# Existence and Global Asymptotical Stability of  Periodic Solution for theT-Periodic logistic

Authors: JinRong Wang; X. Xiang; W. Wei; Qian Chen
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

This paper studies the existence and global asymptotical stability of periodic PC-mild solution for
theT-periodic Logistic system with time-varying generating operators and T0-periodic impulsive perturbations on
Banach spaces. Two sufficient conditions that guarantee the exponential stability of the impulsive evolution operator
corresponding to homogenous well-posed T-periodic system with time-varying generating operators and T0-periodic
impulsive perturbations are given. It is shown that the system have a unique periodic PC-mild solution which is globally asymptotically stable when T and T0 are rational dependent and its period must be nT0 for some n∈N. At last, an example is given for demonstration.


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## Body

## 1. Introduction

For modeling
the dynamics  of an ecological system, Cushing [1] pointed out that it is
necessary and important to consider models with periodic ecological parameters
or perturbations which might be quite naturally exposed (e.g., those due to
seasonal effects of wheatear, food supply, mating habits, etc.). The periodic
solution theory of dynamic equations has been developed over the last decades.
We refer the readers to [2–6] for infinite dimensional cases,
and to [1, 7–9] for finite dimensional
cases. Especially, there are many results of periodic solutions (such as
existence and stability) for impulsive periodic systems on finite dimensional
spaces (see [7–9]). There are also some
relative results of periodic solutions for periodic systems with time-varying
generating operators on infinite dimensional spaces (see [5, 6, 10–12]).On the other hand, the ecological system is often
deeply perturbed by human exploit activities such as planting and harvesting.
Usually, these activities are considered continuously by adding some items
to [13–15], whereas this is not how things stand. It is often
the case that planting and harvesting of the species are seasonal or occur in
regular pulses. These perturbations may also naturally be periodic, for
example, a fisherman may go fishing at the same time once a day or once a week.
Systems with short-term perturbations are often naturally described by
impulsive differential equations, which are found in almost every domain of
applied sciences. For the basic theory on impulsive differential equations on
finite dimensional spaces, the reader can refer to Lakshmikantham's book and
Yang's book (see [9, 16]). For the basic theory on impulsive differential
equations on infinite dimensional spaces, the reader can refer to Ahmed's
paper, Liu's paper, and Xiang's papers (see [17–19]).In this paper, we will study the following generalized
Logistic system with impulsive perturbations:(1.1)∂∂tx(t,y)=A(y,t,D)x(t,y)+f(t,y),y∈Ω,t>0,t≠τk,k∈Z0+,x(t,y)=0,y∈∂Ω,t>0,Δx(t,y)=Bkx(t,y)+ck,y∈Ω,t=πk,k∈Z0+,  where x(t,y) denotes the population
number of isolated species at time t and location y, Ω is an open-bounded domain in R2,
and ∂Ω is smooth enough. The operator(1.2)A(y,t,D)≡∑1≤i,j≤2aij(y,t)DiDj+∑1≤i≤2bi(y,t)Di+c(y,t),(y,t)∈(Ω×[0,+∞)),where all the coefficients are
smooth functions enough and Di denotes the spatial derivative with respect to yi. f is related to the periodic change of the
resources maintaining the evolution of the population. Time sequence 0=τ0<τ1<⋯<τk⋯ and τk→∞ as k→∞, Δx(τk,y)=x(τk+,y)−x(τk,y)=Bkx(τk,y)+ck denote mutation of the isolate species at time τk.Suppose the first equation of (1.1) is T-periodic and the third equation of (1.1) is T0-periodic, that is, A(y,t+T,D)=A(y,t,D), f(t+T,y)=f(t,y), t≥0,
and T0 is the
least-positive constant such that there are δ  τks in the interval (0,T0) and τk+δ=τk+T0,Bk+δ=Bk,ck+δ=ck, k∈Z0+.The first equation of (1.1) describes the variation of
the population number x of the species in T-periodically changing environment. The second
equation of (1.1) shows that the species is isolated. The third equation of
(1.1) reflects the possibility of impulsive effects on the population. As we
assumed, these impulsive perturbations are T0-periodic. Naturally, this period is distinct
from T,
the period of the change of environment. Even when we want to carry out the
perturbations according to the period T,
we cannot do it since we do not know T exactly. Thus, it is interesting how the
dynamics of the first equation of (1.1) are
affected by the periodic changing of environment
together with the periodic impulsive perturbations.DefineA(t)(y)=A(y,t,D), x(⋅)(y)=x(⋅,y), f(⋅)(y)=f(⋅,y),
(1.1) can be abstracted impulsive periodic evolution equations of the
form(1.3)x˙(t)=A(t)x(t)+f(t),t⁢≠τk,(T-periodic)Δx(t)=Bkx(t)+ck,t=τk,(T0-periodic)on the Banach space X.
It is obvious that the investigation of (1.3) cannot only be used to discuss
(1.1) but also provide a foundation for study of the general impulsive periodic
systems.Assume thatT-periodic evolution equation of the form x˙(t)=A(t)x(t),x∈X,t>0,
is well posed, that is, there exists a T-periodic evolutionary process {U(t,θ),t≥θ≥0} which satisfies, among other things, the
conditions (1)–(5) of Definition 2.1 which follows.
Once the evolution equations x˙(t)=A(t)x(t),x∈X,t>0,
and x˙(t)=A(t)x(t)+f(t),x∈X,t>0,
are well posed, the asymptotic behavior of solutions at infinity, such as
stability and periodicity, is of particular interest, which has been a central
topic discussed for the past decade. We refer the reader to the books [20, 21], and the surveys [22], and the references therein
for more complete information on the subject. Because (1.3) can be used to
describe more social and natural phenomena, it is naturally
needed to study the stability and periodicity of
solutions for (1.3).The aim of this paper is to study the existence and
global asymptotical stability of periodic PC-mild solution of (1.3) without compactness
condition. We will show that (i) if γ=T/T0 is rational, that is, T and T0 are rational dependent, then (1.3) may have a
unique periodic PC-mild solution which is globally
asymptotically stable and (ii) if x(t,x¯) is a periodic PC-mild solution of (1.3) with x(0)=x¯,
then its period must be nT0 for some n∈N.This paper is organized as follows. In Section2, the
properties of the impulsive evolution operator are collected, two sufficient
conditions that guarantee the exponential stability of the impulsive evolution
operator are given. In Section 3, the existence of periodic PC-mild solution which is globally
asymptotically stable for (1.3) is obtained. At last, the abstract results are
applied to a special case of (1.1). This work not only
provides the theory basis for managing some single
species but is also fundamental for further
discussion on the existence and stability of periodic solution for nonlinear
impulsive periodic system with time-varying generating operators on infinite
dimensional spaces.


## 2. Exponential Stability of Impulsive Evolution Operator

LetX be a Banach space, £(X) denotes the space of linear operators on X; £b(X) denotes the space of bounded linear operators
on X.
Let £b(X) be the Banach space with the usual supremum
norm. Denote γ is rational, let γ=p/q, p,q∈N, p, q are relatively prime. Denote T˜=pT0=qT, D˜={τ1,…,τpδ}⊂[0,T˜]    and define PC([0,T˜];X)≡{x:[0,T˜]→X∣x is continuous at t∈[0,T˜]∖D˜, x is continuous from left and has right-hand
limits at t∈D˜} , and PC1([0,T˜];X)≡{x∈PC([0,T˜];X)∣x˙∈PC([0,T˜];X)}.
Set(2.1)∥x∥PC=max⁡{sup⁡t∈[0,T˜]∥x(t+0)∥,sup⁡t∈[0,T˜]∥x(t−0)∥},∥x∥PC1=∥x∥PC+∥x˙∥PC.It can be seen that endowed with
the norm ∥⋅∥PC  (∥⋅∥PC1), PC([0,T˜];X)  (PC1([0,T˜];X)) is a Banach space.In order to investigate periodic solution, introduce
the following two spaces:(2.2)LT1([0,+∞);X)≡{f:[0,+∞)→X∣f(t)=f(t+T),∫0T∥f(t)∥dt<∞},PCT˜([0,+∞);X)≡{x∈PC([0,+∞);X)∣x(t)=x(t+T˜),t≥0}.Set(2.3)∥f∥LT1=∫0T∥f(t)∥dt<∞,∥x∥PCT˜=max⁡{sup⁡t∈[0,T˜]∥x(t+0)∥,sup⁡t∈[0,T˜]∥x(t−0)∥}.It can be seen that endowed with
the norm ∥⋅∥LT1  (∥⋅∥PCT˜), LT1([0,+∞);X)  (PCT˜([0,+∞);X)) is a Banach space.The following notation will be used throughout the
paper, we recall these concepts in the following definitions.Definition 2.1.
A family of
bounded linear operators{U(t,θ),t≥θ≥0} from a Banach space X to itself is called strongly continuous
evolutionary process if the following conditions (1)–(4) are satisfied:(1)
U(t,t)=I, t≥0,(2)
U(t,r)U(r,θ)=U(t,θ),t≥r≥θ≥0,(3)
the map(t,θ)→U(t,θ)x is continuous for every fixed x∈X,(4)
∥U(t,θ)∥≤Meω(t−θ) for some M≥1, ω∈R independent of t≥θ≥0,
further, if(5)
U(t+T,θ+T)=U(t,θ) for all t≥θ≥0.
Then the strongly continuous evolutionary process{U(t,θ),t≥θ≥0} is called T-periodic.Definition 2.2.
The
evolutionary process{U(t,θ),t≥θ≥0} is called exponentially bounded if(2.4)ϖ(U)=inf⁡{ω∈R:∃M≥1with∥U(t,θ)∥≤Meω(t−θ)fort≥θ≥0}<∞.Definition 2.3.
The linear
equation(2.5)x˙(t)=A(t)x(t),x∈X,t≥0is said to be well posed if
there exists an evolutionary process {U(t,θ),t≥θ≥0},
which satisfies the conditions (1)–(4) in Definition 2.1, such that for every θ≥0 and x∈D(A(θ)),
the function x(t)=U(t,θ)x¯ is the uniquely determined classical solution
of (2.5) satisfying x(θ)=x¯.Definition 2.4.
The functionx∈X is said to be a mild solution to the linear
equation (2.5) if and only if(2.6)x(t)=U(t,θ)x(θ),t≥θ≥0.
We introduce the following assumption (H1):(H1.1)
A(t):D(A(t))→X, t≥0,
is a family of linear unbounded operators on X, A(t) is T-periodic, that is, for t≥0, A(t+T)=A(t);(H1.2)
the linear equationx˙(t)=A(t)x(t),x∈X,t≥0,
is well posed;(H1.3)
there existsδ∈N such that τk+δ=τk+T0;(H1.4)
for eachk∈Z0+, Bk∈£b(X) and Bk+δ=Bk.
By (H1.1) and (H1.2),{U(t,θ),t≥θ≥0} is T-periodic strongly continuous evolutionary
process.
Under (H1), consider(2.7)x˙(t)=A(t)x(t),t≠τk,Δx(t)=Bkx(t),t=τk,and Cauchy
problem(2.8)x˙(t)=A(t)x(t),t∈[0,T˜]∖D˜,Δx(τk)=Bkx(τk),k=1,2,…,pδ,x(0)=x¯.
For everyx¯∈X, D(A(t)), t≥0,
is an invariant subspace of Bk;
and step by step, one can verify that the Cauchy problem (2.7)
has a unique classical solution x∈PC1([0,T˜];X) represented by x(t)=Φ(t,0)x¯,
where(2.9)Φ(⋅,⋅):Δ={(t,θ)∈[0,T˜]×[0,T˜]∣0≤θ≤t≤T˜}→£(X),given by(2.10)Φ(t,θ)={U(t,θ),τk−1≤θ≤t≤τk,U(t,τk+)(I+Bk)U(τk,θ),τk−1≤θ<τk<t≤τk+1,U(t,τk+)[∏θ<τj<t(I+Bj)U(τj,τj−1+)](I+Bi)U(τi,θ),τi−1≤θ<τi≤⋯<τk<t≤τk+1.The operator {Φ(t,θ),(t,θ)∈Δ} is called impulsive evolution operator
associated with {U(t,θ),(t,θ)∈Δ} and {Bk;τk}k=1∞.
The following lemma on the properties of the impulsive
evolution operator{Φ(t,θ),(t,θ)∈Δ} associated with {U(t,θ),(t,θ)∈Δ} and {Bk;τk}k=1∞ are widely used in this paper.Lemma 2.5.
Assumption
(H1) holds. The impulsive evolution operator{Φ(t,θ),(t,θ)∈Δ} has the following properties:(1)
for0≤θ≤t≤T˜, Φ(t,θ)∈£b(X),
there exists MT˜>0 such that sup⁡0≤θ≤t≤T˜∥Φ(t,θ)∥≤MT˜,(2)
for0≤θ<r<t≤T˜, r⁢≠τk, Φ(t,θ)=Φ(t,r)Φ(r,θ),(3)
for0≤θ≤t≤T˜, n∈Z+, Φ(t+nT˜,θ+nT˜)=Φ(t,θ),(4)
for0≤θ≤t≤T˜, n∈Z+, Φ(t+nT˜,0)=Φ(t,0)[Φ(T˜,0)]n,(5)
for0≤θ<t,
there exits M≥1, ω∈R such that(2.11)∥Φ(t,θ)∥≤Mexp⁡{ω(t−θ)+∑θ≤τn<tln⁡(M∥I+Bn∥)}.Proof.
For (1)–(4), the reader can see
[23, Lemma 2.7]. We only need to
verify (5). Without loss of generality, for τi−1≤θ<τi≤⋯<τk<t≤τk+1,(2.12)∥Φ(t,θ)∥=∥U(t,τk+)∥∥I+Bk∥∥U(τk,τk−1+)∥⋯∥I+Bi∥∥U(τi,θ)∥≤Meω(t−τk+)[∏n=i+1k∥I+Bn∥Me(τn−τn−1+)]∥I+Bi∥Meω(τi−θ)≤Mexp⁡{ω(t−θ)+∑θ≤τn<tln⁡(M∥I+Bn∥)}.This completes the proof.In order to study the asymptotical properties of
periodic solution, it is necessary to discuss the exponential stability of the
impulsive evolution operator{Φ(t,θ),t≥θ≥0}.
We first give the definition of exponentially stable for {Φ(t,θ),t≥θ≥0}.Definition 2.6.
{Φ(t,θ),t≥θ≥0} is called exponentially stable if there exist K≥0 and ν>0 such that(2.13)∥Φ(t,θ)∥≤Ke−ν(t−θ),t>θ≥0.
Assumption (H2):{U(t,θ),t≥θ≥0} is exponentially stable, that is, there exist K0>0 and ν0>0 such that(2.14)∥U(t,θ)∥≤K0e−ν0(t−θ),t>θ≥0.Next, two sufficient conditions that guarantee the
impulsive evolution operator{Φ(t,θ),t≥θ≥0} with rational γ is exponentially stable are given.Lemma 2.7.
Supposeγ is rational and (H1) and (H2) hold. There
exists 0<λ<ν0 such that(2.15)∏k=1δ(K0∥I+Bk∥)γe−λT<1.
Then{Φ(t,θ),t≥θ≥0} is exponentially stable.Proof.
Sinceγ is rational, let γ=p/q, p,q∈N and p, q are relatively prime. Let T˜=pT0  (=qT),
then (2.7) is T˜-periodic. Without loss of generality, for τi−1≤θ<τi≤⋯<τk<t≤τk+1,
by (5) of Lemma 2.5, we have(2.16)∥Φ(t,θ)∥≤K0e−(ν0−λ)(t−θ)(∏θ≤τk<t(K0∥I+Bk∥)e−λ(t−θ)).Suppose t∈(nT˜,(n+1)T˜] and let b=max⁡s∈[0,T˜]Π0<τk<s{K0∥I+Bk∥}. Then(2.17)∏θ≤τk<t(K0∥I+Bk∥)e−λ(t−θ)≤∏0≤τk<nT˜(K0∥I+Bk∥)e−λnT˜⋅∏nT˜≤τk<t(K0∥I+Bk∥)e−λ(t−nT˜)eλθ≤∏0≤τk<npT0(K0∥I+Bk∥)e−λnqTbeλθ≤∏k=1δ(K0∥I+Bk∥)npe−λnqTbeλθ=[∏k=1δ(K0∥I+Bk∥)γe−λT]nqbeλθ<beλθ.Let K=K0beλθ>0 and ν=ν0−λ>0,
then we obtain ∥Φ(t,θ)∥≤Ke−ν(t−θ),t>θ≥0.Remark 2.8.
An
exponentially bounded evolutionary process{U(t,θ),⁢t≥θ≥0} is exponential stability if and only if for
some 1≤p<∞ and all x∈X and t>θ≥0,
there is a constant κ>0 such that(2.18)∫θ∞∥U(t,θ)x∥pdt≤κp∥x∥p.Lemma 2.9.
Assumption
(H1) holds. Suppose(2.19)0<μ1=inf⁡k=1,2,…,δ(τk−τk−1)≤sup⁡k=1,2,…,δ(τk−τk−1)=μ2<∞.If there exists α>0 such that(2.20)ω+1μln⁡(M∥I+Bk∥)≤−α<0,k=1,2,…,δ,where(2.21)μ={μ1,α+ω<0,μ2,α+ω≥0,then {Φ(t,θ),t>θ≥0} is exponentially stable.Proof.
It comes from (2.20)
that(2.22)ln⁡(M∥I+Bk∥)≤−μ(α+ω)<0,k=1,2,…,δ.Further,(2.23)∑θ≤τk<tln⁡(M∥I+Bk∥)≤−∑θ≤τk<tμ(α+ω)=−μ(α+ω)N(θ,t),where N(θ,t) is denoted the number of impulsive points in [θ,t).
Forτi−1≤θ<τi≤⋯<τk<t≤τk+1,
by (2.19), we obtain the following two inequalities:(2.24)t−θ≥(τk−τk−1)+⋯+(τi+1−τi)≥(N(θ,t)−1)μ1,t−θ≤(τk+1−τk)+(τk−τk−1)+⋯+(τi+1−τi)+(τi−τi−1)≤(N(θ,t)+1)μ2.This implies that(2.25)μ1(N(θ,t)−1)≤t−θ≤μ2(N(θ,t)+1),that is,(2.26)1μ2(t−θ)−1≤N(θ,t)≤1μ1(t−θ)+1.Then(2.27)−μ(α+ω)N(θ,t)≤{−μ1(α+ω)[1μ1(t−θ)+1]=−(α+ω)(t−θ)−μ1(α+ω),α+ω<0.−μ2(α+ω)[1μ2(t−θ)−1]=−(α+ω)(t−θ)+μ2(α+ω),α+ω≥0.=−(α+ω)(t−θ)+μ|α+ω|.Thus, we obtain(2.28)ω(t−θ)+∑θ≤τk<tln⁡(M∥I+Bk∥)≤−α(t−θ)+μ|α+ω|.By (5) of Lemma 2.5, let K=Meμ|α+ω|>0, ν=α>0, ∥Φ(t,θ)∥≤Ke−ν(t−θ),⁢t>θ≥0.


## 3. Periodic Solution and Global Asymptotical Stability

Assumption (H3):f∈LT1([0,+∞);X).
For each k∈Z0+, δ∈N,
and ck∈X, ck+δ=ck.Under the (H1) and (H3), consider the
following system(3.1)x˙(t)=A(t)x(t)+f(t),t⁢≠τk,Δx(t)=Bkx(t)+ck,t=τk,and Cauchy
problem(3.2)x˙(t)=A(t)x(t)+f(t),t∈[0,T˜]∖D˜,Δx(τk)=Bkx(τk)+ck,k=1,2,…,pδ,x(0)=x¯.Now we list the PC-mild solution of Cauchy problem (3.2) and T˜-periodic PC-mild solution of (3.1).Definition 3.1.
For everyx¯∈X, f∈L1([0,+∞);X),
the function x∈PC([0,T˜];X),
given by(3.3)x(t)=Φ(t,0)x¯+∫0tΦ(t,θ)f(θ)dθ+∑0≤τk<tΦ(t,τk+)ckfor t∈[0,T˜],
is said to be a PC-mild solution of the Cauchy problem (3.2).Definition 3.2.
A functionx∈PC([0,+∞);X) is said to be a T˜-periodic PC-mild solution of (3.1) if it is a PC-mild solution of Cauchy problem (3.2)
corresponding to some x¯ and x(t+T˜)=x(t) for t≥0.In the sequel, we show the existence and global
asymptotical stability ofT˜-periodic PC-mild solution of (3.1).Theorem 3.3.
Assumptions
(H1) and (H3) hold. Suppose∥Φ(T,0)∥=l<1,
then (3.1) has a unique T˜-periodic PC-mild solution xT˜(t)≡(P(f,ck))(t), t≥0,
given by(3.4)xT˜(t)=Φ(t,0)[I−Φ(T˜,0)]−1z+∫0tΦ(t,θ)f(θ)dθ+∑0≤τk<tΦ(t,τk+)ck≡(P(f,ck))(t),where(3.5)z=∫0T˜Φ(T˜,θ)f(θ)dθ+∑0≤τk<T˜Φ(T˜,τk+)ck.Further,
operator(3.6)P:LT1([0,+∞);X)×Xδ→PCT˜([0,+∞);X)is a bounded linear operator,
that is, there exists B˜>0 such that(3.7)∥P(f,ck)∥PCT˜=∥xT˜∥PCT˜≤B˜(q∥f∥LT1+p∑k=1δ∥ck∥),where B is independent on f and ck.Proof.
By (1) and (4) of Lemma2.5, ∥Φ(T˜,0)∥=∥Φ(T,0)∥q=lq<1,
operator [I−Φ(T˜,0)]−1 exists, is bounded and(3.8)[I−Φ(T˜,0)]−1=∑n=0∞Φ(nT˜,0).Thus,(3.9)∥[I−Φ(T˜,0)]−1∥≤11−∥Φ(T˜,0)∥=11−lq.For(3.10)z=∫0T˜Φ(T˜,θ)f(θ)dθ+∑0≤τk<T˜Φ(T˜,τk+)ck,operator equation [I−Φ(T˜,0)]x¯=z has a unique
solution x¯=[I−Φ(T˜,0)]−1z.
Letx(0)=x¯=[I−Φ(T˜,0)]−1z,
consider the Cauchy problem(3.11)x˙(t)=A(t)x(t)+f(t),t∈[0,T˜]∖D˜,Δx(τk)=Bkx(τk)+ck,k=1,2,…,pδ,x(0)=[I−Φ(T˜,0)]−1z.It has a PC-mild solution xT˜(⋅) given by(3.12)xT˜(t)=Φ(t,0)[I−Φ(T˜,0)]−1z+∫0tΦ(t,θ)f(θ)dθ+∑0≤τk<tΦ(t,τk+)ck≡(P(f,ck))(t).By (2), (3), and (4) of Lemma
2.5, one can easily verify that xT˜(⋅) is just the unique T˜-periodic PC-mild solution of (3.1).
Obviously, operatorP:LT1([0,+∞);X)×Xδ→PCT˜([0,+∞);X) is linear.
Fort∈[0,T˜],(3.13)∥P(f,ck)(t)∥=∥xT˜(t)∥≤(∥Φ(t,0)∥∥[I−Φ(T˜,0)]−1∥+1)×(∫0T˜∥Φ(T˜,θ)∥∥f(θ)∥dθ+∑0≤τk<T˜∥Φ(T˜,τk+)∥∥ck∥)≤MT˜(MT˜1−lq+1)(q∥f∥LT1+p∑k=1δ∥ck∥).Let B˜=MT˜(MT˜/(1−lq)+1),
one can obtain the estimation immediately.Theorem 3.4.
Assumptions
(H1) and (H3) hold. Suppose{Φ(t,θ),t≥θ≥0} is exponentially stable, then (3.1) has a
unique T˜-periodic PC-mild solution xT˜(⋅,x¯) given by(3.14)xT˜(t,x¯)=Φ(t,0)x¯+∫0tΦ(t,θ)f(θ)dθ+∑0≤τk<tΦ(t,τk+)ck,where(3.15)x¯=[I−Φ(T˜,0)]−1z,z=∫0T˜Φ(T˜,θ)f(θ)dθ+∑0≤τk<T˜Φ(T˜,τk+)ck,and there exists B^>0 such that(3.16)∥xT˜(t,x¯)∥≤B^(q∥f∥LT1+p∑k=1δ∥ck∥),where B^>0 is independent on f and ck.
Further, for arbitraryx0∈X,
the PC-mild solution of the Cauchy problem (3.2)
corresponding to the initial value x0 satisfies the following inequality:(3.17)∥x(t,x0)−xT˜(t,x¯)∥≤B1B2e−νt,where xT˜(⋅,x¯) is the T˜-periodic PC-mild solution of (3.1), B1>0 is
independent on x0, f,
and ck; B2=∥x0∥+q∥f∥LT1+p∑k=1δ∥ck∥.Proof.
Consider the operator seriesS=∑n=0∞[Φ(T˜,0)]n. By (4) of Lemma 2.5 and the stability of {Φ(⋅,⋅)},
we have(3.18)∥[Φ(T˜,0)]n∥=∥Φ(nT˜,0)∥≤Ke−νnT˜→0asn→∞.Thus, we obtain(3.19)∥S∥≤∑n=0∞∥[Φ(T˜,0)]n∥≤∑n=0∞Ke−νnT˜.Obviously, the series ∑n=0∞Ke−νnT˜ is convergent, thus operator S∈£b(X).
It comes from(3.20)[I−Φ(T˜,0)]S=S[I−Φ(T˜,0)]=Ithat(3.21)S=[I−Φ(T˜,0)]−1∈£b(X).Similar proof in Theorem 3.3, it
is not difficult to verify that (3.1) has a unique T˜-periodic PC-mild solution xT˜(⋅,x¯) and xT˜(⋅,x¯) can be given by (3.14) and (3.15).
Next, verify the estimation (3.16). In fact, for t∈[0,T˜],
we have(3.22)∥xT˜(t,x¯)∥≤∥Φ(t,0)∥∥x¯∥+∫0t∥Φ(t,θ)∥∥f(θ)∥dθ+∑0≤τk<t∥Φ(t,τk+)∥∥ck∥.On the other
hand,(3.23)∥x¯∥≤∥[I−Φ(T˜,0)]−1∥[∫0T˜∥Φ(T˜,θ)∥∥f(θ)∥dθ+∑0≤τk<T˜∥Φ(T˜,τk+)∥∥ck∥]≤∥S∥[∫0T˜Ke−ν(T˜−θ)∥f(θ)∥dθ+Ke−ν(T˜−τk+)p∑k=1δ∥ck∥]≤K∥S∥(q∥f∥LT1+p∑k=1δ∥ck∥).Let B^=K(K∥S∥+1),
one can obtain (3.16) immediately.
System (3.1) has a unique T˜-periodic PC-mild solution xT˜(⋅,x¯) given by (3.14) and (3.15). The PC-mild solution of the Cauchy problem (3.2)
corresponding to initial value x0 can be given by (3.3) replacing x¯ with x0.
Combining with (3.23), we obtain(3.24)∥x(t,x0)−xT˜(t,x¯)∥≤Ke−νt[∥x0∥+K∥S∥(q∥f∥LT1+p∑k=1δ∥ck∥)].Let B1=max⁡{K,K2∥S∥}>0, B2=∥x0∥+q∥f∥LT1+p∑k=1δ∥ck∥,
one can obtain (3.17) immediately.Definition 3.5.
TheT˜-periodic PC-mild solution xT˜(⋅,x¯) of (3.1) is said to be globally asymptotically
stable in the sense that(3.25)lim⁡t→+∞∥x(t,x0)−xT˜(t,x¯)∥=0,where x(⋅,x0) is any PC-mild solution of the Cauchy problem (3.2)
corresponding to initial value x0∈X.By Theorem3.4 and Lemma 2.7 (Lemma 2.9), one can
obtain the following results.Corollary 3.6.
Assumptions
of Lemma2.7 (Lemma 2.9) and (H3) hold. System (3.1) has a unique T˜-periodic PC-mild solution xT˜(⋅,x¯) which is globally asymptotically stable.Theorem 3.7.
Ifx(t,x¯) is a periodic PC-mild solution of (3.1), then its period must
be nT0 for some n∈N.Proof.
LetT˜ be the period of x(t)=x(t,x¯).
Then(3.26)x((T˜+t)±0)=x(t±0),t≥0.Clearly, T˜ is not an impulsive moment, suppose there are   s⁢τks in the interval (0,T˜).
Let t=τ1.
We have(3.27)x(T˜+τ1)=x(τ1),x((T˜+τ1)+)=x(τ1+)=(I+B1)x(τ1)+c1=(I+B1)x(T˜+τ1)+c1,which means
that T˜+τ1 is one of the impulsive moments. Clearly,
there is no τks in the interval (T˜,T˜+τ1).
For otherwise, suppose t¯∈(T˜,T˜+τ1) is an impulsive moment, then(3.28)x((T˜+(t¯−T˜))+)=x(t¯+)≠x(t¯)=x(T˜+(t¯−T˜))=x(t¯−T˜)=x((t¯−T˜)+),which is a contradiction. Thus, T˜+τ1=τs+1, Bs+1=B1,
and cs+1=c1.
Similarly, we have(3.29)T˜+τk=τs+k,Bs+k=Bk,cs+k=ck,k∈Z0+.Now we can claim that s=nδ for some n∈N.
Otherwise, suppose s=nδ+j for some n∈N∪{0} and 1<j<δ.
As a consequence, nT0<T˜<(n+1)T0.
By (3.29), we have(3.30)T˜+τk=τs+k=τnδ+j+k=nT0+τj+k,Bk=Bs+k=Bnδ+j+k=Bj+k,ck=cs+k=cnδ+j+k=cj+k,k∈Z0+,or(3.31)(T˜−nT0)+τk=τj+k,Bk=Bj+k,ck=cj+k,k∈Z0+.Thus T˜−nT0∈(0,T0) is a period of the impulsive perturbations for
the second equation of (3.1), which contradicts to that T0 is the least-positive period.
Therefore,(3.32)T˜+τ1=τs+1=τnδ+1=nT0+τ1and T˜=nT0 for some n∈N.
The proof is complete.


## 4. Example

Consider a
special case of (1.1)(4.1)∂∂tx(t,y)=((Δ−λI)+Q(t))x(t,y)+sin⁡(t+|y|),|y|=y12+y22∈Ω,t>0,t⁢≠τk,k∈Z0+,x(t,y)=0,y∈∂Ω,t>0,Δx(t,y)={0.05Ix(t,y),k=3m−2,−0.05Ix(t,y),k=3m−1,y∈Ω,t>0,t=τk=k+m−14π,⁢k,m∈Z0+.0.05Ix(t,y),k=3m,where Ω⊂R2 is bounded domain and ∂Ω∈C2, Δ is the Laplace operator in R2, λ is a parameter. For each t∈[0,2π], Q(t)∈£b(L2(Ω),L2(Ω)), sup⁡{∥Q(t)∥,t∈[0,2π]}<∞,
and Q(⋅+2π)=Q(⋅).SetX=L2(Ω),
for fixed λ≥1, D(Aλ)=H2(Ω)∩H01(Ω).
Define operator ℬλx=(Δ−λI)x,x∈D(Aλ). By [24, Theorem 2.5], ℬλ is just the infinitesimal generator of a
contraction C0-semigroup in L2(Ω),
that is, ∥Tλ(t)∥≤1.
Obviously, for λ>1, ℬλ+I=Δ−(λ−1)I can generate a exponentially stable C0-semigroup in L2(Ω) and ∥Tλ(t)∥≤e−(λ−1)t.Define operator series(4.2)Uλ,0(t,θ)=Tλ(t−θ),Uλ,n(t,θ)x=∫θtTλ(t−τ)Q(τ)Uλ,n−1(τ,θ)xdτ,0≤θ≤t≤2π,⁢n=1,2,….One can easily
verify the following results:(i)
the evolution operatorUλ(t,θ)=∑n=0∞∥Uλ,n(t,θ)∥, 0≤θ≤t≤2π is uniformly convergence;(ii)
Uλ(t,θ) is the unique solution of the integral
equation(4.3)Uλ(t,θ)x=Tλ(t−θ)x+∫θtTλ(t−τ)Q(τ)Uλ(τ,θ)xdτ;(iii)
Uλ(t,θ) satisfies:(1)
Uλ(t,t)=I, t≥0,(2)
Uλ(t,τ)Uλ(τ,θ)=Uλ(t,θ),t≥τ≥θ≥0,(3)
for every fixedx∈X, (t,θ)→Uλ(t,θ)x is strongly continuous,(4)
∥Uλ(t,θ)∥≤e−(λ−1−∥Q∥)(t−θ),t≥θ≥0,(5)
Uλ(t+2π,θ+2π)=Uλ(t,θ),t≥θ≥0.Thus, linear equationx˙(t)=Aλ(t)x(t),x∈X,t≥0 is well posed, where Aλ(t)≡Δ−(λ−1)I+Q(t).
Obviously, for λ>1+∥Q∥, 𝒜λ(t)≡Δ−(λ−1−∥Q∥)I+Q(t) can determine an exponentially
stable 2π-periodic strongly continuous evolutionary
process {𝒰λ(⋅,⋅)} in L2(Ω).Definex(⋅)(y)=x(⋅,y), f(⋅)(y)=sin⁡(⋅+|y|),
then (4.1) can be abstracted(4.4)x˙(t)=𝒜λ(t)x(t)+f(t),t>0,t≠τk,⁢k∈Z0+,Δx(t)=Bkx(t),t=τk,⁢k∈Z0+,where(4.5)Bk={0.05I,k=3m−2,−0.05I,k=3m−2,0.05I,k=3m,and τk=((k+m−1)/4)π, k,m∈N.Obviously,𝒜λ(⋅+2π)=𝒜λ(⋅), τk+3=τk+π, Bk+3=Bk and f(⋅+2π)(y)=sin⁡(⋅+|y|+2π)=sin⁡(⋅+|y|)=f(⋅)(y), that is, f(⋅+2π)=f(⋅).
For λ>1+∥Q∥+λ¯,
where λ¯>ln⁡[(1.05)2×0.95]2/2π≈0.0147,
by Lemma 2.7, {Φλ(⋅,⋅)} is exponentially stable. Now, all the
assumptions are met in Theorem 3.4. Thus (4.4) has a unique 2π-periodic PC-mild solution x2π(⋅,y)∈PC2π([0+∞);L2(Ω)) which is globally asymptotically stable.That is, supposex(⋅,y) is the PC-mild solution of the following
initial-boundary value problem:(4.6)∂∂tx(t,y)=((Δ−λI)+Q(t))x(t,y)+sin⁡(t+|y|),|y|=y12+y22∈Ω,t>0,t≠τk,⁢k∈Z0+,x(t,y)=0,y∈∂Ω,⁢t>0,x(0,y)=cos⁡|y|∈L2(Ω),Δx(t,y)={0.05Ix(t,y),k=3m−2,−0.05Ix(t,y),k=3m−1,y∈Ω,t>0,t=τk=k+m−14π,k,m∈Z0+.0.05Ix(t,y),k=3m,Then, for λ>1+∥Q∥+λ¯,
where λ¯>0.0147,(4.7)∥x(t,y)−x2π(t,y)∥L2(Ω)=(∫Ω|x(t,y)−x2π(t,y)|2dy)1/2→0ast→+∞.From the above discussion, it is not difficult to find
that a suitable parameterλ chosen by human, which will guarantee the
model (4.1) has a unique 2π-periodic PC-mild solution which is globally
asymptotically stable. That is, we can use a biological approach to maintain
the balance of a single, isolated species or eradicate pests. It provides us a
reliable method for managing the single and isolated species in the nature.



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*Source: 524945-2008-07-21.xml*