gronwall inequality with impulse and Bnorm given by us, the estimate of P C-mild solutions is established. Therefore, the existence
of T0 -periodic P C-mild solutions for impulsive integrodifferential periodic system is shown.
This extends the study of periodic solutions of integrodifferential periodic system without
impulse to integrodifferential periodic system with impulse on general Banach 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. In Section 3, the new generalized
Gronwall inequality with impulse and B-norm is established and the existence of P C-mild
solutions for integrodifferential impulsive system is presented. In Section 4, the T0 -periodic
P C-mild solutions for integrodifferential impulsive periodic system 1.1 are introduced. We
construct the suitable Poincaré operator P and give the relation between T0 -periodic P C-mild
solution and the fixed point of the Poincaré operator P . After showing the continuity and
JinRong Wang et al.
3
compactness of the Poincaré operator P , we can use Horn’s fixed point theorem to establish the
existence of T0 -periodic P C-mild solutions for integrodifferential impulsive periodic system
when the P C-mild solutions are bounded and ultimate bounded. At last, an example is given
to demonstrate the applicability of our result.
2. Preliminaries
In order to study the integrodifferential impulsive periodic system, we first recall some results
about linear impulsive periodic system here. Let X be a Banach space. £X denotes the
space of linear operators in X, and £b X denotes the space of bounded linear operators in X.
{τ1 , . . . , τδ } ⊂ 0, T0 ,
£b X is the Banach space with the usual supremum norm. Define D
where δ ∈ N denotes the number of impulsive points between 0 and T0 . We introduce
x is continuous from
P C0, T0 ; X ≡ {x : 0, T0 → X | x is continuous at t ∈ 0, T0 \ D,
1
left and has right-hand limits at t ∈ D} and P C 0, T0 ; X ≡ {x ∈ P C0, T0 ; X | ẋ ∈
P C0, T0 ; X}. Set
xP C max
sup xt 0, sup xt − 0 ,
t∈0,T0 t∈0,T0 xP C1 xP C ẋP C .
2.1
It can be seen that a Banach space is endowed with the norm ·P C ·P C1 , P C0, T0 ; X
P C1 0, T0 ; X.
Firstly, we consider the homogeneous linear impulsive periodic system
.
x t Axt,
Δxt Bk xt,
t
/ τk ,
t τk .
2.2
We introduce the following assumption.
Assumption H1.
H1.1 A is the infinitesimal generator of a C0 -semigroup {T t, t ≥ 0} on X with domain DA.
H1.2 There exists δ ∈ N such that τkδ τk T0 .
H1.3 For each k ∈ Z0 , δ ∈ N, Bk ∈ £b X and Bkδ Bk .
In order to study system 2.2, we need to consider the associated Cauchy problem
.
x t Axt, t ∈ 0, T0 \ D,
Δx τk Bk x τk , k 1, 2, . . . , δ,
2.3
x0 x.
If x ∈ DA and DA is an invariant subspace of Bk , using 29, Theorem 5.2.2, page
144, step by step, one can verify that the Cauchy problem 2.3 has a unique classical solution
x ∈ P C1 0, T0 ; X represented by xt St, 0x where
S·, ·: Δ t, θ ∈ 0, T0 × 0, T0 | 0 ≤ θ ≤ t ≤ T0 −→ £b X
2.4
4
Abstract and Applied Analysis
given by
⎧
T t − θ,
τk−1 ≤ θ ≤ t ≤ τk ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
τk−1 ≤ θ < τk < t ≤ τk1 ,
⎪
⎨T t − τk I Bk T τk − θ ,
St, θ ⎪
⎪T t − τk
I Bj T τj − τj−1
I Bi T τi − θ ,
⎪
⎪
⎪
⎪
θ<τj <t
⎪
⎪
⎩
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 the P C-mild solution of Cauchy problem 2.3 and T0 -periodic P Cmild solution of the system 2.2.
Definition 2.1. For every x ∈ X, the function x ∈ P C0, T0 ; X given by xt St, 0x is said
to be the P C-mild solution of the Cauchy problem 2.3.
Definition 2.2. A function x ∈ P C0, ∞; X is said to be a T0 -periodic P C-mild solution of
system 2.2 if it is a P C-mild solution of Cauchy problem 2.3 corresponding to some x and
xt T0 xt for t ≥ 0.
The following lemma gives the properties of the impulsive evolution operator
{St, θ, t, θ ∈ Δ} associated with {Bk ; τk }∞
k1 which are widely used in the sequel.
Lemma 2.3 see 26, Lemma 1. Impulsive evolution operator {St, θ, t, θ ∈ Δ} has the
following properties.
1 For 0 ≤ θ ≤ t ≤ T0 , St, θ ∈ £b X, that is, sup0≤θ≤t≤T0 St, θ ≤ MT0 , where MT0 > 0.
2 For 0 ≤ θ < r < t ≤ T0 , r /
τk , St, θ St, rSr, θ.
3 For 0 ≤ θ ≤ t ≤ T0 and N ∈ Z0 , St NT0 , θ NT0 St, θ.
4 For 0 ≤ t ≤ T0 and N ∈ Z0 , SNT0 t, 0 St, 0ST0 , 0N .
5 If {T t, t > 0} is a compact semigroup in X, then St, θ is a compact operator for 0 ≤ θ <
t ≤ T0 .
Here, we note that system 2.2 has a T0 -periodic P C-mild solution x if and only if
ST0 , 0 has a fixed point. The impulsive evolution operator {St, θ, t, θ ∈ Δ} can be used
to reduce the existence of T0 -periodic P 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 periodic P C-mild solutions for integrodifferential impulsive
periodic system on Banach space.
Consider the nonhomogeneous linear impulsive periodic system
ẋt Axt ft,
t/
τk ,
Δxt Bk xt ck ,
t τk ,
2.6
JinRong Wang et al.
5
and the associated Cauchy problem
ẋt Axt ft, t ∈ 0, T0 \ D,
Δx τk Bk x τk ck , k 1, 2, . . . , δ,
2.7
x0 x,
where f ∈ L1 0, T0 ; X, ft T0 ft, and ckδ ck .
Now we introduce the P C-mild solution of Cauchy problem 2.7 and T0 -periodic P Cmild solution of system 2.6.
Definition 2.4. A function x ∈ P C0, T0 ; X, for finite interval 0, T0 , is said to be a P C-mild
solution of the Cauchy problem 2.6 corresponding to the initial value x ∈ X and input
f ∈ L1 0, T0 ; X if x is given by
xt St, 0x t
St, θfθdθ 0
S t, τk ck .
2.8
0≤τk <t
Definition 2.5. A function x ∈ P C0, ∞; X is said to be a T0 -periodic P C-mild solution of
system 2.6 if it is a P C-mild solution of Cauchy problem 2.7 corresponding to some x and
xt T0 xt for t ≥ 0.
3. The generalized Gronwall inequality and existence of solutions
In order to derive an estimate of the P C-mild solutions, we introduce the following
generalized Gronwall inequality with impulse and B-norm.
Lemma 3.1. Let x ∈ P C0, ∞, X and let it satisfy the following inequality:
xt ≤ a b
t
xθλ1 dθ c
0
t
0
λ2
xθ dθ,
B
3.1
where a, b, c ≥ 0, 0 ≤ λ1 , λ2 ≤ 1 are constants, and
xθ B max
sup xξ 0, sup xξ − 0 .
0≤ξ≤θ
3.2
0≤ξ≤θ
Then one has
xt ≤ a 1ebct .
3.3
Proof. i For 0 ≤ λ1 , λ2 < 1, let λ max{λ1 , λ2 } ∈ 0, 1 and
yt ⎧
⎨1,
xt ≤ 1,
⎩xt, xt > 1.
3.4
6
Abstract and Applied Analysis
Then
xt ≤ yt ≤ a 1 b
t
yθλ dθ c
t
0
0
λ
yθ dθ,
B
∀t ∈ 0, T0 ,
3.5
Using 3.5, we obtain
λ
yt ≤ a 1 b c
B
t
0
λ
yθ dθ.
B
3.6
Define
t
λ
yθ dθ,
3.7
λ
u̇t b cyt B , t /
τk ,
u0 a 1,
u τk 0 u τk .
3.8
ut a 1 b c
0
B
then we get
Since yt λB ≤ ut, we then have
u̇t ≤ b cut, t / τk ,
u0 a 1,
u τk 0 u τk .
3.9
For t ∈ τk , τk1 , by 3.9, we obtain
ut ≤ u τk 0 ebct−τk u τk ebct−τk ,
3.10
ut ≤ a 1ebct ,
3.11
xt ≤ yt B ≤ a 1ebct .
3.12
further,
thus,
ii For λ1 λ2 1, we only need to define
u1 t a b c
t
0
xθ dθ.
B
3.13
JinRong Wang et al.
7
Similar to the proof in i, one can obtain
xt ≤ xt B ≤ aebct .
3.14
Combining i and ii, one can complete the proof.
Now, we consider the following integrodifferential impulsive periodic system:
t
ẋt Axt f t, x, gt, s, xds ,
t
/ τk ,
0
Δxt Bk xt ck ,
3.15
t τk ,
and the associated Cauchy problem
t
ẋt Axt f t, x, gt, s, xds ,
t ∈ 0, T0 \ D,
0
Δx τk Bk x τk ck ,
k 1, 2, . . . , δ,
3.16
x0 x.
By virtue of the expression of the P C-mild solution of the Cauchy problem 2.7, we
can introduce the P C-mild solution of the Cauchy problem 3.16.
Definition 3.2. A function x ∈ P C0, T0 ; X is said to be a P C-mild solution of the Cauchy
problem 3.16 corresponding to the initial value x ∈ X if x satisfies the following integral
equation:
xt St, 0x S
t
θ
St, θf θ, xθ, gθ, s, xsds dθ
0
t, τk
ck ,
0
for t ∈ 0, T0 .
3.17
0≤τk <t
Now, we introduce the T0 -periodic P C-mild solution of system 3.15.
Definition 3.3. A function x ∈ P C0, ∞; X is said to be a T0 -periodic P C-mild solution of
system 3.15 if it is a P C-mild solution of Cauchy problem 3.16 corresponding to some x
and xt T0 xt for t ≥ 0.
Assumption H2.
H2.1 f : 0, ∞ × X × X → X is measurable for t ≥ 0, and for any x1 , x2 , y1 , y2 ∈ X satisfying
x1 , x2 , y1 , y2 ≤ ρ there exists a positive constant Lf ρ > 0 such that
f t, x1 , y1 − f t, x2 , y2 ≤ Lf ρ x1 − x2 y1 − y2 .
3.18
8
Abstract and Applied Analysis
H2.2 There exists a positive constant Mf > 0 such that
ft, x, y ≤ Mf 1 x y,
∀x, y ∈ X.
3.19
H2.3 ft, x, y is T0 -periodic in t, that is,
f t T0 , x, y ft, x, y,
t ≥ 0.
3.20
H2.4 g : 0, ∞ × 0, ∞ × X → X is continuous for t ≥ s ≥ 0 and for any x, y ∈ X satisfying
x, y ≤ ρ there exists a positive constant Lg ρ > 0 such that
gt, s, x − gt, s, y ≤ Lg ρx − y.
3.21
H2.5 There exists a positive constant Mg > 0 such that
gt, s, x ≤ Mg 1 x,
∀x ∈ X.
3.22
H2.6 gt, s, x are T0 -periodic in t and s, that is,
g t T0 , s T0 , x gt, s, x, t ≥ s ≥ 0,
T0
gt, s, xds 0, t ≥ s ≥ 0.
3.23
0
H2.7 For each k ∈ Z0 and ck ∈ X there exists δ ∈ N such that ckδ ck .
It is not difficult to verify the following results.
Lemma 3.4. Under the assumptions [H2.4] and [H2.5], one has the following properties:
·
1 0 g·, s, xsds : P C0, T0 ; X → P C0, T0 ; X;
2 for all x1 , x2 ∈ P C0, T0 ; X and x1 P C0,T0 ;X , x2 P C0,T0 ;X ≤ ρ,
t
t
g t, s, x1 s ds − g t, s, x2 s ds ≤ Lg ρT0 x1 − x2 ;
t
t B
0
3.24
0
3 for x ∈ P C0, T0 ; X,
t
gt, s, xsds ≤ Mg T0 1 xt .
B
0
Now we present the existence of P C-mild solution for system 3.16.
3.25
JinRong Wang et al.
9
Theorem 3.5. Assumptions [H1.1], [H2.1], [H2.2], [H2.4], and [H2.5] hold, and for each k ∈ Z0 ,
Bk ∈ £b X, ck ∈ Xare fixed. Then system 3.16 has a unique P C-mild solution given by
xt, x St, 0x t
θ
St, θf θ, xθ,
0
0
gθ, s, xsds dθ S t, τk ck .
3.26
0≤τk <t
Proof. A similar result is given by Wei et al. in 9. 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
ẋt Axt f t, x,
t
gt, s, xds ,
t ∈ s, τ,
3.27
0
xs x ∈ X.
In order to obtain the local existence of mild solution for system 3.27, we only need to set
up the framework for using the contraction mapping theorem. Consider the ball given by
B x ∈ C s, t1 ; X | xt − x ≤ 1, s ≤ t ≤ t1 ,
3.28
where t1 would be chosen and xt ≤ 1 x ρ, s ≤ t ≤ t1 . B ⊆ Cs, t1 , X is a closed
convex set. Define a map Q on B given by
Qxt T tx t
θ
T t − θf θ, xθ, gθ, s, xsds dθ.
s
3.29
0
Under the assumptions H1.1, H2.1, H2.2, H2.4, H2.5, and Lemma 3.1, one can verify
that map Q is a contraction map on B with chosen t1 > 0. This means that system 3.27 has
a unique mild solution x ∈ Cs, t1 ; X given by
xt T tx t
s
θ
T t − θf θ, xθ, gθ, s, xsds dθ
on s, t1 .
3.30
0
Again, using Lemma 3.1, we can obtain the a priori estimate of the mild solutions for system
3.27 and present the global existence of mild solutions.
10
Abstract and Applied Analysis
Step 2. For t ∈ τk , τk1 , consider the Cauchy problem
ẋt Axt f t, x,
t
gt, s, xds ,
t ∈ τk , τk1 ,
0
x τk xk ≡ I Bk x τk ck ∈ X.
3.31
By Step 1, Cauchy problem 3.31 also has a unique P C-mild solution:
xt T t − τk xk t
θ
T t − θf θ, xθ, gθ, s, xsds dθ.
τk
3.32
0
Step 3. Combining all solutions on τk , τk1 k 1, . . . , δ, one can obtain the P C-mild
solution of the Cauchy problem 3.16 given by
xt, x St, 0x t
0
θ
St, θf θ, xθ, gθ, s, xsds dθ S t, τk ck .
0
3.33
0≤τk <t
This completes the proof.
4. Poincaré operator and existence of periodic solutions
To establish the periodic solutions for the system 3.15, we define a Poincaré operator from
X to X as follows:
P x x T0 , x
S T0 , 0 x T0
θ
S T0 , θ f θ, xθ, x,
0
0
gθ, s, xs, xds dθ S T0 , τk ck ,
0≤τk <T0
4.1
where x·, xdenotes the P C-mild solution of the Cauchy problem 3.16 corresponding to
the initial value x0 x, then examine whether P has a fixed point.
We first note that a fixed point of the Poincaré operator P gives rise to a periodic
solution.
Lemma 4.1. System 3.15 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 P x0. 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·, x0 of the Cauchy problem 3.16 corresponding to the initial value x0 x0 , we can define
y· x· T0 , x0 , then y0 xT0 , x0 P x0 x0 . Now, for t > 0, we can use the properties
JinRong Wang et al.
11
2, 3, and 4 of Lemma 2.3 and assumptions H1.2, H1.3, H2.3, H2.6, and H2.7to
arrive at
yt x t T0 , x0
S t T0 , T0 S T0 , 0 x0
T0
θ S t T0 , T0 S T0 , θ f θ, x θ, x0 , g θ, s, x s, x0 ds dθ
0
0
S t T0 , T0 S T0 , τk ck
0≤τk <T0
tT0
T0
θ S t T0 , θ f θ, x θ, x0 , g θ, s, x s, x0 ds dθ
S t
T0 ≤τkδ <tT0
T0 , τkδ
St, 0 S T0 , 0 x0 T0
0
ckδ
θ S T0 , θ f θ, x θ, x0 , g θ, s, x s, x0 ds dθ
0
t
0
S T0 , τk ckδ
0≤τk <T0
0
S t T0 , θ T0 f θ T0 , x θ T0 , x0 ,
S t T0 , τkδ
ckδ
θT0
g θ T0 , s, x s, x0
ds dθ
0
T0 ≤τkδ <tT0
St, 0x T0
t
θT0 S t T0 , θ T0 f θ T0 , x θ T0 , x0 ,
g θ T0 , s, x s, x0 ds dθ
0
S t
T0 ≤τkδ <tT0
T0 , τkδ
T0
ckδ
St, 0x T0
t
θ St, θf θ, x θ T0 , x0 , g θ T0 , s T0 , x s T0 , x0 ds dθ
0
S t
T0 ≤τkδ <tT0
St, 0y0 S
t
T0 , τkδ
t, τk
ckδ
θ
St, θf θ, yθ, y0, gθ, s, ys, y0ds dθ
0
0
0
ck
0≤τk <t
St, 0y0 t
θ
St, θf θ, yθ, y0, gθ, s, ys, y0ds dθ
0
S t, τk ck .
0≤τk <t
0
4.2
12
Abstract and Applied Analysis
This implies that y·, y0 is a P C-mild solution of Cauchy problem 3.16 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.
Second, we show that the Poincaré operator P defined by 4.1 is a continuous operator.
Lemma 4.2. Assumptions [H1.1], [H2.1], [H2.2], [H2.4], and [H2.5] hold. Then the 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 3.16 corresponding to the initial value x and y ∈ X,
respectively, given by
xt, x St, 0x t
0
xt, y St, 0y t
0
θ
St, θf θ, xθ, x, gθ, s, xs, xds dθ S T0 , τk ck ;
0
0≤τ <t
0
0≤τk <t
k
θ
St, θf θ, xθ, y, gθ, s, xs, yds dθ S T0 , τk ck .
4.3
Thus, by assumption H2.2, property 1 of Lemma 2.3, and property 3 of Lemma 3.4, we
obtain
ck xt, x ≤ MT0 x 1 Mg T0 MT0 Mf T0 MT0
0≤τk <T0
MT0 Mf
t
xθ, xdθ MT0 Mf Mg T0
0
t
0
xθ,x dθ
B
4.4
t
t
≤ a0 MT0 Mf xθ, xdθ MT0 Mf Mg T0 xθ,x B dθ,
0
0 ck xt, y ≤ MT0 y 1 Mg T0 MT0 Mf T0 MT0
0≤τk <T0
MT0 Mf
t
0
≤ b0 MT0 Mf
xθ, ydθ MT0 Mf Mg T0
t
t
0
xθ, ydθ MT0 Mf Mg T0
0
xθ,y dθ
B
t
0
4.5
xθ,y dθ.
B
By Lemma 3.1, one can verify that there exist constants M1∗ and M2∗ > 0 such that
xt, x ≤ M1∗ ,
xt, y ≤ M2∗ .
4.6
Let ρ max{M1∗ , M2∗ } > 0, then x·, x, x·, y ≤ ρ which imply that they are locally
bounded.
JinRong Wang et al.
13
By assumption H2.1, property 1 of Lemma 2.3, and property 2 of Lemma 3.4, we
obtain
xt, x − xt, y
≤ St, 0x − y
θ
t
f
θ,
xθ,
x,
gθ,
s,
xs,
xds
St, θ
0
− f θ, xθ, y,
0
θ
0
gθ, s, xs, yds dθ
4.7
≤ MT0 x − y
MT0 Lf ρ
t
xθ, x − xθ, ydθ MT0 Lf ρLg ρT0
0
t
0
xθ,x − xθ,y dθ.
B
By Lemma 3.1 again, one can verify that there exists a constant M3∗ > 0 such that
xt, x − xt, y ≤ M3∗ MT0 x − y ≡ Lx − y,
∀t ∈ 0, T0 ,
4.8
which implies that
P x − P y x T0 , x − x T0 , y ≤ Lx − y.
4.9
Hence, P is a continuous operator of x on X.
In the sequel, we need to prove the compactness of operator P , so we make the
following assumption.
Assumption H3 The semigroup {T t, t > 0} is a compact semigroup on X.
Now, we are ready to prove the compactness of operator P defined by 4.1.
Lemma 4.3. Assumptions [H1.1], [H2.1], [H2.2], [H2.4], [H2.5], 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 in X.
Let Γ be a bounded subset of X. Define K P Γ {P x ∈ X | x ∈ Γ}.
For 0 < ε ≤ T0 , define
Kε Pε Γ S T0 , T0 − ε x T0 − ε, x | x ∈ Γ .
4.10
14
Abstract and Applied Analysis
Next, we show that Kε is precompact in X. In fact, for x ∈ Γ fixed, we have
x T0 − ε, x ≤ S T0 − ε, 0 x 0
T0 −ε θ
S T0 − ε, θ f θ, xθ, x, gθ, s, xs, xds dθ
0≤τk <T0 −ε
0
S T0 − ε, τ ck k
≤ MT0 x MT0 Mf T0 1 Mg T0 MT0 Mf
ck MT Mf Mg T0
0
MT0
T0
0
0≤τk <T0
T0
4.11
xθ, xdθ
0
xθ,x dθ
B
ck .
≤ MT0 x MT0 Mf T0 1 Mg T0 1 Mg T0 MT0 Mf T0 ρ MT0
0≤τk <T0
This implies that the set {xT0 − ε, x | x ∈ Γ} is totally bounded.
By virtue of the compactness of {T t, t > 0} and property 5 of Lemma 2.3, ST0 , T0 −
ε is a compact operator. Thus, Kε is precompact in X.
On the other hand, for arbitrary x ∈ Γ,
T0 −ε
Pε x S T0 , 0 x 0≤τk <T0 −ε
S
θ
S T0 , θ f θ, xθ, x, gθ, s, xs, xds dθ
0
T0 , τk
0
4.12
ck .
Thus, combined with 4.1, we have
Pε x − P x
θ
T0 −ε
≤
S
T
,
θ
f
θ,
xθ,
x,
gθ,
s,
xs,
xds
dθ
0
0
−
T0
0
0
θ
S T0 , θ f θ, xθ, x, gθ, s, xs, xds dθ
0
S T0 , τk ck −
S T0 , τk ck 0≤τ <T −ε
0≤τ <T
k
≤
T0
T0 −ε
0
k
4.13
0
θ
S T0 , θ f θ, xθ, x, gθ, s, xs, xds dθ MT
0
0
≤ MT0 Mf 1 Mg T0 1 ρε MT0
T0 −ε≤τk <T0
ck .
T0 −ε≤τk <T0
ck JinRong Wang et al.
15
It is showing that the set K can be approximated to an arbitrary degree of accuracy by a
precompact set Kε . Hence K itself is a precompact set in X, that is, P takes a bounded set into
a precompact set in X. As a result, P is a compact operator.
After showing the continuity and compactness of operator P , we can follow and
derive periodic P C-mild solutions for system 3.15. 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 locally
boundedness and ultimate boundedness.
Definition 4.4. One says that P C-mild solutions of Cauchy problem 3.16 are bounded if for
each B1 > 0, there is a B2 > 0 such that x ≤ B1 implies xt, x ≤ B2 for t ≥ 0.
Definition 4.5. One says that P C-mild solutions of Cauchy problem 3.16 are locally bounded
if for each B1 > 0 and k0 > 0, there is a B2 > 0 such that x ≤ B1 implies xt, x ≤ B2 for
0 ≤ t ≤ k0 .
Definition 4.6. One says that P C-mild solutions of Cauchy problem 3.16 are ultimate
bounded if there is a bound B > 0 such that for each B3 > 0, there is a k > 0, such that
x ≤ B3 and t ≥ k imply xt, x ≤ B.
We also need the following results as a reference.
Lemma 4.7 see 21, Theorem 3.1. Local boundedness and ultimate boundedness imply
boundedness and ultimate boundedness.
Lemma 4.8 Horn’s fixed point theorem 20, Lemma 3.1. Let E0 ⊂ E1 ⊂ E2 be convex subsets
of Banach space X, with E0 and E2 compact subsets and E1 open 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,
P j E1 ⊂ E0 , m ≤ j ≤ 2m − 1,
4.14
then P has a fixed point in E0 .
With these preparations, we can prove our main result in this paper.
Theorem 4.9. Let assumptions [H1], [H2], and [H3] hold. If the P C-mild solutions of Cauchy
problem 3.16 are ultimate bounded, then system 3.15 has a T0 -periodic P C-mild solution.
Proof. By Theorem 3.5, 4.4 in Lemma 4.2, and Definition 4.5, the Cauchy problem 3.16
corresponding to the initial value x0 x has a P C-mild solution x·, x which is locally
bounded. From ultimate boundedness and Lemma 4.7, x·, x is bounded. Next, let B > 0 be
the bound in the definition of ultimate boundedness. Then by boundedness, there is a B1 > B
such that x ≤ B implies xt, x ≤ B1 for t ≥ 0. Furthermore, there is a B2 > B1 such that
x ≤ B1 implies xt, x ≤ B2 for t ≥ 0. Now, using ultimate boundedness again, there is a
positive integer m such that x ≤ B1 implies xt, x ≤ B for t ≥ m − 2T0 .
16
Abstract and Applied Analysis
Define y·, y0 x· T0 , x, then y0 xT0 , x P x. From 4.2 in Lemma 4.1,
we obtain P y0 yT0 , y0 x2T0 , x. Thus, P 2 x P P x P y0 x2T0 , x.
Suppose there exists integer m − 1 such that P m−1 x xm − 1T0 , x. By induction we arrive
at
P m x P m−1 P x P m−1 y0 ym − 1T0 , y0 x mT0 , x .
4.15
Thus, we obtain
j−1 P x xj − 1T0 , x < B2 ,
j 1, 2, . . . , m − 1, x < B1 ;
j−1 P x xj − 1T0 , x < B,
j ≥ m, x < B1 .
4.16
It comes from Lemma 4.3 that P x xT0 , x on X is compact. Now let
H x ∈ X : x < B2 ,
E2 cl.convP H,
W x ∈ X : x < B1 ,
E1 W ∩ E2 ,
G x ∈ X : x < B ,
E0 cl.convP G,
4.17
where convY is the convex hull of the set Y defined by convY { ni1 λi yi | n ≥ 1, yi ∈
n
Y, λi ≥ 0, i1 λi 1}, and cl. denotes the closure. Then we see that E0 ⊂ E1 ⊂ E2 are convex
subsets of X with E0 , E2 compact subsets and E1 open relative to E2 , and from 4.16 one has
P j E1 ⊂ P j W P P j−1 W ⊂ P H ⊂ E2 ,
P E1 ⊂ P W P P
j
j
j−1
W ⊂ P G ⊂ E0 ,
j 1, 2, . . . , m − 1;
j m, m 1, . . . , 2m − 1.
4.18
We see that P : E2 → X is a continuous map continuous from Lemma 4.2. Consequently,
from Horn’s fixed point theorem, we know that the operator P has a fixed point x0 ∈ E0 ⊂ X.
By Lemma 4.1, we know that the P C-mild solution x·, x0 of the Cauchy problem 3.16
corresponding to the initial value x0 x0 is just T0 -periodic. Therefore, x·, x0 is a T0 periodic P C-mild solution of system 3.15. This proves the theorem.
At last, an example is given to illustrate our theory. Consider the following problem:
∂
xt, y Δxt, y xt, y ∂t
y ∈ Ω,
t
1 sint − s 3x2 s, y 2ds,
0
t ≥ s,
t, s ∈ 0, 2π \
3
1
π, π, π ,
2
2
JinRong Wang et al.
17
⎧
⎪
⎪
⎨0.05Ixτi , y,
−
Δxτi , y xτi , y − xτi , y −0.05Ixτi , y,
⎪
⎪
⎩0.05Ixτ , y,
i
xt, y|y∈∂Ω 0,
i 1,
i
i 2, y ∈ Ω, τi π, i 1, 2, 3,
2
i 3,
t > 0,
x0, y x2π, y,
4.19
where Ω ⊂ R3 is a bounded domain and ∂Ω ∈ C3 .
Define X L2 Ω, DA H 2 Ω ∩ H01 Ω, and Ax −∂2 x/∂y12 ∂2 x/∂y22 ∂2 x/∂y32 for x ∈ DA. Then, A generates a compact semigroup {T t, t ≥ 0} on L2 Ω. Define
t
·
x·y x·, y, sin·y sin·, y, f·, x·, 0 g·, s, xdsy x·y 0 1 sin· −
s 3x2 · 2dsy, and
⎧
⎪
⎪
⎪0.05I,
⎨
Bi −0.05I,
⎪
⎪
⎪
⎩0.05I,
i 1,
i 2,
4.20
i 3.
Thus problem 4.19 can be rewritten as
t
3
1
π, π, π ,
gt, s, xds , t ≥ s, t, s ∈ 0, 2π \
2
2
0
i
i
π Bi x
π , i 1, 2, 3,
Δx
2
2
ẋt Axt f t, x,
4.21
x0 x2π.
If the P C-mild solutions of Cauchy problem 4.21 are ultimate bounded, then all the
assumptions in Theorem 4.9 are met, and our results can be used to system 4.19, that is,
problem 4.19 has a 2π-periodic P C-mild solution x2π ·, y ∈ P C2π 0 ∞; L2 Ω, where
P C2π 0, ∞; L2 Ω ≡ {x ∈ P C0, ∞; L2 Ω | xt xt 2π, t ≥ 0}.
4.22
Acknowledgments
This work is supported by Natural Science Foundation of Guizhou Province Education
Department no. 2007008 and Guizhou Province no. 20052001. This work is also supported
by the undergraduate carve out project of Department of Guiyang Science and Technology
2008 no. 15-2.
References
1 D. Baı̆nov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66
of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical,
Harlow, UK, 1993.
18
Abstract and Applied Analysis
2 V. Lakshmikantham, D. D. Baı̆nov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6
of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
3 X. Liu, “Impulsive stabilization and applications to population growth models,” The Rocky Mountain
Journal of Mathematics, vol. 25, no. 1, pp. 381–395, 1995.
4 T. Yang, Impulsive Control Theory, vol. 272 of Lecture Notes in Control and Information Sciences, Springer,
Berlin, Germany, 2001.
5 N. U. Ahmed, “Optimal impulse control for impulsive systems in Banach spaces,” International Journal
of Differential Equations and Applications, vol. 1, no. 1, pp. 37–52, 2000.
6 N. U. Ahmed, “Some remarks on the dynamics of impulsive systems in Banach spaces,” Dynamics of
Continuous, Discrete & Impulsive Systems. Series A, vol. 8, no. 2, pp. 261–274, 2001.
7 N. U. Ahmed, K. L. Teo, and S. H. Hou, “Nonlinear impulsive systems on infinite dimensional
spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 5, pp. 907–925, 2003.
8 N. U. Ahmed, “Existence of optimal controls for a general class of impulsive systems on Banach
spaces,” SIAM Journal on Control and Optimization, vol. 42, no. 2, pp. 669–685, 2003.
9 W. Wei, X. Xiang, and Y. Peng, “Nonlinear impulsive integro-differential equations of mixed type and
optimal controls,” Optimization, vol. 55, no. 1-2, pp. 141–156, 2006.
10 X. Xiang and W. Wei, “Mild solution for a class of nonlinear impulsive evolution inclusions on Banach
space,” Southeast Asian Bulletin of Mathematics, vol. 30, no. 2, pp. 367–376, 2006.
11 X. Xiang, W. Wei, and Y. Jiang, “Strongly nonlinear impulsive system and necessary conditions of
optimality,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 12, no. 6, pp. 811–824,
2005.
12 X. Yu, X. Xiang, and W. Wei, “Solution bundle for a class of impulsive differential inclusions on Banach
spaces,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 220–232, 2007.
13 M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of
Contemporary Mathematics and Its Applications, Hindawi, New York, NY, USA, 2006.
14 H. Amann, “Periodic solutions of semilinear parabolic equations,” in Nonlinear Analysis: A Collection
of Papers in Honor of Erich H. Rothe, pp. 1–29, Academic Press, New York, NY, USA, 1978.
15 B. Aulbach and N. Van Minh, “Bounded and almost periodic solutions and evolution semigroups
associated with nonautonomous functional differential equations,” Abstract and Applied Analysis, vol.
5, no. 4, pp. 245–263, 2000.
16 Y. S. Eidelman and I. V. Tikhonov, “On periodic solutions of abstract differential equations,” Abstract
and Applied Analysis, vol. 6, no. 8, pp. 489–499, 2001.
17 J. Mawhin, “Periodic solutions of some semilinear wave equations and systems: a survey,” Chaos,
Solitons & Fractals, vol. 5, no. 9, pp. 1651–1669, 1995.
18 J. H. Liu, “Bounded and periodic solutions of differential equations in Banach space,” Applied
Mathematics and Computation, vol. 65, no. 1–3, pp. 141–150, 1994.
19 J. H. Liu, “Bounded and periodic solutions of semilinear evolution equations,” Dynamic Systems and
Applications, vol. 4, no. 3, pp. 341–350, 1995.
20 J. H. Liu, “Bounded and periodic solutions of finite delay evolution equations,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 34, no. 1, pp. 101–111, 1998.
21 J. H. Liu, T. Naito, and N. Van Minh, “Bounded and periodic solutions of infinite delay evolution
equations,” Journal of Mathematical Analysis and Applications, vol. 286, no. 2, pp. 705–712, 2003.
22 D. Paşca, “Periodic solutions of a class of non-autonomous second-order differential inclusions
systems,” Abstract and Applied Analysis, vol. 6, no. 3, pp. 151–161, 2001.
23 P. Sattayatham, S. Tangmanee, and W. Wei, “On periodic solutions of nonlinear evolution equations
in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 98–108, 2002.
24 X. Xiang and N. U. Ahmed, “Existence of periodic solutions of semilinear evolution equations with
time lags,” Nonlinear Analysis: Theory, Methods & Applications, vol. 18, no. 11, pp. 1063–1070, 1992.
25 D. J. Guo and X. Liu, “Extremal solutions of nonlinear impulsive integrodifferential equations in
Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 177, no. 2, pp. 538–552, 1993.
26 J. R. Wang, “Linear impulsive periodic system on Banach space,” in Proceedings of the 4th International
Conference on Impulsive and Hybrid Dynamical Systems (ICIDSA ’07), vol. 5, pp. 20–25, Nanning, China,
July 2007.
JinRong Wang et al.
19
27 J. R. Wang, X. Xiang, and W. Wei, “Linear impulsive periodic system with time-varying generating
operators on Banach space,” Advances in Difference Equations, vol. 2007, Article ID 26196, 16 pages,
2007.
28 J. R. Wang, X. Xiang, and W. Wei, “Existence and global asymptotical stability of periodic solution
for the T -periodic logistic system with time-varying generating operators and T0 -periodic impulsive
perturbations on Banach spaces,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 524945,
16 pages, 2008.
29 N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, vol. 246 of Pitman Research
Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1991.