Funkcialaj Ekvacioj, 35 (1992) 451-466

Kneser Type Theorems for Functional DifferentialEquations in a Banach Space

By

Jong Son SHIN(Korea University, Japan)

§1. Introduction

Let $E$ be an infinite dimensional Banach space with a norm $|$ $¥cdot|_{E}$ and let$¥ovalbox{¥tt¥small REJECT}$ be the phase space satisfying the fundamental axioms introduced by Haleand Kato [3] (cf. [5, 8]). If $x$ : $(-¥infty, ¥sigma+a)¥rightarrow E$ , $ 0<a¥leq¥infty$ , then for any$t¥in(-¥infty, ¥sigma+a)$ we define $x_{t}$ : ( $-¥infty$ , $¥mathrm{O}]¥rightarrow E$ by $x_{t}(¥theta)=x(t+¥theta)$ , $-¥infty<¥theta¥leq 0$ .

The purpose of this paper is to show Kneser type theorems for thefunctional differential equation (FDE) with infinite delay in a Banach space(for brevity, $¥mathrm{C}¥mathrm{P}(1.1)$ or $¥mathrm{C}¥mathrm{P}(f,$ $¥sigma$ , $¥varphi)$ )

(1.1) $¥frac{d¥mathrm{x}}{dt}=f(t, ¥chi_{t})$ , $¥mathrm{x}_{¥sigma}=¥varphi¥in¥ovalbox{¥tt¥small REJECT}$ ,

under the condition that $f$ : $[¥sigma, ¥sigma+a]¥times¥ovalbox{¥tt¥small REJECT}(¥varphi, r)¥rightarrow E$ is uniformly continuous(cf. [2]), where $¥ovalbox{¥tt¥small REJECT}(¥varphi, r):=$ $¥{¥psi¥in¥ovalbox{¥tt¥small REJECT}||¥varphi-¥psi|_{¥ovalbox{¥tt¥small REJECT}}¥leq r¥}$ .

Kneser type theorems for ordinary differential equations (ODEs) in Banachspaces have been shown by Szufla [15] and Kubiaczyk [7], etc.. For FDEswith infinite delay in finite dimensional space, Kaminogo [4] has provedKneser’s theorem. Recently, the basic theorems on the existence of solutionsand the uniqueness of solutions for $¥mathrm{C}¥mathrm{P}(1.1)$ in Banach spaces have beendeveloped by Shin [9, 10, 11, 12, 13].

In this paper we shall prove Kneser type theorems for $¥mathrm{C}¥mathrm{P}(1.1)$ in aBanach space by using a property (Theorem 2.2) on Kuratowski’s measure ofnoncompactness. In particular, the results given in [10, Theorem 3.1 and 12,Theorem 2.4] are, in some sense, contained in our existence theorem $(/¥mathrm{T}¥mathrm{h}¥mathrm{e}¥mathrm{o}¥mathrm{r}¥mathrm{e}¥mathrm{m}$

$4.3)$ .

§2. A property of the a-measure of noncompactness

In this section, we shall consider a property on Kuratowski’s measure ofnoncompactness (for brevity, a-measure). Let $¥mathrm{Y}$ be a Banach space with anorm $|¥cdot|_{¥mathrm{Y}}$ . For a bounded set $¥Omega$ of $¥mathrm{Y}$ the a-measure of $¥Omega$ is defined as follows:

452 Jong Son SHIN

$¥alpha(¥Omega)=¥inf$ { $ d>0|¥Omega$ has a finite cover of diameter $<d$ }.

For properties of the a-measure refer to [1, 6]. Henceforth, we will use thesame notation $¥alpha(¥cdot)$ for the a-measure of noncompactness in any Banachspace. We denote by $N$ the set of all positive integers. To prove the maintheorem in this section, we need the following result which is well known (cf.[6, Theorem 1.4.1] $)$ .

Lemma 2.1. If $¥{A_{n}¥}$ is a family of nonempty bounded subsets of Banachspace $¥mathrm{Y}$ such that $A_{n+1}¥subset A_{n}$ for every $n¥in N$ and $¥alpha(A_{n})¥rightarrow 0$ as $ n¥rightarrow¥infty$ , then

$¥bigcap_{n=1}¥mathrm{C}1A_{n}$ is $n$nonempty and compact, where $¥mathrm{C}1¥mathrm{A}$ stands for the closure of $A$ .

Using the above result, we can obtain the main result in this section,which plays an essential role in showing Kneser type theorems for $¥mathrm{C}¥mathrm{P}$ $(1.1)$ .

Theorem 2.2. Let $¥{S_{n}¥}$ be a family of nonempty bounded subsets of Banachspace $¥mathrm{Y}$ such that $S_{n+1}¥subset S_{n}$ for every $n¥in N$. If the set $S_{n}$ is connected forevery $n¥in N$ and if $¥alpha(S_{n})¥rightarrow 0$ as $ n¥rightarrow¥infty$ , then th $e$ set

$¥bigcap_{n=1}¥mathrm{C}1S_{n}$ is $n$nonempty,

compact and connected.

Proof. Set $S=¥bigcap_{n=1}¥mathrm{C}1S_{n}$ . Then it follows from Lemma 2.1 that the set

$S$ is nonempty and compact. Thus it remains to show that $S$ isconnected. Suppose that $S$ is not connected in Banach space Y. Then thereexist nonempty closed sets $F^{1}$ and $F^{2}$ such that $S=F^{1}¥cup F^{2}$ and $ F^{1}¥cap F^{2}=¥emptyset$ .

Hence there exist two disjoint open sets $U^{1}$ and $U^{2}$ such that $F^{1}¥subset U^{1}$ and$F^{2}¥subset U^{2}$ . Put $U=U^{1}¥cup U^{2}$ . Clearly, we have $S¥subset U$ . Now, we shall provethat there is an $m¥in N$ such that $Cl$ $S_{m}¥subset U$ . Suppose, for a contradiction, that$¥mathrm{C}1S_{n}¥backslash U¥neq¥phi$ for every $n¥in N$. Since $¥alpha(¥mathrm{C}1S_{n}¥backslash U)¥rightarrow 0$ as $ n¥rightarrow¥infty$ and $¥mathrm{C}1S_{n+1}¥subset$

$¥mathrm{C}-1S_{n}$ , we have $¥phi=S¥backslash U=¥bigcap_{n=1}(¥mathrm{C}1S_{n}¥backslash U)¥neq¥phi$ by Lemma 2.1. This is a

contradiction. Hence we have $S¥subset ClS_{m}¥subset U$. It is obvious that $U^{1}¥cap ¥mathrm{C}1S_{m}$

$¥neq¥phi$ and $ U^{2}¥cap ¥mathrm{C}1S_{m}¥neq¥phi$. Set $P_{m}=(U^{1})^{c}¥cap ¥mathrm{C}1S_{m}$ and $Q_{m}=(U^{2})^{c}¥cap ¥mathrm{C}1S_{m}$ , where$(U^{i})^{c}=¥mathrm{Y}¥backslash U^{i}$ , $i=1,2$. Then $ClS_{m}=P_{m}¥cup Q_{m}$ , $P_{m}¥neq¥phi¥neq Q_{m}$ , and $ P_{m}¥cap Q_{m}=¥emptyset$ .

Thus $¥mathrm{C}1S_{m}$ is disconnected, which yields a contradiction with the connectednessof $S_{m}$ . Therefore the proof of the theorem is completed.

§3. The connectedness of the set of approximate solutions

Let $R^{-}=(-¥infty, 0], R^{+}=[0, ¥infty)$ and $R=$ $(-¥infty, ¥infty)$ . We denote by$¥ovalbox{¥tt¥small REJECT}^{X}=¥ovalbox{¥tt¥small REJECT}(R^{-}, X)$ a linear space of functions from $R^{-}$ into $X$ with semi-norm

Kneser Type Theorems for FDE 453

$|¥cdot|_{¥ovalbox{¥tt¥small REJECT} ¥mathrm{x}}$ , where $X=E$ or $R$ . Let $X_{¥sigma}$ $[¥delta]$ (resp. $¥ovalbox{¥tt¥small REJECT}_{¥sigma}(¥delta)$ ) be the set of all functions$x$ from ( $-¥infty$ , $¥sigma+¥delta]$ (resp. $(-¥infty,$ $¥sigma+¥delta)$ ), $¥delta>0$ , to $X$ such that $¥chi_{¥sigma}¥in¥ovalbox{¥tt¥small REJECT}^{X}$ and$x(t)$ is continuous on $[¥sigma, ¥sigma+¥delta]$ (resp. [$¥sigma$ , $¥sigma+¥delta$ ) $)$ . In particular, for a given$¥varphi¥in¥ovalbox{¥tt¥small REJECT}^{X}$ we denote by $¥ovalbox{¥tt¥small REJECT}_{¥sigma}^{¥varphi}$ $[¥delta]$ the set of all $x¥in X_{¥sigma}$ $[¥delta]$ satisfying $ x_{¥sigma}=¥varphi$ . Also,we will use the notation $e_{¥sigma}$ $[¥delta]$ for $X_{¥sigma}$ $[¥delta]$ if $X=E$, etc.. Throughout thispaper we assume that the following axioms on the phase space $¥ovalbox{¥tt¥small REJECT}^{X}$ are alwayssatisfied:

$(¥mathrm{B}_{1})$ If $x¥in X_{¥sigma}(a)$ , $ 0<a¥leq¥infty$ , then $x_{t}$ lies in $¥ovalbox{¥tt¥small REJECT}^{X}$ for all $t¥in[¥sigma,$ $¥sigma+a$) and$x_{t}$ is continuous in $t$ on $[¥sigma,$ $¥sigma+a)$ .

$(¥mathrm{B}_{2})$ There exist a constant $L>0$ and functions $K(t)>0$ and $M(t)¥geq 0$

with the following properties:

(i) $K(t)$ is continuous for $t$ in $R^{+}$ ,

(ii) $M(t)$ is locally bounded on $R^{+}$

(iii) For every function $x¥in¥ovalbox{¥tt¥small REJECT}_{¥sigma}(a)$ , it holds that

$¥frac{1}{L}|x(t)|_{X}¥leq|¥chi_{t}|_{¥ovalbox{¥tt¥small REJECT}^{X}}¥leq K(t-¥sigma)¥sup¥{|x(s)|_{X}|¥sigma¥leq s¥leq t¥}+M(t-¥sigma)|x_{¥sigma}|_{¥ovalbox{¥tt¥small REJECT}^{X}}$

for $ t¥in$ $[¥sigma,$ $¥sigma+a)$ .$(¥mathrm{B}_{3})$ The quotient space $¥tilde{¥ovalbox{¥tt¥small REJECT}}^{X}:=¥ovalbox{¥tt¥small REJECT}^{X}/|¥cdot|_{¥ovalbox{¥tt¥small REJECT} ¥mathrm{x}}$ is a Banach space.

If $E$ is understood from the context, we shall write $¥ovalbox{¥tt¥small REJECT}$ for $¥ovalbox{¥tt¥small REJECT}^{E}$ and $|¥cdot|$ for$|¥cdot|_{E}$ . Set $K_{a}=¥sup¥{K(t)|0¥leq t¥leq a¥}$ and $M_{a}=¥sup¥{M(t)|0¥leq t¥leq a¥}$ in Axiom$(¥mathrm{B}_{2})$ .

Let $f$ : $[¥sigma, ¥sigma+a]¥times¥ovalbox{¥tt¥small REJECT}(¥varphi, r)¥rightarrow E$ in $¥mathrm{C}¥mathrm{P}(1.1)$ be a continuous function suchthat $|f|¥leq M-1$ , $M>1$ . Then for each $n¥in N$ a function $u$ : $(-¥infty,$ $¥sigma+¥gamma]¥rightarrow E$ ,

$0<¥gamma¥leq a$ , is said to be an $¥frac{1}{n}-$ approximate solution for $¥mathrm{C}¥mathrm{P}(1.1)$ if it satisfiesthe following conditions:

(1) $u$ belongs to $¥ovalbox{¥tt¥small REJECT}_{¥sigma}^{¥varphi}[¥gamma]$ ;(2) $u$ has the right hand derivative $D_{+}u$ such that $|(D_{+}u)(t)|¥leq M$ on

[$¥sigma$, $¥sigma+¥gamma)$ , and satisfies

$u(t)=¥varphi(0)+¥int_{¥sigma}^{t}(D_{+}u)(s)ds$ for $ t¥in$ $[¥sigma, ¥sigma+¥gamma]$ ; and

(3) $|(D_{+}u)(t)-f(t, u_{t})|¥leq¥frac{1}{n}$ for all $ t¥in$ $[¥sigma,$ $¥sigma+¥gamma)$ .

Let $C([a, b], X)$ be the set of all continuous functions $x:[a, b]¥rightarrow X$ withsupremum norm. For a subset $e$ of $¥ovalbox{¥tt¥small REJECT}_{¥sigma}$ $[¥gamma]$ the following notations will be

454 Jong Son SHIN

used in the present paper;

$d(t)=¥{x(t)¥in E|x¥in¥ovalbox{¥tt¥small REJECT}¥}$ , $¥ovalbox{¥tt¥small REJECT}_{t}=¥{x_{t}¥in¥#|x¥in e¥}$ for $ t¥in$ $[¥sigma, ¥sigma+¥gamma]$ ,

$d|[c, d]=¥{x|[c, d]¥in C([c, d], E)|x¥in ¥mathcal{B}¥}$

and

$¥ovalbox{¥tt¥small REJECT}|_{[c,d]}=$ { $¥zeta|[c,$ $d]¥in C$ ( $[c,$ $d]$ , $¥ovalbox{¥tt¥small REJECT}$ ) $|¥zeta(t):=x_{t}$ for $t¥in[¥sigma,$ $¥sigma+¥gamma]$ and $x¥in¥ovalbox{¥tt¥small REJECT}$ },

where $c$ , $ d¥in$ $[¥sigma, ¥sigma+¥gamma]$ and $x|[c, d]$ is the restriction of $x$ to $[c, d]$ .

Define a linear operator $S_{X}(t):¥ovalbox{¥tt¥small REJECT}^{X}¥rightarrow¥ovalbox{¥tt¥small REJECT}^{X}$, $t$ $¥geq 0$, by

$[S_{X}(t)¥varphi](¥theta)=¥left¥{¥begin{array}{l}¥varphi(t+¥theta)¥mathrm{i}¥mathrm{f}t+¥theta<0¥¥¥varphi(0)¥mathrm{i}¥mathrm{f}t+¥theta¥geq 0.¥end{array}¥right.$

We shall now prove the connectedness of the set of all $¥frac{1}{n}$ -approximate

solutions for $¥mathrm{C}¥mathrm{P}$ $(1.1)$ , which corresponds to Proposition 2 in [7].

Lemma 3.1. Suppose the phase space $¥ovalbox{¥tt¥small REJECT}^{E}$ satisfies Axioms $(¥mathrm{B}_{1})$ and$(¥mathrm{B}_{2})$ . Assume that $f$ : $[¥sigma, ¥sigma+a]¥times¥ovalbox{¥tt¥small REJECT}(¥varphi, r)¥rightarrow E$ is uniformly continuous and$|f|¥leq M¥_ 1$ , $M>1$ , on $[¥sigma, ¥sigma+a]¥times¥ovalbox{¥tt¥small REJECT}(¥varphi, r)$ . Let $¥delta_{0}$ be the largest numbersuch that $|S_{E}(t)¥varphi¥_¥varphi|_{¥ovalbox{¥tt¥small REJECT}}¥leq r/2$ for all $t¥in[0, ¥delta_{0}]$ . Then for each $n¥in N$ the set

$Q^{n}$ of all $¥frac{1}{n}$ -approximate solutions, defined on ( $-¥infty$ , $¥sigma+¥gamma]$ , for $¥mathrm{C}¥mathrm{P}(1.1)$ is

nonempty, and $Q^{n}|$ $[¥sigma, ¥sigma+¥gamma]$ and $Q^{n}|_{[¥sigma,¥sigma+¥gamma]}$ are $connecte,d$ in $C([¥sigma, ¥sigma+¥gamma], E)$

and in $C([¥sigma, ¥sigma+¥gamma], ¥ovalbox{¥tt¥small REJECT})$ , respectively, where $¥gamma=¥min¥{a, r/¥mathit{2}K_{a}M, ¥delta_{0}¥}$ .

Proof. The proof will be divided into three parts as follows.Step 1. We shall show that for any $n¥in N$ the set $Q^{n}$ is nonempty. By the

method of Euler-Cauchy polygons [13, Lemma 2.3], we define a function $x^{¥epsilon}$

for each $¥epsilon$ , $¥gamma>¥epsilon>0$, as follows:

$¥mathrm{x}^{¥epsilon}(t)=¥left¥{¥begin{array}{l}¥varphi(t-¥sigma)¥¥x^{¥epsilon}(¥epsilon(i))+(t-¥epsilon(i))f(¥epsilon(i),x_{¥epsilon(i)}^{¥epsilon})¥end{array}¥right.$ $tt¥leq¥sigma¥in[¥epsilon(i),$

$¥epsilon(i+1))$ ,

where $¥epsilon(i)=¥sigma+i¥epsilon$ , $i=0,1$ , $¥ldots,n(¥epsilon)$ , $n(¥epsilon)=[¥frac{¥gamma}{¥epsilon}]$ , $¥epsilon(n(¥epsilon)+1)=¥sigma+¥gamma$ (if $ n(¥epsilon)¥epsilon<¥gamma$).

Define a mapping $¥tau_{¥epsilon}$ : $[¥sigma, ¥sigma+¥gamma]¥rightarrow$ $[¥sigma, ¥sigma+¥gamma]$ by $¥tau_{¥epsilon}(t)=¥epsilon(i)$ for $ t¥in$

[$¥epsilon(i)$ , $¥epsilon(i+1))$ , where $ i=0,1,¥ldots$ , $n(¥epsilon)$ . Then the above function $x^{¥epsilon}$ can beexpressed in the form

Kneser Type Theorems for FDE 455

$¥mathrm{x}^{¥epsilon}(t)=¥{¥varphi¥varphi(t-¥sigma)(0)+¥int_{¥sigma}^{t}f(¥tau_{¥epsilon}(s), X_{¥tau_{¥epsilon}(s)}^{¥epsilon})ds$

$tt¥leq¥sigma¥in[¥sigma, ¥sigma+¥gamma]$

.

It is easy to see that $¥chi_{t}^{¥epsilon}¥in¥ovalbox{¥tt¥small REJECT}(¥varphi, r)$ for all $t¥in J:=$ $[¥sigma, ¥sigma+¥gamma]$ . Since $f$ is uniformly

continuous, there is a $¥delta(n)>0$ such that $|f(t, ¥varphi_{1})-f(s, ¥varphi_{2})|¥leq¥frac{1}{n}$ if $|t$ $-s|¥leq¥delta(n)$

and $|¥varphi_{1}-¥varphi_{2}|_{¥ovalbox{¥tt¥small REJECT}}¥leq¥delta(n)$ for $(t, ¥varphi_{1})$ , $(s, ¥varphi_{2})¥in J¥times¥ovalbox{¥tt¥small REJECT}(¥varphi, r)$ . Since $¥{S_{E}(t)¥}_{t¥geq 0}$ is astrongly continuous semigroup in $¥ovalbox{¥tt¥small REJECT}$ , we can choose a $¥delta_{0}(n)>0$ such that$|S_{E}(¥tau_{¥epsilon}(t)-¥sigma)¥varphi-S_{E}(t-¥sigma)¥varphi|_{¥ovalbox{¥tt¥small REJECT}}<¥delta(n)/2$ for all $¥epsilon¥in(0, ¥delta_{0}(n))$ . Put $¥beta(n)=$

$¥min$ $¥{¥delta(n)/2(K_{a}+1)M, ¥delta_{0}(n)¥}$ . By Lemma 2.1 in [12], we have

$|¥mathrm{x}_{¥tau_{¥epsilon}(t)}^{¥epsilon}-¥mathrm{x}_{t}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}¥leq K_{a}M|¥tau_{¥epsilon}(t)-t|+|S_{E}(¥tau_{¥epsilon}(t)-¥sigma)¥varphi-S_{E}(t-¥sigma)¥varphi|$

$¥leq K_{a}M¥beta(n)+¥delta(n)/2¥leq¥delta(n)$ if $0<¥epsilon<¥beta(n)$ ,

which imlies that $x^{¥epsilon}(¥cdot)¥in Q^{n}$ for all $¥epsilon¥in(0, ¥beta(n))$ . Put

$V_{¥beta}^{n}=¥{x^{¥epsilon}(¥cdot)¥in Q^{n}|0<¥epsilon<¥beta(n)¥}$ .

Step 2. We shall prove that $V_{¥beta}^{n}|J$ is connected in $C(J, E)$ . To do this,it is sufficient to prove the continuity of the mapping $¥epsilon¥rightarrow x^{¥epsilon}|J$ , $x^{¥epsilon}¥in V_{¥beta}^{n}$ ; thatis, for any fixed $¥epsilon$ , $0<¥epsilon<¥beta(n)$ , we must prove that $x^{¥delta}(t)¥rightarrow x^{¥epsilon}(t)$ uniformly on

$J$ as $¥delta¥rightarrow¥epsilon$ . Without loss of generality, we may assume that $¥mathrm{H}¥frac{¥gamma}{¥delta}=[¥frac{¥gamma}{¥epsilon}]$ and

$¥epsilon(n(¥epsilon)+1)=¥sigma+¥gamma$ . Set $¥delta(i)=¥sigma+i¥delta$ , $i=0,1,¥ldots,n(¥epsilon)$ , $¥delta(n(¥epsilon)+1)=¥sigma+¥gamma$.

Clearly, $¥zeta(¥delta, ¥epsilon):=¥max$ $¥{|¥epsilon(i)-¥delta(i)||i =1,2, ¥ldots,n(¥epsilon)+1¥}¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$ .

We shall at first prove that for each $j$ , $1¥leq j¥leq n(¥epsilon)+1$ ,

(3.1) $|x_{¥epsilon(j)}^{¥delta}-x_{¥epsilon(j)}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}¥rightarrow 0$ and $|x_{¥delta(j)}^{¥delta}-x_{¥epsilon(j)}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$.

The proof is by induction on $j$ . Ifj $=1$ , then we have, by Axiom $(B_{2})$ ,

$|¥chi_{¥epsilon(1)}^{¥delta}-¥chi_{¥epsilon(1)}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}$

$¥leq K_{a}¥sup_{¥sigma¥leq t¥leq¥epsilon(1)}|¥int_{¥sigma}^{t}[f(¥tau_{¥delta}(s), ¥mathrm{x}_{¥tau_{¥delta}(s)}^{¥delta})-f(¥tau_{¥epsilon}(s), x_{¥tau_{¥epsilon}(¥mathrm{s})}^{¥epsilon})]ds|$

$¥leq K_{a}¥int_{¥sigma}^{¥epsilon(1)}|f(¥tau_{¥delta}(s), ¥mathrm{x}_{¥tau_{¥delta}(s)}^{¥delta})-f(¥tau_{¥epsilon}(s), x_{¥tau_{¥epsilon}(s)}^{¥epsilon})|ds$

$¥leq K_{a}|¥delta(1)-¥epsilon(1)||f(¥delta(1), x_{¥delta(1)}^{¥delta})-f(¥sigma, ¥varphi)|$

$¥leq 2K_{a}M¥zeta(¥delta, ¥epsilon)¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$ .

Since $|x^{¥delta}(t)-x^{¥delta}(s)|¥leq M|t-s|$ for all $t$ , $s¥in J$ and for all $¥delta¥in(0, ¥beta(n))$ and$¥{x_{¥mathrm{r}^{¥delta}}¥}_{0<¥delta<¥beta(n)}$ is uniformly equicontinuous on $¥mathrm{J}$ , we see that $¥mathrm{x}^{¥delta}(¥delta(1))¥rightarrow x^{¥epsilon}(¥epsilon(1))$

456 Jong Son SHIN

and $¥mathrm{x}_{¥delta(1)}^{¥delta}¥rightarrow X_{¥epsilon(1)}^{¥epsilon}$ as $¥delta¥rightarrow¥epsilon$ . Assume that (3.1), $ j=1,¥ldots$ , $i$ , hold. Then, we havethe estimate

$¥sup$ $¥{|x^{¥delta}(s)-x^{¥epsilon}(s)||¥epsilon(i)¥leq s¥leq¥epsilon(i+1)¥}$

$¥leq¥sup¥{|¥int_{¥epsilon(i)}^{s}[f(¥tau_{¥delta}(s), ¥mathrm{x}_{¥tau_{¥delta}(s)}^{¥delta})-f(¥tau_{¥epsilon}(s), ¥mathrm{x}_{¥tau_{¥epsilon}(s)}^{¥epsilon})]ds|$ $|¥epsilon(i)¥leq s¥leq¥epsilon(i+1)¥}$

$+|x^{¥delta}(¥epsilon(i))-x^{¥epsilon}(¥epsilon(i))|$

$¥leq|I|+|¥chi^{¥delta}(¥epsilon(i))-x^{¥epsilon}(¥epsilon(i))|$

where

$I$ $=¥int_{¥epsilon(i)}^{¥tilde{t}}[f(¥tau_{¥delta}(s), x_{¥tau_{¥delta}(¥mathrm{s})}^{¥delta})-f(¥tau_{¥epsilon}(s), x_{¥tau_{¥epsilon}(s)}^{¥epsilon})]ds$ for some $¥tilde{t}¥in(¥epsilon(i),$ $¥epsilon(i+1)]$ .

(1) Case $¥epsilon(i)¥leq¥delta(i)¥leq¥tilde{t}$. Then

$|I|¥leq M|¥delta(i)-¥epsilon(i)|+|(¥tilde{t}-¥delta(i))f(¥delta(i), x_{¥delta(i)}^{¥delta})-(¥tilde{t}-¥epsilon(i))f(¥epsilon(i), ¥mathrm{x}_{¥epsilon(i)}^{¥epsilon})|$

$¥leq 2M|¥delta(i)-¥epsilon(i)|+|¥tilde{t}-¥epsilon(i)||f(¥delta(i), x_{¥delta(i)}^{¥delta})-f(¥epsilon(i), x_{¥epsilon(i)}^{¥epsilon})|$.

(2) Case $¥tilde{t}<¥delta(i)<¥epsilon(i+1)$ . Then

$|I|¥leq|¥tilde{t}-¥epsilon(i)||f(¥delta(i-1), x_{¥delta(i-1)}^{¥delta})-f(¥epsilon(i), x_{¥epsilon(i)}^{¥epsilon})|$

$¥leq 2M|¥delta(i)-¥epsilon(i)|$ .

(3) Case $¥epsilon(i)<¥delta(i+1)¥leq¥tilde{t}$ . Then

$|I|¥leq M|¥delta(i+1)-¥epsilon(i+1)|$

$+|(¥delta(i+1)-¥epsilon(i))f(¥delta(i), ¥chi_{¥delta(i)}^{¥delta})-(¥tilde{t}-¥epsilon(i))f(¥epsilon(i), ¥mathrm{x}_{¥epsilon(i)}^{¥epsilon})|$

$¥leq 2M|¥delta(i+1)-¥epsilon(i+1)|$

$+|¥tilde{t}-¥epsilon(i)||f(¥delta(i), x_{¥delta(i)}^{¥delta})-f(¥epsilon(i), x_{¥epsilon(i)}^{¥epsilon})|$ .

(4) Case $¥tilde{t}<¥delta(i+1)<¥epsilon(i+1)$ . Then

$|I|¥leq|¥tilde{t}-¥epsilon(i)||f(¥delta(i), x_{¥delta(i)}^{¥delta})-f(¥epsilon(i), ¥chi_{¥epsilon(i)}^{¥epsilon})|$ .

Summarizing the above cases, we have

$|I|¥leq 2M¥zeta(¥delta, ¥epsilon)+¥epsilon|f(¥delta(i), x_{¥delta(i)}^{¥delta})-f(¥epsilon(i), ¥chi_{¥epsilon(i)}^{¥epsilon})|$

$¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$,

which implies that

(3.2) $¥sup$ $¥{|x^{¥delta}(s)-¥mathrm{x}^{¥epsilon}(s)||¥epsilon(i)¥leq s¥leq¥epsilon(i+1)¥}¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$ .

Kneser Type Theorems for FDE 457

Thus we get, together with Axiom $(¥mathrm{B}_{2})$ ,

$¥frac{1}{L}|x^{¥delta}(¥epsilon(i+1))-x^{¥epsilon}(¥epsilon(i+1))|¥leq|x_{¥epsilon(i+1)}^{¥delta}-¥chi_{¥epsilon(i+1)}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}$

$¥leq K_{a}¥sup$ $¥{|x^{¥delta}(s)-x^{¥epsilon}(s)||¥epsilon(i)¥leq s¥leq¥epsilon(i+1)¥}+M_{a}|x_{¥epsilon(i)}^{¥delta}-x_{¥epsilon(i)}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}$

$¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$.

Consequently, we see that $¥max$ $¥{|¥mathrm{x}_{¥delta(j)}^{¥delta}-x_{¥epsilon(j)}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}|j=1,2,¥ldots,n(¥epsilon)+1¥}¥rightarrow 0$ and$¥max$ $¥{|x^{¥delta}(¥delta(j))-x^{¥epsilon}(¥epsilon(j))||j=1,2,¥ldots,n(¥epsilon)+1¥}¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$.

Next, we shall show that for each $t¥in J$, $|¥mathrm{x}_{t}^{¥delta}-¥chi_{t}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$ . For each$ t¥in(¥sigma$ , $¥sigma+¥gamma$], there exists an $i¥in N$ such that $¥epsilon(i)<t¥leq¥epsilon(i+1)$ . Using Axiom$(¥mathrm{B}_{2})$ , (3.1) and (3.2), we can obtain

$|¥chi_{t}^{¥delta}-¥chi_{t}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}$

$¥leq K_{a}¥sup$ $¥{|¥chi^{¥delta}(s)-X^{¥epsilon}(s)||¥epsilon(i)¥leq s¥leq¥epsilon(i+1)¥}+M_{a}|x_{¥epsilon(i)}^{¥delta}-x_{¥epsilon(i)}^{¥epsilon}|_{¥ovalbox{¥tt¥small REJECT}}$

$¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$,

and hence

$|¥chi^{¥delta}(t)-x^{¥epsilon}(t)|¥rightarrow 0$ as $¥delta¥rightarrow¥epsilon$ .

Step 3. We shall prove that the set $Q^{n}|J$ is connected in $C(J, E)$ . Forany fixed $¥epsilon_{0},0<¥epsilon_{0}<¥beta(n)$ , and for each $z(¥cdot)¥in Q^{n}$ we define a set $ T_{z}^{n}:=¥{z^{p}(¥cdot)¥in$

$Q^{n}|¥sigma¥leq p¥leq¥sigma+¥gamma¥}$ consisting of the function

$z^{p}(t)=¥{z(t)$$t$ $¥leq p$

$z(p)+¥int_{p}^{t}f(¥tau(p, s), z_{¥tau(p,s)}^{p})ds$ $t¥in(p,$ $¥sigma+¥gamma]$ ,

where $¥tau(p, t)=¥tau_{¥epsilon_{0}}(t)$ , if $t¥in[p, ¥sigma+¥gamma]¥backslash [p,$ $¥epsilon_{0}(j+1))$ , $p¥in[¥epsilon_{0}(j),$ $¥epsilon_{0}(j+1))$ , while$¥tau(p, t)=p$ if $t¥in[p,$ $¥epsilon_{0}(j+1))$ . It is not difficult to see that $z^{¥sigma}(¥cdot)=¥chi^{¥epsilon_{0}}(¥cdot)¥in V_{¥beta}^{n}$

and $z^{p}(¥cdot)¥in Q^{n}$ . Now we will prove that $z^{q}(t)¥rightarrow z^{p}(t)$ as $q¥rightarrow p$ for every$t¥in J$ . Without loss of generality, we may assume that $p<q<t$, $p$ , $ q¥in$ $[¥epsilon_{0}(j)$ ,$¥epsilon_{0}(j+1))$ , $t¥in[¥epsilon_{0}(k),$ $¥epsilon_{0}(k+1))$ , where $k¥geq j$. Clearly, $|z^{q}(q)-z^{p}(p)|¥rightarrow 0$ and$|z_{q}^{q}-z_{p}^{p}|_{¥ovalbox{¥tt¥small REJECT}}¥rightarrow 0$ as $q¥rightarrow p$ . Furthermore, we have

$|z_{¥epsilon_{0}(j+1)}^{q}-z_{¥epsilon_{0}(j+1)}^{p}|_{¥ovalbox{¥tt¥small REJECT}}¥leq K_{a}¥sup_{p¥leq s¥leq¥epsilon_{0}(j+1)}|z^{q}(s)-z^{p}(s)|$

$¥leq K_{a}¥max¥{¥sup_{p¥leq s¥leq q}|z^{q}(s)-z^{p}(s)|,¥sup_{q¥leq s¥leq¥epsilon_{0}(j+1)}|z^{q}(s)-z^{p}(s)|¥}$

$¥leq K_{a}¥max¥{2M|p-q|$ , $|z(p)-z(q)|+|p-q|$ $|f(p, z_{p}^{p})|$

458 Jong Son SHIN

$+¥sup_{q¥leq s¥leq¥epsilon ¥mathrm{o}(j+1)}¥int_{q}^{s}|f(q, z_{q}^{q})-f(p, z_{p}^{p})|ds¥}$

$¥leq K_{a}¥{2M|p-q|+¥epsilon_{0}|f(p, z_{p}^{p})-f(q, z_{q}^{q})|¥}$

$¥rightarrow 0$ as $q¥rightarrow p$ .

Assume that $|z_{¥epsilon_{0}(i)}^{q}-z_{¥epsilon_{0}(i)}^{p}|_{¥ovalbox{¥tt¥small REJECT}}¥rightarrow 0$ as $q¥rightarrow p$ , where $i=j+1,¥ldots,k-1$ .

Then we have

$¥sup_{¥epsilon_{¥mathrm{O}}(k-1)¥leq s¥leq¥epsilon_{0}(k)}|z^{q}(s)-z^{p}(s)|$

$¥leq¥int_{¥epsilon ¥mathrm{o}(k-1)}^{¥epsilon_{¥mathrm{O}}(k)}|f(¥epsilon_{0}(k-1), z_{¥epsilon_{¥mathrm{O}}(k-1)}^{q})-f(¥epsilon_{0}(k-1), z_{¥epsilon_{0}(k-1)}^{p})|ds$

$+|z^{q}(¥epsilon_{0}(k-1))-z^{p}(¥epsilon_{0}(k-1))|$

$¥leq¥epsilon_{0}|f(¥epsilon_{0}(k-1), z_{¥epsilon ¥mathrm{o}(k-1)}^{q})-f(¥epsilon_{0}(k-1), z_{¥epsilon_{0}(k-1)}^{p})|$

$+|z^{q}(¥epsilon_{0}(k-1))-z^{p}(¥epsilon_{0}(k-1))|$

$¥rightarrow 0$ as $q¥rightarrow p$ ,

and hence

$|z_{¥epsilon_{0}(k)}^{q}-z_{¥epsilon_{0}(k)}^{p}|_{¥ovalbox{¥tt¥small REJECT}}$

$¥leq K_{a}¥sup_{¥epsilon_{0}(k-1)¥leq s¥leq¥epsilon_{0}(k)}|z^{q}(s)-z^{p}(s)|+M_{a}|z_{¥epsilon ¥mathrm{o}(k-1)}^{q}-z_{¥epsilon_{0}(k-1)}^{p}|_{¥ovalbox{¥tt¥small REJECT}}$

$¥rightarrow 0$ as $q¥rightarrow p$ .

Thus we get

$|z_{¥mathrm{r}^{q}}-z_{t}^{p}|_{¥ovalbox{¥tt¥small REJECT}}¥leq K_{a}¥sup_{¥epsilon_{¥mathrm{O}}(k)¥leq s¥leq t}|z^{q}(s)-z^{p}(s)|+M_{a}|z_{¥epsilon_{0}(k)}^{q}-z_{¥epsilon_{0}(k)}^{p}|_{¥ovalbox{¥tt¥small REJECT}}$

$¥leq K_{a}|t-¥epsilon_{0}(k)||f(¥epsilon_{0}(k), z_{¥epsilon_{0}(k)}^{q})-f(¥epsilon_{0}(k), z_{¥epsilon_{0}(k)}^{p})|$

$+M_{a}|z_{¥epsilon_{0}(k)}^{q}-z_{¥epsilon_{0}(k)}^{p}|_{¥ovalbox{¥tt¥small REJECT}}+K_{a}|z^{q}(¥epsilon_{0}(k))-z^{p}(¥epsilon_{0}(k))|$

$¥rightarrow 0$ as $q¥rightarrow p$,

and hence, by Axiom $(¥mathrm{B}_{2})$ ,

(3.3) $|z^{q}(t)-z^{p}(t)|¥rightarrow 0$ as $q¥rightarrow p$ .

Since $T_{Z}^{n}¥subset Q^{n}$ and $Q^{n}|J$ is uniformly equicontinuous, (3.3) holds uniformly on$J$ . Thus $T_{Z}^{n}|J$ is connected in $C(J, E)$ . Since $z^{¥sigma}(¥cdot)=¥chi^{¥epsilon_{0}}(¥cdot)¥in V_{¥beta}^{n}¥cap T_{Z}^{n}$, the set

$V_{¥beta}^{n}|J¥cup T_{z}^{n}|J$ is connected in $C(J, E)$ . It is obvious that$Q^{n}|J=¥bigcup_{z¥in Q^{n}}V_{¥beta}^{n}|J¥cup T_{z}^{n}|J$

.

Kneser Type Theorems for FDE 459

Thus the set $Q^{n}|J$ is also connected in $C(J, E)$ . It is not difficult to provethat $Q^{n}|_{J}$ is connected in $C(J, ¥ovalbox{¥tt¥small REJECT})$ . Hence the proof of the lemma is completed.

§4. Main result

In this section, we shall prove Kneser type theorems for $¥mathrm{C}¥mathrm{P}$ $(1.1)$ .

A function $¥eta$ : ( $¥sigma$ , $¥sigma+a]¥times[0,2r]¥rightarrow R$ is said to be a Kamke-type functionif the following conditions hold:

$(¥eta_{1})$ $¥eta=¥eta(t, s)$ is a real-valued function, defined on $(¥sigma,$ $¥sigma+a]¥times[0,2r]$ ,which is Lebesgue measurable in $t$ for each fixed $s¥in[0,¥mathit{2}r]$ and is continuousin $s$ for $a.a$ . $ t¥in$ $[¥sigma, ¥sigma+a]$ .

$(¥eta_{2})$ There exists a function $¥alpha$ , defined on ( $¥sigma$ , $¥sigma+a]$ and locally integrablethere, such that $|¥eta(t, s)|¥leq¥alpha(t)$ for $a.a$ . $t¥in(¥sigma,$ $¥sigma+a$] and all $s¥in[0,2r]$ .

To state Kneser type theorem, we need the following.

Hypothesis $A$ ; For each subset $¥ovalbox{¥tt¥small REJECT}$ of $¥ovalbox{¥tt¥small REJECT}_{¥sigma}$ $[¥gamma]$ such that $¥alpha(¥ovalbox{¥tt¥small REJECT}_{¥sigma})=0$ and$¥ovalbox{¥tt¥small REJECT}|$ $[¥sigma, ¥sigma+¥gamma]$ is $¥mathrm{i}/$-Lipschitzian for some $H>0$, the inequality $¥alpha(¥theta_{t}))¥leq|¥hat{¥alpha}_{t}|_{¥ovalbox{¥tt¥small REJECT}^{R}}$

holds for $ t¥in$ $[¥sigma, ¥sigma+¥gamma]$ , where $¥hat{¥alpha}(t)=¥alpha(¥ovalbox{¥tt¥small REJECT}(t))$ for $ t¥in$ $[¥sigma, ¥sigma+¥gamma]$ and $¥hat{¥alpha}(t)=0$ for$t¥in(-¥infty,$ $¥sigma]$ .

Here we note that the a-measure of noncompactness on $¥ovalbox{¥tt¥small REJECT}$ , which is nota Banach space, is well defined and has usual properties (cf. [12]).

Example 4. 1. Let $¥gamma¥in R$ . The space $¥varphi_{¥gamma}$ is the space of continuousfunctions $¥varphi:R^{-}¥rightarrow E$ having the limit $¥lim_{s¥rightarrow-¥infty}e^{¥gamma s}¥varphi(s)$ with the norm$|¥varphi|_{q_{¥gamma}}=¥sup_{s¥in R}¥_ e^{¥gamma s}|¥varphi(s)|$ . Then, by Proposition 1.7 in [12], we have

$¥alpha(¥ovalbox{¥tt¥small REJECT}_{t})=e^{-¥gamma t}¥sup_{¥sigma¥leq s¥leq t}e^{¥gamma s}¥alpha(¥ovalbox{¥tt¥small REJECT}(s))$

$=|¥hat{¥alpha}_{t}|e_{¥gamma}^{R}$ .

Example 4. 2. Let

$¥ovalbox{¥tt¥small REJECT}=¥{¥varphi:R^{-}¥rightarrow E|$measurable in ( $-¥infty$ , ? $r$], continuous on$[- r, 0]$ and $|¥varphi|_{¥ovalbox{¥tt¥small REJECT}}<¥infty$ },

where $ 0¥leq r<¥infty$ and

$|¥varphi|_{¥ovalbox{¥tt¥small REJECT}}=¥sup_{-r¥leq s¥leq 0}|¥varphi(s)|+¥int_{-¥infty}^{0}e^{s}|¥varphi(s)|ds$ .

Then we have, by Lemma 2.2 in [10],

460 Jong Son SHIN

$¥alpha(¥ovalbox{¥tt¥small REJECT}_{t})¥leq¥sup_{-r¥leq s¥leq 0}¥alpha(¥ovalbox{¥tt¥small REJECT}(t+s))+¥int_{¥sigma-t}^{0}e^{s}¥alpha(¥ovalbox{¥tt¥small REJECT}(t+s))ds$

$=|¥hat{¥alpha}_{t}|_{¥ovalbox{¥tt¥small REJECT}^{R}}$ .

We are now in a position to prove the main theorem in this paper.

Theorem 4.3. Suppose that $¥ovalbox{¥tt¥small REJECT}^{E}$ and $¥ovalbox{¥tt¥small REJECT}^{R}$ satisfy Axioms $(¥mathrm{B}_{1})-(¥mathrm{B}_{3})$ andHypothesis $A$ is satisfied. Assume that $f$ : $[¥sigma, ¥sigma+a]¥times¥ovalbox{¥tt¥small REJECT}(¥varphi, r)¥rightarrow E$ is uniformlycontinuous and $|f|¥leq M-1$ , $M>1$ , on $[¥sigma, ¥sigma+a]¥times¥ovalbox{¥tt¥small REJECT}(¥varphi, r)$ and that thereexists a Kamke-type function $¥omega$ : ( $¥sigma$ , $¥sigma+a]¥times[0,2r]¥rightarrow R^{+}$ such that

1) $¥alpha(f(t, B))¥leq¥omega(t, ¥alpha(B))$ for $a.a$. $t¥in(¥sigma,$ $¥sigma+a$] and

for all $B¥subset¥ovalbox{¥tt¥small REJECT}(¥varphi, r)$ ;2) $¥omega(t, s)$ is nondecreasing in $s$ ; and3) $z¥equiv 0$ is the unique absolutely continuous function from $(-¥infty,$ $¥sigma+a]$

into $R^{+}$ , which satisfies the initial condition $(D^{+}z)$ $(¥sigma):=¥lim_{t¥rightarrow¥sigma+}¥frac{z(t)}{t-¥sigma}=0$ and$z(t)=0$ for $t$ $¥leq 0$ and the scalar differential inequality

(1.1) $|¥frac{dz}{dt}|¥leq¥omega(t, |z_{t}|_{¥ovalbox{¥tt¥small REJECT}^{R}})$ for $a.a$ . $ t¥in$ $(¥sigma,$ $¥sigma+a]$ .

Then the set $S$ of all solutions, defined on ( $-¥infty$ , $¥sigma+¥gamma]$ , for $¥mathrm{C}¥mathrm{P}(1.1)$ isnonempty and the set $S|J$, $J=$ $[¥sigma, ¥sigma+¥gamma]$ , is compact and connected in $C(J, E)$ ,where $¥gamma$ is as in Lemma 3.1.

Proof. Let $Q^{n}$ be the same set as in Lemma 3.1. We note that$x$ : ( $-¥infty$ , $¥sigma+¥gamma]¥rightarrow E$ belongs to $S$ if and only if $x$ satisfies the conditions

$x|J¥in¥bigcap_{n=1}$ Cl $(Q^{n}|J)$ and $ x_{¥sigma}=¥varphi$ . From Theorem 2.2 and Lemma 3.1 it is

sufficient to see that $¥alpha(Q^{n}|J)¥rightarrow 0$ as $ n¥rightarrow¥infty$ . Since $Q^{n}|J$ is an equicontinuoussubset of $C(J, E)$ , we have $¥alpha(Q^{n}|J)=¥sup$ $¥{¥alpha(Q^{n}(t))|t¥in J¥}$ by Lemma 1.4.1 in[6]. Thus we must prove that $¥alpha(Q^{n}(t))¥rightarrow 0$ uniformly on $J$ as $ n¥rightarrow¥infty$ . Put$z^{n}(t)=¥alpha(Q^{n}(t))$ for $t¥in J$ , while $z^{n}(t)=0$ for $t¥in(-¥infty,$ $¥sigma$]. Then for $t$ , $s¥in J$ andany $n¥in N$ we have

$z^{n+1}(t)¥leq z^{n}(t)$ and $|z^{n}(t)-z^{n}(s)|¥leq M|t-s|$ ,

which implies that the sequence $¥{z^{n}(t)¥}$ converges to a function $z^{0}(t)$ uniformlyon ( $-¥infty$ , $¥sigma+¥gamma]$ . Clearly, we have $|z^{0}(t)-z^{0}(s)|¥leq M|t-s|$ for $t$ , $s¥in J$.

Next, we shall show that $z^{0}(t)¥equiv 0$ on $J$ . From the definition of thefamily $Q^{n}$ , properties (cf. [6. Theorem 1.4.1]) of the a-measure and Theorem1.3.1 in [6] it follows that for $h>0$,

Kneser Type Theorems for FDE 461

$¥frac{1}{h}|¥alpha(Q^{n}(t+h))-¥alpha(Q^{n}(t))|$

$¥leq¥frac{1}{h}¥alpha(¥{u(t+h)-u(t)|u¥in Q^{n}¥})$

$¥leq¥alpha(¥{D_{+}u)(s)|s¥in[t,$ $t+h)$ , $u¥in Q^{n}¥})$

$¥leq¥alpha(¥{(D_{+}u)(s)-f(s, u_{s})+f(s, u_{s})|s¥in[t, t+h),$ $u¥in Q^{n}¥})$

$¥leq¥alpha(¥{(D_{+}u)(s)-f(s, u_{s})|s¥in[t, t+h),$ $u¥in Q^{n}¥})$

$+¥alpha(¥{f(s, u_{s})|s¥in[t, t+h),$ $u¥in Q^{n}¥})$

$¥leq¥alpha(¥{f(s, u_{s})|s¥in[t, t+h),$ $u¥in Q^{n}¥})+¥frac{2}{n}$ .

Since $Q^{n}|_{J}$ is an equicontinuous subset of $C(J, ¥ovalbox{¥tt¥small REJECT})$ and $f$ is uniformlycontinuous, for any $¥epsilon>0$ we have

$0¥leq¥alpha(¥{f(s, u_{s})|s¥in[t, t+h),$ $ u¥in Q^{n}¥})-¥alpha(f(t, Q_{t}^{n}))<¥epsilon$,

provided $h$ is sufficiently small and $n$ is sufficiently large. Hence we have

$|¥frac{dz^{n}(t)}{dt}|¥leq¥alpha(f(t, Q_{t}^{n}))+¥frac{2}{n}+¥epsilon$ for $a.a$ . $t¥in J$ .

By Hypothesis A and the assumptions 1) and 2), we get

$|¥frac{dz^{n}(t)}{dt}|¥leq¥omega(t, ¥alpha(Q_{t}^{n}))+¥frac{2}{n}+¥epsilon$ for $a.a$ . $t¥in J$

$¥leq¥omega(t, |z_{t}^{n}|_{¥varpi^{R}})+¥frac{2}{n}+¥epsilon$ for $a.a$ . $t¥in J$ .

Letting $ n¥rightarrow¥infty$ , we have

$|¥frac{dz^{0}(t)}{dt}|¥leq¥omega(t, |z_{t}^{0}|_{¥ovalbox{¥tt¥small REJECT}^{R}})+¥epsilon$ for $a.a$ . $t¥in J$ ,

because of Lemma 3.1 in [13]. Since $¥epsilon$ is arbitrary, we obtain finally

$|¥frac{dz^{0}(t)}{dt}|¥leq¥omega(t, |z_{t}^{0}|_{¥ovalbox{¥tt¥small REJECT}^{R}})$ for $a.a$ . $t¥in J$ .

Moreover, it is easy to see that

$z^{0}(¥sigma+h)¥leq z^{n}(¥sigma+h)¥leq¥frac{2h}{n}+¥epsilon h$ ,

462 Jong Son SHIN

which implies that $(D^{+}z^{0})(¥sigma)=z^{0}(¥sigma)=0$. Thus we have $z^{0}(t)¥equiv 0$ by theassumption 3). This completes the proof of the theorem.

Corollary 4.4. The differential inequality (4.1) in Theorem 4.3 can bereplaced by the differential equation

(4.2) $¥frac{dz}{dt}=K(0)¥omega(t, z(t))+¥beta_{¥mathrm{v}}^{R}z(t)$ for $a.a$ . $ t¥in$ $(¥sigma,$ $¥sigma+a]$

provided $¥beta_{¥mathrm{v}}^{R}¥in R$ , or

(4.3) $¥frac{dz}{dt}=¥omega(t, K(t-¥sigma)z(t))$ for $a.a$ . $ t¥in$ $(¥sigma,$ $¥sigma+a]$ ,

where $¥beta_{v}^{R}=¥lim_{¥tau¥rightarrow}¥sup_{0+}$ $¥{||S_{R}(t)||-1¥}/t$, $-¥infty¥leq¥beta_{¥mathrm{v}}^{R}<¥infty$ , and $||S_{R}(t)||$ stands for the

operator norm of $S_{R}(t)$ .

For the proof refer to [13, Proposition 4.3 and 11, Theorem 3.1].

Remark 4. 5. From Corollary 4.4 we can see that Theorem 4.3 generalizes,in some sense, Theorem 2.4 in [12] and Theorem 3.1 in [10]. For the caseof phase space $¥ovalbox{¥tt¥small REJECT}=e_{¥gamma}$ , refer to [14].

§5. Related results

We make the following hypothesis on a mapping $f:¥Omega¥rightarrow E$ , $¥Omega¥subset R¥times¥ovalbox{¥tt¥small REJECT}$ .

Hypothesis $B$ . For every point $(¥tau, ¥psi)¥in¥Omega$ , there exist positive numbers $a$, $r$

and a Kamke-type function $¥omega(t, s)$ : ( $¥tau$ , $¥tau+a]¥times[0,2r]¥rightarrow R^{+}$ satisfying thefollowing conditions:

(1) $[¥tau-a, ¥tau+a]¥times¥ovalbox{¥tt¥small REJECT}(¥psi, r)¥subset¥Omega$.

(2) $f$ is uniformly continuous and bounded on $[¥tau-a, ¥tau+a]¥times¥ovalbox{¥tt¥small REJECT}(¥psi, r)$ .

(3) $¥alpha(f(t, B))¥leq¥omega(t, ¥alpha(B))$ for $a.a$ . $ t¥in(¥tau$ , $¥tau+a$] and every subset $ B¥subset$

$¥ovalbox{¥tt¥small REJECT}(¥psi, r)$ .

(4) $u(t)¥equiv 0$ is the unique absolutely continuous function from $(-¥infty,$ $¥tau+a]$

into $R^{+}$ , which satisfies the initial condition $(D^{+}u)(¥tau)=0$ and $u(t)=0$ for$ t¥leq¥sigma$ and the scalar differential inequality (4.1)

The following result is a modification of Theorem 3.2 in [12]. Since theproof is essentially based on the argument used in the proof of Theorem 3.2in [12], it is omitted.

Lemma 5. 1. Suppose that $¥ovalbox{¥tt¥small REJECT}^{E}$ and $¥ovalbox{¥tt¥small REJECT}^{R}$ satisfy Axioms $(¥mathrm{B}_{1})-(¥mathrm{B}_{3})$ and

Kneser Type Theorems for FDE 463

Hypothesis $A$ is satisfied Let $f^{n}$ : $¥Omega¥rightarrow E$, $n¥in N$, be a continuous function, where$¥Omega$ is an open subset of $R¥times¥ovalbox{¥tt¥small REJECT}$, and let $x^{n}$ : $(-¥infty, ¥tau_{n})¥rightarrow E$ be a noncontinuablesolution of $¥mathrm{C}¥mathrm{P}$ $(f^{n}, ¥sigma_{n}, ¥varphi^{n})$ , $¥sigma_{n}¥in(-¥infty, ¥tau_{n})$ . Assume that

1) for every point $(¥tau, ¥psi)¥in¥Omega$, $¥mathrm{C}¥mathrm{P}$ $(f^{n}, ¥tau, ¥psi)$ has a solution;2) $f$ satisfies Hypothesis $B$ ;3) if $¥{f^{n}¥}_{n¥in N}$ is uniformly bounded on a closed bounded set $F$ of $¥Omega$ , then

$¥alpha(¥{f^{n}(t, ¥psi)-f(t, ¥psi)|(t, ¥psi)¥in F, ¥mathrm{n}¥geq k¥})¥rightarrow 0$ as $ k¥rightarrow¥infty$ ;

4) $¥{f^{n}¥}_{n¥in N}$ converges to the function $f$ uniformly on every compact subsetof $¥Omega$ ; and

5) $¥sigma_{n}¥rightarrow¥sigma$ and $¥varphi^{n}¥rightarrow¥varphi$ as $ n¥rightarrow¥infty$ and $(¥sigma, ¥varphi)¥in¥Omega$ .

Then there exist a subsequence $¥{x^{n(i)}¥}$ of $¥{¥mathrm{x}^{n}¥}$ and a noncontinuable solution$x$ , defined on $(-¥infty, ¥tau)$ , of $¥mathrm{C}$ $(/, ¥sigma, ¥varphi)$ such that the following conditions hold:

(i) $¥tau¥leq¥lim_{i¥rightarrow}¥sup_{¥infty}¥tau_{n(i)}$ .

(ii) $¥sup$ $¥{|¥mathrm{x}^{n(i)}(t)-x(t)||s(n(i))¥leq t¥leq d¥}¥rightarrow 0$ as $ i¥rightarrow¥infty$ for every $ d¥in$ $(¥sigma, ¥tau)$ ,where $ s(n)=¥max$ $¥{¥sigma, ¥sigma_{n}¥}$ .

Set

$S_{E}(¥varphi)=$ { $x:I$ $¥rightarrow E|x$ is a solution of $¥mathrm{C}¥mathrm{P}(f,$ $¥sigma$ , $¥varphi)$ },$S_{¥ovalbox{¥tt¥small REJECT}}(¥varphi)=$ { $¥zeta:I$ $¥rightarrow¥ovalbox{¥tt¥small REJECT}|¥zeta(t)=x_{t}$ for $t¥in I$ and $x¥in S_{E}(¥varphi)$ },

where I is a subinterval of [ $¥sigma$ , $¥infty)$ , and

$S_{Z}(Q)=¥bigcup_{¥varphi¥in Q}S_{Z}(¥varphi)$ ,

where $Q¥subset¥ovalbox{¥tt¥small REJECT}$ and $Z=E$ or $¥ovalbox{¥tt¥small REJECT}$ . Clearly, we have $S_{Z}(Q)¥subset C(I, Z)$ .

Proposition 5.2. Let $f$ in $¥mathrm{C}¥mathrm{P}(1.1)$ be a mapping defined on $I¥times¥ovalbox{¥tt¥small REJECT}$,$I:=$ $[¥sigma, ¥sigma+a]$ . Suppose that all the conditions in Theorem A3 are satisfied.Then for any compact and connected subset $Q$ in $¥ovalbox{¥tt¥small REJECT}$ the set $S_{Z}(Q)$ is $a$

compact and connected subset in $C(I, Z)$ .

Proof. We shall prove that the set $S_{E}(Q)$ is compact in $C(I, E)$ . Let$¥{u^{n}¥}_{n¥in N}$ be any sequence in $S_{E}(Q)$ . Then for each $u^{n}$ there exists a solution$x^{n}$ : ( $-¥infty$ , $¥sigma+a]¥rightarrow E$ of $¥mathrm{C}$ $(/, ¥sigma, ¥varphi^{n})$ such that $x^{n}|I=u^{n}$ and $¥chi_{¥sigma}^{n}=¥varphi^{n}¥in Q$ . ByLemma 5.1 we can choose a subsequence $¥{x^{n_{i}}¥}$ of $¥{¥chi^{n}¥}$ and a solution $x^{0}$ of$¥mathrm{C}$ $(/, ¥sigma, ¥varphi^{0})$ such that $X_{¥sigma}^{n_{¥mathrm{i}}}¥rightarrow¥varphi^{0}$ in $¥ovalbox{¥tt¥small REJECT}$ and $x^{n_{i}}(t)¥rightarrow x^{0}(t)$ uniformly on I as$ i¥rightarrow¥infty$ . Thus the set $S_{E}(Q)$ is compact in $C(I, E)$ . By Theorem 2.1 in [10]we have $¥alpha(S_{¥ovalbox{¥tt¥small REJECT}}(Q))¥leq K_{a}¥alpha(S_{E}(Q))+M_{a}¥alpha(Q)$ and hence $¥alpha(S_{¥ovalbox{¥tt¥small REJECT}}(Q))=0$. Clearly,

464 Jong Son SHIN

the set $S_{¥ovalbox{¥tt¥small REJECT}}(Q)$ is compact in $C(I, ¥ovalbox{¥tt¥small REJECT})$ . The proofs of the connectedness for$S_{E}(Q)$ and $S_{¥ovalbox{¥tt¥small REJECT}}(Q)$ follow from Kaminogo [4]. Hence the proof is complete.

Corollary 5.3. If all the assumptions in Proposition 5.2 are satisfied, then

$¥sum(Q)=$ { $(t,$ $x_{t})|x¥in S_{E}(Q)$ and $t¥in I$ }

is compact and connected in $R¥times¥ovalbox{¥tt¥small REJECT}$ .

Proof. The mapping on $I¥times S_{¥ovalbox{¥tt¥small REJECT}}(Q)$ defined by $(t, ¥zeta)¥rightarrow(t, ¥zeta(t))$ is continuousonto $¥sum(Q)$ . Since $I¥times S_{¥ovalbox{¥tt¥small REJECT}}(Q)$ is compact and connected, the set $¥sum(Q)$ is alsocompact and connected in $R¥times¥ovalbox{¥tt¥small REJECT}$.

The following result extends Theorem 1 in [7]. Since the proof is similarto that of Theorem 4.3 (or [14, Theorem 3.4]), it is omitted.

Theorem 5.4. Suppose that ME and $¥mathrm{M}¥mathrm{R}$ satisfy Axioms $(¥mathrm{B}_{1})-(¥mathrm{B}_{3})$ andHypothesis $A$ is satisfied. Assume that $f$ : $[¥sigma, ¥sigma+a]¥times¥ovalbox{¥tt¥small REJECT}(¥varphi, r)¥rightarrow E$ is $a$

uniformly continuous function such that $|f|¥leq M-1$ , $M>1$ , on $[¥sigma, ¥sigma+a]¥times$

$¥ovalbox{¥tt¥small REJECT}(¥varphi, r)$ , and that there exists a Kamke-type function $¥omega(t, s)$ : $(¥sigma,$ $¥sigma+a]¥times[0,¥mathit{2}r]$

$¥rightarrow R^{+}$ such that

1) $¥lim_{h¥rightarrow 0}¥inf_{-}¥frac{1}{h}$ $[¥alpha(A(t))-¥alpha(¥{x(t)-hf(t, x_{t}):x¥in A¥})]$

$¥leq¥omega(t, ¥alpha(A_{t}))$

for $a.a$ . $t¥in(¥sigma,$ $¥sigma+a$] and for $awy$ subset $A¥subset¥ovalbox{¥tt¥small REJECT}_{¥sigma}^{¥varphi}[a]$ such that $A|$ $[¥sigma, ¥sigma+a]$ isequicontinuous and $A_{t}¥subset¥ovalbox{¥tt¥small REJECT}(¥varphi, r)$ for all $ t¥in$ $[¥sigma, ¥sigma+a]$ ;

2) $¥omega(t, s)$ is nondecreasing in $s$ for $a.a$ . $ t¥in$ $[¥sigma, ¥sigma+a]$ ; and3) the condition 3) of Theorem 4.3 is satisfied, provided (4.1) is replaced

by the scalar differential inequality

(5.1) $¥frac{dz}{dt}¥leq¥omega(t, |z_{t}|_{¥ovalbox{¥tt¥small REJECT}^{R}})$ for $a.a$ . $ t¥in$ $(¥sigma,$ $¥sigma+a]$ .

Then the conclusion of Theorem 4.3 remains valid.

Remark 5.5 [13, Proposition 4.3]. The differential inequality (5.1) inTheorem 5.4 can be replaced by the differential equation (4.2) or (4.3) underan additional condition:

4) $¥omega(t, |z_{t}|_{¥ovalbox{¥tt¥small REJECT}^{R}})¥rightarrow 0$ as $t$ $¥rightarrow¥sigma+0$ and $¥int_{¥sigma}^{t}¥omega(s, |z_{s}|_{¥ovalbox{¥tt¥small REJECT}^{R}})ds<¥infty$ ,

whenever $z$ : ( $-¥infty$ , $¥sigma+a]¥rightarrow[0,¥mathit{2}r]$ is an absolutely continuous functionsatisfying the conditions $(D^{+}z)(¥sigma)=0$ and $z(t)=0$ for $ t¥leq¥sigma$ .

Kneser Type Theorems for FDE 465

Remark 5.6. (1) The condition 1) in Theorem 5.4 is weaker than thecondition 1) in Theorem 4.3 (cf. [6, p. 48]).

(2) Theorem 5.4 is a generalization of Theorem 4.3 if $¥omega$ in Theorem 5.4is the Nagumo-type function $¥omega(t, s)=s/(t-¥sigma)$ . Indeed, the function $¥omega$

satisfies the condition 4) in Remark 5.5 (cf. [9. Proposition 6.1]), which meansthat the condition 3) in Theorem 5.4 and the condition 3) in Theorem 4.3are equivalent to each other.

(3) The condition 3) in Theorem 5.4 is weaker than the condition (iii)in [7, Definition 3] even for ODEs.

References

[I] Banas, J. and Goebel, K., Measures of Noncompactness in Banach Spaces, Marcal Dekker,New York and Basel, 1980.

[2] Deimling, K., Some open problems in ordinary differential equations and fixed points of(weakly inward) maps in Banach spaces, in “Nonlinear Analysis and Applications”(V. Lakshmikanth, Ed), pp. 153-159, Dekker, New York and Basel, 1987.

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[14]?, On Kneser type theorem for functional differential equations with the phasespace (

$¥mathscr{C}_{¥gamma}$ in Banach spaces, in “The Study of Dynamical Systems” vol. 7. (N. Aoki Ed), pp.157-170, World Scientific, Singapore, 1989.

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Jong Son SHIN

nuna adreso:Department of MathematicsKorea UniversityKodaira, Tokyo 187Japan

(Ricevita la 7-an de junio, 1990)(Reviziita la 12-an de februaro, 1991)