# Periodic Solutions of Semilinear Impulsive Periodic System with Time-Varying Generating Operators on Banach Space

Authors: JinRong Wang; X. Xiang; W. Wei
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

A class of semilinear impulsive periodic systems with time-varying generating operators
on Banach space is considered. Using impulsive periodic evolution operator given by us, theT0-periodic PC-mild
solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of
Poincaré operator
and using a new generalized Gronwall  inequality with mixed type integral operators given by us, we utilize
Leray-Schauder fixed point theorem to prove the existence of T0-periodic PC-mild solutions. Our method is an innovation and it is much different from methods of other papers. At last, an example is given for demonstration.


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## Body

## 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. No autonomous periodic systems with applications on
finite dimensional spaces have been extensively studied. Particularly, no
autonomous impulsive periodic systems on finite dimensional spaces are
considered and some important results (such as the existence and stability of
periodic solutions, the relationship between bounded solution and periodic
solution, and robustness by perturbation) are obtained (see [1–5]).Since the end of last century, many authors including
us pay great attention on impulsive systems with time-varying generating
operators on infinite dimensional spaces. Particulary, Dr. Ahmed investigated
optimal control problems of system governed by artificial heart model,
uncertain systems, impulsive system with time-varying generating operators,
access control mechanism model, computer network traffic controllers model, and
active queue management (AQM) system (see [6–14]). We also gave a series of results for semilinear
(strongly nonlinear) impulsive systems with time-varying generating operators
and optimal control problems (see [15–18]).Although, there are some papers on periodic solutions
of periodic system with time-varying generating operators on infinite
dimensional spaces (see [19–22]), to our knowledge, nonlinear impulsive periodic
systems with time-varying generating operators on infinite dimensional (with
unbounded operator) have not been extensively investigated. Recently, we
consider impulsive periodic system on infinite dimensional spaces. For linear
impulsive evolution operator is constructed and T0-periodic PC-mild solution is introduced. The existence of
periodic solutions and alternative theorem, criteria of Massera type, as well
as asymptotical stability and robustness by perturbation are established (see
[23–25]).Herein, we go on studying the semilinear impulsive
periodic system with time-varying generating operators(1.1)x˙(t)=A(t)x(t)+f(t,x),t≠τk,Δx(t)=Bkx(t)+ck,t=τk,in the parabolic case on
infinite dimensional Banach space X,
where {A(t),t∈[0,T0]} is a family of closed densely defined linear
unbounded operators on X and the resolvent of the unbounded operator A(t) is compact. Time sequence 0=τ0<τ1<τ2<⋯<τk⋯, limk→∞τk=∞, τk+δ=τk+T0, Δx(τk)=x(τk+)−x(τk−), k∈ℤ0+, T0 is a fixed positive number and δ∈ℕ denoted the number of impulsive points between 0 and T0. f is a measurable function from [0,∞)×X to X and is T0-periodic in t, Bk+δ=Bk, ck+δ=ck.
This paper is mainly concerned with the existence of periodic solution for
semilinear impulsive periodic system with time-varying generating operators on
infinite dimensional Banach space X.In this paper, we use Leray-Schauder fixed point
theorem to obtain the existence of periodic solutions for semilinear impulsive
periodic system with time-varying generating operators (1.1). First, by virtue
of impulsive evolution operator corresponding to linear homogeneous impulsive
system with time-varying generating operators, we construct a new Poincaré operator P for semilinear impulsive periodic system with
time-varying generating operators (1.1), then overcome some difficulties to
show the compactness of Poincaré operator P which is very important. By a new generalized
Gronwall's inequality with mixed-type integral operators given by us, the
estimate of fixed point set {x=λPx,λ∈[0,1]} is established. Therefore, the existence of T0-periodic PC-mild solutions for semilinear impulsive
periodic system with time-varying generating operators is shown.In order to obtain the existence of periodic
solutions, many authors use Horn's fixed point theorem or Banach fixed point
theorem. In [26, 27], by virtue of
Horn's fixed point theorem and Banach fixed point theorem, respectively, we
also obtain the existence of periodic solutions for impulsive periodic systems.
However, the conditions for Horn's fixed point theorem are not easy to be
verified sometimes and the conditions for Banach's fixed point theorem are too
strong. Here, a new way to show the existence of periodic solutions is given by
us, which is much different from our previous works, and other related results
in the literature. In addition, the conditions are easier to be verified and
more weak compared with some related papers (see [20, 26]). Of course, the new generalized Gronwall's
inequality with mixed-type integral operators given by us which can be used in
other problems have played an essential role in the study of nonlinear problems
on infinite dimensional spaces.This paper is organized as follows. In Section2, some
results of linear impulsive periodic system with time-varying generating
operators and properties of impulsive periodic evolution operator corresponding
to homogeneous linear impulsive periodic system with time-varying generating
operators are recalled. In Section 3, first, the new generalized Gronwall's
inequality with mixed-type integral operator is shown and the T0-periodic PC-mild solution for semilinear impulsive
periodic system with time-varying generating operators (1.1) is introduced. We
construct the suitable Poincaré operator P and give the relation between T0-periodic PC-mild solution and the fixed point of P.
After showing the compactness of the Poincaré operator P and obtaining the boundedness of the fixed
point set {x=λPx,λ∈[0,1]} by virtue of the generalized Gronwall's
inequality, we can use Leray-Schauder fixed point theorem to establish the
existence of T0-periodic PC-mild solutions for semilinear impulsive
periodic system with time-varying generating operators. At last, an example is
given to demonstrate the applicability of our result.


## 2. Linear impulsive periodic system with
time-varying generating operators

In order to study the semilinear impulse periodic
system with time-varying generating operators, we first recall some results
about linear impulse periodic system with time-varying generating operators
here. LetX be a Banach space. £(X) denotes the space of linear operators in X; £b(X) denotes the space of bounded linear operators
in X. £b(X) is the Banach space with the usual supremum
norm. Define D˜={τ1,…,τδ}⊂[0,T0],
where δ∈ℕ denotes the number of impulsive points between [0,T0].
We introduce PC([0,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 PC1([0,T0];X)≡{x∈PC([0,T0];X)∣x˙∈PC([0,T0];X)}. Set(2.1)∥x∥PC=max{supt∈[0,T0]∥x(t+0)∥,supt∈[0,T0]∥x(t−0)∥},∥x∥PC1=∥x∥PC+∥x˙∥PC.It can be seen that endowed with
the norm ∥·∥PC(∥·∥PC1), PC([0,T0];X)(PC1([0,T0];X)) is a Banach space.Consider the following homogeneous linear impulsive
periodic system with time-varying generating operators:(2.2)x˙(t)=A(t)x(t),t≠τk,Δx(τk)=Bkx(τk),t=τkon Banach space X,
where Δx(τk)=x(τk+)−x(τk−), {A(t),t∈[0,T0]} is a family of closed densely defined linear
unbounded operators on X satisfying the following assumption.Assumption (see [28, page 158]).
Fort∈[0,T0] one has
(P1)
The domainD(A(t))=D is independent of t and is dense in X.(P2)
Fort≥0,
the resolvent R(λ,A(t))=(λI−A(t))−1 exists for all λ with Reλ≤0,
and there is a constant M independent of λ and t such that(2.3)∥R(λ,A(t))∥≤M(1+|λ|)−1for  Reλ≤0.(P3)
There exist constantsL>0 and 0<α≤1 such that(2.4)∥(A(t)−A(θ))A−1(τ)∥≤L|t−θ|αfort,θ,τ∈[0,T0].Lemma 2.1 (see [28, page 159]).
Under
the Assumption A1, the Cauchy problemEq.1x˙(t)+A(t)x(t)=0,t∈(0,T0]withx(0)=x0has a unique evolution system {U(t,θ)∣0≤θ≤t≤T0} in X satisfying the following properties:
(1)
U(t,θ)∈£b(X) for 0≤θ≤t≤T0;(2)
U(t,r)U(r,θ)=U(t,θ) for 0≤θ≤r≤t≤T0;(3)
U(·,·)x∈C(Δ,X) for x∈X, Δ={(t,θ)∈[0,T0]×[0,T0]∣0≤θ≤t≤T0};(4)
For0≤θ<t≤T0, U(t,θ) : X→D and t→U(t,θ) is strongly differentiable in X.
The derivative (∂/∂t)U(t,θ)∈£b(X) and it is strongly continuous on 0≤θ<t≤T0.
Moreover,(2.5)∂∂tU(t,θ)=−A(t)U(t,θ)for0≤θ<t≤T0,∥∂∂tU(t,θ)∥£b(X)=∥A(t)U(t,θ)∥£b(X)≤Ct−θ,∥A(t)U(t,θ)A(θ)−1∥£b(X)≤Cfor0≤θ≤t≤T0.(5)
For everyv∈D and t∈(0,T0],U(t,θ)v is differentiable with respect to θ on 0≤θ≤t≤T0(2.6)∂∂θU(t,θ)v=U(t,θ)A(θ)v,and, for each x0∈X,
the Cauchy problem (Eq.1) has a unique classical solution x∈C1([0,T0];X) given by(2.7)x(t)=U(t,0)x0,t∈[0,T0].In addition to Assumption A1, we introduce the
following assumptions.Assumption 2.
There
exitsT0>0 such that A(t+T0)=A(t) for t∈[0,T0].Assumption 3.
Fort≥0,
the resolvent R(λ,A(t)) is compact.Then, we
have the following lemma.Lemma 2.2.
Assumptions A1, A2, and A3 hold. Then evolution system{U(t,θ)∣0≤θ≤t≤T0} in X also satisfying the following two properties:
(6)
U(t+T0,θ+T0)=U(t,θ) for 0≤θ≤t≤T0;(7)
U(t,θ) is compact operator for 0≤θ<t≤T0.In order to introduce an impulsive evolution operator
and give it's properties, we need the following assumption.Assumption 4.
For eachk∈ℤ0+, Bk∈£b(X),
there exists δ∈ℕ such that τk+δ=τk+T0andBk+δ=Bk.Consider the following Cauchy's
problem(2.8)x˙(t)=A(t)x(t),t∈[0,T0]∖D˜,Δx(τk)=Bkx(τk),k=1,2,…,δ,x(0)=x0.For everyx0∈X, D is an invariant subspace of Bk,
using Lemma 2.1, step by step, one can verify that the Cauchy problem (2.8) has
a unique classical solution x∈PC1([0,T0];X) represented by x(t)=𝒮(t,0)x0 where 𝒮(·,·):Δ→£(X) given by (2.9)𝒮(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 𝒮(t,θ)((t,θ)∈Δ) is called impulsive evolution operator
associated with {Bk;τk}k=1∞.The following lemma on the properties of the impulsive
evolution operator𝒮(t,θ)((t,θ)∈Δ) associated with {Bk;τk}k=1∞ are widely used in this paper.Lemma 2.3 (see [24, Lemma 1]).
Assumptions A1, A2,
A3, and B hold. Impulsive evolution operator{𝒮(t,θ),(t,θ)∈Δ} has the following properties.
(1)
For0≤θ≤t≤T0, 𝒮(t,θ)∈£b(X),
that is, sup0≤θ≤t≤T0∥𝒮(t,θ)∥≤MT0,
where MT0>0.(2)
For0≤θ<r<t≤T0, r≠τk, 𝒮(t,θ)=𝒮(t,r)𝒮(r,θ).(3)
For0≤θ≤t≤T0 and N∈Z0+, 𝒮(t+NT0,θ+NT0)=𝒮(t,θ).(4)
For0≤t≤T0 and M∈Z0+, 𝒮(MT0+t,0)=𝒮(t,0)[𝒮(T0,0)]M.(5)
𝒮(t,θ) is compact operator for 0≤θ<t≤T0.Here, we note that system (2.2) has a T0-periodic PC-mild solution x if and only if S(T0,0) has a fixed point. The impulsive evolution
operator {𝒮(t,θ),(t,θ)∈Δ} can be used to reduce the existence of T0-periodic PC-mild solutions for linear impulsive periodic
system with time-varying generating operators to the existence of fixed points
for an operator equation. This implies that we can build up the new framework
to study the periodic PC-mild solutions for the semilinear impulsive
periodic system with time-varying generating operators on Banach space.Now we introduce thePC-mild solution of Cauchy's problem (2.8) and T0-periodic PC-mild solution of the system (2.2).Definition 2.4.
For
everyx0∈X,
the function x∈PC([0,T0];X) given by x(t)=𝒮(t,0)x0 is said to be the PC-mild solution of the Cauchy problem (2.8).Deffinition 2.2.
A
functionx∈PC([0,+∞);X) is said to be a T0-periodic PC-mild solution of system (2.2) if it is a PC-mild solution of Cauchy's problem (2.8) corresponding
to some x0 and x(t+T0)=x(t) for t≥0.Secondly, we recall the following nonhomogeneous
linear impulsive periodic system with time-varying generating
operators(2.10)x˙(t)=A(t)x(t)+f(t),t≠τk,Δx(τk)=Bkx(τk)+ck,t=τk,where f∈L1([0,T0];X), f(t+T0)=f(t) for t≥0 and ck satisfies the following assumption.Assumption 5.
For eachk∈ℤ0+ and ck∈X,
there exists δ∈ℕ such that ck+δ=ck.In order to study system (2.10), we need to consider
the following Cauchy problem(2.11)x˙(t)=A(t)x(t)+f(t),t∈[0,T0]∖D˜,Δx(τk)=Bkx(τk)+ck,k=1,2,…,δ,x(0)=x0,and introduce the PC-mild solution of Cauchy's problem (2.11) and T0-periodic PC-mild solution of system (2.10).Definition 2.6.
A
functionx∈PC([0,T0];X),
for finite interval [0,T0],
is said to be a PC-mild solution of the Cauchy problem (2.10)
corresponding to the initial value x0∈X and input f∈L1([0,T0];X) if x is given byEq.2x(t)=𝒮(t,0)x0+∫0t𝒮(t,θ)f(θ)dθ+∑0≤τk<t𝒮(t,τk+)ck.Definition 2.7.
A
functionx∈PC([0,+∞);X) is said to be a T0-periodic PC-mild solution of system (2.10) if it satisfies
the expression
(Eq.2) and x(t+T0)=x(t) for t≥0.


## 3. Periodic solutions of semilinear
impulsive periodic system with time-varying generating operators

In order to use
Leray-Schauder theorem to show the existence of periodic solutions, we need the
following generalized Gronwall's inequality with mixed-type integral operator
which is much different from the classical Gronwall's inequality and can be
used in other problems (such as discussion on integral-differential equation of
mixed type, see [15]).
It will play an essential role in the study of nonlinear problems on infinite
dimensional spaces.Lemma 3.1.
Leta≥0, b≥0, c≥0, 0≤λ1≤1, 0≤λ2<1.
If x∈PC([0,T0];X) satisfies(3.1)∥x(t)∥≤a+b∫0t∥x(θ)∥λ1dθ+c∫0T0∥x(θ)∥λ2dθ,∀t∈[0,T0],then there exists a constant M*=M*(a,b,c,λ2,T0)>0 such that(3.2)∥x(t)∥≤M*,∀t∈[0,T0].Proof.
Let(3.3)y(t)=∥x(t)∥+1,∀t∈[0,T0],M=maxt∈[0,T0]∥y(t)∥.Then,(3.4)1≤y(t)≤1+a+b∫0t∥y(θ)∥dθ+cT0Mλ2,∀t∈[0,T0].By Gronwall's inequality, we
obtain(3.5)y(t)≤(1+a+cT0Mλ2)ebt,∀t∈[0,T0].Thus,(3.6)M≤(1+a+cT0Mλ2)ebT0≤(1+a+cT0)ebT0Mλ2.Therefore,(3.7)∥x(t)∥≤M≤((1+a+cT0)ebT0)1/(1−λ2)≡M*(a,b,c,λ2,T0),∀t∈[0,T0].This completes the proof.Now, we consider the following semilinear impulsive
periodic system with time-varying generating operators(3.8)x˙(t)=A(t)x(t)+f(t,x),t≠τk,Δx(t)=Bkx(t)+ck,t=τk,and introduce Poincaré operator and study the T0-periodic PC-mild solution of system (3.8).In order to study the system (3.8), we first consider
Cauchy's problem(3.9)x˙(t)=A(t)x(t)+f(t,x),t∈[0,T0]∖D˜,Δx(τk)=Bkx(τk)+ck,k=1,2,…,δ,x(0)=x¯.By virtue of the expression of
the PC-mild solution of the Cauchy problem (2.11), we
can introduce the PC-mild solution of the Cauchy problem
(3.9).Definition 3.2.
A
functionx∈PC([0,T0];X) is said to be a PC-mild solution of the Cauchy problem (3.9)
corresponding to the initial value x¯∈X if x satisfies the following integral
equation:(3.10)x(t)=𝒮(t,0)x¯+∫0t𝒮(t,θ)f(θ,x(θ))dθ+∑0≤τk<t𝒮(t,τk+)ckfort∈[0,T0].Now, we introduce theT0-periodic PC-mild solution of system (3.8).Definition 3.3.
A
functionx∈PC([0,+∞);X) is said to be a T0-periodic PC-mild solution of system (3.8) if it is a PC-mild solution of Cauchy's problem (3.9)
corresponding to some x¯ and x(t+T0)=x(t) for t≥0.In order to prove the existence of thePC-mild solution of Cauchy's problem (3.9), we
need the following assumption.Assumption 6.
(F1):f:[0,∞)×X→X is measurable for t≥0 and for any x,y∈X satisfying ∥x∥,∥y∥≤ρ there exists a positive constant Lf(ρ)>0 such that(3.11)∥f(t,x)−f(t,y)∥≤Lf(ρ)∥x−y∥.
(F2): There exists a positive
constantMf>0 such that(3.12)∥f(t,x)∥≤Mf(1+∥x∥)∀x∈X.
(F3):f(t,x) is T0-periodic in t.
That is, f(t+T0,x)=f(t,x), t≥0.Then, we
have the following theorem.Theorem 3.4.
Assumptions A1, F(F1), and F(F2) hold. Cauchy's problem (3.9) has a
unique PC-mild solution given by(3.13)x(t,x¯)=𝒮(t,0)x¯+∫0t𝒮(t,θ)f(θ,x(θ,x¯))dθ+∑0≤τk<t𝒮(t,τk+)ck.Proof.
Under the Assumptions A1, F(F1),
and F(F2), using the similar method of Theorem 5.3.3 (see [28, page 169]), Cauchy's problem
(3.14)x˙(t)=A(t)x(t)+f(t,x),t∈[s,τ],x(s)=x¯∈Xhas a unique mild
solution(3.15)x(t)=U(t,s)x¯+∫stU(t,θ)f(θ,x(θ))dθ.
In general, fort∈(τk,τk+1],
Cauchy's problem(3.16)x˙(t)=A(t)x(t)+f(t,x),t∈(τk,τk+1],x(τk)=xk≡(I+Bk)x(τk)+ck∈Xhas a unique PC-mild solution(3.17)x(t)=U(t,τk)xk+∫τktU(t,θ)f(θ,x(θ))dθ.
Combining all of solutions on[τk,τk+1] (k=1,…,δ), one can obtain the PC-mild solution of the Cauchy problem (3.9)
given by(3.18)x(t,x¯)=𝒮(t,0)x¯+∫0t𝒮(t,θ)f(θ,x(θ,x¯))dθ+∑0≤τk<t𝒮(t,τk+)ck.In order to study the periodic solutions of the system
(3.8), we define Poincaré operator from X to X as following:(3.19)P(x¯)=x(T0,x¯)=𝒮(T0,0)x¯+∫0T0𝒮(T0,θ)f(θ,x(θ,x¯))dθ+∑0≤τk<T0𝒮(T0,τk+)ck,where x(·,x¯) denote the PC-mild solution of the Cauchy problem (3.9)
corresponding to the initial value x(0)=x¯.
We note that a fixed point of P gives rise to a periodic solution.Lemma 3.5.
System (3.8) has a T0-periodic PC-mild solution if and only if P has a fixed point.Proof.
Supposex(·)=x(·+T0),
then x(0)=x(T0)=P(x(0)).
This implies that x(0) is a fixed point of P.
On the other hand, if Px0=x0, x0∈X,
then for the PC-mild solution x(·,x0) of the Cauchy problem (3.9) corresponding to
the initial value x(0)=x0,
we can define y(·)=x(·+T0,x0),
then y(0)=x(T0,x0)=Px0=x0.
Now, for t>0,
we can use the (2), (3), and (4) of Lemma 2.3 and Assumptions A2, B, C,
F(F3) to obtain(3.20)y(t)=x(t+T0,x0)=𝒮(t+T0,T0)𝒮(T0,0)x0+∫0T0𝒮(t+T0,T0)𝒮(T0,θ)f(θ,x(θ,x0))dθ+∑0≤τk<T0𝒮(t+T0,T0)𝒮(T0,τk+)ck+∫T0t+T0𝒮(t+T0,θ)f(θ,x(θ,x0))dθ+∑T0≤τk+δ<t+T0𝒮(t+T0,τk+δ+)ck=𝒮(t,0){𝒮(T0,0)x0+∫0T0𝒮(T0,θ)f(θ,x(θ,x0))dθ+∑0≤τk<T0𝒮(T0,τk+)ck}+∫0t𝒮(t+T0,s+T0)f(s+T0,x(s+T0,x0))ds+∑0≤τk<t𝒮(t,τk+)ck=𝒮(t,0)y(0)+∫0t𝒮(t,s)f(s,y(s,y(0)))ds+∑0≤τk<t𝒮(t,τk+)ck.This implies that y(·,y(0)) is a PC-mild solution of Cauchy's problem (3.9) with
initial value y(0)=x0.
Thus, the uniqueness implies that x(·,x0)=y(·,y(0))=x(·+T0,x0),
so that x(·,x0) is a T0-periodic.Next, we show thatP defined by (3.19) is a continuous and compact
operator.Lemma 3.6.
Assumptions A1, A3, F(F1), and F(F2) hold. Then,P is a continuous and compact operator.Proof.
(1) Show thatP is a continuous operator on X.
Letx¯, y¯∈Ξ⊂X,
where Ξ is a bounded subset of X.
Suppose x(·,x¯) and x(·,y¯) are the PC-mild solutions of Cauchy's problem (3.9)
corresponding to the initial value x¯ and y¯∈X,
respectively, given by(3.21)x(t,x¯)=𝒮(t,0)x¯+∫0t𝒮(t,θ)f(θ,x(θ,x¯))dθ+∑0≤τk<t𝒮(T0,τk+)ck,x(t,y¯)=𝒮(t,0)y¯+∫0t𝒮(t,θ)f(θ,x(θ,y¯))dθ+∑0≤τk<t𝒮(T0,τk+)ck.Thus, we obtain(3.22)∥x(t,x¯)∥≤MT0∥x¯∥+MT0MfT0+MT0∑0≤τk<T0∥ck∥+MT0∫0t∥x(θ,x¯)∥dθ,∥x(t,y¯)∥≤MT0∥y¯∥+MT0MfT0+MT0∑0≤τk<T0∥ck∥+MT0∫0t∥x(θ,y¯)∥dθ.By Gronwall's inequality with
impulse [5, Lemma 1.7.1], one can verify that there exist constants M1* and M2*>0 such that(3.23)∥x(t,x¯)∥≤M1*,∥x(t,y¯)∥≤M2*.Let ρ=max{M1*,M2*}>0,
then ∥x(·,x¯)∥, ∥x(·,y¯)∥≤ρ which imply that they are locally bounded. By
Assumption F(F1), we obtain(3.24)∥x(t,x¯)−x(t,y¯)∥≤∥𝒮(t,0)∥∥x¯−y¯∥+∫0t∥𝒮(t,θ)∥f(θ,x(θ,x¯))−f(θ,x(θ,y¯))∥dθ≤MT0∥x¯−y¯∥+MT0Lf(ρ)∫0t∥x(θ,x¯)−x(θ,y¯)∥dθ.By Gronwall's inequality with
impulse [5, Lemma 1.7.1] again, one can verify that there exists constant M>0 such that(3.25)∥x(t,x¯)−x(t,y¯)∥≤MMT0∥x¯−y¯∥≡L∥x¯−y¯∥,∀t∈[0,T0],which implies
that(3.26)∥P(x¯)−P(y¯)∥=∥x(T0,x¯)−x(T0,y¯)∥≤L∥x¯−y¯∥.Hence, P is a continuous operator on X.
(2) Verifies thatP takes a bounded set into a precompact set in X.
LetΓ be a bounded subset of X.
Define K=PΓ={P(x¯)∈X∣x¯∈Γ}.
For0<ε<t≤T0,
define(3.27)Kε=PεΓ=𝒮(T0,T0−ε){x(T0−ε,x¯)∣x¯∈Γ}.
Next, we show thatKε is precompact in X.
In fact, for x¯∈Γ fixed, we have(3.28)∥x(T0−ε,x¯)∥=∥𝒮(T0−ε,0)x¯+∫0T0−ε𝒮(T0−ε,θ)f(θ,x(θ,x¯))dθ+∑0≤τk<T0−ε𝒮(T0−ε,τk+)ck∥≤MT0∥x¯∥+MT0MfT0+∫0T0∥x(θ,x¯)∥dθ+MT0∑0≤τk<T0∥ck∥≤MT0∥x¯∥+MT0MfT0+T0ρ+MT0∑0≤τk<T0∥ck∥.This implies that the set {x(T0−ε,x¯)∣x¯∈Γ} is bounded.
By (5) of Lemma2.3, 𝒮(T0,T0−ε) is a compact operator. Thus, Kε is precompact in X.
On the other hand, for arbitraryx¯∈Γ,(3.29)Pε(x¯)=𝒮(T0,0)x¯+∫0T0−ε𝒮(T0,θ)f(θ,x(θ,x¯))dθ+∑0≤τk<T0−ε𝒮(T0,τk+)ck.Thus, combined with (3.19), we
have(3.30)∥Pε(x¯)−P(x¯)∥≤∥∫0T0−ε𝒮(T0,θ)f(θ,x(θ))dθ−∫0T0𝒮(T0,θ)f(θ,x(θ))dθ∥+∥∑0≤τk<T0−ε𝒮(T0,τk+)ck−∑0≤τk<T0𝒮(T0,τk+)ck∥≤∫T0−εT0∥𝒮(T0,θ)∥∥f(θ,x(θ))∥dθ+MT0∑T0−ε≤τk<T0∥ck∥≤2MT0Mf(1+ρ)ε+MT0∑T0−ε≤τk<T0∥ck∥.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 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.In order to use Leary-Schauder fixed pointed theorem
to examine that the operatorP has a fixed point, we have to make the
Assumption F(F2) a little strong as following.(F2′): there exist constantNf>0 and 0<λ<1 such that(3.31)∥f(t,x)∥≤Nf(1+∥x∥λ)∀x∈X.Now, we can give the main results in this paper.Theorem 3.7.
Assumptions A1, A2, A3, B, C, F(F1), F(F2′), and F(F3) hold. Then system (3.8) has a T0-periodic PC-mild solution on [0,+∞).Proof.
By (5) of Lemma2.3, 𝒮(T0,0) is a compact operator on infinite dimensional
space X.
Thus, 𝒮(T0,0)≠αI, α∈ℝ.
Then, there exists β>0 such that(3.32)∥[σ𝒮(T0,0)−I]x¯∥≥β∥x¯∥forσ∈[0,1].In fact, define Πσ=I−σ𝒮(T0,0), σ∈[0,1],
and(3.33)Πσ:[0,1]→£b(X),h(σ)=∥Πσ∥:[0,1]→ℝ+.It is obvious that h∈C([0,1];ℝ+).
Thus, there exist σ*∈[0,1] and β>0 such that(3.34)h(σ*)=min{h(σ)∣σ∈[0,1]}≥β>0.If not, there exits σ¯∈[0,1] such that h(σ¯)=0.
We can assert that σ¯≠0 unless h(σ¯)=1.
Thus, for σ¯∈(0,1](3.35)𝒮(T0,0)=1σ¯Iwhere1σ¯≥1,which is a contradiction with 𝒮(T0,0)≠αI, α∈ℝ.
By Theorem3.4, for fixed x¯∈X,
the Cauchy problem (3.9) corresponding to the initial value x(0)=x¯ has the PC-mild solution x(·,x¯).
By Lemma 3.6, the operator P defined by (3.19), is compact.
According to Leray-Schauder fixed point theory, it
suffices to show that the set{x¯∈X∣x¯=σPx¯,σ∈[0,1]} is a bounded subset of X.
In fact, let x¯∈{x¯∈X∣x¯=σPx¯,σ∈[0,1]},
we have(3.36)β∥x¯∥≤∥[σ𝒮(T0,0)−I]x¯∥=σ∫0T0∥𝒮(T0,θ)∥∥f(θ,x(θ,x¯))∥dθ+σ∑0≤τk<T0∥𝒮(T0,τk+)∥∥ck∥.
By Assumption F(F2′),(3.37)∥x¯∥≤σβ∫0T0∥𝒮(T0,θ)∥∥f(θ,x(θ,x¯))∥dθ+σβ∑0≤τk<T0∥𝒮(T0,τk+)∥∥ck∥≤σβMT0(NfT0+Nf∫0T0∥x(θ,x¯)∥λdθ+∑0≤τk<T0∥ck∥).
Fort∈[0,T0],
we obtain(3.38)∥x(t,x¯)∥≤MT0∥x¯∥+MT0NfT0+MT0Nf∫0t∥x(θ,x¯)∥λdθ+MT0∑0≤τk<t∥ck∥≤σβMT02(NfT0+Nf∫0T0∥x(θ,x¯)∥λdθ+∑0≤τk<T0∥ck∥)+MT0NfT0+MT0Nf∫0t∥x(θ,x¯)∥λdθ+MT0∑0≤τk<t∥ck∥≤MT0NfT0+σβMT02NfT0+σβMT02∑0≤τk<T0∥ck∥+MT0∑0≤τk<T0∥ck∥+MT0Nf∫0t∥x(θ,x¯)∥λdθ+σβMT02Nf∫0T0∥x(θ,x¯)∥λdθ.By Lemma 3.1, there exists M*>0 such that(3.39)∥x(t,x¯)∥≤M*fort∈[0,T0].This implies that ∥x(0,x¯)∥=∥x¯∥≤M* for all x¯∈{x¯∈X∣x¯=σPx¯,σ∈[0,1]}.
Thus, by Leray-Schauder fixed point theory, there
exitsx0∈X such that Px0=x0.
By Lemma 3.5, we know that the PC-mild solution x(·,x0) of Cauchy's problem (3.9) corresponding to the
initial value x(0)=x0,
is just T0-periodic. Therefore x(·,x0) is a T0-periodic PC-mild solution of system (3.8).


## 4. Application

In this
section, an example is given to illustrate our theory. Consider the following
problem:(4.1)∂∂tx(t,y)=sintΔx(t,y)+3x2/3(t,y)+2+sin(t,y),y∈Ω,t∈(0,2π]∖{12π,π,32π},Δx(τi,y)=x(τi+0,y)−x(τi−0,y)={0.05Ix(τi,y),i=1,−0.05Ix(τi,y),i=2,0.05Ix(τi,y),i=3,y∈Ω,τi=i2π,i=1,2,3,x(t,y)|y∈∂Ω=0,t>0,x(0,y)=x(2π,y),where Ω⊂R3 is bounded domain and ∂Ω∈C3.DefineX=L2(Ω), D(A)=H2(Ω)⋂H01(Ω),
and A(t)x=−sint(∂2x/∂y12+∂2x/∂y22+∂2x/∂y32) for x∈D(A),
which satisfies Assumptions A1, A2, and A3. Define x(·)(y)=x(·,y), sin(·)(y)=sin(·,y), f(·,x(·))(y)=3x2/3(·,y)+2+sin(·,y) and(4.2)Bi={0.05I,i=3m−2,−0.05I,i=3m−1,τi=iπ2,i,m∈ℕ.0.05I,i=3m.It is obvious thatτi+3=τi+2π, Bi+3=Bi∈£b(L2(Ω)) which satisfy the Assumption B.For anyx,z∈X satisfying 1≤∥x∥,∥z∥≤ρ,(4.3)∥f(t,x)−f(t,z)∥≤∥3x2/3+2−3z2/3+2∥≤(∥x∥1/3+∥z∥1/3)∥x1/3−z1/3∥3∥x∥2/3+2+3∥z∥2/3+2≤ρ1/32∥x−z∥.Meanwhile,(4.4)∥f(t,x)∥≤∥3x2/3+2∥+∥sint∥≤3∥x∥2/3+2+1≤3(1+∥x∥2/3),f(·+2π,x)=3x2/3+2+sin(·+2π)=3x2/3+2+sin(·)=f(·,x).These imply that Assumptions
F(F1), F(F2′), and F(F3) hold. It comes
from(4.5)𝒮(2π,0)=U(2π,τ3)(I+B3)U(τ3,τ2)(I+B2)U(τ2,τ1)(I+B1)U(τ1,0)=(0.952·1.05)U(2π,0)that 𝒮(2π,0)≠αI, α>1.
In fact, if (0.952·1.05)T(2π)=αI,
then U(2π,0)=(α/(0.952·1.05))I cannot be compact in L2(Ω), which is a contradiction with U(t,θ)which is compact
operator for 0≤θ<t≤T0.
Thus problem (4.1) can be rewritten as(4.6)x˙(t)=A(t)x(t)+f(t,x),t∈(0,2π]∖{12π,π,32π},Δx(i2π)=Bix(i2π),i=1,2,3,x(0)=x(2π).It satisfies all the assumptions
given in Theorem 3.7, our results can be used to problem (4.1). That is,
problem (4.1) has a 2π-periodic PC-mild solution x2π(·,y)∈PC2π([0+∞);L2(Ω)),
where(4.7)PC2π([0,+∞);L2(Ω))≡{x∈PC([0,+∞);L2(Ω))∣x(t)=x(t+2π),t≥0}.



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