# Existence of Periodic Solutions for Integrodifferential Impulsive Periodic System on Banach Space

Authors: JinRong Wang; X. Xiang; W. Wei
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

This paper deals with a class of integrodifferential impulsive periodic systems on Banach space.
Using impulsive periodic evolution operator given by us, theT0-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.


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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. 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 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 PC-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 PC-mild solutions and optimal controls are
presented (see [9]).Herein, we go on studying the following
integrodifferential impulsive periodic system:(1.1)x˙(t)=Ax(t)+f(t,x,∫0tg(t,s,x)ds),t≠τk,Δx(t)=Bkx(t)+ck,t=τk,on infinite dimensional Banach
space X,
where 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 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 Bk+δ=Bk, ck+δ=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 B-norm given by us, the estimate of PC-mild solutions is established. Therefore, the
existence of T0-periodic PC-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 Section2, 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 PC-mild solutions for integrodifferential
impulsive system is presented. In Section 4, the T0-periodic PC-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 PC-mild solution and the fixed point of the Poincaré operator P.
After showing the continuity and compactness of the Poincaré operator P,
we can use Horn's fixed point theorem to establish the existence of T0-periodic PC-mild solutions for integrodifferential
impulsive periodic system when the PC-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. LetX 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. £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 and 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
a Banach space is endowed with the norm ∥⋅∥PC(∥⋅∥PC1), PC([0,T0];X)(PC1([0,T0];X)).Firstly, we consider the homogeneous linear impulsive
periodic system(2.2)x.(t)=Ax(t),t≠τk,Δx(t)=Bkx(t),t=τk.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 D(A).[H1.2]
There existsδ∈N such that τk+δ=τk+T0.[H1.3]
For eachk∈ℤ0+, δ∈N, Bk∈£b(X) and Bk+δ=Bk.In order to study system (2.2), we need to consider
the associated Cauchy problem(2.3)x.(t)=Ax(t),t∈[0,T0]∖D˜,Δx(τk)=Bkx(τk),k=1,2,…,δ,x(0)=x¯.Ifx¯∈D(A) and D(A) 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∈PC1([0,T0];X) represented by x(t)=S(t,0)x¯ where
(2.4)S(⋅,⋅):Δ={(t,θ)∈[0,T0]×[0,T0]∣0≤θ≤t≤T0}→£b(X)given by(2.5)S(t,θ)={T(t−θ),τk−1≤θ≤t≤τk,T(t−τk+)(I+Bk)T(τk−θ),τk−1≤θ<τk<t≤τk+1,T(t−τk+)[∏θ<τj<t(I+Bj)T(τj−τj−1+)](I+Bi)T(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∞.Now we introduce thePC-mild solution of Cauchy problem (2.3) and T0-periodic PC-mild solution of the system (2.2).Definition 2.1.
For everyx¯∈X,
the function x∈PC([0,T0];X) given by x(t)=S(t,0)x¯ is said to be the PC-mild solution of the Cauchy problem (2.3).Definition 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 problem (2.3)
corresponding to some x¯ and x(t+T0)=x(t) for t≥0.The following lemma gives the properties of the
impulsive evolution operator{S(t,θ),(t,θ)∈Δ} associated with {Bk;τk}k=1∞ which are widely used in the sequel.Lemma 2.3 (see [26, Lemma 1]).
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 N∈Z0+, S(NT0+t,0)=S(t,0)[S(T0,0)]N.(5)
If{T(t),t>0} is a compact semigroup in X,
then S(t,θ) is a 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 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 integrodifferential
impulsive periodic system on Banach space.Consider the nonhomogeneous linear impulsive periodic
system(2.6)x˙(t)=Ax(t)+f(t),t≠τk,Δx(t)=Bkx(t)+ck,t=τk,and the associated Cauchy
problem(2.7)x˙(t)=Ax(t)+f(t),t∈[0,T0]∖D˜,Δx(τk)=Bkx(τk)+ck,k=1,2,…,δ,x(0)=x¯,where f∈L1([0,T0];X), f(t+T0)=f(t), and ck+δ=ck.Now we introduce thePC-mild solution of Cauchy problem (2.7) and T0-periodic PC-mild solution of system (2.6).Definition 2.4.
A functionx∈PC([0,T0];X),
for finite interval [0,T0],
is said to be a PC-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(2.8)x(t)=S(t,0)x¯+∫0tS(t,θ)f(θ)dθ+∑0≤τk<tS(t,τk+)ck.Definition 2.5.
A functionx∈PC([0,+∞);X) is said to be a T0-periodic PC-mild solution of system (2.6) if it is a PC-mild solution of Cauchy problem (2.7)
corresponding to some x¯ and x(t+T0)=x(t) for t≥0.


## 3. The Generalized Gronwall Inequality and Existence of Solutions

In order to
derive an estimate of thePC-mild solutions, we introduce the following
generalized Gronwall inequality with impulse and B-norm.Lemma 3.1.
Letx∈PC([0,∞),X) and let it satisfy
the following inequality:(3.1)∥x(t)∥≤a+b∫0t∥x(θ)∥λ1dθ+c∫0t∥xθ∥Bλ2dθ,where a, b, c≥0,0≤λ1,λ2≤1 are constants, and(3.2)∥xθ∥B=max{sup0≤ξ≤θ∥x(ξ+0)∥,sup0≤ξ≤θ∥x(ξ−0)∥}.
Then one
has(3.3)∥x(t)∥≤(a+1)e(b+c)t.Proof.
(i) For0≤λ1, λ2<1,
let λ=max{λ1,λ2}∈[0,1) and(3.4)y(t)={1,∥x(t)∥≤1,x(t),∥x(t)∥>1.Then(3.5)∥x(t)∥≤∥y(t)∥≤(a+1)+b∫0t∥y(θ)∥λdθ+c∫0t∥yθ∥Bλdθ,∀t∈[0,T0],
Using (3.5), we obtain(3.6)∥yt∥Bλ≤(a+1)+(b+c)∫0t∥yθ∥Bλdθ.Define(3.7)u(t)=(a+1)+(b+c)∫0t∥yθ∥Bλdθ,then we get(3.8)u˙(t)=(b+c)∥yt∥Bλ,t≠τk,u(0)=a+1,u(τk+0)=u(τk).Since ∥yt∥Bλ≤u(t),
we then have(3.9)u˙(t)≤(b+c)u(t),t≠τk,u(0)=a+1,u(τk+0)=u(τk).
Fort∈(τk,τk+1],
by (3.9), we obtain(3.10)u(t)≤u(τk+0)e(b+c)(t−τk)=u(τk)e(b+c)(t−τk),further,(3.11)u(t)≤(a+1)e(b+c)t,thus,(3.12)∥x(t)∥≤∥yt∥B≤(a+1)e(b+c)t.
(ii) Forλ1=λ2=1,
we only need to define(3.13)u1(t)=a+(b+c)∫0t∥xθ∥Bdθ.Similar to the proof in (i), one
can obtain(3.14)∥x(t)∥≤∥xt∥B≤ae(b+c)t.Combining (i) and (ii), one can
complete the proof.Now, we consider the following integrodifferential
impulsive periodic system:(3.15)x˙(t)=Ax(t)+f(t,x,∫0tg(t,s,x)ds),t≠τk,Δx(t)=Bkx(t)+ck,t=τk,and the associated Cauchy
problem(3.16)x˙(t)=Ax(t)+f(t,x,∫0tg(t,s,x)ds),t∈[0,T0]∖D˜,Δx(τk)=Bkx(τk)+ck,k=1,2,…,δ,x(0)=x¯.By virtue of the expression of thePC-mild solution of the Cauchy problem (2.7), we
can introduce the PC-mild solution of the Cauchy problem
(3.16).Definition 3.2.
A functionx∈PC([0,T0];X) is said to be a PC-mild solution of the Cauchy problem (3.16)
corresponding to the initial value x¯∈X if x satisfies the following integral
equation:(3.17)x(t)=S(t,0)x¯+∫0tS(t,θ)f(θ,x(θ),∫0θg(θ,s,x(s))ds)dθ+∑0≤τk<tS(t,τk+)ck,fort∈[0,T0].
Now, we introduce theT0-periodic PC-mild solution of system (3.15).Definition 3.3.
A functionx∈PC([0,+∞);X) is said to be a T0-periodic PC-mild solution of system (3.15) if it is a PC-mild solution of Cauchy problem (3.16)
corresponding to some x¯ and x(t+T0)=x(t) 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(3.18)∥f(t,x1,y1)−f(t,x2,y2)∥≤Lf(ρ)(∥x1−x2∥+∥y1−y2∥).[H2.2]
There exists a positive constantMf>0 such that (3.19)∥f(t,x,y)∥≤Mf(1+∥x∥+∥y∥),∀x,y∈X.[H2.3]
f(t,x,y) is   T0-periodic in t, that is,(3.20)f(t+T0,x,y)=f(t,x,y),t≥0.[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(3.21)∥g(t,s,x)−g(t,s,y)∥≤Lg(ρ)∥x−y∥.[H2.5]
There exists a positive constantMg>0 such that(3.22)∥g(t,s,x)∥≤Mg(1+∥x∥),∀x∈X.[H2.6]
g(t,s,x) are T0-periodic in t and s,
that is,(3.23)g(t+T0,s+T0,x)=g(t,s,x),t≥s≥0,∫0T0g(t,s,x)ds=0,t≥s≥0.[H2.7]
For eachk∈ℤ0+ and ck∈X there exists δ∈ℕ such that ck+δ=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,x(s))ds:PC([0,T0];X)→PC([0,T0];X);(2)
for allx1,x2∈PC([0,T0];X) and ∥x1∥PC([0,T0];X),∥x2∥PC([0,T0];X)≤ρ,(3.24)∥∫0tg(t,s,x1(s))ds−∫0tg(t,s,x2(s))ds∥≤Lg(ρ)T0∥(x1)t−(x2)t∥B;(3)
forx∈PC([0,T0];X),(3.25)∥∫0tg(t,s,x(s))ds∥≤MgT0(1+∥xt∥B).Now we present the existence ofPC-mild solution for system (3.16).Theorem 3.5.
Assumptions
[H1.1], [H2.1], [H2.2], [H2.4], and [H2.5] hold, and for eachk∈ℤ0+, Bk∈£b(X), ck∈X are fixed. Then
system (3.16) has a unique PC-mild solution given by(3.26)x(t,x¯)=S(t,0)x¯+∫0tS(t,θ)f(θ,x(θ),∫0θg(θ,s,x(s))ds)dθ+∑0≤τk<tS(t,τk+)ck.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(3.27)x⋅(t)=Ax(t)+f(t,x,∫0tg(t,s,x)ds),t∈[s,τ],x(s)=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(3.28)𝔅={x∈C([s,t1];X)∣∥x(t)−x¯∥≤1,s≤t≤t1},where t1 would be chosen and ∥x(t)∥≤1+∥x¯∥=ρ¯, s≤t≤t1. 𝔅⊆C([s,t1],X) is a closed convex set. Define a map Q on 𝔅 given by(3.29)(Qx)(t)=T(t)x¯+∫stT(t−θ)f(θ,x(θ),∫0θg(θ,s,x(s))ds)dθ.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∈C([s,t1];X) given by(3.30)x(t)=T(t)x¯+∫stT(t−θ)f(θ,x(θ),∫0θg(θ,s,x(s))ds)dθon[s,t1].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.Step 2.
Fort∈(τk,τk+1],
consider the Cauchy problem(3.31)x˙(t)=Ax(t)+f(t,x,∫0tg(t,s,x)ds),t∈(τk,τk+1],x(τk)=xk≡(I+Bk)x(τk)+ck∈X.
By Step 1, Cauchy problem (3.31)
also has a unique PC-mild solution:(3.32)x(t)=T(t−τk)xk+∫τktT(t−θ)f(θ,x(θ),∫0θg(θ,s,x(s))ds)dθ.Step 3.
Combining all solutions on(τk,τk+1] (k=1,…,δ), one can obtain the PC-mild solution of the Cauchy problem (3.16)
given by(3.33)x(t,x¯)=S(t,0)x¯+∫0tS(t,θ)f(θ,x(θ),∫0θg(θ,s,x(s))ds)dθ+∑0≤τk<tS(t,τk+)ck.
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:(4.1)P(x¯)=x(T0,x¯)=S(T0,0)x¯+∫0T0S(T0,θ)f(θ,x(θ,x¯),∫0θg(θ,s,x(s,x¯))ds)dθ+∑0≤τk<T0S(T0,τk+)ck,where x(⋅,x¯) denotes the PC-mild solution of the Cauchy problem (3.16)
corresponding to the initial value x(0)=x¯,
then examine whether P has a fixed point.We first note that a fixed point of thePoincaré operator P gives rise to a periodic solution.Lemma 4.1.
System (3.15)
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 (3.16) 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 properties (2), (3), and (4) of
Lemma 2.3 and assumptions [H1.2], [H1.3], [H2.3], [H2.6], and [H2.7] to arrive
at(4.2)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),∫0θg(θ,s,x(s,x0))ds)dθ+∑0≤τk<T0S(t+T0,T0)S(T0,τk+)ck+∫T0t+T0S(t+T0,θ)f(θ,x(θ,x0),∫0θg(θ,s,x(s,x0))ds)dθ+∑T0≤τk+δ<t+T0S(t+T0,τk+δ+)ck+δ=S(t,0){S(T0,0)x0+∫0T0S(T0,θ)f(θ,x(θ,x0),∫0θg(θ,s,x(s,x0))ds)dθ+∑0≤τk<T0S(T0,τk+)ck+δ}+∫0tS(t+T0,θ+T0)f(θ+T0,x(θ+T0,x0),∫0θ+T0g(θ+T0,s,x(s,x0))ds)dθ+∑T0≤τk+δ<t+T0S(t+T0,τk+δ+)ck+δ=S(t,0)x(T0)+∫0tS(t+T0,θ+T0)f(θ+T0,x(θ+T0,x0),∫T0θ+T0g(θ+T0,s,x(s,x0))ds)dθ+∑T0≤τk+δ<t+T0S(t+T0,τk+δ+)ck+δ=S(t,0)x(T0)+∫0tS(t,θ)f(θ,x(θ+T0,x0),∫0θg(θ+T0,s+T0,x(s+T0,x0))ds)dθ+∑T0≤τk+δ<t+T0S(t+T0,τk+δ+)ck+δ=S(t,0)y(0)+∫0tS(t,θ)f(θ,y(θ,y(0)),∫0θg(θ,s,y(s,y(0)))ds)dθ+∑0≤τk<tS(t,τk+)ck=S(t,0)y(0)+∫0tS(t,θ)f(θ,y(θ,y(0)),∫0θg(θ,s,y(s,y(0)))ds)dθ+∑0≤τk<tS(t,τk+)ck.
This implies that y(⋅,y(0)) is a PC-mild solution of Cauchy problem (3.16) 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.Second, we show that thePoincaré 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 operatorP is a continuous operator of x¯ on X.Proof.
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 problem (3.16)
corresponding to the initial value x¯ and y¯∈X, respectively, given by(4.3)x(t,x¯)=S(t,0)x¯+∫0tS(t,θ)f(θ,x(θ,x¯),∫0θg(θ,s,x(s,x¯))ds)dθ+∑0≤τk<tS(T0,τk+)ck;x(t,y¯)=S(t,0)y¯+∫0tS(t,θ)f(θ,x(θ,y¯),∫0θg(θ,s,x(s,y¯))ds)dθ+∑0≤τk<tS(T0,τk+)ck.Thus, by assumption [H2.2],
property (1) of Lemma 2.3, and
property (3) of Lemma 3.4, we obtain(4.4)∥x(t,x¯)∥≤MT0∥x¯∥+(1+MgT0)MT0MfT0+MT0∑0≤τk<T0∥ck∥+MT0Mf∫0t∥x(θ,x¯)∥dθ+MT0MfMgT0∫0t∥xθ,x¯∥Bdθ≤a0+MT0Mf∫0t∥x(θ,x¯)∥dθ+MT0MfMgT0∫0t∥xθ,x¯∥Bdθ,(4.5)∥x(t,y¯)∥≤MT0∥y¯∥+(1+MgT0)MT0MfT0+MT0∑0≤τk<T0∥ck∥+MT0Mf∫0t∥x(θ,y¯)∥dθ+MT0MfMgT0∫0t∥xθ,y¯∥Bdθ≤b0+MT0Mf∫0t∥x(θ,y¯)∥dθ+MT0MfMgT0∫0t∥xθ,y¯∥Bdθ.
By Lemma3.1, one can verify that there exist constants M1* and M2*>0 such that(4.6)∥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 [H2.1],
property (1) of Lemma2.3, and property
(2) of Lemma 3.4, we obtain(4.7)∥x(t,x¯)−x(t,y¯)∥≤∥S(t,0)∥∥x¯−y¯∥+∫0t∥S(t,θ)∥∥f(θ,x(θ,x¯),∫0θg(θ,s,x(s,x¯))ds)−f(θ,x(θ,y¯),∫0θg(θ,s,x(s,y¯))ds)∥dθ≤MT0∥x¯−y¯∥+MT0Lf(ρ)∫0t∥x(θ,x¯)−x(θ,y¯)∥dθ+MT0Lf(ρ)Lg(ρ)T0∫0t∥xθ,x¯−xθ,y¯∥Bdθ.By Lemma 3.1 again, one can verify
that there exists a constant M3*>0 such that(4.8)∥x(t,x¯)−x(t,y¯)∥≤M3*MT0∥x¯−y¯∥≡L∥x¯−y¯∥,∀t∈[0,T0],which implies
that(4.9)∥P(x¯)−P(y¯)∥=∥x(T0,x¯)−x(T0,y¯)∥≤L∥x¯−y¯∥.Hence, P is a continuous operator of x¯ on X.In the sequel, we need to prove the compactness of
operatorP,
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 operatorP defined by (4.1).Lemma 4.3.
Assumptions
[H1.1], [H2.1], [H2.2], [H2.4], [H2.5], and [H3] hold. Then the operatorP is a compact operator.Proof.
We only need to verify 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≤T0,
define(4.10)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(4.11)∥x(T0−ε,x¯)∥≤∥S(T0−ε,0)x¯∥+∫0T0−ε∥S(T0−ε,θ)f(θ,x(θ,x¯),∫0θg(θ,s,x(s,x¯))ds)∥dθ+∑0≤τk<T0−ε∥S(T0−ε,τk+)ck∥≤MT0∥x¯∥+MT0MfT0(1+MgT0)+MT0Mf∫0T0∥x(θ,x¯)∥dθ+MT0∑0≤τk<T0∥ck∥+MT0MfMgT0∫0T0∥xθ,x¯∥Bdθ≤MT0∥x¯∥+MT0MfT0(1+MgT0)+(1+MgT0)MT0MfT0ρ+MT0∑0≤τk<T0∥ck∥.This implies that the set {x(T0−ε,x¯)∣x¯∈Γ} is totally bounded.
By virtue of the compactness of{T(t),t>0} and property (5)
of Lemma 2.3, S(T0,T0−ε) is a compact operator. Thus, Kε is precompact in X.
On the other hand, for arbitraryx¯∈Γ,(4.12)Pε(x¯)=S(T0,0)x¯+∫0T0−εS(T0,θ)f(θ,x(θ,x¯),∫0θg(θ,s,x(s,x¯))ds)dθ+∑0≤τk<T0−εS(T0,τk+)ck.Thus, combined with (4.1), we
have(4.13)∥Pε(x¯)−P(x¯)∥≤∥∫0T0−εS(T0,θ)f(θ,x(θ,x¯),∫0θg(θ,s,x(s,x¯))ds)dθ−∫0T0S(T0,θ)f(θ,x(θ,x¯),∫0θg(θ,s,x(s,x¯))ds)dθ∥+∥∑0≤τk<T0−εS(T0,τk+)ck−∑0≤τk<T0S(T0,τk+)ck∥≤∫T0−εT0∥S(T0,θ)∥∥f(θ,x(θ,x¯),∫0θg(θ,s,x(s,x¯))ds)∥dθ+MT0∑T0−ε≤τk<T0∥ck∥≤MT0Mf(1+MgT0)(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 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
operatorP,
we can follow and derive periodic PC-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 PC-mild solutions here, so we only define the
(locally) boundedness and ultimate boundedness.Definition 4.4.
One says thatPC-mild solutions of Cauchy problem (3.16) are
bounded if for each B1>0,
there is a B2>0 such that ∥x¯∥≤B1 implies ∥x(t,x¯)∥≤B2 for t≥0.Definition 4.5.
One says thatPC-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 ∥x(t,x¯)∥≤B2 for 0≤t≤k0.Definition 4.6.
One says thatPC-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 ∥x(t,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]).
LetE0⊂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(4.14)Pj(E1)⊂E2,1≤j≤m−1,Pj(E1)⊂E0,m≤j≤2m−1,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 thePC-mild solutions of Cauchy problem (3.16) are
ultimate bounded, then system (3.15) has a T0-periodic PC-mild solution.Proof.
By Theorem3.5, (4.4) in Lemma 4.2, and
Definition 4.5, the Cauchy problem (3.16) corresponding to the initial value x(0)=x¯ has a PC-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 ∥x(t,x¯)∥≤B1 for t≥0.
Furthermore, there is a B2>B1 such that ∥x¯∥≤B1 implies ∥x(t,x¯)∥≤B2 for t≥0.
Now, using ultimate boundedness again, there is a positive integer m such that ∥x¯∥≤B1 implies ∥x(t,x¯)∥≤B for t≥(m−2)T0.
Definey(⋅,y(0))=x(⋅+T0,x¯),
then y(0)=x(T0,x¯)=P(x¯). From (4.2) in Lemma 4.1, we obtain P(y(0))=y(T0,y(0))=x(2T0,x¯). Thus, P2(x¯)=P(P(x¯))=P(y(0))=x(2T0,x¯). Suppose there exists integer m−1 such that Pm−1(x¯)=x((m−1)T0,x¯). By induction we arrive at(4.15)Pm(x¯)=Pm−1(P(x¯))=Pm−1(y(0))=y((m−1)T0,y(0))=x(mT0,x¯).Thus, we obtain(4.16)∥Pj−1(x¯)∥=∥x((j−1)T0,x¯)∥<B2,j=1,2,…,m−1,∥x¯∥<B1;∥Pj−1(x¯)∥=∥x((j−1)T0,x¯)∥<B,j≥m,∥x¯∥<B1.
It comes from Lemma4.3 that P(x¯)=x(T0,x¯) on X is compact. Now let(4.17)H={x¯∈X:∥x¯∥<B2},E2=cl.(conv(P(H))),W={x¯∈X:∥x¯∥<B1},E1=W∩E2,G={x¯∈X:∥x¯∥<B},E0=cl.(conv(P(G))),where conv(Y) is the convex hull of the set Y defined by conv(Y)={∑i=1nλiyi∣n≥1,yi∈Y,λi≥0,∑i=1nλi=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(4.18)Pj(E1)⊂Pj(W)=PPj−1(W)⊂P(H)⊂E2,j=1,2,…,m−1;Pj(E1)⊂Pj(W)=PPj−1(W)⊂P(G)⊂E0,j=m,m+1,…,2m−1.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 PC-mild solution x(⋅,x0) of the Cauchy problem (3.16) corresponding to
the initial value x(0)=x0 is just T0-periodic. Therefore, x(⋅,x0) is a T0-periodic PC-mild solution of system (3.15). This proves
the theorem.At last, an example is given to illustrate our theory.
Consider the following problem:(4.19)∂∂tx(t,y)=Δx(t,y)+x(t,y)+∫0t(1+sin(t−s))3x2(s,y)+2ds,y∈Ω,t≥s,t,s∈(0,2π]∖{12π,π,32π},Δx(τi,y)=x(τi+,y)−x(τi−,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 a bounded domain and ∂Ω∈C3.DefineX=L2(Ω), D(A)=H2(Ω)∩H01(Ω),
and Ax=−(∂2x/∂y12+∂2x/∂y22+∂2x/∂y32) for x∈D(A).
Then, A generates a compact semigroup {T(t),t≥0} on L2(Ω).
Define x(⋅)(y)=x(⋅,y), sin(⋅)(y)=sin(⋅,y), f(⋅,x(⋅),∫0⋅g(⋅,s,x)ds)(y)=x(⋅)(y)+∫0t(1+sin(⋅−s))3x2(⋅)+2ds(y),
and(4.20)Bi={0.05I,i=1,−0.05I,i=2,0.05I,i=3.Thus problem (4.19) can be rewritten as(4.21)x˙(t)=Ax(t)+f(t,x,∫0tg(t,s,x)ds),t≥s,t,s∈(0,2π]∖{12π,π,32π},Δx(i2π)=Bix(i2π),i=1,2,3,x(0)=x(2π).If the PC-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 PC-mild solution x2π(⋅,y)∈PC2π([0+∞);L2(Ω)),
where(4.22)PC2π([0,+∞);L2(Ω))≡{x∈PC([0,+∞);L2(Ω))∣x(t)=x(t+2π),t≥0}.



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