数理解析研究所講究録
900 巻 1995 年 71-90

71

PERIODIC SOLUTIONS OF SOME DIFFUSIVEFUNCTIONAL DIFFERENTIAL EQUATIONS
Satoru Murakami (村上悟)(Okayama University of Science, 1-1 Ridai-cho, Okayama 700, Japan)\S 1.

Introduction.

In this paper, we shall consider the diffusive functional differential equationin$\frac{\partial u}{\partial t}(t, x)=D(x)\triangle u(\dagger, x)+f(t, x, ut(\cdot, X))$$(0, \infty)\cross\Omega$

(1.1)

,

together with the boundary condition$\frac{\partial u}{\partial n}(t, .x)=\kappa(x)(I’\mathrm{t}-v(t, X))$

on$(0, \infty)\cross\partial\Omega$

(1.2)

,

and discuss the existence of periodic solutions of (1.1) and (1.2). Here $u=(u_{1}, \cdots, u_{N}),$is a bounded domain in$\alpha\in(0,1))$ ,and$\triangle$

and$R^{p}$

with smooth boundary$\partial/\partial n$$D(x)=diag(d_{1}(x),$$di_{\mathit{0}}g(f_{\tau}’(1X),$$\cdots$,$\kappa_{N}(x)\mathrm{I}$$\cdots,$

with(e.g.,$\partial\Omega\in C^{2+\alpha}$

for some

respectively denote the Laplacian operator in

the exterior normal derivative at$R^{N},$$\partial\Omega$$\partial\Omega$$dN(x)\mathrm{I}$. Moreover,

with$R^{N}$

and$R^{\ell}$andis a (positive) constant vector in$d_{i}\in C^{\alpha}(\overline{\Omega})$$\kappa_{i}\in C^{1+\alpha}(\partial\Omega)$function mapping $R_{-}:=(-\infty.0$ ] into

$Ii^{\Gamma}$$\Omega$and $d_{i}(x)>0$ on$\kappa_{i}(x)\geq 0$on$\partial\Omega$and$\kappa(x)=$$\overline{\Omega},$$u_{f}(\cdot, x)$

defined by $u_{t}(\theta, x)=u(t+\theta, x)$ foris a$\theta\in R_{-}$.

The subject is intimately related to the work of Zhang [14], as well as the one of Burton
and Zhang [2]. In [14], Zhang has treated the equation (1.1) together with the Dirichletboundary condition, and by using an a priori$H^{1}(\Omega)$-bound for periodic solutions he has

deduced the existence of periodic solutions which satisfy (1.1) in the sense ofThe$\mathrm{p}_{\mathrm{U}1}\cdot \mathrm{P}^{\mathrm{o}\mathrm{S}\mathrm{e}}$$L^{2}(\Omega)$.

of this paper is to discuss the existence of periodic solutions which satisfy(1.1) and (1.2) in the classical sense and whose values are in a bounded region in$R^{N}$.

Especiallv. we feel interest ill periodic classical solutions whose range is contained in the

72positive cone in$R^{N}$, which are called positive periodic solutions of (1.1) and (1.2). In aclear reason, the existence of positive periodic solutions would be an important subject

in connection with biology, ecology or other fields. In the analysis of the subject, we-bound. Roughly speaking, in this-bound rather than anneed to have a$H^{1}(\Omega)$$C(\overline{\Omega})$

paper we shall employ the following strategy to deduce the existence of positive periodic
equipped
solutions of (1.1) and (1.2). First we consider the Banach space$X=C(\overline{\Omega})$with the supremum norm and the (unbounded) linear operator

in $X$ of the operator$\partial\Omega\}$$D\triangle$

with domain

which is the closure$A$$D(D\triangle)=\{\xi\in C^{2}(\overline{\Omega};RN)$:$\partial\xi/\partial n+\kappa\xi=0$

on, and then reformulate (1.1) and (1.2) as an abstract functional differential equation$\frac{dv}{dt}=Av(t)+G(t, v_{t})$,$t>0$ ,

(1.3)

on $X$ , where $v(t)=u(t, \cdot)-K$ and $G(t, v_{t})(x)=f(t, x, ut(\cdot, X))$ . Moreover, following
an idea in [2] and [14] we consider a functional differential equation with a parameter$k$

together with an associated map $H$ corresponding to (1.3). Next, observing that a fixed
point of the map $H$ for $k=1$ yields a periodic solution of (1.3), we deduce the existenceof positive periodic solutions of (1.3) from an a priori bound on all possible positive
fixed point of the map $H$ for $0<k\leq 1$ . Consequently, one can obtain positive periodic
solutions of (1.1) and (1.2) by assuming a$C_{\text{ノ}}(\overline{\Omega})$-bound for all possible positive periodic

solutions of the parametrized diffusive functional differential equation corresponding to(1.1) and (1.2) (Theorem 3.2). We provide also two examples to illustrate how ourtheorem is effectively applicable (Theorems 3.3 and 3.4). In the examples, we derive-bound for possible positive periodic solutions of the parametrized equation bya$C(\overline{\Omega})$

applying the maximum principle. Our approach in this paper would be advantageous
in several ways. Among others, it should be noted that the intermediate space definedby the fractional power $(-A)^{1/2}$ is not needed in the analysis of concrete problems(Theorems 3.3 and 3.4), while it played an important role in [14]. We emphasize thatthe structure of the intermediate space is well-known in case $X=L^{2}(\Omega)$ , but it is not

so in case

$X=C(\overline{\Omega})$

.

73

\S 2.

Abstract results.

Let $X$ be a Banach space with norm$||\cdot||$, and letbe a (unbounded) linear operator$A$

which generates an analytic compact semigroup $T(t)$ of bounded linear operators on $X$
with $\sup_{t\geq 0}||T(t)||<\infty$ . We consider the (abstract) functional differential equation$\frac{du}{dt}=A(u(t)-a)+F(t, u_{t})$

where

$a$

$(-\infty, 0]$$Y$$t>0$ ,,

(2.1)

is a fixed element in $X$ and $F$ is a function mapping $R\cross BC(R_{-}; X),$

, into $X$ . Here and hereafter, for any topological space$O$$R_{-}$

and any Banach space

we denote by $C(O;Y)$ the space of all continuous functions mapping

$O$

into

$Y$

, and by

$BC(O;Y)$ the space of all $\varphi\in C(O;Y)$ whose supremum norm $|| \varphi||:=\sup\{||\varphi(\theta)||$ :$O\}$

is finite. Moreover, for any function $u\in BC(R;Y)$ and any $t\in R,$

element in $BC(R_{-} ; Y)$ defined by $u_{t}(\theta)=u(t+\theta)$ for$\theta\in R_{-}$$:=$$\theta\in$denotes the$u_{t}$. We impose the following

condition on $F$ .
(H1) (i) For some $\omega>0,$ $F$ is$(t, \varphi)\in R\cross BC(R_{-} ; X)$$\omega$-periodic in , that is, $F(t+\omega, \varphi)=F(t, \varphi)$ for all$t$;(ii) for any $\varphi\in BC(R;x),$$F(t, \varphi t)$

is continuous in $t\in R$ ;(iii) for any $r>0$ there exist constants $L>0$ and $\theta\in(0,1]$ such that$||F(t, \varphi)-F(s, ?l’)||\leq L\{|t-S|^{\theta}+||\varphi-\psi||^{\theta}\}$for all$(t, \varphi),$$(s, \psi)\in[0, r]\cross BC(R_{-;x})$For any $c>0$ , we set $A_{c}=A-cI$ , where

$I$

with

$||\varphi||\leq r$

and$||\psi||\leq r$.

is the identity operator on $X$ . Clearly$A_{c}$

generates the analytic compact semigroup $T_{c}(t):=T(t)e^{-C}t$ . Since the semigroup $T(t)$ is
uniformly bounded by the assumption, one can derive that for each $z\in C$ with $\Re z>0$ ,the bounded inverse $(A-zI)-1$ exists and it is given by the formula $(A-\sim\gamma I)-1x=$$- \int_{0}^{\infty}e^{-}T(t)_{X}ztdt,$$x\in X$ .

Combining this with the fact that the semigroup $T(t)$ is

74

analytic, one can see that there exist $M>0$ and $\pi/2<\eta<\pi$ such that
invertible for $\lambda\in\Sigma:=\{\lambda : |\arg\lambda|<\eta\}\cup\{0\}$ and$||(A_{c}- \lambda I)^{-1}||\leq\frac{M}{1+|\lambda|}$Therefore the fractional powers$(-A_{c})^{\alpha}\mathrm{o}\mathrm{f}-A_{c}$,$\lambda\in\Sigma$$A_{c}-\lambda I$

is.

is defined for(e.g., [8, Section$\alpha\geq 0$2.6]), and the estimate$||(-A_{c})\alpha\tau_{C}(t)||\leq C_{\alpha}t^{-\alpha}e^{-\delta t}$holds, here

2.6.13$(\mathrm{c})])$$\delta$

and

,

(2.2)

$t>0$ ,

are some positive constants (independent of ) (e.g., [8, Theorem$t$$C_{\alpha}$.

Now we consider the space$\mathcal{X}=c_{\omega}(R;x)=${ $\varphi\in C(R;x)$ :$\varphi(t+\omega)=\varphi(t)$

on

$R$

}.

Clearly X endowed with the norm$|| \varphi||=\sup\{||\varphi(t)|| :is a Banach space. For any$\varphi\in \mathcal{X}$0\leq t\leq\omega\}$, we set,$( \mathcal{H}\varphi)(t)=a+\int_{-\infty}^{t}T_{\mathrm{c}}(t-\theta)[c(\varphi(\theta)-a)+F(\theta, \varphi_{\theta})]d\theta$

By the periodicity of the function$F(\theta.\varphi_{\theta})$$t\in R$

(2.3).

and (2.2) with $\alpha=0$ , one can see that$\mathcal{H}$isa well-defined mapping from X into X.

Lemma 2.1.

The map$\mathcal{H}$:$\mathcal{X}\mapsto \mathcal{X}$is compact.

Proof. First we establish the continuity of$\mathcal{H}$. Let$\varphi$and$\psi$

with$||\varphi||\leq r$

and$||\psi||\leq r$

be given. By (Hl-iii) and (2.2). we have$||\mathcal{H}\varphi-\mathcal{H}\psi||$$=$$\sup_{0\leq t\leq\omega}||\int_{-\infty}^{t}\tau_{c}(t-\theta)[c(\varphi(\theta)-\psi,(\theta))+F(\theta, \varphi\theta)-F(\theta, \psi\theta)]d\theta||$$\leq$$\sup_{0\leq t\leq\omega}.\int_{-}^{t}.\infty\{c_{0}e^{-}-\theta)d\delta(t\theta c||\varphi-\psi||+L||\varphi-\psi||^{\theta}\}$$\leq$$(C_{0}/\delta)\{c||\varphi-\psi||+L||\varphi-\psi||\theta\}$.

75This shows the continuity ofis relatively compact in$\mathcal{L}S$$\mathcal{H}$$\mathcal{X}$. Next we prove that for any bounded set $S$ in, where$\mathcal{L}\varphi:=\mathcal{H}\varphi-a$$\sup\{||C(\varphi(\theta)-a)+F(\theta, \varphi\theta)|| :$\mathcal{X}$the set. By (Hl-iii), we get\varphi\in S, \theta\in R\}(=:Q)<\infty$.

Then (2.2) yields$||(-A_{c})^{/}12(\mathcal{L}\varphi)(t)||$for all$\varphi\in S$$Q \int_{-\infty}^{t}||(-Ac)^{1}/2\tau_{c}(t-\mathcal{T})||d_{\mathcal{T}}$$\leq$$QC_{1/2} \int_{-\infty}^{t}(t-\mathcal{T})^{-}1/2(\mathrm{t}-\tau)e^{-s}d_{\mathcal{T}}$$=$$QC_{1/2} \int_{0}^{\infty}S^{-1/2-\delta s}edS<\infty$, which shows that the set $[((-A_{c})1/2\mathcal{L})s](t)$ is bounded in $X$ for all

Therefore the set$t\in R$ .

$\leq$$(\mathcal{L}S)(t)$is relatively compact in $X$ , because the operator$(-A_{\mathrm{c}})^{-1}/2= \frac{1}{\Gamma(1/2)}.\int_{0}^{\infty}t^{-}/2T1(Ct)dt$functionsset$\mathcal{L}S$and$\{(\mathcal{L}\varphi)(\cdot) :\varphi\in S\}$is equicontinuous onis relatively compact in$\varphi\in S$: $X\mapsto X$ is compact. We claim that the family of$\mathcal{X}$$R$. If the claim holds true, then theby the Ascoli-Arz\’ela theorem, as required. Let $h>0$

be given, and set $g(f_{\text{ノ}})=C(\varphi(t)-a)+F(t, \varphi_{t})$ . Then$||(\mathcal{L}\varphi)(t+h)-(\mathcal{L}\varphi)(t)||$$=$$\leq$$||.

\int^{\mathrm{t}+h}t\int\tau_{C}(t+h-\mathcal{T})g(\tau)d\tau+.\infty-(t(T_{\mathrm{C}}(t+h-\tau)-T\mathrm{C}(t-\tau))g\mathcal{T})d_{\mathcal{T}}||$$c_{\text{ノ}}\mathrm{o}Qh+Q \int_{-\infty}^{\dagger}||T_{C}(t+h-\mathcal{T})-\tau_{c}(t-\mathcal{T})||d\tau$by (2.2). Since$||\tau_{c}(t+h-\tau)-\tau_{c}(t-\mathcal{T})||\leq 2(_{-0}^{\gamma}\mathit{1}e^{-\delta()}t-\mathcal{T}$$||T_{c}(t+h-\mathcal{T})-\tau_{c}(t-\mathcal{T})||$

for $\tau<t$ , we get$||(\mathcal{L}\varphi)(t+h)-(\mathcal{L}\varphi)(t)||$

$=$

and

$|| \int_{t-\overline{j}}^{t-T}+h|A_{C}\tau c(\theta)d\theta|$$\leq$$C_{1} \int_{t-\tau}^{t-\tau}+h\theta^{-1}e-\delta\theta d\theta$$\leq$$C_{1}h(t-\mathcal{T})^{-1}e-\delta(t-\mathcal{T})$76$\leq$$C_{0}Qh+Q \{\int_{-\infty}^{t-h}||\tau_{c}(t+h-\tau)-\tau_{c}(t-\tau)||d\tau+.\int_{t-h}^{t}||\tau_{c}(t+h-\tau)-T_{c}(t-\mathcal{T})||d\mathcal{T}\}$$\leq$$C_{0}Qh+Q \{\int_{-}t-h(\infty C1ht-\mathcal{T})-1e-\delta(t-\mathcal{T})d_{\mathcal{T}}+\int_{t-h}^{t}2C\text{ノ}0e^{-\delta}-t\mathcal{T})d\mathrm{t}\mathcal{T}\}$$\leq$$C_{0}Qh+Q \{C1h\int_{h}^{\infty}\theta^{-}1e-\delta\theta d\theta+2C0h\}$$3C_{0}Qh+Qc1h(\delta^{-1}+|\log h|)$ .

$\leq$

Thus$\sup\{||(\mathcal{L}\varphi)(t+h)-(\mathcal{L}\varphi)(t)|| :\varphi\in S, t\in R\}\leq 3C_{0}Qh+Qc_{\mathit{1}}1h(\delta^{-1}+|\log h|)$for $h>0$ , and consequently $\sup\{||(\mathcal{L}\varphi)(t+h)-(\mathcal{L}\varphi)(t)|| : \varphi\in S, t\in R\}arrow 0$ as $harrow 0^{+}$

This proves the equicontinuity, as required.
For any $k\in R$ we consider the equation$\frac{du}{dt}=A_{c}(u(t)-a)+k[c(u(t)-a)+F(t, u_{t})]$ ,and moreover, we define the mapping$H:R\mathrm{x}\mathcal{X}\text{ト}arrow \mathcal{X}$(2.4)$t>0$ ,by$[H(k, \varphi)](t)=a+k\int_{-\infty}^{t}\tau_{c}(t-\theta)\{c(\varphi(\theta)-a)+F(\theta, \varphi_{\theta})\}d\theta$

for any$(k, \varphi)\in R\cross \mathcal{X}$. In case of $k=1,$ $(2.4)$ and $H(k, \cdot)$ are identical with (2.1) andrespectively. $u\in C(R;^{x})$ is called a solution of (2.4) if
on$(0, \infty)$

(2.5)

$u$

$\mathcal{H}$,is continuously differentiable

and it satisfies (2.4) together with $u(t)-a\in D(A_{c})=D(A)$ for $t>0$ .

Lemma 2.2. Let$H(k, \varphi)=\varphi$$\varphi\in \mathcal{X}$. Then$\varphi$

is an$\omega$-periodic solution of (2.4) if and only if.

Proof. In order to prove the “if part, we suppose”$H(k, \varphi)=\varphi$$\varphi(t)-a=\tau_{C}(t)(\varphi(0)-a)+k\int_{0}^{t}T_{C}(t-S)g(s)dS$,. We first assert that$t\geq 0$

,

(2.6)

77

where $g(t):=c(\varphi(t)-a)+F(t, \varphi_{t})$ . Indeed, sincesequence of continuously differentiable functions
and $||g_{n}(t)-g(t)||<1/n$ on$(1/h)\{\tau_{c}(h)-I\}v_{n}(t)$$R$$g\in C_{\omega}\text{ノ}(R;X)$$\{g_{n}\}\subset C_{\omega}(R;x)$, one can choose asuch that $||g_{n}||\leq||g||$. Set $v_{n}(t)= \int_{-\infty}^{t}T_{C}(f-s)gn(S)dS$ for $t\in R$ . Then$=$$(1/h) \mathrm{f}\int^{t}-\infty\tau_{C}(t+h-s)gn(s)ds-\int_{-\infty}^{t}\tau_{C}(t-S)gn(s)ds\}$$=$$. \int_{-\infty}^{t}T_{c}(t-\theta)\frac{g_{n}(\theta+h)-g_{n}(\theta)}{h}d\theta-\frac{1}{h}\int_{t-h}^{t}T_{\mathrm{C}}(t-\theta)gn(\theta+h)d\theta$

for $h>0$ , and hence$\lim_{harrow 0+}(1/h)\{\tau_{C}(h)-I\}v_{n}(t)$$=$$\int_{-\infty}^{t}\tau_{c}(t-\theta)g_{n}(’\theta)d\theta-g_{n}(t)$$=$$v_{n}’(t)-gn(t)$by the convergence theorem. We thus get$v_{n}(t)\in D(A_{C})$and$A_{\mathrm{c}}\iota_{\eta}’(t)=v_{n}’(t,)-g_{\eta}(t)$

(2.7)

.

Making use of this, one can derive the relation$v_{n}(t)=TC(t)v_{n}( \mathrm{o})+.\int_{0}^{t}T_{C}(t-S)g_{\eta}(s)d_{S}$,$t\geq 0$

.

(2.8)

Since $\lim_{narrow\infty}[kv_{n}(t)]=k\int_{-\infty}^{t}T_{C}(t-s)g(S)dS=[H(k, \varphi)](t)-a=\varphi(t)-a,$ $(2.6)$ follows
from (2.8). Now, by [8, Theorem 4.3.1], (2.6) implies thaton$(0, \infty)$, and in particular, it is H\"older continuous on$\varphi$

is locally H\"older continuous$[\omega, 2\omega]$. Therefore$\varphi$

is uniformly

H\"older continuous on $R$ because of the periodicity. From this observation and (Hl-iii) it

follows that the function$g(t)$$F(t, \varphi_{t})$of is H\"older continuous on$t$. Thus, by (2.6) and [8, Theorem 4.3.2], we see that

Next we prove the ‘only if ” part. Suppose that$\varphi$$\zeta$$\varphi\in \mathcal{X}$$R$, and so is the function

is a solution of (2.4).
is a solution of (2.4). Then$(d/dt)(\varphi(t)-a)=(d/dt)\varphi(t)=A_{c}(\varphi(t)-a)+kg(t)$, and hence$\varphi(t)-a=T_{c}(t)(\varphi(0)-a)+k.\int_{0}^{t}T_{C}(t-\theta)g(\theta)d\theta$

78

for $t\geq 0$ , where $g(t)=c(\varphi(t)-a)+F(t, \varphi_{t})$ . Since$g$$\varphi$

is -periodic, so is the function$\omega$. Then$\varphi(t)$

$=$

$=$

$a+ \tau_{c}(t+n\omega)(\varphi(0)-a)+k\int_{0}^{t+n\omega}T(Ct+n\omega-\theta)g(\theta)d\theta$$a+^{\tau_{c}}(t+n \omega)(\varphi(0)-a)+k\int_{-n\omega}^{t}T_{C}(t-\theta)g(\theta \mathrm{I}^{d\theta}\cdot$

Note that $||\tau_{c}(t+n\omega)(\varphi(0)-a)||arrow 0$ as $narrow\infty$ . Letting $narrow\infty$ in the above, we get$\varphi(t)=a+k\int_{-\infty}^{t}\tau_{c}(t-\theta)g(\theta)d\theta=[H(k, \varphi)](t)$ , as required.

Proposition 2.3. Assume $(Hl).$ Moreoverf let$a$$\in G_{f}$be a bounded open set in$\mathcal{X}$withand suppose that$\varphi\not\in\partial G$

whenever$\omega$$G$$\varphi\in\overline{G}$(Proof. Consider the operator
then there exist

boundary of$G$)is a solution of (2.4) with $k\in(0,1]$ . Then the equation (2.1) has an-periodic solution which belongs to

We assert that$:=\mathrm{t}\mathrm{h}\mathrm{e}$.

$\mathcal{I}=\mathcal{H}|_{\overline{G}}:\overline{C_{X}}\mapsto \mathcal{X}$$\mathcal{T}\varphi\neq a+\tau(\varphi-a)$$\varphi\in\partial G$$G$

for all $\tau>1$ andand $\tau>1$ such$\varphi(t)=a+\frac{1}{\tau}[(\mathcal{H}\varphi)(t)-a]$, where$\mathcal{H}$is the one defined by (2.3).

$\varphi\in\partial G$. Indeed, if this is false,$\mathrm{t}_{c}\mathrm{h}\mathrm{a}\mathrm{t}\tau\varphi=a+\tau(\varphi-a)$, and hence$=$$a+ \frac{1}{\tau}\int_{-\infty}^{t}\tau_{c}(t-\theta)[C(\varphi(\theta)-a)+F(\theta, \varphi\theta)]d\theta$$=$$[H(1/\tau, \varphi)](t)$.

is a solution of (2.4) with $k=1/\tau$ by Lemma 2.2, which contradicts
our assumption. Thus the assertion must be true. Now the operator is compact byThen$\varphi\in\partial G$$\mathcal{T}$Lemma 2.1. Therefore, by the fixed point principle of oInitted rays (e.g., [13, Theorem. Such is a solution of (2.4) with $k=1$
such that13. ]), there exists a$\mathrm{A}$$\varphi\in\overline{G}$by Lemma 2.2, and hence$\varphi\in G$$\mathcal{T}\varphi=\varphi$$\varphi$by our assumption. Thus$\varphi$is the desired one.

79

\S 3.

Periodic solutions of some diffusive functional differential

equations.
Throughout this section, we will employ the following notation. Let

dimensional Euclidean space with norm$|\cdot|$$R^{N}$

be the N-. For any diagonal matrix $B=diag(b_{1}, \cdots, b_{N})$and any vector $u=(u_{1}, \cdots, u_{N})$ , we denote by $Bu$ the vector. For any$(b_{1}u_{1}, \cdots , b_{N}u_{N})$vectors $u=(u_{1}, \cdots, u_{N})$ and $v=(v_{1}, \cdots, v_{N})$ , we write as $u\leq v$ (resp. $u<v$ ) when-ever$u_{i}\leq v_{i}$(resp.

$\cdots,$$N$(e.g.,$\partial\Omega$$\partial\Omega\in C^{2+\alpha}$exterior normal derivative at$\triangle v$

anddenote$\partial v/\partial n$$\partial\Omega$. If $u,$ $v\in R^{N}$ with $u\leq v$ , we set

and call it an interval in

the set $\{u\in R^{N} : 0:=(0, \cdot\cdot’, 0)\leq u\}$ . Letsmooth boundaryand) for all $\dot{l}=1,$u\leq w\leq v\}$ ,$[u, v]=\{w\in R^{N} :$R_{+}^{N}$$u_{i}<v_{i}$$\Omega$$R^{N}$. Also, we denote by

be a bounded domain infor some $\alpha\in(0,1)$ ), and denote by. Furthermore,$\triangle$$\triangle v=(\triangle v_{1}, \cdots, \triangle v_{N})$$R^{l}$

with$\partial/\partial n$denotes the Laplacian operator inandthe$R^{\ell}$$\partial v/\partial n=(\partial v_{1}/\partial n, \cdots, \partial v_{N}/\partial n)$,,.
respectively, for any (smooth) mapping $v=(v_{1}, \cdots, v_{N})$ :In this section, we discuss the existence of periodic solutions of the diffusive functional$\overline{\Omega}\mapsto R^{N}$

differential equation$\frac{\partial u}{\partial t}(t, .\mathrm{r})=D(x)\triangle u(t, X)+f(t, x, u_{t}(\cdot, X))$

in$(0, \infty)\cross\Omega$

(3.1)

.

(3.2)

satisfying the boundary condition$\frac{\partial u}{\partial n}(t, x)=\kappa(X)(I\zeta-u(t, x))$Here $D(x)=d_{l}ag(d_{1}(x), \cdots , d_{N}(X))$ with$diag(\kappa_{1}(x), \cdots, \kappa_{N}(x))$

with

on

$(0, \infty)\cross\partial\Omega$$d_{i}\in C^{\alpha}(\overline{\Omega})$$\kappa_{i}\in C^{1+\alpha}(\partial\Omega)$

andwith $d_{i}(x)>0$ on$\kappa_{i}(x)\geq 0$on$\partial\Omega$$\overline{\Omega},$$\kappa(x)=$, and $K\in R^{N}$ is a

(fixed) constant vector such that $0<K$ . We assume the following condition on the

mapping$f$:$R\cross\overline{\Omega}\cross BC(R_{-} ; R^{N})\mapsto R^{N}$(H2) (i) $f(t.X, \xi)$ is(ii) for each$\omega$.

-periodic in ;$t$$\psi\in BC(R\cross\overline{\Omega};R^{\mathit{1}\backslash ^{\tau}})$. the function. $f(t, x, \sqrt,t(\cdot, .r))$ is continuous in

80$(t, x)\in R\mathrm{X}\overline{\Omega}$;(iii) for any $r>0$ there exist constants $L>0$ and $\theta\in(0,1]$ such that$|f(t, X, \xi)-f(_{S}, y, \mathrm{x})|\leq L\{|t-s|\theta|_{X}+-y|^{\theta}+||\xi-\chi||\theta\}$

for all $(t, x, \xi),$$(s, y, \chi)\in[0, r]\mathrm{x}\overline{\Omega}\mathrm{x}BC(R_{-} ; RN)$

with$||\xi||\leq r$

and$||\chi||\leq r$.

In order to apply the results in the previous section, we take the Banach space$C(\overline{\Omega};R^{N})$equipped with the supremum norm as $X$ , and define the map$BC(R_{-} ; X)\mapsto X$$t\in R,$$($

Clearly $F$ satisfies the condition (H1). For each $i=1,$(unbounded) linear operator$\tilde{A}_{i}$with the omain$\overline{\mathrm{d}}$$d_{i}\triangle$$R\cross$by$F(t, \varphi)(x)=f$ ( $t.$ I. $\tau\hat’\cdot.X)$ ).

operator:$F$

virtue of [11, Theorem 2],$\tilde{4}$in$-\tilde{\lambda’}=C_{\text{ノ}}(\overline{\Omega};R)$$D(d_{i}\triangle)=${$x\in\overline{\Omega}$$\cdots$.

, $N$ , we next consider thewhich is the closed:$\xi\in C_{\text{ノ}^{}2}(\overline{\Omega};R)$$\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}\backslash$$\partial\xi/\partial n+\kappa_{i}\xi=0$

generates an analytic semigroup$\tilde{T}_{i}(t)$on$\tilde{X}$onof the$\partial\Omega$}. By. Moreover, by

the estimate$|z|^{-}1/2||\xi||_{C^{1}(}\overline{\Omega})\leq C||(\tilde{A}_{i}-z)\xi||c(\overline{\Omega})$

for all complex$z$

in a truncat,ed sector

that each resolvent operators of$\tilde{A}_{i}$$| \arg_{Z}|\leq\frac{1}{2}\pi+\epsilon,$is compact on$\tilde{X}$$|z|\geq\lambda_{0}$(cf. (1.1) in [11]), we see, and hence the semigroupcompact. It is a direct consequence of the ma,ximum principle thatthat is,$||\tilde{T}_{i}||\leq 1$

$A$

is

is nonexpansive;. For any $\varphi=(\varphi_{1}, \cdots, \varphi_{N})\in D(\tilde{A}_{1})\cross\cdots\cross D(\tilde{A}_{N})=:D(A)$ , we set$A\varphi=$Then$\tilde{T}_{i}(t)$$\tilde{T}_{i}(t)$$(\tilde{A}_{1\varphi 1}$, .. ,$\cdot$$\tilde{A}_{N}\varphi_{N})$generates the analytic compact semigroup.

$T(t):=(\tilde{\tau}_{1}(t), \cdots , \tilde{T}_{N}(t))$

pansive bounded linear operators on $X$ . Take any (small) constant$c>0_{\tau}$of nonex-

and consider

the operator $H(k, \varphi)$ defined by (2.5) with $a=K$ . By Lemma 2.2, $\varphi=H(k, \varphi)$ means
that$\varphi(t)$is an$\omega$-periodic solution of the (abstract) equation (2.4) on$X=C(\overline{\Omega};R^{N})$.

81

Now we certify that $u(t, x):=[\varphi(t)](x)$ satisfies the diffusive functional differential
equation$\frac{\partial u}{\partial t}(t, x)=D(x)\triangle u(t, x)+c(1-k)(K-u(t, x))+kf(t, x, v_{t}(\cdot, x))$in$(0, \infty)\cross\Omega$(3.3), together with the boundary condition (3.2), whenever $\varphi=H(k, \varphi)$ .

Lemma 3.1. Let$(H\mathit{2})$hold, and suppose that $H(k, \varphi)=\varphi$ for somethe function $u(t, x):=[\varphi(t)](x)$ is an$\omega$$\varphi\in \mathcal{X}$-periodic (classical) solution of (3.3) and. Then$(\mathit{3}_{\sim}^{\mathit{6})}.)$.

Proof. The lemma can be proved by the standard regularity argument (e.g., [4, pp.7576]). For completeness we contain the proof. Set $v(t)=\varphi(t)-a$ and $g(t)=cv(t)+$$F(t, \varphi_{t})$

.

Since

$v$

is

$\omega$-periodic, from the fact that $v\in C^{1}((0, \infty);x_{)}$ it follows that$v(t, x):=[v(t)](X)$ is continuously differentiable with respect to $t\in R$ uniformly for$x\in\overline{\Omega}$$\infty$$C$.

, together with $\sup_{t\in R}||(d/dt)v(t)||=\sup\{|(\partial/\partial t)v(t, x)| : t\in R, x\in\overline{\Omega}\}<$. Then (H2-iii) yields that $||g(t)-g(s)||\leq C,$$|t-S|^{\theta}$

$

$t,$

$s\in R$ ,for some constant

Let $0<\delta<\beta<\theta$ , and take $p>0$ so large that $\delta+(N/p)<\min\{2\beta, 1\}$ .

Since $(d/dt)v(t)=A_{c}v(t)+kg(t)$ , it follows from [4, Lemma 3.5.1] that the function
is locally H\"alder continuous, and consequently$t\in Rrightarrow A_{C}^{\beta}(dv/dt)\in X(\subset L^{p}(\Omega))$

the function$t\in R\mapsto(\partial v/\partial t)(t, \cdot)\in C^{\mathit{5}}(\overline{\Omega})$standard argument in$L^{p}$

also is locally H\"older continuous by the-theory (e.g., [4, p.75], [8, Chapter 8]). Also, from the fact thatthe function $t\in R\mapsto A_{c}v(t)\in X$ is continuous, it follows that the function $t\in R\mapsto$$v(t, \cdot)\in c^{1+\mathit{5}}(\overline{\Omega})$

is continuous. Thus

$A_{c}v(t)=dv/dt-kg(t)\in C^{\mathit{5}}(\overline{\Omega})$$g(t)(\cdot)\in C^{\theta}(\overline{\Omega})$. Henceby (H2-iii), and consequently$v(t, \cdot)\in C^{2+\mathit{5}}(\overline{\Omega})$theorem for elliptic equations (cf. [4, p.10]), and consequently
iii). Also, by the standard argument in$\partial v/\partial n+\kappa v=0$

on$(0, \infty)\cross\partial\Omega$$g\in C’(\delta/2,\delta R\cross\overline{\Omega})$-theory (e.g., [4, p.75]) it follows that$v$

by (H2-

satisfies. Consequently, $\partial u/\partial n=\partial v/\partial n=-\kappa v=\kappa(K-u)$ on

satisfies (3.2). Since the compatibility condition of order$(0, \infty)\mathrm{x}\partial\Omega$, and hence

satisfied for$v(0, \cdot)\in C^{2+\delta}(\overline{\Omega})$$u$$L^{p}$

by a classical regularity

$0$

is

, by [5, Theorem 5.3, p.320] there exists a unique function

82$\overline{v}\in C^{1+\delta/+}2,2\delta(R\cross\overline{\Omega})$onsatisfying$\partial\overline{v}/\partial t=D\triangle\overline{v}-C\overline{v}+kg$

and $\overline{v}(0, x)=v(0, x)$ on$(0, \infty)\mathrm{x}\partial\Omega$$\overline{\Omega}$in$(0, \infty)\mathrm{x}\Omega,$$\partial\overline{v}/\partial n+\kappa\overline{v}=0$. Then $(d/dt)\overline{v}(t)=A_{c}\overline{v}(t)+kg(t)$ andin $X$ , and hence one gets $\overline{v}(t)=T_{\mathrm{c}}(t)v(0)+k\int_{0}^{t}Tc(t-s)g(s)dS=v(t)$ or$\overline{v}(0)=v(0)$$\overline{v}(t, x)\equiv v(t, x)$. Consequently, the function $u(t, x)$ is continuously differentiable in ,$t$

twice continuously differentiable in , and satisfies (3.3) on$x$$R\cross\Omega$. This completes theproof.
Combining Lemma 3.1 with Proposition 2.3, we obtain the following result:
Theorem 3.2. Let$\mu_{1},$$\mu_{2},$$\nu_{1}$and$\nu_{2}$in$R^{N}$$(H\mathit{2})$hold, and assume that there exist some constant vectors

such that$\mu_{1}<\mu_{2}\leq K\leq\nu_{2}<\iota \text{ノ_{}1}$$\mu_{2}\leq u(t, X)\leq\nu_{2}$whenever $u(t, x)$ is an$u(t, x)\leq\nu_{1}$on$R\cross\overline{\Omega}$$\omega$on

and that$R\cross\overline{\Omega}$-periodic solution of (3.3) with $k\in(0,1]$ satisfying

together$u$ ) $ith(\mathit{3}.\mathit{6}\mathit{2})$. Then there exists anof (3.1) and (3.2) of which the range is contained in the interval$\omega$$\mu_{1}\leq$-periodic solution$[\mu_{2}, \nu_{2}]$.

Now we provide two examples which show how Theorem 3.2 is effectively applicable.

Example 1. Together with the boundary condtion (3.2) (with $N=1$ ), we consider
the scalar diffusive functional differential equation$\frac{\partial u}{\partial t}(t, x)=d(x)\triangle u(t, x)-\lambda(t, X)u(t, X)+g(t, x, ut(\cdot, x))$where

$d\in C^{\alpha}(\overline{\Omega})$with $d(x)>0$ on$\overline{\Omega}$in$(0, \infty)\cross\Omega$(3.4). Assume that:(H3) (H2) is satisfied for the function $f(t, x, \xi)=-\lambda(t, x)\xi(\mathrm{o})+g(t,$(H4) (i),$X,$ $\xi$

I with $N=1$ .

;$0< \underline{\lambda}:=\inf_{t,x}\lambda(t, x)\leq\sup_{t,x}\lambda(t, x)=:\overline{\lambda}<\infty$(ii) $\xi\in BC(R_{-} ; R+),$ $R_{+}:=[0, \infty)$ , implies $\inf_{t,x}g(t, x, \xi)>0$ . Moreover,$\in BC(R_{-;}R_{+})$

with$\xi(\theta)\leq\chi(\theta)$

on

$R$

-implies $g(t, x, \xi)\geq g(t, x, \chi)$ on$\xi,$$\lambda$$R\cross\overline{\Omega}$(iii) there exists a constant $\nu_{1}>0$ such that $\nu_{1}>\nu_{2}:=(1/\underline{\lambda})\sup_{t,x}g(t, x, 0)$;

83

and thaton$g(t, X, \nu_{1})\leq K\lambda(t, x)\leq g(t, x, 0)$$R\cross\overline{\Omega}$,

where $K$ is the one in (3.2).
Equation (3.4) describes a mathematical model for the survival of red blood cells inan animal (cf. [6, 12]). It is easy to see that (H4-ii) is satisfied whenever$g(t, X, \xi)=\int_{0}^{\infty}e^{-}-\ulcorner)d\gamma(_{\mathcal{T})\xi}(\cdot(p\mathcal{T})$, where $\gamma\in BC(R_{+}; R+)$ and

$p$

$g$

is given by: $R_{+}\mapsto R$ is bounded

and nondecreasing. We set$\mu_{2}=(1/\overline{\lambda})\inf_{t,x}g(t, x, \mathcal{U}_{1})$.

Then $0<\mu_{2}\leq K\leq\nu_{2}$ by (H4).

Theorem 3.3. Assume

above. Then there exists an
contained in the interval

Proof. Take a$u(t, x)$ be an$\omega$$(H\mathit{3})$$\omega$$(H\mathit{4})$,and let$\mu_{2}$and$\nu_{2}$be the constants cited-periodic solution of (3.4) and (3.2) of which the range is$[\mu_{2,2}\nu]$$\mu_{1}\in(0, \mu_{2})$and.

. Then $0<\mu_{1}<\mu_{2}\leq K\leq\nu_{2}<\nu_{1}$ . For any$k\in(\mathrm{O}, 1]$, let-periodic solution of the equation$\frac{\partial u}{\partial t}(t_{\mathrm{y}}.x)=d(x)\triangle u(t, X)+c(1-k)(K-u(t., x))+k[-\lambda(t, x)u(t, X)+g(t, x, ut(\cdot, x))]$in$(0, \infty)\cross\Omega$satisfying $\mu_{1}\leq v(t, x)\leq\nu_{1}$ on$R\mathrm{x}\overline{\Omega}$

together with the condition (3.2).

In order to establish the theorem, it suffices to prove that$\mu_{2}\leq u(t, x)\leq\nu_{2}$Now, let$m(t)$$\epsilon>0$

be any number such thaton

$R\cross\overline{\Omega}$.

$. \wedge<\min\{\mu_{1}.\overline{\lambda}\mu 2\}$, and consider the solutionof the ordinary differential equation$\frac{d}{d,t}m(t)=[-\lambda\cdot(\lambda-c)-c]r7l(t)+I\iota’c+k(\overline{\lambda}\mu_{\mathit{2}^{-}}I\mathrm{t}^{F}C)$,$t>0$ ,

84

with$m(\mathrm{O})=\mu_{1}-\epsilon$. Clearly $m(t)$ is given by$m(t)=( \mu_{1}-\epsilon)e^{-}-+c)t+(k\overline{\lambda}kC\frac{\mathrm{A}^{r_{C+}}k(\overline{\lambda}\mu_{2}-Ic_{c)}}{k^{\wedge}(\overline{\lambda}-c)+c}\{1-e-(k\overline{\lambda}-kc+c)t\}$

It is easy to check that $0<m(t)<I\mathrm{t}^{r}$ on$R_{+}$$u(t, x)>m(t)$ ,

.

(3.5)

. We assert that$(t, x)\in R_{+}\mathrm{x}\overline{\Omega}$

(3.6)

.

To establish the assertion by a contradiction, we suppose that (3.6) is false. Then there

exists some$(t_{0}, x\mathrm{o})\in(0, \infty)\cross\overline{\Omega}$$(t, x)\in[0, t_{0})\cross\overline{\Omega}$all. Set $w(t, x)=m(t)-u(t, x)$ . Then $w(t_{0}, X_{0})=0$ and $w(t, x)<0$ for$(t, x)\in[\mathrm{o}, t_{0})\cross\overline{\Omega}$$\frac{\partial w}{\partial t}(t, x)$

$=$

$=$

such that $u(t_{0,0}x)=m(t_{0)}$ and $u(t, x)>m(t)$ for all. Moreover, we get$\frac{d}{dt}m(t)-\frac{\partial u}{\partial t}(t, x)$$d(x)\triangle w(\dagger,, x)-(k\lambda(t, X)+c-kc)u)(t, x)+km(t)(\lambda(t, x)-\overline{\lambda})$$+k(\overline{\lambda}\mu_{2}-g(t, x, u_{t}(\cdot, x))$on$(0, t_{0}]\cross\Omega$. Since $0<\mu_{1}\leq u(t, x)\leq\nu_{1}$ on$g(t, x, u_{t}(\cdot, x))\geq g(t, x, \mathcal{U}_{1})\geq\overline{\lambda}\mu_{2}$$\underline{\lambda}\nu_{2}\geq g(t, x, 0)\geq$. This would lead to a contradiction. Indeed,$(\mathrm{o}, t_{0}]\cross\Omega$, then $w(t, x)\equiv 0$ on$[0, t_{0}]\cross\overline{\Omega}$$x_{0}\in\Omega$, we getby (H4-ii), and hence $d(x)\triangle w(t, x)-(\partial/\partial t)w(t, x)-$$(k(\lambda(t, x)+c-kC)w(t, X)\geq 0$ on

if$R\mathrm{x}\overline{\Omega}$by the strong maximum principle

Theorems 3.3.5, 3.3.6 and 3.3.7]), which is a contradiction because of$m(\mathrm{O})$. We thus obtain$(t_{0}, x\mathrm{o})$$x_{0}\in\partial\Omega$

and $w(t, x)<0$ on$[0, t_{0}]\mathrm{x}\Omega$$(\mathrm{e}.\mathrm{g}.,[9$,$u(\mathrm{O}, x)\geq\mu_{1}>$, and hence$\partial w/\partial n>0$

atby the strong maximum principle, again. However, this is impossible because of.

$(\partial w/\partial n)(t_{0,x\mathrm{o})-}=(\partial v_{\text{ノ}}/\partial n)(t_{0,\mathit{0}^{)}}X=\kappa(x0)(u(t_{0}, xo)-I\{\mathrm{i})=\kappa(xo)(m(t_{\mathit{0}})-K)\leq 0$

Now we obtain $\lim_{tarrow\infty}m(t)=[Kc+k(\overline{\lambda}\mu_{2^{-}}Kc)]/[k(\overline{\lambda}-C)+c]$ by (3.5), and con-sequently$\lim_{tarrow\infty}m(t)\geq\mu_{2}$because of $k\in(0,1]$ . Therefore, from (3.6) it follows that$\varliminf_{tarrow\infty}u(t, x)\geq\lim_{tarrow\infty}m(t)\geq\mu_{2}$on$\overline{\Omega}$, and consequently

the periodicity of $u(t, x)$ . Similarly, one can prove that$\min_{\overline{\Omega}}u(t, \cdot)\geq\mu_{2}$$\max_{\overline{\Omega}}u(t, \cdot)\leq\nu_{2}$

on

$R$

on

$R$

by

. Indeed,in place of $m(t)$ , we may consider the solution $M(t)$ of the ordinary differential equation$\frac{d}{dt}M(t)=[-(\underline{\lambda}-C)k-C]M(t)+Kc+k(\underline{\lambda}_{\mathcal{U}_{2^{-}}}K_{C)},$$t>0$ ,

85with $M(0)=\nu_{1}+\epsilon$ , and repeat the argument$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{r}$to the one for $;71(t)$ . This completes

the proof of t,he theorem.
Example 2. We next consider a system of diffusive functional differential equations$\frac{\partial\uparrow r_{i}}{\partial t}(t, x)=d_{i}(X)\triangle u_{i}(t, x)+ui(t, X)${$o_{i}(r.X)-l)i(\dagger,$$X \mathrm{I}u_{\mathrm{t}}(t..Y)-j1\sum_{=}^{N}Cij(t,$$x)h_{i}\dot{J}(t,$x.

$\mu t(\cdot,$

$x))$

}.

(3.7)
$i=1,$

in$(0, \infty)\cross\Omega$where $u=(v_{1},$$\cdots,$

continuous and(H5), $N$ .

$N_{7}$$\omega$$v_{N\mathrm{I}}$on, and the functions-periodic in$t$,$a_{?}\cdot$$(0, \infty)\mathrm{x}\partial\Omega$.

$b_{i}.c_{ij}$and moreover

:

(3.8)

,

$R\cross\overline{\mathrm{Q}}_{-}\mapsto R,$$d_{i}\in C^{\alpha}(\overline{\Omega})$$i,j=1,$$\cdots,$

with $d_{i}(x)>0$ on

$N$

, are

$\overline{\Omega}$

for

Note that (3.8) is a special case of (3.2) (with $\kappa=0$ ). We assume that:(H2) is satisfied for the function $f(t, x_{\}\xi):=(f_{1}(t, X, \xi), \cdots , f_{\mathrm{A}^{7}}(t, X, \xi))$ , where$f_{i}$

for$, together with the homogeneous Neumann boundary condition$\frac{\partial v}{\partial n}(t, x)=^{\mathrm{o}}$$\iota=1,$ $\cdots$$\cdots$($t,$x. ) $= \xi_{i}(0)\{a_{?}(t, x)-b_{i}(t.X)\xi_{\mathrm{i}}(0)-.\sum_{1J=}cij(t.x)h_{i_{\dot{J}}}(t, x.\xi)\}N$$\xi=(\xi_{1}, \cdots, \xi_{N})$

$\xi$

.
$0< \underline{a}_{i}:=\inf_{t,x}a_{i}(t, x)\leq\sup_{t,x}a_{i}(t, X)=:\overline{a}_{i}<\infty$(H6) (i)$0<\underline{b}_{i}$,,$:= \inf_{t,x}b_{i}(t, x)\leq\sup_{t,\alpha}.b_{i}(t, x)=:\overline{b}_{i}<\infty$$0 \leq \mathrm{r}jC:=\inf_{t,x}c_{i_{J}}\cdot(t, x)\leq\sup_{t,x^{\mathrm{Q}_{j}}}(t, X)=:\overline{c}_{i_{\dot{J}}}<\infty$

for$\dot{\iota},\dot{J}=1,$$\cdots,$$N$ ,and moreover$\underline{a}_{i}>\sum_{j=1}\overline{C}_{i}j(N\overline{a}_{i}/\underline{b},\cdot)$

86

for $i=1,$$\cdots,$$N$;(ii) for any $i,j=1,$$\cdots,$$N,$ $\xi,$ $\chi\in BC(R_{-} ; R_{+}^{N})$with$\xi(\theta)\leq\chi(\theta)$on$R_{-}$

implies $0\leq h_{ij}(t, x, \xi)\leq h_{ij}(t, x, x)$ , and moreover $h_{ij}(t, x,p)=pj$(normalized) whenever $p=(p_{1}, \cdots,p_{N})$ is a constant.
$N$
In mathematical ecology, (3.7) describes the growth of competing -species whosei-th population density at time and place is $u_{i}(t, x)$ , and $h_{ij}(t, x, ut(\cdot, X))$ represents
$x$

$t$

the effect of the past history on the present growth rate (cf. [1, 3, 6, 10]). It is easy
for $\xi=(\xi 1, \cdots,\xi_{N})\in$
to see that (H6-ii) is satisfied if$h_{ij}(t, x, \xi)=\int_{0}^{\infty}K_{ij}(\theta)\xi_{j(-}\theta)d\theta$$BC(R_{-;)}R^{N}$, where$K_{ij}(\theta)\geq 0$

and$\int_{0}^{\infty}K_{\mathrm{i}j}(\theta)d\theta=1$. The condition (H6-i) is the one

considered by Gopalsamy in [3] to derive the existence of a globally stable -periodic
solution for integrodifferential equations (without diffusion). Ahmad and Lazer [1, The$\omega$

depends on the variable as wellandorem 4.1] have treated the case where
as , and established the existence of an -periodic solution under the condition (H6-i); that is, (3.7) does notwith $N=2$ . We remark that the case where$\mathit{1}\mathrm{V}=2$$t$$d_{i}$$\omega$$x$$h_{ij}(t, x_{\text{ノ}}.\xi)=\xi_{j}(0)$not, applicablecontain delay-terms, has been treated in [1], and the method in [1] isto delay-equations. In the following theorem we also impose the condition (H6-i) anddeduce the existence of an -periodic solution of (3.7) and (3.8).
$\omega$

Theorem 3.4. Assunt$e(H\mathit{5})and(H\mathit{6})$(3.7) and (3.8) of which the$rang\epsilon$. Then there exists an$\omega$-periodic solution ofis contained in the product.

$\prod_{i=1}^{N}[\frac{\underline{a}_{i}-\Sigma_{\dot{J}}^{\backslash r}=1^{\overline{C}_{i}}j(\overline{a}j/\underline{b}_{j})}{\overline{b}_{i}}, \frac{\overline{a}_{i}}{\underline{b}_{i}}]$

Proof. Take a$\delta>0$

so that$0<2 \delta\overline{b}_{i}<\underline{a}_{i^{-}}\sum_{j=1}^{N}\overline{C}ij(\overline{a}_{j}/\underline{b}_{\dot{J}}\mathrm{I},$which is possible by (H6-i), and consider the vectors$i=1,$$f^{l_{1}},$$\cdots,$$\mu_{2},$$\mathcal{U}_{1}$$N$ ,.l ノ 2and$I\iota’$in$R^{N}$

whose

87

i-th components are defined by$\mu_{1,i}$$\nu_{2,i}$

respectively. Since

$=$

$=$

$\delta$,$\mu_{2,i}=[\underline{a}_{i}-\sum_{=j1}^{N}\overline{C}ii(\overline{a}_{j}/\underline{b})j]/\overline{b}_{i}$$\overline{a}_{i}/\underline{b}_{i}$,$\nu_{1,i}=2\overline{a}_{i}/\underline{b}i$

’

,

,

$K_{i}=( \overline{a}_{i}-2\delta\sum^{N}c_{j})j=1\infty/\underline{b}_{i}$for all , we get $0<\mu_{1}<\mu_{2}\leq K\leq\nu_{2}<\nu_{1}$ . Now, for$\dot{\iota}$$2\delta<\overline{a}_{i}/\underline{b}_{i}$any $k\in(0,1]$ , let $u(t, x)=(u_{1}(t, x),$$\cdots,$$uN(t, x))$ be an$\omega$-periodic solution of a systemof diffusive functional differential equations$\frac{\partial u_{i}}{\partial t}(t, x)$$=$$d_{i}(x)\triangle?r_{i}(t,$$x\mathrm{I}+c(1-k\mathrm{I}(\mathrm{A}’-u_{i}i(t, x))$$+ku_{i}(t, X) \{ai(t, x)-bi(t, x)ui(t, x)-j\sum_{=1}Cij(t, x)h_{i}j(t, xN, ut(\cdot, X))\}$$i=1,$

in$(0, \infty)\cross\Omega$satisfying $\mu_{1}\leq u(t, x)\leq\nu_{1}$ on$R\cross\overline{\Omega}$$\cdots,$$N$,

together with (3.8), where $c>0$ isa constant. In order to establish the theorem, it suffices to certify that$\mu_{2}\leq u(t, x)\leq\nu_{2}$Consider the solutionon$M(t)=(M_{1}(t), \cdots.-lI\wedge\backslash ^{-(t}))$$R\cross\overline{\Omega}$

(3.9)

.

of a system of ordinary differential

equations$\frac{d}{d\mathrm{f}}M_{i}(t)$$=$,$k \cdot\{(M_{i}(t)-I^{r_{?}}\backslash )(C-\frac{c}{\lambda^{\wedge}})+M_{i}(t)(\overline{a}i-\underline{b}_{i}M_{i}(t)-J’.\sum_{1=}^{N}\underline{c}ij\mu_{1},j)\}$$t>0$ ,

with$M(\mathrm{O})=\nu_{1}+\epsilon$, whereglobally and $0<M(\dagger)$ on$\epsilon$$i=1,$$\cdots,$$N$,

is any positive nurnber. It is easy to see that $M(t)$ exists$R_{+}$. Furthermore, by employing the same manner as in the

proof of (3.6), one can deduce that$\iota\ell(t, X)<\wedge 1l(\dagger)$,$(t..\mathrm{t}\cdot)\in R_{+}\cross\overline{\Omega}$.

(3.10)

88

Each component$M_{\dot{l}}(t)$of$M(\dagger)$is a )$\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{v}\mathrm{e}$$1$solution of$01^{\cdot}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{r}\mathrm{y}$differential equation$N$$\dot{y}=-By^{2}+Ay+C(\equiv C_{\mathrm{T}}(y))$ ,$C=I\mathrm{t}_{i}’c(1-k)$

where$A=c(k-1)+k \overline{a}_{i}-k\sum\underline{c}ij\mu_{1},j\dot{J}=1’ B=\underline{b}_{i}k$and. Since $B>0$ and $C\geq 0$ . $M_{i}(t)$ tends to the unique positive root of thequadratic equation $G(y)=0$ (say.

$\gamma^{\mathit{1}}(k\cdot,$$?)$), as $tarrow\infty$ . The root$\gamma(k, i)$

is given by theequation

2$\underline{b}_{i}\gamma(k, i)=\overline{a}_{i}-\mathcal{T}-.\sum_{=J1}\underline{C}_{\iota}j\mu 1,j+\{(\overline{o}_{i}-\tau-\sum\underline{c}j\mu\dot{x}1,j\mathrm{I}24\underline{b}_{i}\wedge’/\cdot=N1+K_{i}\mathcal{T}\}1/2$

with $\tau=c(1/k-1)\geq 0$ . Consider the right hand side of the above equation as a functionof$\tau\geq 0$, and write it by$\prime \mathrm{r}(\tau)$, simply. Then $l(\tau)$ is nonincreasing in$\tau$, because of$\wedge \mathrm{f}’(\tau)\{(\overline{a}_{i}-\tau-j=1\sum^{N}\underline{C}_{i}j\mu 1,j)2+4\underline{b}iKi^{\mathcal{T}}\}^{1}/2$$\cross[\{(\overline{a}_{i}-\tau-\sum_{=j1}\underline{c}_{j}\mu 1,j)24+\underline{b}iI’\iota \mathcal{T}\}1/2-(\overline{\zeta 4}_{i}-\tau-\sum l?l\mu_{1},j)+2\underline{b}_{i}K_{i}]Nj=1N\underline{c}_{j}$

$=$

$=$

$4\underline{b}_{i}^{2}I\mathrm{t}_{i}^{r}(K_{\dot{l}}-^{\frac{\overline{a}_{i}-\Sigma^{N}j=1\underline{C}_{i}j\mu 1,j}{\underline{b}_{l}}})$$4 \underline{b}_{i}K_{i}\sum_{=j1}N\underline{C}ij(\mu 1,j-, 2\delta)\leq 0$.

Thus we get$\gamma(k, i)\leq^{r}\mathrm{r}(0)/(2\underline{b}i)=(\overline{a}_{i}-j=\sum_{1}^{N}\underline{c}ij\mu_{1,j})/\underline{b}_{i}\leq \mathcal{U}_{2,i}$for all $k\in$ $(0,1]$ and all $i=1,$ $\cdots N$ . By the periodicity of , this fact and (3.10) yield)

that$u(t, x)\leq_{I}\nu_{2}$on$R\cross\overline{\Omega}$$u$, which proves the half part of (3.9). To prove the remainderof (3.9), we consider the solution $m(t)=(m_{1}(t), \cdots , m_{N}(t))$ of a system of ordinary

differential equations$\frac{d}{dt}m_{i}(t)$$=$$k \{(m_{i}(t)-K_{i})(C-\frac{c}{k})+m_{i}(t)(\underline{a}_{i^{-}}\overline{b}_{i}m_{i}(t)-\sum_{j=1}\overline{c}_{ij2}N\nu,j)\}$$t>0$ ,

$i=1,$$\cdots,$

,

$N$

,

89
with

$m(\mathrm{O})=\mu_{1}-(\epsilon, \cdots, \epsilon)$$\dot{l}=1,$ $\cdots,$ $N$$u(t, x)\leq\nu_{2}$, where$\epsilon$

is any positive numberthat$\mathrm{s}\mathrm{u}\mathrm{c}\prime \mathrm{h}$$\epsilon<\mu_{1,i}$. It is easy to see that $m(t)$ exists globally and $0<m(t)$ on

on$R\cross\overline{\Omega}$. Since, by the same reasoning as for $M(t)$ one can deduce that$m(t)<u(f_{}, .x)$ ,Observe that each

$R_{+}$

for all$m_{i}(t)$$(t, x)\in R_{+}\cross\overline{\Omega}$

tends to the a positive number.

$\tilde{\gamma}(k, i)$(3.11)

as$tarrow\infty$, where$\tilde{\gamma}(k, i)$

is given by the equation$\underline{9}\overline{b}_{\dot{l}}\tilde{\gamma}(k, i)=\underline{a}_{i}-\mathcal{T}-.\sum_{/=1}^{N}\overline{C}i\dot{J},j\mathcal{U}_{2}+\{(\underline{a}l-\tau-j\sum_{=1}^{N}\overline{c}i\dot{J}U_{2,j})^{2}+4\overline{b}iK_{i}\tau\}^{1}/2$,

where $\tau=c(1/k-1)\geq 0$ . It is straightforward to see that the right hand side of theabove equation is nondecreasing in$\tau\geq 0$.

Then we get$\tilde{\gamma}(h, \iota)\geq\frac{arrow a-\Sigma_{\dot{J}-}^{N}-1^{\overline{\mathrm{Q}}}j\nu 2,j}{\overline{b}_{i}}=\mu_{2,i}$for all$k\in(\mathrm{O}, 1]$

and all $i=1,$$\cdots$, $N$ . By the periodicity of , the above observation and$1l$(3.11) yield the remainder part of (3.9). This completes the proof.

References[1] S. Ahmad and A. C. Lazer, Asymptotic behavior of solutions of periodic competition

diffusion system. Nonlinear Analysis, T. M. A., 13 (1989), 263-284.
[2] T. A. Burton and B. Zhang,$Pe\uparrow\cdot iodiC$

solutions of abstract differential equations with

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