Funkcialaj Ekvacioj, 12 (1969), 23-40

Asymptotically Almost Periodic Solutions

of an Almost Periodic System

By Taro YOSHIZAWA(Tohoku University)

1. Introduction.Assuming the existence of a bounded solution with some stability property,

several authors discussed the existence theorem for almost periodic solutions.In this direction, there are many interesting results for ordinary differentialequations and for functional-differential equations. Fixed point theorems, Lia-punov’s second method and the theory of dynamical systems etc. are applied to

obtain existence theorems for periodic solutions and almost periodic $¥mathrm{s}¥mathrm{o}¥mathrm{l}¥mathrm{u}¥mathrm{t}¥mathrm{i}¥mathrm{o}¥dot{¥mathrm{n}}¥mathrm{s}$ .

Hale [8] and the author [16, 17] have discussed the existence of periodic solu-tions and almost periodic solutions by using Liapunov functions. Seifert [11, 12]has applied Amerio’ $¥mathrm{s}$ result [1]. In these papers, they have assumed that abounded solution is asymptotically stable. Under a weaker condition, propertiesof dynamical systems have been applied to the existence of almost periodic solu-tions. For a periodic system, Deysach and Sell [4] have shown the existenceof an almost periodic solution under the assumption that the system has abounded uniformly stable solution. MiUer [9] has considered almost periodic

differential equations as dynamical systems and assumed that the system has abounded solution which is totally stable. Seifert [13] also has applied a resultof Deysach and Sell under the assumption that a bounded solution has a kind ofstability property called $¥Sigma$-stability which is weaker than total stability. Sell[14, 15] has extended a result of Deysach and Sell to the existence of periodicsolutions for a periodic system and to a more general nonautonomous system.

All of them required the uniqueness of solutions for the initial value problem,

because of dynamical systems.Recently, Coppel [3] has used properties of an asymptotically almost periodic

function introduced by Frechet [5] and proved Miller’s result without the as-

sumption that the solution is unique. To obtain existence theorems for analmost periodic solution, properties of an asymptotically almost periodic functionwere utilized by Reuter [10] for a second order differential equation and by

Halanay [6, 7] for a quasi-linear system. In this paper, more generally, weshall discuss functional-differential equations and we shall see that all of resultsmentioned above will be obtained by using properties of an asymptotically

24 T. $¥mathrm{Y}_{¥mathrm{o}¥mathrm{S}¥mathrm{H}¥mathrm{I}¥mathrm{Z}¥mathrm{A}¥mathrm{W}¥mathrm{A}}$

almost periodic function, and hence we are not required to assume the uniquenessof solutions except for the last theorem.

2. Asymptotically almost periodic function.Let $f(t)$ be a continuous vector function defined on $ a¥leqq t<¥infty$ . $f(t)$ is said

to be asymptotically almost periodic if it is a sum of a continuous almost periodicfunction $p(t)$ and a continuous function $q(t)$ defined on $ a¥leqq t<¥infty$ which tendsto zero as $ t¥rightarrow¥infty$ , that is,

(1) $f(t)=p(t)+q(t)$ .

For an asymptotically almost periodic function $f(t)$ , its decomposition (1) isunique and $f(t)$ is bounded and uniformly continuous on $ a¥leqq t<¥infty$ . Moreover,$f(t)$ is asymptotically almost periodic if and only if for any sequence $¥{¥tau_{k}¥}$ suchthat $¥tau_{k}¥rightarrow¥infty$ as $ k¥rightarrow¥infty$ there exists a subsequence $¥{¥tau_{k_{j}}¥}$ for which $f(t+¥tau_{k_{j}})$ con-

verges uniformly on $ a¥leqq t<¥infty$ . For the details, see [5]. For our purpose,the following property is important.

Lemma 1. Suppose that an asymptotically almost periodic function $f(t)$ is

differentiate and that its derivative $f^{f}(t)$ is also asymptotically almost periodic.Then the decomposition of $f^{¥prime}(t)$ is

(2) $f^{¥prime}(t)=p^{r}(t)+q^{¥prime}(t)$ ,

where $p^{J}(t)$ and $q^{¥prime}(t)$ are the derivatives of $F(t)$ and $q(t)$ in (1), respectively.In this article, we shall consider a system of almost periodic functional-

differential equations. For a given $h¥geqq 0$ , let $C$ denote the space of continuousfunctions defined on $[-h_{;}0]$ . For an $x¥in R^{n}$ , let $|x|$ be any norm. For a $¥varphi¥in C$ ,

we define the norm $||¥varphi||$ of $¥varphi$ by $||¥varphi||=¥sup¥{|¥varphi(¥theta)|; -h¥leqq¥theta¥leqq 0¥}$ . Moreover, wedenote by $C_{a}$ the set of $¥varphi$ such that $||¥varphi||<$ a and $¥overline{C}_{¥alpha}$ is the closure of $C_{a}$ .

Letting $i(t)$ be the right-hand derivative of $x(t)$ , consider a system of functional-differential equations

(3) $¥dot{x}(t)=F(t, x_{t})$ ,

where $x_{t}$ will denote the function $x(t+¥theta)$ , $-h¥leqq¥theta¥leqq 0$ , that is, $x_{t}¥in C$ . For the$¥mathrm{d}¥dot{¥mathrm{e}}$finition of the solution and the existence theorem, see [ $17¥mathrm{Q}$ . We shalldenote by $x(t_{0}, ¥varphi)$ a solution of (3) such that $ x_{t_{0}}(t_{¥dot{0}}, ¥varphi)=¥varphi$ and denote by $x(t$ ;$t_{0}$ , $¥varphi)$ the value at $t$ of $x(t_{0}, ¥varphi)$ .

Let $F(t, ¥varphi)$ be a continuous function defined on $RXD$, where $R=(-¥infty, ¥infty)$

and $D=C_{a}$ or $D=¥overline{C}_{a}$ .

Definition. $F(t, ¥varphi)$ is said to be almost periodic in $t$ uniformly for $¥varphi¥in D,$ .

if for $¥mathrm{a}¥dot{¥mathrm{n}}¥mathrm{y}$$¥mathrm{e}>0$ and any compact set $S¥subset D$, there exists an $l(¥mathrm{e}, S)>0$ . such $¥mathrm{t}¥mathrm{h}¥dot{¥mathrm{a}}¥mathrm{t}$ ,

every interval of length $l(¥mathrm{e}, S)$ contains a $¥tau$ for which

Asymptotically Almost Periodic Solutions of an Almost Periodic System 25

(4) $|F(t+¥tau, ¥varphi)-F(t, ¥varphi)|¥leqq¥epsilon$ for all $t¥in R$ and $¥varphi¥in S$ .

Lemma 2. If $F(t, ¥varphi)$ is almost periodic in $t$ uniformly for $¥varphi¥in D$, for any

real sequence $¥{¥tau_{k}^{¥prime}¥}$ there exists a subsequence $¥{¥tau_{k}¥}$ of $¥{¥tau_{k}^{¥prime}¥}$ and a continuousfunction $G(t, ¥varphi)$ such that

(5) $F(t+¥tau_{k}, ¥varphi)¥rightarrow G(t, ¥varphi)$

uniformly on $R¥times S$ as $ k¥rightarrow¥infty$ , where $S$ is any compact set in D. Moreover,$G(t, ¥varphi)$ is almost periodic in $t$ uniformly for $¥varphi¥in D$.

By using Lemma 2, we can see the following property for an almost periodicfunction. Namely, let $F(t, ¥varphi)$ be almost periodic in $t$ uniformly for $¥varphi¥in D$.

Then, for any real sequences $¥{¥alpha k’¥}$ and $¥{¥beta^{¥prime}k¥}$ there exist subsequences {a$k$ } and$¥{¥beta_{k}¥}$ such that

$¥lim_{rn¥rightarrow¥infty}¥{¥lim_{l¥rightarrow¥infty}F(t+¥alpha_{l}+¥beta_{m}, ¥varphi¥}=¥lim_{k¥rightarrow¥infty}F(t+¥alpha_{k}+¥beta_{k}, ¥varphi)$

uniformly on $R¥times S$, where $S$ is any compact set in $D$. This property was givenby Bochner [2] for an almost periodic function $f(t)$ .

Definition. Let $F(t, ¥varphi)$ be almost periodic in $t$ uniformly for $¥varphi¥in D$. Weshall denote by $T(F)$ the function space consisting of all translates of $F$, that is,$F^{¥tau}¥in T(F)$ where $F^{¥tau}(t, ¥varphi)=F(t+¥tau,¥varphi)$ , $¥tau¥in R$ . Let $H(F)$ be the uniform closure,in the sense of (5), of $T(F)$ . $H(F)$ is called the hull of $F(t, ¥varphi)$ .

Now we assume that $F(t, ¥varphi)$ is continuous on $R¥times¥overline{C}_{B}$ and $F(t, ¥varphi)$ is almostperiodic in $t$ uniformly for $¥varphi¥in¥overline{C}_{B}$ . We shall denote by 1 the interval $ 0¥leqq t<¥infty$ .

Theorem 1. Suppose that the system (3) has a solution $¥xi(t)$ defined onI such that $||¥xi_{t}||¥leqq B$ for $t¥geqq 0$ . If the solution $¥xi(t)$ is asymptotically dmostperiodic, then the system (3) has an almost periodic solution $p(t)$ .

Proof. Since $¥xi(t)$ is asymptotically almost periodic, it has the decomposi-tion $¥xi(t)=p(t)+q(t)$ , where $p(t)$ is almost periodic and $q(t)¥rightarrow 0$ as $ t¥rightarrow¥infty$ . Let$¥{¥tau_{k}¥}$ , $¥tau_{k}>0$ , be a sequence such that $¥tau_{k}¥rightarrow¥infty$ as $ k¥rightarrow¥infty$ and $p(t+¥tau_{k})¥rightarrow p(t)$ as

$ k¥rightarrow¥infty$ . Then we have $¥xi(t+¥tau_{k})=p(t+¥tau_{k})+¥dot{q}(t+¥tau_{k})$ for $t+¥tau_{k}¥geqq 0$ . This impliesthat $|p(t)|¥leqq B$ for all $t¥in R$ . Since $|¥xi(t)|¥leqq B$ for $t¥geqq 0$ and $|p(t)|¥leqq B$ for all$t¥in R$ and since $¥xi(t),p(t)$ are uniformly continuous, there exists a compact set$s¥subset¥overline{c}_{B}$ such that $¥xi_{t}¥in S$ for $t¥in I$ and $p_{t}¥in S$ for $t¥in R$ . Therefore we can showthat $F(t,p_{t})$ is almost periodic in $t$ . Since $¥xi(t)$ is a solution of (3), we have

(6) $¥dot{¥xi}(t)=F(t,p_{t})+F(t, ¥xi_{t})-F(t,p_{t})$ .

On the other hand, it is clear that $F(t, ¥xi_{t})-F(t,p_{t})¥rightarrow 0$ as $ t¥rightarrow¥infty$ , because $¥xi_{t}=$

$p_{t}+q_{t}$ for $t¥geqq h$ . Thus (6) shows that $¥dot{¥xi}(t)$ also is asymptotically almost periodic,and $¥mathrm{h}¥dot{¥mathrm{e}}¥mathrm{n}¥mathrm{c}¥mathrm{e}$ it follows from Lemma 1 that

(7) $¥dot{p}(t)=F(t,p_{t})$ for $t¥in R$ ,

26 T. YoSHIZAWA

which shows that $p(t)$ is an almost periodic solution of (3). This completesthe proof.

Thus, when an almost periodic system has an asymptotically almost periodicsolution, we always can see the existence of an almost periodic solution. Here,it is noticed that we do not require the uniqueness of solutions.

3. Periodic systems.We shall begin $¥grave{¥mathrm{n}}¥mathrm{y}$ discussing the existence of an asymptotically almost

periodic solution of a periodic system

(8) $i(t)=F(t, x_{t})$ ,

where $F(t, ¥varphi)$ is continuous on $R¥times¥overline{C}_{B}*$ and $F(t+¥omega, ¥varphi)=F(t, ¥varphi)$ , $¥omega>0$ . We as-

sume that the system (8) has a solution $¥xi(t)$ defined on I such that $||¥xi_{t}||¥leqq B$

for $t¥geqq 0$ , where $B<B^{*}$ , and moreover, we assume that $|F(t, ¥varphi)|¥leqq L$ on $ R¥times$

$¥overline{C}_{B}*$ .

Theorem 2. If the solution $¥xi(t)$ is uniformly stable for $t¥geqq 0$ , then $¥xi(t)$

is an asymptotically almost periodic solution of (8).Proof. Since $||¥xi_{f}||¥leqq B$ and $|¥dot{¥xi}(t)|¥leqq L$ for $t¥geqq 0$ , there is a compact set

$s¥subset CB*$ such that $¥xi_{t}¥in S$ for all $t¥geqq 0$ . Let $¥{¥tau_{k}¥}$ be a sequence such that $¥tau_{k}>0$

and $¥tau_{k}¥rightarrow¥infty$ as $ k¥rightarrow¥infty$ . For each $¥tau_{k}$ , there exists an positive integer such that$ N_{k}¥omega¥leqq¥tau_{B}<(N_{k}+1)¥omega$. If we set $¥tau_{k}=N_{k}¥omega+¥sigma_{k}$ , then $ 0¥leqq¥sigma_{k}<¥omega$ . Setting $¥xi^{k}(t)=$

$¥xi(t+¥tau_{k})$ , $¥xi^{k}(t)$ is a solution of the system

(9) $¥dot{x}(t)=F(t+¥sigma_{k}, x_{t})$

through $(0, ¥xi_{¥tau_{k}})$ , and clearly $¥xi_{f}^{k}¥in S$ for all $t¥geqq 0$ and any $k$ . Since $¥xi(t)$ is auniformly stable solution of (8), $¥xi^{k}(t)$ is also a uniformly stable solution of (9)with the same pair $(¥epsilon, ¥delta(¥epsilon))$ as the one for $¥xi(t)$ . Since $F(t, ¥varphi)$ is periodic,$ 0¥leqq¥sigma_{k}<¥omega$ and $¥xi_{0}^{k}¥in S$, there exists a subsequence of $¥{¥tau_{k}¥}$ , which we shall denoteby $¥{¥tau_{k}¥}$ again, such that as $ k¥rightarrow¥infty$ , $F(t+¥tau_{k}, ¥varphi)$ converges uniformly for all$t¥in R$ and $¥varphi¥in S$ , $¥sigma_{k}¥rightarrow¥sigma$ , $ 0¥leqq¥sigma¥leqq¥omega$ , and $¥xi^{k_{0}}$ is convergent. Then, for a given$¥epsilon>0$ , there is an positive integer $n_{0}(¥mathrm{e})>0$ such that if $m¥geqq k¥geqq n_{0}(¥mathrm{e})$ ,

(10) $||¥xi_{0}^{k}-¥xi^{m_{0}}||<¥frac{1}{2}¥delta(¥epsilon)$

and

(11) $|¥xi^{m}(t+¥theta)-¥xi^{m}(t+¥theta+¥sigma_{k}-¥sigma_{m})|<¥frac{1}{2}¥delta(¥mathrm{e})$ for $t¥geqq 0$ and $-h¥leqq¥theta¥leqq 0$ ,

because $¥{¥xi^{k}(t)¥}$ is equicontinuous on $-h¥leqq t<¥infty$ . Since (11) implies that if$m¥geqq k¥geqq n_{0}(¥mathrm{e})$ ,

Asymptotically Almost Periodic Solutions of an Almost Periodic System 27

(12) $||1¥xi^{m_{0}}-¥xi^{m_{¥sigma_{k}-¥sigma_{m}}}||<¥frac{1}{2}¥delta(¥mathrm{e})$ ,

we have

(13) $||¥xi_{0}^{k}-¥xi^{m_{¥sigma_{k}-¥sigma_{m}}}||¥leqq||¥xi_{0}^{k}-¥xi_{¥mathrm{o}_{1}^{1}}^{m}|+||¥xi^{m_{0}}-¥xi^{m_{¥sigma_{k}-¥sigma_{n}}}||<¥delta(¥epsilon)$.

Since $¥xi^{m}(t)=¥xi(t+¥tau_{m})$ is a solution of $i(t)=F(t+¥sigma_{m}, x_{t})$ through $(0, ¥xi_{¥tau_{n}})$ ,

$¥xi^{m}(t+¥sigma_{k}-¥sigma_{m})$ is a solution of (9) through $(0, ¥xi^{m_{¥sigma_{k^{¥_}}¥sigma_{m}}})$ . From uniform-stability

of $¥xi^{k}(t)$ and (13), it follows that if $m¥geqq k¥geqq n_{0}(¥mathrm{e})$ , then $||¥xi_{t}^{k}-¥xi^{m_{t+¥sigma_{k}¥sigma_{m}}}¥_||<¥epsilon$ forall $t¥geqq 0$ . On the other hand, (11) implies that if $m¥geqq k¥geqq n_{0}(¥mathrm{e})$ , $||¥xi_{t}^{m}-¥xi^{m_{t+¥sigma_{k}-¥sigma_{m}}}||$

$<¥mathrm{e}$ for all $t¥geqq 0$ . Thus we have

$||¥xi_{t}^{k}-¥xi_{t_{1}^{1}}^{m}|<2¥mathrm{e}$ for all $t$ $¥geqq 0$ and $m¥geqq k¥geqq n_{0}(¥epsilon)$ ,

which implies that

$|¥xi^{k}(t)-¥xi^{m}(t)|<2¥mathrm{e}$ for all $t¥geqq 0$ and $m¥geqq k¥geqq n_{0}(¥epsilon)$ .

Thus we see that for any sequence $¥{¥tau_{k}¥},¥tau_{k}>0$, such that $¥tau_{k}¥rightarrow¥infty$ as $ k¥rightarrow¥infty$, thereexists a subsequence $¥{¥tau_{h_{j}}¥}$ for which $¥xi(t+¥tau_{k_{j}})¥mathrm{c}¥mathrm{o}¥mathrm{n}¥mathrm{v}¥mathrm{e}¥tau ¥mathrm{g}¥mathrm{e}¥mathrm{s}$ uniformly on I as $ j¥rightarrow$

$¥infty$ . This shows that $¥xi(t)$ is asymptotically almost periodic.Remark. Halanay [6, 7] proved this result under the assumption that the

solution of (8) is unique for the initial value problem.

Lemma 3. Under the assumptions in Theorem 2, let $¥{¥tau_{k}¥}$ be a sequencesuch that $¥tau_{k}>0$ , $¥tau_{k}¥rightarrow¥infty$ , $¥xi(t+¥tau_{k})$ converges to a function $¥eta(t)$ uniform $ly$ on $I$

and $F(t+¥tau_{k}, ¥varphi)¥rightarrow G(t, ¥varphi)$ as $ k¥rightarrow¥infty$. T&n $¥eta(t)$ is a solution of the system

(14) $i(t)=G(t, x_{t})$ ,

and $¥eta(t)$ is uniformly stable for $t¥geqq 0$ . Moreover, if $¥xi(t)$ is uniformly asymp-totically stab& for $t¥geqq 0$, then $¥eta(t)$ also is uniformly asymptotically stable for$t¥geqq 0$ .

Proof. Let $S$ denote the set of functions $¥varphi(¥theta)$ defined on $[-h, 0]$ such that$|_{1}^{¥mathrm{i}}¥varphi||¥leqq B^{*}$ and $|¥varphi(¥theta)-¥varphi(¥theta^{¥prime})|¥leqq L|¥theta-¥theta^{¥prime}|$ for any $¥theta$ , $¥theta^{¥prime}¥in[-h,0]$ . Then, $S$ is a

compact set in $¥overline{C}B*$ . Clearly, $¥xi_{f}¥in S$ for all $t¥geqq h$ and $¥xi_{t}^{k}¥in S$ for all $t¥geqq 0$ andfor large $k$ , where $¥xi^{k}(t)=¥xi(t+¥tau_{k})$ . Thus we can assume that $¥xi(t+¥tau_{k})$ converges

90 $¥eta(t)$ uniformly on [ $-h$ , $¥infty)$ , and hence $¥xi_{0}^{k}¥rightarrow¥eta_{0}$ as $ k¥rightarrow¥infty$ . Moreover, clearly$¥eta_{t}¥in S$ for all $t$ $¥geqq 0$ . By Lemma 2, we can select a subsequence $¥{¥tau_{k_{j}}¥}$ of $¥{¥tau_{k}¥}$ for

which $F(t+¥tau_{k_{j}},¥varphi)¥rightarrow G(t, ¥varphi)$ uniformly on $R¥times S$ as $ j¥rightarrow¥infty$ . Since $¥xi^{k}jt¥in S$ and$¥eta_{t}¥in S$ and $F(t,¥varphi)$ is uniformly continuous, we have

$ j!¥rightarrow ¥mathrm{i}¥mathrm{m}¥int_{0}^{f}F(s+¥tau_{k_{j}}, ¥xi^{k}j_{S})ds=¥int_{0}^{f}G(s, ¥eta_{s})ds¥infty$ .

28 T. v○ SHIZAWA

Thus we can see that $¥eta(t)$ is a solution of (14).Now we shall see that $¥eta(t)$ is uniformly stable for $t¥geqq 0$ . Set $¥tau_{k}=N_{k}¥omega+¥sigma_{k}$

as before. Then $ 0¥leqq¥sigma_{k}<¥omega$ . Let $¥{¥tau_{k_{j}}¥}$ be a subsequence of $¥{¥tau_{k}¥}$ such that $¥sigma_{k_{j}}$

$¥rightarrow¥sigma$ as $ j¥rightarrow¥infty$ . Then $ 0¥leqq¥sigma¥leqq¥omega$ and$¥xi^{k_{j}}(t)=¥xi(t+¥tau_{k_{j}})¥rightarrow¥eta(t)$ uniformly on $I$.

Since $F(t+¥tau_{k}, ¥varphi)¥rightarrow G(t, ¥varphi)$ as $ k¥rightarrow¥infty$ , we have $G(t, ¥varphi)=F(t+¥sigma, ¥varphi)$ . For any$¥epsilon>0$ , let $¥delta(¥epsilon)$ be the one for uniform-stability of $¥xi(t)$ , where we can assumethat $¥epsilon<¥frac{B^{*}-B}{2}$ . For any $t_{0}¥in I$, let $x(t)$ be a solution of (14) such that $||¥eta_{f_{0}}-x_{f_{0}}||$

$<¥delta(¥epsilon)$ . For a fixed $x(t)$ , we have $||¥eta_{t_{0}}-x_{f_{0}}^{1}|=¥gamma<¥delta(¥epsilon)$ . If $j$ is sufficiently large,

$||¥xi^{k_{j}}t_{0}-¥eta_{t_{0}}||<¥frac{¥delta(¥epsilon)-¥gamma}{2}$

and

$||¥xi_{t_{0}}+¥sigma¥dagger N_{k_{j}}¥omega-¥xi_{t_{0}}+¥sigma_{k_{j}}¥dagger N_{k_{j}}¥omega||<¥frac{¥delta(¥epsilon)-¥gamma}{2}-$ ,

and hence$¥xi||t_{0}+¥sigma¥dagger N_{k_{¥mathrm{j}}}¥omega-x_{t_{0}}||<||¥xi_{¥mathrm{f}_{¥mathrm{O}}}+¥sigma¥dagger N_{k_{j}}¥omega-¥xi_{t_{0}}+¥sigma_{¥mathrm{k}_{j}}+N_{k_{j}}¥omega||+||¥xi_{t_{0}}+¥sigma_{k_{j}}+N_{k_{¥mathrm{j}}}¥omega-¥eta_{f_{0}}||$

$+||¥eta_{t_{0}}-x_{t_{0}}||<¥delta¥zeta ¥mathrm{e})$ .

Since $¥xi(t+¥sigma+Nk_{j}¥omega)$ is a uniformly stable solution of (14), that is, $¥dot{x}(t)=F(t+$

$¥sigma$ , $x_{t})$ , we have$|_{1}^{1}¥xi r+¥sigma+N_{k_{j}}¥omega-x_{f}||<¥mathrm{e}$ for all $t¥geqq t_{0}$ .

On the other hand, for an arbitrary $¥gamma>0$, if $j$ is sufficiently large,$||¥eta_{t_{0}}-¥xi_{t_{0}}+¥sigma+N_{k_{j}}¥omega||¥leqq||¥eta_{t_{0}}-¥xi_{t_{0}¥dagger¥sigma_{k_{j}}¥dagger N_{k_{j}}¥omega|}||+||¥xi_{t_{0}}+¥sigma_{k_{j}}¥dagger N_{k_{j}}¥omega-¥xi_{t_{0}}+¥sigma¥dagger N_{k_{j}}¥omega||$

$<¥delta(¥gamma)$ ,

and hence $||¥eta_{t}-¥xi_{t}+¥sigma+N_{k_{j}}¥omega||<¥gamma$ for all $t¥geqq t_{0}$ . Thus we have $||¥eta_{t}-x_{f}||<¥mathrm{e}+¥gamma$ forall $t¥geqq t_{0}$ . Since $¥gamma$ is arbitrary, we have

$||¥eta_{t}-x_{t}||¥leqq ¥mathrm{e}$ for all $t¥geqq t_{0}$ if $||¥eta_{t_{0}}-x_{f_{0}}||<¥delta(¥mathrm{e})$ .

This proves that $¥eta(t)$ is uniformly stable for $t¥geqq 0$ .

Next, we assume that $¥xi(t)$ is uniformly asymptotically stable for $t¥geqq 0$ . Let$x(t)$ be a solution of (14) such that $||¥eta_{t_{0}}-x_{t_{0}}||<¥delta_{0}$ , where $¥delta_{0}$ is the one foruniformly asymptotic stability of $¥xi(t)$ . $¥xi(t+¥sigma+N_{k_{j}}¥omega)$ is a uniformly asymp-totically stable solution of (14) with the same $¥delta_{0}$ as the one for $¥xi(t)$ . For afixed $x(t)$ , $||¥eta_{t_{0}}-x_{t_{0}}||=¥delta_{1}<¥delta_{0}$ and

$||¥xi t_{0}+¥sigma¥dagger N_{k_{j}}¥omega-x_{t_{0}}||¥leqq||¥xi_{t_{0}}+¥sigma¥dagger N_{k_{j}}¥omega-¥xi_{t_{0}}+¥sigma_{k_{j}}¥dagger N_{k_{j}}¥omega||+||¥xi_{t_{0}}+¥sigma_{k_{j}}+N_{k_{j}}¥omega-¥eta_{t_{0}}||$

$+|_{¥mathrm{I}}^{1}¥eta_{t_{0}}-x_{t_{0}}||<¥delta_{0}$

Asymptotically Almost Periodic Solutions of $a¥prime i$ Almost Periodic System 29

if $j$ is sufficiently large, and hence, for sufficiently large $j$

$||¥xi_{t}+¥sigma¥dagger N_{h_{j}}¥omega-x_{t}||<¥epsilon$ for $t¥geqq t_{0}+T(¥epsilon)$ .

On the other hand,

$||¥xi t_{0}+¥sigma+N_{k¥mathrm{j}}¥omega-¥eta_{t_{0}}||¥leqq||¥mathrm{I}|¥xi t_{0}+¥sigma+N_{kj}¥omega-¥xi t_{0}¥dagger¥sigma_{kj}+N_{kj}¥omega||+||¥xi t_{0}+¥sigma_{kj}+N_{k_{j}}¥omega-¥eta_{P¥mathfrak{n}}||$

$<¥delta_{0}$

if $j$ is sufficiently large, and hence, for sufficiently large $j$

$||¥xi_{t}+¥sigma+N_{k_{j}}¥omega-¥eta_{t}||<¥epsilon$ for $t¥geqq t_{0}+T(¥epsilon)$ .

Thus we have$||¥eta_{t}-x_{t}||<2¥epsilon$ for $t¥geqq t_{0}+T(¥epsilon)$ ,

if $||¥eta_{t0}-x_{t0}||<¥delta_{0}$ . This shows that $¥eta(t)$ is uniformly asymptotically stable for$t¥geqq 0$ .

By using Lemma 3, we have a corollary of Theorem 2.Corollary. If the periodic system (8) has a bounded uniformly stab& solu-

tion $¥xi(t)$ for $t¥geqq 0$ such that $||¥xi_{t}||¥leqq B$ for $t¥geqq 0$ , then the system (8) has analmost periodic solution which is uniformly stable for $t¥geqq 0$ .

Proof. By Theorem 2, $¥xi(t)$ is asymptotically almost periodic, and hence,$¥xi(t)=p(t)+q(t)$ , where $p(t)$ is almost periodic and $q(t)¥rightarrow 0$ as $ t¥rightarrow¥infty$ . Let$¥tau_{k_{j}}=k_{j}¥omega$ such that $p(t+¥tau_{k_{j}})¥rightarrow p^{*}(t)$ uniformly on $R$ . Then $p^{*}(t)$ is almostperiodic and $¥tau_{k_{j}}>0$ , $¥tau_{k_{¥mathrm{j}}}¥rightarrow¥infty$ as $ j¥rightarrow¥infty$ . Moreover, $F(t+¥tau_{k_{j}},¥varphi)=F(t,¥varphi)$ on $ R¥times$

$¥overline{C}B*$ and $¥xi(t+¥tau_{k_{j}})¥rightarrow p^{*}(t)$ uniformly on I as $ j¥rightarrow¥infty$ . Therefore, by Lemma 3,$p^{*}(t)$ is a uniformly stable almost periodic solution of (8).

Remark. Deysach and Sell [4] obtained this result for ordinary differentialequations by using properties of dynamical systems and consequently, assuming

the uniqueness of solutions.Theorem 3. Under the assumptions in Theorem 2, $i$] $¥xi(t)$ is uniformly

asymptotically stable for $t¥geqq 0$ , then the system (8) has a periodic solution ofperiod $ m¥omega$ for some integer $m¥geqq 1$ which is uniformly asymptotically stable for$t$ $¥geqq 0$ .

Proof. Set $¥xi^{k}(t)=¥xi(t+k¥omega)$ . Then, as was seen in Theorem 2, a sub-sequence $¥{¥xi^{k}j(t)¥}$ converges to a solution $¥eta(t)$ of (8) uniformly on $[-h,¥infty)$ .

Since $¥xi^{k}j0$ is convergent, there is an integer $k_{p}$ such that $||¥xi^{k_{p}}¥mathrm{o}0-¥xi^{k_{p+1}}||<¥delta_{0}$ , where$¥delta_{0}$ is the one for uniformly asymptotic stability of $¥xi(t)$ . Set $m=k_{p+l}-k_{p}$ andconsider the solution $¥xi(t+m¥omega)$ of (8). Then we have

$||¥xi^{m_{k_{p}¥omega}}-¥xi_{k_{p}¥omega}|^{¥iota}|=||¥xi_{k_{p+1}¥omega}-¥xi_{k_{¥mathrm{p}}¥omega}1|=||¥xi^{k_{p+}}¥iota_{0}-¥xi^{k_{p_{0}}}||<¥delta_{0}$,

and hence

30 T. $¥mathrm{Y}_{¥mathrm{o}¥mathrm{S}¥mathrm{H}¥mathrm{I}¥mathrm{Z}¥mathrm{A}¥mathrm{W}¥mathrm{A}}$

(15) $||¥xi_{f}^{m}-¥xi_{t}||¥rightarrow 0$ as $ t¥rightarrow¥infty$ ,

because $¥xi(t)$ is uniformly asymptotically stable. On the other hand, by Theorem2, $¥xi(t)$ is asymptotically almost periodic, and hence

(16) $¥xi(t)=p(t)+q(t)$ ,

where $p(t)$ is almost periodic and $q(t)¥rightarrow 0$ as $ t¥rightarrow¥infty$ . From (15) and (16), itfollows that

$|p(t)-p(t+m¥omega)|¥rightarrow 0$ as $ t¥rightarrow¥infty$ .

Therefore $p(t)=p(t+m¥omega)$ for all $t¥in R$ , because $p(t)$ is almost periodic. Thisshows that the system (8) has a periodic solution $p(t)$ of period $ m¥omega$ , because$p(t)$ also is a solution of (8).

If we consider a sequence $¥{¥tau_{k}¥}$ such that $¥tau_{k}=km¥omega$ , we have

$¥xi(t+km¥omega)=p(t)+q(t+km¥omega)$ ,

and hence $¥xi(t+¥tau_{k})¥rightarrow p(t)$ uniformly on I as $ k¥rightarrow¥infty$ . Since $F(t+¥tau_{k}, ¥varphi)=F(t,¥varphi)$

for $t¥in R$ and $¥varphi¥in¥overline{C}B*$ , $p(t)$ is uniformly asymptotically stable by Lemma 3.Remark. If $¥xi(t)$ is uniformly asymptotically stable in the large, we have

$||¥xi_{f+¥omega}-¥xi_{t}||¥rightarrow 0$ as $ t¥rightarrow¥infty$ , and hence, clearly $p(t)=p(t+¥omega)$ (cf. [18]).Remark. Sell [14] proved this theorem for ordinary differential equations

by considering dynamical systems and consequently, assuming the uniqueness ofsolutions. Halanay [6,7] proved the existence of a periodic solution of period$ m¥omega$ under a weaker condition on $¥xi(t)$ . He also assumed the uniqueness ofsolutions, but we can drop this assumption by changing his statements in hisproof. Since Halanay does not assume the stability of $¥xi(t)$ , the periodic solu-tion is not necessarily stable in his case.

4. Almost Periodic systems.We shall discuss the existence of an asymptotically almost periodic solution

of an almost periodic system

(17) $i(t)=F(t,x_{f})$ .

We assume that $F(t, ¥varphi)$ is defined and continuous on $R¥times¥overline{C}B*$ and $F(t,¥varphi)$ isalmost periodic in $t$ uniformly for $¥varphi¥in¥overline{C}B*$ and that there exists an $L>0$ suchthat $|F(t, ¥varphi)|¥leqq L$ on $R¥times¥overline{C}B*$ . Moreover, we assume that the system (17) hasa bounded solution $¥xi(t)$ defined for $t¥geqq 0$ such that $||¥mathrm{I}¥xi_{t}||¥leqq B$ for $t¥geqq 0$, where$B<B^{*}$ .

Theorem 4. If the bounded solution $¥xi(t)$ of (17) is asymptotically almostperiodic, then for any $G¥in H(F)$ there exists a sequence $¥{¥tau_{k}¥}$ such that $¥xi(t+¥tau_{k})$

tends to an almost periodic solution of the system

Asymptotically Almost Periodic Solutions of an Almost Periodic System 31

(18) $i(t)=G(t, x_{t})$

uniformly on I as $ k¥rightarrow¥infty$ .

Proof. Since $¥xi(t)$ is asymptotically almost periodic, $¥xi(t)=p(t)+q(t)$ , where$p(t)$ is almost periodic and $q(t)¥rightarrow 0$ as $ t¥rightarrow¥infty$ . Clearly, $¥xi(t)$ and $p(t)$ are uni-formly continuous and $||¥xi_{t}||¥leqq B$ , $||p_{t}||¥leqq B$, and hence there exists a compact set$s¥subset CB*$ such that $¥xi_{f}¥in S$ and $p_{t}¥in S$ for $t¥geqq 0$ . Since $G¥in H(F)$ , there is a sequence$¥{¥tau_{k}¥}$ such that $¥tau_{k}¥rightarrow¥infty$ as $ k¥rightarrow¥infty$ and $F(t+¥tau_{k}, ¥varphi)¥rightarrow G(t, ¥varphi)$ uniformly on $R¥times S$ as

$ k¥rightarrow¥infty$ and that $p(t+¥tau_{k})¥rightarrow p^{*}(t)$ as $ k¥rightarrow¥infty$ . Clearly, $¥xi(t+¥tau_{k})¥rightarrow p^{*}(t)$ uniformlyon I and $p^{*}(t)$ is almost periodic, and hence we can see that $p^{*}(t)$ is an almostperiodic solution of (18). This completes the proof.

Remark. Clearly, for this theorem, we do not need the assumption thatthere is an $L>0$ such that $|F(t, ¥varphi)|¥leqq L$ .

Coppel [3] and Miller [9] have obtained an existence theorem for an almostperiodic solution of ordinary differential equations under the assumption that abounded solution of (17) is totally stable.

Definition. The bounded solution $¥xi(t)$ of (17) is said to be totally stablefor $t¥geqq 0$ , if for any $¥epsilon>0$ there exists a $¥delta(¥epsilon)>0$ such that if $t_{0}¥in L||¥xi_{f_{0}}-¥phi||<¥delta(¥epsilon)$

and $|R(t)|<¥delta(¥epsilon)$ for $t¥geqq t_{0}$ , then $||¥xi_{t}-x_{t}(t_{0},¥phi)||<¥epsilon$ for all $t¥geqq t_{0}$ , where $R(t)$ iscontinuous for $t¥geqq t_{0}$ and $x(t_{0}, ¥phi)$ is a solution of

(18) $¥dot{x}(t)=F(t,x_{t})+R(t)$ .

Moreover, we have the following stability property. Let $K$ denote thespace of functions $¥varphi(¥theta)$ defined on $[-h,0]$ such that $||¥varphi||¥leqq B^{*}$ and $|¥varphi(¥theta)-¥varphi(¥theta^{¥prime})|$

$¥leqq L|¥theta-¥theta^{¥prime}|$ , $-h¥leqq¥theta$ , $¥theta^{¥prime}¥leqq 0$ . Then, clearly $K$ is a compact set. For $G¥in H(F)$

and $P¥in H(F)$ , define $¥rho(G, P)$ by

$¥rho(G, P)=¥sup$ $¥{|G(t, ¥varphi)-P(t,¥varphi)|; t¥in R, ¥varphi¥in K¥}$ .

Definition. The bounded solution $¥xi(t)$ of (17) is said to be stable underdisturbances from $H(F)$ , if for any $¥epsilon>0$ there exists a $¥delta(¥epsilon)>0$ such that $||¥xi_{t+¥tau}$

$-x_{t}(0,¥psi, G)||¥leqq¥epsilon$ for $t$ $¥geqq 0$ , whenever $G¥in H(F)$ , $||¥xi_{¥tau}-¥phi||¥leqq¥delta(¥epsilon)$ , $¥phi¥in K$, and $¥rho(F^{¥tau}$ ,$G)¥leqq¥delta(¥epsilon)$ for some $¥tau¥geqq 0$ , where $x(0, ¥phi, G)$ denotes a solution of (18) such that$ x_{0}(0,¥phi, G)=¥phi$ .

Lemma 4. Suppose that the bounded solution $¥xi(t)$ of (17) is stab $fe$ underdisturbances from $H(F)$ and let $c$ be a positive constant. Then $¥eta(t)=¥xi(t+c)$ ,$¥eta_{0}=¥xi_{c}$ , is a solution of

$i(t)=F(t+c, x_{t})$ ,

which is stable under disturbances from $H(F^{c})$ with the same pair $(¥epsilon, ¥delta(¥epsilon))$ asthe one for $¥xi(t)$ .

32 T. YoSHIZAWA

Theorem 5. If the bounded solution $¥xi(t)$ of (17) is stable under distur-bances from $H(F)$ , then $¥xi(t)$ is an asymptotically afmost periodic solution of(17). Thus the system (17) has an almost periodic solution

Proof. Let $¥{¥tau_{k}¥}$ be any sequence such that $¥tau_{k}>0¥mathrm{a}¥dot{¥mathrm{n}}¥mathrm{d}¥tau_{k}¥rightarrow¥infty$ as $ k¥rightarrow¥infty$ .

Set $¥xi^{k}(t)=¥xi(t+¥tau_{k})$ . Then $¥xi^{k}(t)$ is a solution of the system

(20) $¥dot{x}(t)=F(t+¥tau_{k}, x_{t})$

through $(0, ¥xi_{¥tau_{k}})$ and $||¥xi_{t}^{k}||¥leqq B$ for all $t¥geqq 0$ and all $k$ . By Lemma 4, $¥xi^{k}(t)$ is stableunder disturbances from $H(F^{¥tau_{k}})$ with the same pair $(¥epsilon, ¥delta(¥epsilon))$ as the one for $¥xi(t)$ .Clearly

$¥xi_{t}^{k}¥in K$ for all $t¥geqq 0$ ,

if $k$ is sufficiently large. Since $F(t,¥varphi)$ is almost periodic in $t$ uniformly for$¥varphi¥in¥overline{C}B*$ and $K$ is a $¥mathrm{c}¥dot{¥mathrm{o}}¥mathrm{m}¥mathrm{p}¥mathrm{a}¥mathrm{c}¥mathrm{t}$ set in $¥overline{C}B*$ , $¥{¥tau_{h}¥}$ has a subsequence, which we shalldenote by $¥{¥tau_{k}¥}$ again, such that $F(t+¥tau_{k},¥varphi)$ converges uniformly on $R¥times K$ as$ k¥rightarrow¥infty$, and hence there is an integer $k_{0}(¥epsilon)>0$ such that if $m¥geqq k¥geqq k_{0}(¥epsilon)$ ,

(21) $|F(t+¥tau_{k}, ¥varphi)-F(t+¥tau_{m},¥varphi)|¥leqq¥delta(¥epsilon)$ on $R¥times K$,

where $¥delta(¥mathrm{e})$ is the one for stability under disturbances. Therefore, if $ m¥geqq k¥geqq$

$k_{0}(¥epsilon)$ , we have $¥rho(F^{¥tau_{k}}, F^{¥tau_{n}})¥leqq¥delta(¥epsilon)$ . Moreover, since $¥xi^{k_{0}}¥in K$ for large $k$ , we canassume that if $m¥geqq k¥geqq k_{0}(¥epsilon)$ ,

(22) $||¥xi_{0}^{k}-¥xi_{0}^{n¥mathrm{t}}||¥leqq¥delta(¥epsilon)$ ,

taking a subsequence again, if necessary. Since $¥xi^{m}(t)$ is a solution of

(23) $¥dot{x}(t)=F(t+¥tau_{m}, x_{t})$

and $F^{¥tau_{n}}¥in H(F^{¥tau_{k}})$ and $¥xi^{k}(t)$ is stable under disturbances from $H(F^{¥tau}k)$ , we have

$||¥xi_{t}^{k}-¥xi_{t}^{n}||¥leqq¥epsilon$ for $t¥geqq 0$ ,

if $m¥geqq k¥geqq k_{0}(¥epsilon)$ . This implies that $|¥xi(t+¥tau_{k})-¥xi(t+¥tau_{m})|¥leqq¥epsilon$ for all $t$ $¥geqq 0$ , if$m¥geqq k¥geqq k_{0}(¥epsilon)$ . Thus we see that $¥xi(t)$ is asymptotically almost periodic. Theexistence of an almost periodic solution follows immediately from Theorem 1.

Lemma 5. If the bounded solution $¥xi(t)$ of (17) is totally stable for $t¥geqq 0$ ,

then it is stable $u¥dot{n}$der disturbances from $H(F)$ .

Proof. For $G¥in H(F)$ and $P¥in H(F)$ , since $¥rho(G, P)=¥sup$ { $|G(t, ¥varphi)-P(t,¥varphi)|$ ;$t¥in R$ , $¥varphi¥in K¥}$ , $¥rho(F^{r}, G)¥leqq¥delta^{¥prime}(¥epsilon)<¥delta(¥epsilon)$ implies that

(24) $|F(t+¥tau, ¥varphi)-G(t, ¥varphi)|<¥delta(¥epsilon)$ on $R¥times K$,

where $¥delta(¥epsilon)$ is the one for total stability of $¥xi(t)$ . Let $¥phi$ satisfy $||¥xi_{¥tau}-¥psi||¥leqq¥delta^{¥prime}(¥epsilon)$

and $¥psi¥in K$. Let $¥eta(t)$ be a solution of (18) through $(0, ¥phi)$ . Then $y(t)=¥eta(t-¥tau)$ ,

$Asyrnptotica/ly$ $Aln¥iota ost$ Periodic $Solutior/s$ of an Al7I2oSt Periodic $ Syster/¥iota$ 33

$t$ $¥geqq¥tau$, is a solution of

$¥dot{x}(t)=G(t-¥tau, x_{t})$

through $(¥tau, ¥psi)$ . In other words, $y(t)$ is a solution of

(25) $¥dot{x}(t)=F(t, x_{t})+G(t-¥tau, y_{t})-F(t, y_{t})$

such that $ y_{¥tau}=¥psi$ . Clearly, $y_{t}¥in K$ for $ t¥geqq¥tau$ , and hence, as long as $y(t)$ exists,

(26) $|G(t-¥tau, y_{t})-F(t, y_{t})|<¥delta(¥epsilon)$

by (24). Since $¥xi(t)$ is a totally stable solution of (17) and $||¥xi_{¥tau}-¥psi||<¥delta(¥epsilon)$ andwe have (26), $]|y_{t}-¥xi_{t}||<¥epsilon$ as long as $y(t)$ exists, which implies that $y(t)$ existsfor all $ t¥geqq¥tau$ and $||y_{t}-¥xi_{t}||<¥epsilon$ for $ t¥geqq¥tau$ . Replacing $t$ by $ t+¥tau$ , we have $|_{1}¥eta_{t}-¥xi_{t+¥tau}||<¥epsilon$

for all $t¥geqq 0$ . This proves that $¥xi(t)$ is stable under disturbances from $H(F)$ .By Theorem 5 and Lemma 5, we have the following corollary.

Corollary. If the bounded solution $¥xi(t)$ is totally stable for $t$ $¥geqq 0$ , $¥xi(t)$ isan asymptotically almost periodic solution of (17). Thus the system (17) hasan almost periodic solution.

The following theorem corresponds to a result of Amerio [1] for ordinarydifferential equations.

Theorem 6. Suppose that for each $G¥in H(F)$ there exists a $¥lambda(G)>0$ suchthat if $x(t)$ and $y(t)$ are distinct solutions of (18) which satisfy $|x(t)|¥leqq B$ and$|y(t)|¥leqq B$ for $alf$ $t¥in R$ , then $||x_{t}-y_{t}||¥geqq¥lambda(G)$ for all $t¥in R$. T&n the boundedsolution $¥xi(t)$ of (17) such that $||¥xi_{t_{1}}!|¥leqq B$ for $t¥geqq 0$ is asymptotically almostperiodic, and consequently, the system (17) has an dmost periodic solution.

This theorem can be proved by the same argument used by Amerio.Corollary. Under the assumptions in Theorem 6, if $¥xi(t)$ is a solution of

(17) defined for all $t¥in R$ and $|||¥xi_{t}||¥leqq B$ for all $t¥in R$ , then $¥xi(t)$ is an almost peri-odic solution of (17).

Proof. For $t$ $¥geqq 0$ , $¥xi(t)$ is asymptotically almost periodic, and hence $¥xi(t)=$

$p(t)+q(t)$ , where $p(t)$ is almost periodic and $q(t)¥rightarrow 0$ as $ t¥rightarrow¥infty$ . Since $|¥xi(t)|¥leqq B$,

we have $|p(t)|¥leqq B$. Thus, if $¥xi(t)$ and $p(t)$ are distinct solutions of (17),$||¥xi_{t}-p_{t}||¥geqq¥lambda(F)>0$ for all $t¥in R$ . However, $¥xi(t)-p(t)¥rightarrow 0$ as $ t¥rightarrow¥infty$ . Thus therearises a contradiction. Therefore $¥xi(t)¥equiv p(t)$ for $t¥in R$ .

In the following lemmas and theorem, we assume that for every $G¥in H(F)$

the solution of (18) is unique for the initial condition. Let $K=¥{¥varphi;$ $||¥varphi||¥leqq B^{*}$ ,$|¥varphi(¥theta)-¥varphi(¥theta^{¥prime})|¥leqq L|¥theta-¥theta^{¥prime}|$ , $-h¥leqq¥theta$ , $¥theta^{¥prime}¥leqq 0¥}$ again.

Lemma 6. For the system (17), let $T>0$ be given. Then, for any $¥epsilon>0$

there exists a $¥delta(¥epsilon)>0$ such that if $t_{0}¥in I$, $|||¥xi_{t_{0}}-¥psi|_{1}^{1}<¥delta(¥epsilon)$ and $|g(t)|<¥delta(¥epsilon)$ , wehave

34 T. $¥mathrm{Y}_{¥mathrm{o}¥mathrm{S}¥mathrm{H}¥mathrm{I}¥mathrm{Z}¥mathrm{A}¥mathrm{W}¥mathrm{A}}$

$||¥xi_{t}-x_{f}(t_{0}, ¥psi)|^{1}<¥epsilon$ on $t_{0}¥leqq t¥leqq t_{0}+T$,

ivhere $x(t_{0},¥phi)$ is a solution through $(t_{0}, ¥psi)$ of the system

(27) $¥dot{x}(t)=F(t, x_{t})+g(t)$

and $g(t)$ is continuous on $L$

Proof. For each fixed interval, there exists such a $¥delta>0$ which may dependon the interval, because $¥xi(t)$ is unique to the right for the initial condition.Now suppose that for some $¥epsilon>0$ there is no $¥delta$ which satisfies the condition inLemma. Then there exist sequences $¥{¥delta_{k}¥}$ , $¥{t_{k}¥}$ , $¥{¥tau_{k}¥}$ and $g_{k}(t)$ such that $¥delta_{k}¥rightarrow 0$ ,

$ t_{k}¥rightarrow¥infty$ as $ k¥rightarrow¥infty$ , $¥tau_{k}¥in(t_{k},$ $t_{k}+T$], $|g_{k}(t)|<¥delta_{k}$ , $||¥xi_{t_{k}}-x_{t_{k}}^{k}||<¥delta_{k}$ and

$|¥xi(¥tau_{k})-x^{k}(¥tau_{k})|=¥frac{¥epsilon}{2}$, $|¥xi(t)-x^{k}(t)|<¥frac{¥epsilon}{2}$ on $t_{k}¥leqq t<¥tau_{k}$ ,

where $x^{k}(t)$ is a solution of the system

(28) $x¥dot{(}t)=F(t, x_{t})+g_{k}(t)$

and we can assume that $¥epsilon<B^{*}-B$. If we set $¥xi^{k}(t)=¥xi(t+t_{k})$ and $y^{k}(t)=$

$x^{k}(t+t_{k})$ , $¥xi^{k}(t)$ is a solution of the system

(29) $¥dot{x}(t)=F(t+t_{k}, x_{t})$

and $y^{k}(t)$ is a solution of the system

(30) $i(t)=F(t+t_{k}, x_{t})+g_{k}(t+t_{k})$

through $(0, x^{k_{f_{k}}})$ . If we set $¥tau_{k}=t_{k}+¥sigma_{k}$ , then $0<¥sigma_{k}¥leqq T$, where we can assumethat $¥sigma_{k}¥rightarrow¥sigma$ , $0<¥sigma¥leqq T$, as $ k¥rightarrow¥infty$. If $k$ is sufficiently large, we have $|¥sigma_{k}-¥sigma|$

$<¥frac{¥epsilon}{8L}$ . In case $¥sigma_{k}¥geqq¥sigma$, clearly $y^{k}(t)$ exists on[-h, $¥sigma$ ]. For $¥sigma_{k}<¥sigma$ , as long as$y^{k}(t)$ exists,

$y^{k}(t)=y^{k}(¥sigma_{k})+¥int_{¥sigma}^{f}kF(s+t_{k}, y_{s}^{k})ds+¥int_{¥sigma}^{t}kg_{k}(s¥dagger t_{k})ds$ , $t¥geqq¥sigma_{k}$ .

Thus, as long as $y^{k}(t)$ exists for $¥sigma_{k}¥leqq t¥leqq¥sigma$ , $|y^{k}(t)-y^{k}(¥sigma_{k})|<L¥frac{¥epsilon}{8L}+¥delta_{k}¥frac{¥epsilon}{8L}=¥frac{¥epsilon}{8}$

$+¥delta_{k}¥frac{¥epsilon}{8L}$ , and hence

$|y^{k}(t)-¥xi^{k}(t)|¥leqq|y^{k}(t)-y^{k}(¥sigma_{k})|+|y^{k}(¥sigma_{k})-¥xi^{k}(¥sigma_{k})|+|¥xi^{k}(¥sigma_{k})-¥xi^{k}(t)|$

$<¥frac{¥epsilon}{8}+¥delta_{k}¥frac{¥epsilon}{8L}+¥frac{¥epsilon}{2}+¥frac{¥epsilon}{8}$ ,

because $|y^{k}(¥sigma_{k})-¥xi^{k}(¥sigma_{k})|=|x^{k}(t_{k}+¥sigma_{k})-¥xi(t_{k}+¥sigma_{k})|=|x^{k}(¥tau_{k})-¥xi(¥tau_{k})|=¥frac{¥epsilon}{2}$ . Thus, if$k$ is sufficiently large, we have $|y^{k}(t)-¥xi^{k}(t)|<¥epsilon$ on $[-h, ¥sigma]$ as long as $y^{k}(t)$

exists, and hence we can conclude that if $k$ is sufficiently large, $y^{k}(t)$ exists on

Asymptotically Almost Periodic Solutions of an Almost Periodic System 35

$[-h, ¥sigma]$ , and we have

(31) $ y^{k}(t)=¥{x^{k}x^{k}(t_{k})+¥int_{t)}¥mathrm{o}t-h¥leqq t<0F(s+t_{k}, y_{s}^{k})ds+¥int_{0}^{t}g_{k}(s+t_{k})ds(t_{k}+,.’ 0¥leqq t¥leqq¥sigma$

For the sequence $¥{t_{k}¥}$ , there is a subsequence, which we shall denote by $¥{t_{k}¥}$

again, for which $F(t+t_{k}, ¥varphi)$ converges to some $G¥in H(F)$ uniformly on $R¥times S$, $s$ :

any compact set, and $¥xi_{t_{k}}¥rightarrow¥psi$ as $ k¥rightarrow¥infty$ , because $F(t, ¥varphi)$ is almost periodic and$¥xi_{t_{k}}¥in K$ for large $k$ . Since $y_{¥mathit{0}}^{k}=x^{k_{P_{k}}}$ and $||¥xi_{t_{h}}-x^{k_{t_{k}}}||<¥delta_{k}$ , clearly $ y_{0}^{k}¥rightarrow¥phi$ as $ k¥rightarrow¥infty$ .Therefore, $¥{y^{k}(¥theta)¥}$ , $-h¥leqq¥theta¥leqq 0$ , is uniformly bounded and equicontinuous on$-h¥leqq¥theta¥leqq 0$ . Moreover, if $k$ is sufficiently large, $|y^{k}(t)-¥xi^{k}(t)|<¥epsilon$ on $-h¥leqq t¥leqq¥sigma$

implies $|y^{k}(t)|<B+¥epsilon<B^{*}$ . Thus $¥{y^{k}(t)¥}$ is uniformly bounded and equiconti-

nuous on $-h¥leqq t¥leqq¥sigma$ . Therefore there is a subsequence of $¥{y^{k}(t)¥}$ , which weshall denote by $¥{y^{k}(t)¥}$ again, such that $y^{k}(t)¥rightarrow x(t)$ uniformly on $[-h,¥sigma]$ as$ k¥rightarrow¥infty$. Clearly, $x_{t}$ and $y_{t}^{k}$ belong to a compact set for $t¥geqq 0$, if $k$ is large,and hence, letting $ k¥rightarrow¥infty$ , from (31) it follows that

$x(t)=¥{¥psi¥psi(t),-¥ulcorner h¥leqq t<0(0)^{1}¥int_{0}^{t}G(s,x_{s}).ds$

, $ 0¥leqq t¥leqq¥sigma$

This shows that $x(t)$ is the solution of (18) through $(0, ¥phi)$ . On the other hand,clearly $¥xi^{k}(t)$ tends to the solution $¥eta(t)$ of (18) which also passes through $(0, ¥emptyset)$ .

By the uniqueness, $x(t)¥equiv¥eta(t)$ on $-h¥leqq t¥leqq¥sigma$ . However, $|x^{k}(¥tau_{k})-¥xi(¥tau_{k})|=$

$|y^{k}(¥sigma_{h})-¥xi^{k}(¥sigma_{k})|=¥frac{¥epsilon}{2}$ implies $|x(¥sigma)-¥eta(¥sigma)|=¥frac{¥epsilon}{2}$. Thus we have a contradiction.

This proves the lemma.Remark. In the case where $F$ is periodic, that is, $F(t+¥omega,¥varphi)=F(t,¥varphi)$ , if

$¥xi(t)$ is uniformly stable for $t¥geqq 0$ , $¥eta(t)$ is $¥mathrm{u}¥mathrm{n}¥check{¥mathrm{i}}¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{m}¥mathrm{l}¥mathrm{y}$ stable by Lemma 3, and

hence $¥eta(t)$ is a unique solution through $(0, ¥emptyset)$ . Thus we have $x(t)¥equiv¥eta(t)$ .Therefore, in this case, Lemma 6 holds good without the assumption that thesolution is unique.

Lemma 7. Let $¥{¥tau_{k}¥}$ be a sequence such that $¥tau_{k}>0$ , $¥tau_{k}¥rightarrow¥infty$ , $¥xi_{¥tau_{k}}¥rightarrow¥psi$ and$F(t+¥tau_{h}, ¥varphi)¥rightarrow G(t, ¥varphi)$ , $G¥in H(F)$ , uniformly on $R¥times K$ as $ k¥rightarrow¥infty$ . If the boundedsolution $¥xi(t)$ of (17) is uniformly stable for $t¥geqq 0$ , then the solution $¥eta(t)$ of(18) through $(0, ¥phi)$ is uniformly stable $t¥geqq 0$ . Moreover, if $¥xi(t)$ is uniformly

asymptotically stable for $t$ $¥geqq 0$ , then $¥eta(t)$ afso is uniformly asymptotically stable

for $t$ $¥geqq 0$ .

Proof. Since $¥xi(t)$ is uniformly stable, $¥xi^{k}(t)=¥xi(t+¥tau_{k})$ is a uniformly stablesolution of (20) through $(0, ¥xi_{¥tau_{k}})$ with the same pair $(¥epsilon,¥delta(¥epsilon))$ as the one for $¥xi(t)$ ,

and $||¥xi_{t}^{k}||¥leqq B$ for $t¥underline{¥geq}0$ and $|¥dot{¥xi}^{k}(t)|¥leqq L$ for $t¥geqq-h$ and large $k$ . Therefore $¥{¥xi^{k}(t)¥}$

36 T. $¥mathrm{Y}_{¥mathrm{o}¥mathrm{S}¥mathrm{H}¥mathrm{I}¥mathrm{Z}¥mathrm{A}¥mathrm{W}¥mathrm{A}}$

is uniformly bounded and equicontinuous on [ $-h,$ $¥infty)$ , if $k$ is sufficiently large.

Moreover, if $k$ is sufficiently large, $¥xi_{t}^{k}¥in K$ for all $t$ $¥geqq 0$ , and clearly $¥psi¥in K$.

Therefore we can see that there exists a subsequence, which we shall denoteby $¥{¥xi^{k}(t)¥}$ again, such that $¥xi^{k}(t)$ converges to the solution $¥eta(t)$ of (18) through$(0, ¥phi)$ uniformly on any compact interval in $[-h,$ $¥infty)$ .

For a fixed $t_{0}¥in I$, if $k$ is sufficiently large, we have

$|^{1}¥xi_{t_{0}}^{k}-¥eta_{t_{0}}||<¥frac{1}{2}¥delta(¥frac{¥epsilon}{2})$ ,

where $¥delta(¥epsilon)$ is the one for uniform stability of $¥xi(t)$ and we can assume that$¥epsilon<B^{*}-B$. Let $¥varphi$ be such that

(32) $||¥varphi-¥eta_{t_{0}}||<¥frac{1}{2}¥delta(¥frac{¥epsilon}{2})$ and $¥varphi¥in K$,

and let $x(t)$ be the solution of (17) such that $ x_{f_{0}+¥tau_{k}}=¥varphi$ . Then $x^{k}(t)=x(t+¥tau_{k})$

is the solution of (20) and $ x_{t_{0}}^{k}=¥varphi$ . Since $¥xi^{k}(t)$ is uniformly stable and’$||¥xi^{k_{f_{0}}}-x_{t_{0}}^{k}||<¥delta(¥frac{¥epsilon}{2})$, we have

(33) $||¥xi_{t}^{k}-x_{t^{1}}^{k}|<¥frac{¥epsilon}{2}$ for all $t¥geqq t_{0}$ .

On the other hand, since $x_{t}^{k}¥in K$ for all $t¥geqq 0$ and large $k$ , a subsequence of$¥{x^{k}(t)¥}$ converges to the solution $y(t)$ of (18) through $(t_{0}, ¥varphi)$ , which is uniquelydetermined, uniformly on any compact interval $[t_{0}, t_{0}+N]$ . We shall denoteby $¥{x^{k}(t)¥}$ this subsequence again. Thus, if $k$ is sufficiently large, say $ k¥geqq$

$k_{0}(¥epsilon, N)$ ,

(34) $||x_{f}^{k}-y_{t}||<¥frac{¥epsilon}{4}$ and $||¥xi_{t}^{k}-¥eta_{t}||<¥frac{¥mathrm{e}}{4}$ on $[t_{0}, t_{0}+N]$ .

From (33) and (34), it follows that

$||¥eta_{t}-y_{t}||<¥epsilon$ on $[t_{0}, t_{0}+N]$ .

Since $N$ is arbitrary, $||¥eta_{t}-y_{f}(t_{0}, ¥varphi)|_{1}^{1}<¥epsilon$ for all $t¥geqq t_{0}$ if $||¥varphi-¥eta_{t_{0}}|^{1}<¥frac{1}{2}¥delta(¥frac{¥epsilon}{2})$ and$¥varphi¥in K$, where $y(t_{0}, ¥varphi)$ is the solution of (18) through $(t_{0}, ¥varphi)$ .

For any $t_{0}¥in I$, as was seen above, $¥mathrm{i}_{1}^{¥mathrm{r}}||¥eta_{t_{0}+h}-¥varphi||<¥frac{1}{2}¥delta(¥frac{¥epsilon}{2})$ and $¥varphi¥in K$, then$||¥eta_{f}-y_{t}(t_{0}+h, ¥varphi)||<¥epsilon$ for all $t¥geqq t_{0}+h$ , where $y$ is the solution of (18). ByLemma 6, there exists a $¥delta^{*}(¥epsilon)>0$ such that if $t_{0}¥in I$ and $||¥eta_{t_{0}}-¥psi||<¥delta^{*}(¥epsilon)$ , $¥psi¥in C$ ,

we have $||¥eta_{t}-x_{t}(t_{0}, ¥psi)||<¥frac{1}{2}¥delta(¥frac{¥epsilon}{2})$ on $t_{0}¥leqq t_{-}¥leq t_{0}+h$ , where $x(t_{0}, ¥psi)$ is the solution

of (18). Clearly, $||¥eta_{t_{0}+h}-x_{t_{0}+h}(t_{0}, ¥psi)||<¥frac{1}{2}¥delta(¥frac{¥epsilon}{2})$ and $x_{t_{0}+h}(t_{0},¥psi)¥in K$, and hence

Asymptotically Almost Periodic Solutions of an Almost Periodic System 37

$||¥eta_{t}-x_{t}(t_{0}, ¥phi)||<¥epsilon$ for all $t¥geqq t_{0}$ . This proves that $¥eta(t)$ is uniformly stable for$t¥geqq 0$ .

Now we assume that $¥xi(t)$ is uniformly asymptotically stable for $t¥geqq 0$ . Then$¥xi^{k}(t)$ is a uniformly asymptotically stable solution of (20) with the same pair$(¥delta_{0}, ¥epsilon, T(¥epsilon))$ as the one for $¥xi(t)$ . As was seen, $¥eta(t)$ is uniformly stable with$(¥epsilon, ¥delta^{*}(¥epsilon))$ . For a fixed $¥epsilon_{0}$ such that $0<¥epsilon_{0}<B^{*}-B$, let $¥delta_{0}^{*}=¥delta^{*}(¥epsilon_{0})$ . For a fixed$t_{0}¥in I$, if $k$ is sufficiently large, we have $||¥xi_{t_{0}}^{k}-¥eta_{t_{0}}||<¥frac{1}{2}¥delta_{0}$ , where $¥delta_{0}$ is the one

for uniformly asymptotic stability of $¥xi(t)$ and we can assume that $¥frac{1}{2}¥delta_{0}¥leqq¥delta_{0}^{*}$ .

Let $¥varphi$ be such that $¥varphi¥in K$ and $||¥eta_{t_{0}}-¥varphi||<¥frac{1}{2}¥delta_{0}$ and let $x(t)$ be the solution of (17)

such that $ x_{t_{0}+¥tau_{k}}=¥varphi$ . Then, $x^{k}(t)=x(t+¥tau_{k})$ is the solution of (20) and $ x^{k_{f_{0}}}=¥varphi$ .

Since $||¥xi_{P_{0}}^{k}-¥varphi||<¥delta_{0}$ and $¥xi^{k}(t)$ is uniformly asymptotically stable, we have

$||¥xi_{t}^{k}-x_{t_{1}^{1}}^{k}|<¥frac{¥epsilon}{2}$ for $t$ $¥geqq t_{0}+T(¥frac{¥epsilon}{2})$.

A subsequence of $¥{x^{k}(t)¥}$ converges to the solution $y(t)$ of (18) through $(t_{0}, ¥varphi)$

uniformly on any compact interval $t_{0}+T(¥frac{¥epsilon}{2})¥leqq t¥leqq t_{0}+T(¥frac{¥epsilon}{2})+N$. We shalldenote by $¥{x^{k}(t)¥}$ this subsequence again. Thus, if $k$ is sufficiently large, say$k$ $¥geqq k_{0}(¥epsilon, N)$ ,

$||x_{t}^{k}-y_{f}|||<¥frac{¥epsilon}{4}$ and $||¥xi_{t}^{k}-¥eta_{t}||<¥frac{¥epsilon}{4}$ on $t_{0}+T(¥frac{¥epsilon}{2})¥leqq t¥leqq t_{0}+T(¥frac{¥epsilon}{2})+N$,

and hecne, $||y_{t}-¥eta_{t}||<¥epsilon$ on $t_{0}+T(¥frac{¥epsilon}{2})¥leqq t¥leqq t_{0}+T(¥frac{¥epsilon}{2})+N$. Since $N$ is arbitrary,

we have

$||¥eta_{t}-y_{t}(t_{0}, ¥varphi)||<¥epsilon$ for all $t¥geqq t_{0}+T(¥frac{¥epsilon}{2})$,

if $||¥eta_{t_{0}}-¥varphi||<¥frac{1}{2}¥delta_{0}$ and $¥varphi¥in K$. By Lemma 6, there exists a $¥delta_{0}^{**}>0$ such that if

$t_{0}¥in I$ and $||¥eta_{t_{0}}-¥psi||<¥delta_{0}^{**}$ , $¥phi¥in C$ , we have $||¥eta_{t_{0}+h}-x_{t_{0}+h}(t_{0}, ¥psi)_{1}^{1}|<¥frac{1}{2}¥delta_{0}$ and $x_{t_{0}+h}(t_{0}$ ,

$¥psi)¥in K$, where $x(t_{0}, ¥psi)$ is the solution of (18). Therefore we have $||¥eta_{t}-x_{t}(t_{0}$ ,

$¥psi)||<¥epsilon$ for all $t$ $¥geqq t_{0}+h+T(¥frac{¥epsilon}{2})$ , if $||¥eta_{t_{0}}-¥psi||<¥delta_{0}^{**}$ . This proves that $¥eta(t)$ isuniformly asymptotically stable for $t$ $¥geqq 0$ .

By using these lemmas, we shall prove the following theorem.

Theorem 7. For the system (17), we assume that for every $G¥in H(F)$ thesolution of (18) is unique for the initial condition. If the bounded solution $¥xi(t)$

is uniformly asymptotically stable for $t$ $¥geqq 0$ , then $¥xi(t)$ is asymptotically almost

38 T. Y○ SHIZAWA

periodic. Thus the system (17) has an almost periodic solution which is uni-formly asymptotically stable for $t¥geqq 0$ .

Proof. Since $¥xi(t)$ is uniformly asymptotically stable, for each $¥epsilon>0$ thereexists a $¥delta(¥epsilon)>0$ and a $T(¥epsilon)>0$ such that if $x(t)$ is a solution of (17) such that$||¥xi_{t_{0}}-x_{t_{0}}||<¥delta(¥epsilon)$ for some $t_{0}¥geqq 0$ , then $||¥xi_{t}-x_{t}||<¥frac{1}{2}¥epsilon$ for all $t¥geqq t_{0}$ and $||¥xi_{t}-x_{t1}^{¥mathrm{t}}|<$

$¥frac{1}{2}¥delta_{1}(¥epsilon)$ for all $t¥geqq t_{0}+T(¥epsilon)$ . Let $¥{¥tau_{k}¥}$ be a sequence such that $¥tau_{k}>0$ , $¥tau_{h}¥rightarrow¥infty$ as$ k¥rightarrow¥infty$ . If we set $¥xi^{h}(t)=¥xi(t+¥tau_{k})$ , $¥xi^{k}(t)$ is a solution of

(35) $¥dot{x}(t)=F(t+¥tau_{k}, x_{f})$

and $¥xi^{k}(t)$ is uniformly asymptotically stable with the same $¥delta(¥epsilon)$ and $T(¥epsilon)$ asthose for $¥xi(t)$ . By Lemma 6, for $¥delta_{1}(¥epsilon)$ and $T(¥epsilon)$ , there is a $¥delta_{2}(¥epsilon)>0$ such thatif $t_{0}¥in I$, $||¥xi_{t_{0}}-¥psi||<¥delta_{2}(¥epsilon)$ and $|g(t)|<¥delta_{2}(¥epsilon)$ , then

$||¥xi_{f}-x_{t}(t_{0}, ¥psi)||<¥frac{1}{2}¥delta_{1}(¥epsilon)$ on $t_{0}¥leqq t¥leqq t_{0}+T(¥epsilon)$ ,

where $x(t_{0}, ¥psi)$ is a solution of $¥dot{x}(t)=F(t,x_{f})+g(t)$ . For any $k$ , we can see that

if $t_{0}¥in I$, $||¥xi^{k}¥iota_{0}-¥psi||<¥delta_{2}(¥epsilon)$ and $|g_{k}(t)|<¥delta_{2}(¥epsilon)$ , then $||¥xi_{f}^{k}-x^{k_{f}}(t_{0},¥phi)||<¥frac{1}{2}¥delta_{1}(¥epsilon)$ on $t_{0}$

$¥leqq t¥leqq t_{0}+T(¥epsilon)$ with the same $¥delta_{2}(¥mathrm{e})$ , where $¥mathrm{x}^{¥mathrm{k}}(¥mathrm{t}_{0},¥psi)$ is a solution of the system

(36) $¥dot{x}(t)=F(t+¥tau_{k},x_{f})+g_{k}(t)$ .

Since $¥xi_{0}^{k}¥in K$ for large $k$ and $F(t,¥varphi)$ is almost periodic in $i$ uniformly for$¥varphi¥in¥overline{C}B*$ , there exists a subsequence of $¥{¥tau_{k}¥}$ , which we shall denote by $¥{¥tau_{k}¥}$

again, such that $¥xi_{0}^{k}$ converges and $F(t+¥tau_{k},¥varphi)$ converges uniformly on $R¥times K$ as$ k¥rightarrow¥infty$ . Therefore there is a positive integer $k_{0}(¥epsilon)$ such that if $m¥geqq k¥geqq k_{0}(¥epsilon)$ ,then

(37) $|F(t+¥tau_{k},¥varphi)-F(t+¥tau_{m}, ¥varphi)|<¥delta_{2}(¥epsilon)$ for all $t¥in R$ and $¥varphi¥in K$

and

(38) $||¥xi_{0}^{k}-¥xi^{m_{0}}|_{1}^{1}<¥delta_{1}(¥epsilon)$ .

$¥xi^{m}(t)$ is a uniformly asymptotically stable solution of the system

(38) $¥dot{x}(t)=F(t+¥tau_{m}, x_{t})$ ,

and hence the solution $¥eta(t)$ of (39) through $(0, ¥xi_{0}^{k})$ satisfies

(40) $||¥xi_{t}^{m}-¥eta_{t}||<¥frac{1}{2}¥epsilon$ for all $t¥geqq 0$

and

(41) $||¥xi^{m_{T(¥epsilon)}}-¥eta_{T(¥mathrm{g})}||<¥frac{1}{2}¥delta_{1}(¥epsilon)$.

Asymptotically Almost Periodic Solutio $r¥iota s$ of an Almost Periodic System 39

On the other hand, $¥eta(t)$ is a solution of

(42) $i(t)=F(t+¥tau_{k}, x_{t})+F(t+¥tau_{m}, ¥eta_{t})-F(t+¥tau_{k}, ¥eta_{t})$ ,

and $|F(t+¥tau_{m},¥eta_{t})-F(t+¥tau_{k}, ¥eta_{¥mathrm{f}})|=|g(t)|<¥delta_{2}(¥epsilon)$ by (37), because $¥eta_{t}¥in K$. Thus, if$m¥geqq k¥geqq k_{0}(¥epsilon)$ , we have

(43) $||¥xi_{t}^{k}-¥eta_{f}||<¥frac{1}{2}¥delta_{1}(¥epsilon)$ on $¥mathrm{o}¥leqq t¥leqq T(¥epsilon)$ .

From (40) and (43), it follows that if $m¥geqq k¥geqq k_{0}(¥epsilon)$ ,

(44) $||¥xi_{t}^{k}-¥xi_{f}^{m}||<¥epsilon$ on $0¥leqq t¥leqq T(¥epsilon)$ .

By (43), $||¥xi_{T(¥epsilon)}^{k}-¥eta_{T(¥mathrm{g})}||<¥frac{1}{2}¥delta_{1}(¥epsilon)$ , and hence $||¥xi_{T(¥epsilon)}^{k}-¥xi^{m_{T(¥epsilon)}}||<¥delta_{1}(¥epsilon)$ by (41).

Therefore, by the same argument as the above, we can see that if $m¥geqq k¥geqq k_{0}(¥epsilon)$ ,

$||¥xi_{t}^{k}-¥xi_{t}^{m}||<¥epsilon$ on $T(¥epsilon)¥leqq t¥leqq 2T(¥epsilon)$ ,

and, in general,

(45) $||¥xi_{t}^{k}-¥xi_{t}^{m}||<¥epsilon$ on $pT(¥epsilon)¥leqq t¥leqq(p+1)T(¥epsilon)$ , $p=0,1,2$, $¥cdots$

if $m¥geqq k¥geqq k_{0}(¥epsilon)$ . Since (45) implies that $|¥xi^{k}(t)-¥xi^{m}(t)|<¥epsilon$ for all $t¥geqq 0$ , we canconclude that $¥xi(t)$ is asymptotically almost periodic. Therefore the system (17)has an almost periodic solution.

Let $¥xi(t)=p(t)+q(t)$ , where $p(t)$ is almost periodic and $q(t)¥rightarrow 0$ as $ t¥rightarrow¥infty$ ,

and let $¥{¥tau_{k}¥}$ be a sequence such that $¥tau_{k}>0$ , $¥tau_{k}¥rightarrow¥infty$ , $p(t+¥tau_{k})¥rightarrow p(t)$ uniformlyon $R$ and $F(t+¥tau_{k}, ¥varphi)¥rightarrow F(t, ¥varphi)$ uniformly on $R¥times K$ as $ k¥rightarrow¥infty$. Then, $¥xi(t+¥tau_{k})$

$¥rightarrow p(t)$ uniformly on $L$ Therefore, by Lemma 7, the almost periodic solution$p(t)$ is uniformly asymptotically stable for $t¥geqq 0$ . This completes the proof.

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(Ricevita la 18-an de decembro, 1968)