Funkcialaj Ekvacioj, 16 (1973), 195-211
Appoxim-ation and Existence of Solutions
to Ordinary Differential Equations in
Banach Spaces*
By Robert H. MARTIN, JR.
(North Carolina State University)
Introduction.
Let $E$ be a real or complex Banach space with norm denoted by
; and let
$D$ be a locally closed subset of
$
¥
in
D$
$R_{z}>0$
, for each
there is an
such
$E$
$D$
$
¥
{x
¥
in
D:|x-z|
¥
leq
R_{z}
¥
}$
equivalently,
in
or,
that
is closed
is a relatively closed
subset of an open set in $E$ ). Also, let $A$ be a continuous function from $D$ into
$E$ .
In this paper we are interested in studying both the existence of approximate
solutions and the existence of solutions to the initial value problem
§1.
$|¥cdot|$
$¥_$
$E(¥mathrm{i}.¥mathrm{e}.$
(IVP)
$u^{¥prime}(t)=$
Au(t),
$z$
$u(0)=z¥in D$,
and
$t¥geq 0$
.
There are two main aims of this paper. First, in §2, we characterize and
derive some fundamental properties of hose continuous functions $A$ for which
(IVP) has -approximate solutions for each §>0. Second, in §3 and §4, we
show, with the additional assumptions that $A$ is uniformly continuous and $D$ is
bounded, that one may use one-sided derivatives involving the measure of noncompactness to parallel very closely the results involving continuous dissipative
operators on Banach spaces.
$¥dot{¥mathrm{t}}$
$¥mathrm{e}$
Approximate Solutions.
Throughout this secton we assume that $D$ is a locally closed subset of $E$ and
that $A$ is a continuous function from $D$ into $E$ . The objective of this section
is to characterize those continuous functions $A$ such that (IVP) has an e-approand
ximate solution for each $z¥in D$. If
is a function from $[0, T]$ into $E$ ,
then
is said to be an -approximate solution to (IVP) on $[0, T]$ if each of
the following is satisfied: (a)
is continuous, $¥phi(0)=z$ and $¥phi(T)¥in D;(¥mathrm{b})$ if
$[0, T]$ there is a $¥tau(t)¥in[0, t]$ such that
and $¥phi(¥tau(t))¥in D;(¥mathrm{c})$ there is
exists
a countable subset $C$ of $[0, T]$ such that if $t¥in[0, T]-C$ then
¥
¥
¥ ¥
; and (d)
and ¥ ¥
is integrable and
§2.
$¥epsilon>0$
$¥phi$
$¥phi$
$e$
$ t¥in$
$¥phi$
$t-¥tau(t)¥leq ¥mathrm{e}$
$¥phi^{¥prime}(t)$
$| phi^{ prime}(t)-A phi( tau(t))| leq epsilon$
$¥phi^{¥prime}$
$¥phi(t)=z+¥int_{0}^{t}¥phi^{¥prime}(s)ds$
*
This work partially supported by the U. S. Army Research Office, Durham, N. C.
R. H. MARTIN, JR.
196
for all $t¥in[0, T]$ .
Remark 1. In part (a) of the definition of -approximate solution to (IVP)
, it is crucial to require that $¥phi(T)¥in D$ . If we omit this requirement,
it is easy to see that (IVP) has an -approximate solution for any continuous
function $A$ from $D$ into $E$ .
in $E$ let $d(y;D)=¥inf¥{|y-x|:x¥in D¥}$ and denote by $A_{+}(D)$ the
For each
of all continuous function $A$ from $D$ into $E$ such that
$¥mathrm{e}$
$¥langle)¥mathrm{n}[0, T]$
$¥mathrm{e}$
$y$
$|¥mathrm{c}¥mathrm{l}¥mathrm{a}¥mathrm{s}¥mathrm{s}$
(1)
$¥lim_{h¥rightarrow}¥inf_{0+}d(x+hAx;D)/h=0$
for all
$x¥in D$
.
Our fundamental result is given by the following:
Theorem 1. Suppose that $A$ is a continuous function from $D$ into E. Then
these are equivafent:
(i)A is in $A_{+}(D)$ .
(ii) The initial value problem (IVP) has an upproximate solution for
each
and each $z¥in D$ .
Before proving Theorem 1 we first establish two preliminar.v lemmas. The
first lemma, which immediately implies that (i) implies (ii) in Theorem 1, is
proved in [17, Proposition 1].
Lemma 1. Suppose that $A¥in A_{+}(D)$ . Let be in $D$ and let the numbers
:
$R_{z}>0$ and $M_{z}¥geq 1$ be such that $S_{z}=¥{x¥in D:|x-z|¥leq R_{z}¥}$ is closed and $¥sup$ {
$x¥in S_{z}¥}¥leq M_{z}-1$ .
has an
Also, let $T_{z}=R_{z}/M_{z}$ . Then for each
upproximate solution
from $[0, T_{z}]$ into $E$ which satisfies each of
$Jhe$ following: there is a nondecreasing sequence
in $[0, T_{z}]$ such that
;
, and
(i) $t_{0}=0$ ,
if $t_{i}<T_{z}$ ,
$¥epsilon-$
$¥epsilon>0$
$z$
$|A_{x}|$
$¥epsilon>0(¥mathrm{I}¥mathrm{V}¥mathrm{P})$
$¥epsilon-$
$¥phi=¥phi(¥cdot ; z, ¥epsilon)$
$¥{t_{i}¥}_{0}^{¥infty}$
$ t_{i+1}-t_{i}<¥epsilon$
$t_{i}<t_{i+1}$
$i¥lim_{¥rightarrow¥infty}t_{i}=T_{z}$
(ii) $¥phi(0)=z$ and $|¥phi(t)-¥phi(s)|¥leq M_{z}|t-s|$ for $alf$ , $s¥in[0, T_{z}]$ ;
;
is linear on
and
(iii)
¥
; and
, then
and
(iv) if
(v) if $y¥in D$ with $|y-¥phi(t_{i})|¥leq(t_{i+1}-t_{i})M_{z}$ , then $|Ay-A¥phi(t_{i})|¥leq¥epsilon$ .
Actually, the limit infimum in $(*)$ is replaced by the limit in Proposition 1
of [17]; however, there is no change in proof with this modification. Our next
lemma is concerned with the behavior of -approximate solutions.
Lemma 2. Suppose that $A$ is continuous from $D$ into $E$ and (IVP) has an
and each $z¥in D$. Let be in $D$ and let the
upproximate sofution for each
$t$
$¥phi(t_{i})¥in S_{z}$
$[t_{i}, t_{i+1}]$
$¥phi$
$t in(t_{i}, t_{i+1})$
$t_{i}<T_{z}$
$|¥phi^{¥prime}(t)-A¥phi(t_{i})|¥leq¥epsilon$
$¥mathrm{e}$
$¥epsilon>0$
$¥epsilon-$
numbers
$:¥sup$
$R_{z}>¥mathit{0}$
and
$M_{z}¥geq 1$
$¥{|Ax|:x¥in S_{z}¥}¥leq M_{z}-1$
solution
$¥phi=¥phi(¥cdot ; z, ¥epsilon)$
.
from
$z$
be such that $S_{z}=¥{x¥in D:|x-z|¥leq R_{z}¥}$ is closed and
approximate
has an
Then for each
$[0, T_{z}]$ into $E$ where $T_{z}=R_{z}/M_{z}$ .
Moreover.
$¥epsilon>0(¥mathrm{I}¥mathrm{V}¥mathrm{P})$
$|¥phi(t;z, ¥epsilon)-¥phi(s;z, ¥epsilon)|¥leq M_{z}|t-s|$
for
$afl$
$t$
, $s¥in[0, T_{z}]$ and
$¥epsilon>0$
.
$¥epsilon-$
Approximation and Existence
of Solutions
197
Proof. Let
be the class of all e-approximate solu1] and let
to (IVP) on $[0, T]$ where $T¥leq T_{z}$ and $|¥phi(t)-¥phi(s)|¥leq M_{z}|t-s|$ for all
tions
, $s¥in[0, T]$ . If
and
is defined on $[0, T_{i}]$ for $i=1,2$ , write
if
and $¥phi_{1}(t)=¥phi_{2}(t)$ for all $t¥in[0, T_{1}]$ . Then
is a partial order on
and it follows in the usual manner that
has a maximal element.
is a sequence in
(Note that if
with
defined on $[0, T_{i}]$ and
for $i=1,2$ , ; then
exists and, since $|¥phi_{j}(T_{j})-¥phi_{i}(T_{i})|¥leq M_{z}$
$¥epsilon¥in(0,$
$P_{z}(¥epsilon)$
$¥phi$
$¥phi_{i}¥in P_{z}(¥epsilon)$
$t$
$¥phi_{1}¥leq¥phi_{2}$
$¥phi_{i}$
$T_{1}¥leq T_{2}$
$‘‘¥leq’$
$P_{z}(¥epsilon)$
’
$P_{z}(¥epsilon)$
$P_{z}(¥epsilon)$
$(¥phi_{i})_{0}^{¥infty}$
$¥phi_{i+1}$
$¥cdots$
$|T_{¥dot{¥mathcal{J}}}-T_{i}|$
,
$¥phi_{i}¥leq$
$¥phi_{i}$
$¥lim_{i¥rightarrow¥infty}T_{i}=T¥leq T_{z}$
exists.
$w=¥lim_{i¥rightarrow¥infty}¥phi_{i}(T_{i})$
Also
$w¥in S_{z}$
since
$S_{z}$
is closed.
Hence if
$¥phi(t)=$
for all $t¥in[0, T_{i}]$ and $¥phi(T)=w$ , then
and
for all ) Let
be a maximal element in
and assume, for contradiction, that
is
defined on $[0, T]$ with $T<T_{z}$ . Since $¥phi(T)¥in S_{z}$ and $|¥phi(T)-z|¥leq TM_{z}<R_{z}$ , it
follows that if $¥delta=T_{z}-T$ and $y¥in D$ with $|y-¥phi(T)|¥leq¥delta M_{z}$ , then
. Let
be such that
and, by hypothesis, let be an ’-approximate solution
to (IVP) on
with replaced by
, $¥chi(0)=¥phi(T)$ . If
let
and if
let
is
such that
(where
and
$¥phi(¥tau(¥delta))¥in D)$ .
it follows that $¥eta¥in(0,$ ], and since
Since
it follows
¥
that
is an -approximate solution to (IVP) on
replaced by
(with
$¥phi(T))$ .
$
¥
psi(t)=
¥
phi(t)$
Thus if
for $t¥in[0, T]$ and $¥psi(t)=¥chi(t-T)$ for $t¥in[T, T+¥eta]$ ,
it follows that
is an -approximate solution to (IVP) on $[0, T+¥eta]$ . To show
that
we need to show that $|¥chi(t)-¥chi(s)|¥leq M_{z}|t-s|$ for all , $s¥in[0, ¥eta]$ . To
show that this holds we show that $|¥chi(t)-z|¥leq R_{z}$ for all $t¥in[0, ¥eta]$ (since this
¥
¥
¥
¥
would imply that ¥ ¥ ¥ ¥ ¥ ¥
for all but a countable
number of $t¥in[0, ¥eta])$ . By continuity there is a largest number $t_{1}¥in(0,$ ] such
that $|¥chi(s)-z|¥leq R_{z}$ for all $s¥in[0, t_{1}]$ , and since $|¥chi;(s)|¥leq M_{z}$ for all but a countable
number of $s¥in[0, t_{1}]$ we have that
$¥phi_{i}(t)$
$¥phi¥in P_{z}(¥epsilon)$
$¥phi_{i}¥leq¥phi$
$i.$
$P_{z}(¥epsilon)$
$¥phi$
$¥phi$
$y¥in S_{z}$
$¥epsilon^{¥prime}<-¥min$
$¥{¥epsilon, ¥delta¥}$
$[0, ¥delta_{1}]$
$¥eta=¥delta_{1}$
$¥chi$
$¥delta_{1}¥leq¥delta$
$¥tau(¥delta)¥in[0, ¥delta]$
$¥epsilon^{¥prime}<¥delta$
$¥chi$
$¥mathrm{e}$
$¥phi(T)-¥mathrm{i}.¥mathrm{e}.$
$z$
$¥eta=¥tau(¥delta)$
$¥delta_{1}>¥delta$
$¥delta^{¥tau}-¥tau(¥delta)¥leq¥epsilon^{¥prime}$
$¥delta$
$¥epsilon^{¥prime}<¥epsilon$
$[0,
$¥mathrm{e}$
$¥psi$
$¥epsilon^{¥prime}>0$
eta]$
$z$
$¥mathrm{e}$
$¥emptyset¥in P_{z}(¥epsilon)$
$t$
$| chi^{ gamma}( tau)| leq|A chi( tau(t))|+ epsilon leq M_{z}-1+ epsilon leq M_{z}$
$¥eta$
$|¥chi(t_{¥mathrm{I}})-z|¥leq|¥chi(t_{1})-¥chi(0)|+|¥phi(T)-z|¥leq t_{1}M_{z}+TM_{z}=(t_{1}+T)M_{z}$
.
Thus if
then $|¥chi(t_{1})-z|<R_{z}$ which is impossible. Consequently
and we have a contradiction to the maximality of . Thus
is defined on
$[0, T_{z}]$ and the proof of Lemma 2 is complete.
Proof of Theorem 1. As noted above, the fact that (i) implie (ii) is
immediate from Lemma 1. Now suppose that (ii) holds and that $z¥in D$. By
Lemma 2 we can assume that there are numbers $T>0$ and $M¥geq 1$ such that, for
each
, (IVP) has an -approximate solution
on $[0, T]$ with
$¥phi(s)|¥leq M|t-s|$ for all
$s
¥
in[0,
T]$
,
. Now let
]. Since $A$ is continuous,
$T$
¥ ¥
¥
let
] be such that
whenever $y¥in D$ and $|y-z|¥leq¥delta M$. We
¥ ¥
show that if
, then $h^{-l}d(z+hAz; D)¥leq¥epsilon^{¥prime}$ . Choose $h¥in(0,$ ] and let
. Also, for each $s¥in[0, h]$ let $¥tau(s)¥in[0, s]$ be such that
and
. If $s¥in[0, h]$ then $|¥phi_{¥epsilon}(¥tau(s))-z|¥leq¥tau(s)M¥leq¥delta M$ ; so $|A¥phi_{¥mathrm{e}}(¥tau(s))-Az|¥leq$
$¥psi¥in P_{z}(¥epsilon)$
$ t_{¥mathrm{I}}<¥eta$
$¥phi$
$¥epsilon>0$
$¥mathrm{e}$
$|Ay-Az| leq epsilon^{ prime}/3$
$ 0<h leq delta$
$h¥epsilon^{¥prime}/(3M)$
$¥phi_{¥epsilon}(¥tau(s))¥in D$
$|¥phi_{¥epsilon}(t)-$
$¥phi_{¥epsilon}$
$¥epsilon^{¥prime}¥in(0,1$
$t$
$¥delta¥in(0,$
$¥phi$
$¥delta$
$ s-¥tau(s)¥leq¥epsilon$
$¥epsilon=$
R. H. MARTIN, JR.
198
$¥epsilon^{¥prime}/3$
.
Hence for all but a countable number of
$s¥in[0, h]$
we have that
$|¥phi_{¥epsilon}^{¥prime}(s)-Az|¥leq|¥phi_{¥epsilon}^{¥prime}(s)-A¥phi_{¥epsilon}(¥tau(s))|+|A¥phi_{¥epsilon}(¥tau(s))-Az|$
$¥leq¥epsilon+¥frac{¥epsilon^{¥prime}}{3}=¥frac{h¥epsilon^{¥prime}}{3M}+¥frac{¥mathrm{e}^{¥prime}}{3}¥leq¥frac{2¥epsilon^{f}}{3}$
.
Consequently,
$h^{-l}d(z+hAz; D)¥leq h^{-1}|z+hAz-¥phi_{¥epsilon}(¥tau(h))|$
$¥leq h^{-1}|z+hAz-¥phi_{¥epsilon}(h)|+h^{-1}|¥phi_{¥epsilon}(h)-¥phi_{¥epsilon}(¥tau(h))|$
$¥leq h^{-1}|hAz-¥int_{0}^{h}¥phi_{¥epsilon}^{¥prime}(s)ds|+h^{-1}M|h-¥tau(h)|$
$¥leq h^{-1}¥int_{0}^{h}|Az-¥phi_{¥epsilon}^{¥prime}(s)|ds+h^{-1}M¥epsilon$
$¥leq¥frac{2¥epsilon^{¥prime}}{3}+¥frac{¥epsilon^{¥prime}}{3}=¥epsilon^{¥prime}$
.
This completes the proof of Theorem 1.
It follows from the proof of Theorem 1 that if
$A¥in A_{+}(D)$
then
$¥lim d(x+$
$h¥rightarrow 0+$
$hAx$ ;
) $=0$ for all $x¥in D$ . In fact, we have the following stronger result:
Theorem 2. If $A¥in A_{+}(D)$ then
$D$
for all
$¥lim_{h¥rightarrow 0+}d(z+hAz;D)/h=0$
$z¥in D$
,
and this limit is uniform on each compact subset of $D$ .
]. For each $z¥in K$
Proof. Let $K$ be a compact subset of $D$ and let
be the -approximate solution to (IVP) on [0,
and in (0, 1] let
$M_{z}]$
By the compactness of $K$ we may
which is constructed in Lemma 1.
assume that if $M=¥sup¥{M_{z}: z¥in K¥}$ and $T=¥inf¥{R_{z}/M_{z}: z¥in K¥}$ , then $ M<¥infty$ and
$T>0$ . Thus $|¥phi(t;z, ¥epsilon)-¥phi(s;z, ¥epsilon)|¥leq M|t-s|$ for all , $s¥in[0, T]$ , $z¥in K$, and
$T$ ] be such that
( , 1]. Since $K$ is compact and $A$ is continuous, let
$|Ay-Az|¥leq¥epsilon^{¥prime}/3$ whenever $z¥in K$,
$y¥in D$ and $|y-z|¥leq¥delta M$ .
It follows exactly as in
$ 0<h¥leq¥delta$ ,
$h^{-l}d(z+hAz;
D)
¥
leq
¥
epsilon^{
¥prime}$
for all $z¥in K$.
then
the proof Theorem 1 that if
Since is independent of $z¥in K$, the assertion of Theorem 2 follows.
Remark 2. In [9, Lemma 1 and Remark 2], M. Crandall proves Theorem
2 directly (without the use of approximate solutions) in the case that $E$ is a Hilbert
space and $D$ is locally weakly closed. Also, using Theorem 2, one can show
that the approximate solutions constructed in Lemma 1 are actually piecewise
, $t_{i}=T_{z}$ for all sufficiently large ).
linear (
Our next result considers the algebraic structure of $A_{+}(D)$ .
Theorem 3. Suppose that $A$ and $B$ are in $A_{+}(D)$ and $r>0$ . Then $rA$ and
$A+B$ are in $A_{+}(D)$ .
as
Proof. It follows immediately from (1) that $rA¥in A_{+}(D)$ since
$¥epsilon^{¥prime}¥in(0,1$
$¥phi(¥cdot ; ¥mathrm{z}, ¥epsilon)$
$¥epsilon$
$R_{z}/$
$¥mathrm{e}$
$¥epsilon¥in$
$t$
$¥delta¥in(0,$
$0$
$¥delta$
$¥mathrm{i}.¥mathrm{e}.$
$i$
$rh¥rightarrow 0+$
Approximation and Existence
$ h¥rightarrow 0¥dotplus$
all
. Now let
with
$y¥in D$
such that
$¥epsilon>0$
$¥delta_{1}>0$
$|y-z|¥leq¥delta_{1}$
$¥lim h_{k}=0$
199
and
. First choose
such that $|Ay-Az|¥leq¥epsilon/3$ for
. Next choose
such that
and
for all $h¥in(0,$ ]. Now let
be a sequence in (0, ]
$z¥in D$
$h^{-l}d(z+hBz; D)¥leq¥epsilon/4$
of Solutions
$¥delta_{2}>0$
$¥delta_{2}¥epsilon/3+¥delta_{2}|Bz|¥leq¥delta_{1}$
$(h_{k})_{1}^{¥infty}$
$¥delta_{2}$
, and let
be a sequence in
$(x_{k})_{1}^{¥infty}$
$¥delta_{2}$
$D$
such that
$|z+hBz-x_{k}|¥leq$
$ k¥rightarrow¥infty$
for all . Note that $|z-x_{k}|¥leq h_{k}¥epsilon/3+h_{k}|Bz|¥leq¥delta_{1}$, and hence $|Ax_{k}-Az|¥leq¥epsilon/3$ for
all and $¥lim x_{k}=z$ . Thus $¥{x_{k} : k=1,2, ¥cdots¥}$ is contained in a compact subset of
$D$, and we have from Theorem 2 that there is a
such that $h^{-1}kd(x_{k}+h_{k}$
; $D$)
for all $k¥geq N$ where $N$ is such that
for all $k¥geq N$. Thus for
each $k¥geq N$ there is a $y_{k}¥in D$ such that
. Combining each
of these inequalities we have that if $k¥geq N$,
$k$
$h_{k}¥epsilon/3$
$k$
$ k¥rightarrow¥infty$
$¥delta_{3}>0$
$Ax_{le}$
$¥leq¥epsilon/4$
$ h_{k}¥leq¥delta$
$h^{-1}k|x_{h}+h_{l¥epsilon}Ax_{k}-y_{k}|¥leq¥epsilon/3$
$h^{-1}kd(z+h_{k}[Az+Bz];D)¥leq h^{-1}k|z+h_{k}Az+h_{k}Bz-y_{k}|$
$¥leq h^{-1}k|z+h_{k}Bz-x_{k}|+h^{-1}k|x_{k}+h_{k}Az-y_{k}|$
$¥leq¥frac{¥epsilon}{3}+h^{-1}k|x_{k}+h_{k}Ax_{k}-y_{k}|+|Az-Ax_{k}|$
$¥leq¥frac{¥epsilon}{3}+¥frac{¥epsilon}{3}+¥frac{¥epsilon}{3}=¥epsilon$
Thus
$¥lim¥inf h^{-1}d(z+h[Az+Bz];D)¥leq¥epsilon$
.
for each
$h¥rightarrow 0+$
$¥epsilon>0$
, and we conclude that
$A+B¥in A_{+}(D)$ .
This completes the proof of Theorem 3.
If $C(D)$ denotes the vector space of all continuous functions from $D$ into
, then Theorem 3 asserts that $A_{+}(D)$ is a cone in $C(D)$ . If one defines the
topology on $C(D)$ by uniform convergence on compact subsets of , then it is
easy to see that $A_{+}(D)$ is closed in $C(D)$ . Note that if $A¥in A_{+}(D)$ it is not
necessarily the case that $-A¥in A_{+}(D)$ . However, if $A(D)$ denotes the class of
all members $A$ of $C(D)$ such that $¥lim d(z+hAz; D)/h=0$ for each $z¥in D$, then
$A¥in A(D)$ only in case both $A$ and $-A$ are in $A_{+}(D)$ . Hence $A(D)$ is a vector
space over , and is closed uniform convergence on compact sets. Moreover,
by Theorem 1, $A(D)$ is precisely the class of all continuous functions such that
the initial value problem (IVP) has -approximate solutions on intervals of the
, $T_{2}>0$ . Note also that if $D$ is open, then $A_{+}(D)=$
form $[- T_{1}, T_{2}]$ where
$A(D)=C(D)$ .
$oE$
$¥mathrm{D}$
$h¥rightarrow 0$
$¥mathrm{R}$
$¥mathrm{e}$
$T_{1}$
Let
be in $D$ and let
be a sequence of positive numbers with
. Using Lemma 1, construct an -approximate solution
to (IVP)
$(¥epsilon_{n})_{1}^{¥infty}$
$z$
$¥lim¥epsilon_{n}=0$
$¥phi_{n}$
$¥mathrm{e}$
$ n¥rightarrow¥infty$
on
$[0, T_{z}]$
for each
$¥mathrm{n}$
.
By Proposition 2 of [17] we have that if
$u(t)=¥lim¥phi_{n}(t)$
$ n¥rightarrow¥infty$
exists for each $t¥in[0, T_{z}]$ , then
is a solution to (IVP) on $[0, T_{z}]$ . Note in
particular that by (ii) of Lemma 1 the sequence
is equicontinuous. Thus
if $E$ is finite dimensional we have from Ascoli’s Theorem that (IVP) has a
$u$
$(¥phi_{n})^{¥infty}1$
R. H. MARTIN, JR.
200
solution for each in $D$. Conversely, if (IVP) has a solution for each in $D$,
then $A$ is in $A_{+}(D)$ by Theorem 1. Consequently, when $E$ is finite dimensional,
$A$ is in $A_{+}(D)$ if and only if (IVP) has a local solution (to the right) for each
in $D$. Professor J. A. Yorke has recently informed the author by letter that
this result is due to Nagumo [19]. This type of problem in the finite dimensional case has been studied more recently by several authors: Bony [2], Brezis
[3]; Crandall [9]; Hartman [11]; Redheffer [21] . and Yorke [23]; [24]. In the
finite dimensional case, the condition (1) is clearly related to the existence of
invariant sets for autonomous differential equations. See, in particular, the paper
of Yorke [23]. Note also, by Theorem 3, if $E$ is finite dimensional, $A$ and $B$
Au(t), $u(0)=z$
are continuous from $D$ into $E$ , and the initial value problem
and $u^{¥prime}(t)=Bu(t)$ , $u(0)=z$ have solutions to the right for each $z¥in D$, then
Au(t) $+Bu(t)$ , $u(0)=z$ also has a solution to the right for each $z¥in D$.
In the infinite dimensional case the existence of approximate solutions to
(IVP) does not necessarily imply the existence of solutions. However, Lasota
and Yorke [14] have shown that for “most” continuous functions $A$ , (IVP) has
a solution when $D$ is open. There are several additional conditions which one
can assume on $A$ which do guarantee existence. These type of conditions usually involve Lipschitz conditions, compactness conditions, Lyapunov functions, or
measure of noncompactness conditions. See, for example, Browder [4]; Cellina
[5], [6]; Crandall [8]; Hartman [10]; Li [15]; Martin [16], [17]; and Murakami [18]. Each of these references except [8] and [17] assume that $D$ is open.
Remark 3. These techniques also apply to nonautonomous equations (see
[17] . However, in many cases, one may reduce the nonautonomous case to the
is a locally
autonomous case in the usual manner. In particular, suppose that
$F$ and $B$
$[a, b]¥times$
space
function
from
is a continuous
closed subset of a Banach
into $F$ such that
$z$
$z$
$z$
$)$
$u^{¥prime}(t)=$
$u^{¥prime}(t)=$
$)$
$D^{¥prime}$
$D^{¥prime}$
$¥varliminf_{h}¥inf_{0+}h^{-1}d(x+hB(t, x);D^{¥prime})=0$
for all
$(t, x)¥in[a, b]¥times D^{¥prime}$
.
, and defines
is locally closed and $A¥in
If one takes
$E=R¥times F$
$A(t, x)=(1, B(t, x))$
$(t, x)¥in E$
A_{+}(D)$
for all
with
for all
, then $D$
$|(t, x)|=|t|+|x|$
$(t, x)¥in[a,$
$b$
)
$¥times D^{t}=D$
.
Existence Criteria.
In this section we use the measure of noncompactness of bounded subsets of
$E$ to establish existence criteria for solutions to (IVP).
For each bounded subset
$[¥Omega]-$ is defined to be the
$E$
a
?denoted
noncompactness
of the measure of
of
can be covered by a finite number
infimum of positive numbers such that
is in the field over
are subsets of $E$ and
and
of sets of diameter . If
§3.
$¥mathrm{Q}$
$¥Omega$
$¥Omega$
$¥epsilon$
$¥epsilon$
$¥Omega_{1}$
$¥Omega_{2}$
$a$
Approximation and Existence
of Solutions
201
, define
J2 $2=$ { $x+y:x¥in¥Omega_{1}$ and
} and $a¥Omega_{1}=¥{ax:x¥in¥Omega_{1}¥}$ . Some of the
fundamental properties of a are given by the following lemma:
Lemma 3. Suppose that
and
are bounded subsets of $E$ and
is in
the fiefd over E. Then
;
then
(i) if
, where
(ii)
;
is the closure of
(iii)
if and only if is relatively compact;
;
(iv)
; and
(v)
where
is the convex hull of .
(vi)
We also have the following important property of the measure of noncom$E$
$¥Omega_{1}+$
$y¥in¥Omega_{2}$
$¥Omega_{1}$
$¥Omega_{2}$
$a$
$¥mathrm{a}[¥Omega_{1}]¥leq¥alpha[¥Omega_{2}]$
$¥Omega_{1}¥subset¥Omega_{2}$
$¥mathrm{a}[¥Omega_{1}]=¥alpha[¥mathrm{c}1[¥Omega_{1})]$
$¥mathrm{c}1(¥Omega_{1})$
$¥alpha[¥Omega_{1}]=0$
$¥Omega_{1}$
$¥Omega_{1}$
$¥alpha[a¥Omega_{1}]=|a|¥mathrm{a}[¥Omega_{1}]$
$¥mathrm{a}[¥Omega_{1}+¥Omega_{2}]¥leq¥alpha[¥Omega_{1}]+¥mathrm{a}[¥Omega_{2}]$
$¥alpha[¥mathrm{c}¥mathrm{o}(¥Omega_{1})]=¥mathrm{a}[¥Omega_{1}]$
$¥mathrm{c}¥mathrm{o}(¥Omega_{1})$
$¥Omega_{1}$
pactness:
Lemma 4.
subsets
of
$E$
Suppose that
such that
$¥{¥Omega_{n} :
$¥Omega_{n+1}¥subset¥Omega_{n}$
and
n=1,2, ¥cdots¥}$
is a family
$¥lim_{n¥rightarrow¥infty}¥alpha[¥Omega_{n}]=0$
.
of bounded,
Then
$¥bigcap_{n=1}¥mathrm{c}1(¥Omega_{n})$
nonempt.
$f$
is noncom-
pty and compact.
The notion of the measure of noncompactness is due to Kuratowski (see [13,
p. 412]) and has been employed in connection with the existence of solutions to
differential equations in Banach spaces by several authors?see Ambrosetti [1];
Cellina [5], [6]; Corduneanu [7]; Li [15]; and Szufla [22]. In most cases
the measure of noncompactness is used to extend Lipschitz and compactness
criteria on the function $A$ to guarantee existence of solutions to (IVP). In [6],
Cellina shows that one may also extend dissipative and compactness criteria by
using the measure of noncompactness and the duality mapping on $E$ , in the
case that dual space
of $E$ is uniformly convex. This is further extended
by Li [15, Theorem 1]. However, for our existence criteria we use a onesided
derivative type estimate on $A$ involving the measure of noncompactness, which is
equivalent to a condition introduced by Li [15, Theorem 2]. Although these
results are valid in general Banach spaces, the techniques employed depend strongly on the assumption that $A$ is uniformly continuous as opposed to simply
continuous. The results in this section were motivated by the paper of Li [15],
and the author is most appreciative to Professor Li for sending a )
of his
paper.
Let $D$ be a bounded subset of $E$ and let $A$ be a function from $D$ into $E$
with bounded range. If
let $¥{Ax¥}_{¥Omega}=¥{Ax:x¥in¥Omega¥}$ . Then, if
, $h>0$ ,
and $0<¥beta<1$ ,
$E^{*}$
$¥mathrm{r}¥mathrm{e}¥mathrm{p}¥mathrm{r}¥mathrm{i}¥mathrm{n}¥mathrm{t}$
$¥mathrm{I}$
$¥Omega¥subset D$
$¥Omega¥subset D$
$¥alpha[¥{x+¥beta hAx¥}_{¥Omega}]=¥alpha[¥{¥beta(x+hAx)+(1-¥beta)x¥}_{¥Omega}]$
$¥leq¥alpha[¥beta¥{x+hAx¥}_{¥Omega}]+¥alpha[(1-¥beta)¥Omega]$
$=¥beta¥alpha[¥{x+hAx¥}_{¥Omega}]+(1-¥beta)¥alpha[¥Omega]$
.
202
R. H. MARTIN, JR.
Thus, for each
$h>0$
and
$0<¥beta<1$
,
$(¥beta h)^{-1}(¥alpha[¥{x+¥beta hAx¥}_{¥Omega}]-¥alpha[¥Omega])¥leq h^{-1}(¥alpha[¥{x+hAx¥}_{¥Omega}]-¥alpha[¥Omega])$
and it follows that
(2)
$m_{+}[¥alpha, ¥Omega, A]=¥lim_{h¥rightarrow 0+}(¥alpha[¥{x+hAx¥}_{¥Omega}]-¥alpha[¥Omega])/h$
exists (and is finite since each term in the limit is bounded below by
$[¥{Ax¥}_{¥Omega}])$
.
$-¥alpha$
Similarly
(3)
$m_{¥_}[¥alpha, ¥Omega, A]=¥lim_{h¥rightarrow 0-}(¥alpha[¥{x+hAx¥}_{¥Omega}]-¥alpha[¥Omega])/h$
exists and is finite. We list below (without proofs) some of the elementary
:
properties of
and
Lemma 5. Suppose that $D$ is a bounded subset of $E$ , $A$ and $B$ are functions
. Then
from $D$ into $E$ with bounded range, and
$m_{-}[¥alpha, ¥Omega, A]=-m_{+}[¥alpha, ¥Omega, -A]$ ;
(i)
(ii) $m_{¥pm}[¥alpha, ¥Omega, rA]=rm_{¥pm}[¥alpha, ¥Omega, A]$ for each ¥ ;
(iii) $m_{+}[¥alpha, ¥Omega, A+B]¥leq m_{+}[¥alpha, ¥Omega, A]+m_{+}[¥alpha, ¥Omega, B]$ ;
(iv) $m_{¥_}[¥alpha, ¥Omega, A+B]¥geq m_{¥_}[¥alpha, ¥Omega, A]+m_{¥_}[¥alpha, ¥Omega, B]$ ;
¥
¥
¥
¥
¥
;
(v)
in the fiefd
(vi)
for each
$afl$ $x¥in D$
$E$
$Ix=x$
over
).
(where
for
$D$
locally
suppose
closed, bounded subset of $E$ . Denote by
is
a
Now
that
the class of all functions $A$ from $D$ into $E$ such that (a) $A$ is
$A$ is in $A_{+}(D)$ ; and
uniformly continuous and bounded on
(c) there is
for each
. The least number
a real number such that
. Since the terms in the limit defining
such that (c) holds is denoted by
, we see that (c) is equivalent
in (3) are nonincreasing as
$
¥
alpha[
¥
{x-hAx
¥
}_{
¥
Omega}]
¥
geq(1-h
¥
omega)
¥
alpha[
¥
Omega]$
.
for each $h>0$ and
to requiring that
,
Our first result is an existence theorem for (IVP) when $A$ is in
and is analogous to Theorem 2 of Li [15].
Theorem 4. Suppose that $D$ is a locally closed, bounded subset of $E$ , $A$ is
, and $z¥in D$.
in
Then (IVP) has a local solution (to the right).
$D$
Moreover, if
is closed, each noncontinuable solution to (IVP) is defined on
[0, ).
For the proof of Theorem 4 we let
be a decreasing sequence of positive
, ,
, and let
numbers
and
be as in Lemma 1. Also,
$m_{+}$
$m_{¥_}$
$¥Omega¥subset D$
$r geq 0$
$m_{-}[ alpha,
Omega, A] leq m_{+}[ alpha,
Omega, A]$
$m_{¥pm}[¥alpha, ¥Omega, A+aI]=m_{¥pm}[¥alpha, ¥Omega, ¥mathrm{A}]+¥mathrm{R}¥mathrm{e}$ $(a)¥alpha[¥Omega]$
$a$
$¥alpha-D_{+}^{u}(D)$
$D;(¥mathrm{b})$
$¥omega$
$¥Omega¥subset D$
$m_{¥_}[¥alpha, ¥Omega, A]¥leq¥omega¥alpha[¥Omega]$
$¥mathcal{T}[A]$
$¥omega$
$h¥rightarrow 0-$
$m_{¥_}[¥alpha, ¥Omega, A]$
$¥Omega¥subset D$
$¥alpha-D_{+}^{u}(D)$
$¥alpha-D_{+}^{u}(D)$
$¥infty$
$¥{¥epsilon_{n}¥}_{1}^{¥infty}$
$¥mathrm{w}¥mathrm{i}¥mathrm{h}$
$¥lim¥epsilon_{n}=0$
$M_{z}$
$R_{z}$
$T_{z}$
$S_{z}$
$ n¥rightarrow¥infty$
for each positive integer
$n$
, let
$¥phi_{n}$
be an
$¥epsilon_{¥mathrm{n}}$
-approximate solution to (IVP)
Approximati0i? and Existence
203
of Solutions
. To establish
and
of Lemma 1 with
has a uniformly
existence of a solution on $[0, T_{z}]$ , we need only show that
is
Since the sequence
convergent subsequence by [17, Proposition 2].
1,
Theofrom
the
will
follow
Ascoli’
by
result
of
Lemma
equicontinuous
(ii)
rem if it is shown that $¥alpha[¥{¥phi_{n}(t):n¥geq 1¥}]=0$ for each $t¥in[0, T_{z}]$ . To establish
this fact, we need the following lemma (see also [15, Lemma 3. 3]):
Lemma 6. With the suppositions of Theorem 4 and the notations of the
is
above paragraph, $fet$ $p(t)=¥alpha[¥{¥phi_{n}(t):n¥geq 1¥}]$ for each $t¥in[0, T_{z}]$ . Then
$qf$
$D_{
¥
_}p(t)$
$[0,
T_{z}]$
lower
derivative
the
Dini
denotes
, and if
continuous on
left
$t
¥
in(0,
T_{z})$
$D_{
¥
_}p(t)
¥
leq
¥
mathcal{T}[A]p(t)$
.
each
at , then
for
Proof. Using (v) of Lemma 3 and (ii) of Lemma 1 it is easy to see that
satisfying
$(¥mathrm{i})-(¥mathrm{v})$
$¥{t_{i}¥}_{0}^{¥infty}=¥{t^{n}i¥}_{0}^{¥infty}$
$¥epsilon=¥epsilon_{n}$
$¥{¥phi_{n}¥}_{1}^{¥infty}$
$¥{¥phi_{n}¥}_{1}^{¥infty}$
$¥mathrm{s}$
$p$
$p$
$t$
$|p(t)-p(s)|¥leq¥alpha[¥{¥phi_{n}(t)-¥phi_{n}(s):n¥geq 1¥}]¥leq M_{z}|t-s|$
is continuous. For notational convenience, if
for , $s¥in[0, T_{z}]$ , and hence
$(0, T_{z})$ and
where
is the (unique) member
is a positive integer, let
of
such that
). It then follows from (iii) and (iv) of Lemma
for all but a countable number
and
1 that
of in $(0, T_{z})$ . Note also that $p(t)=¥alpha[¥{¥phi_{n}(t):n¥geq N¥}]$ for each positive integer
¥
¥
¥
¥
¥
¥
¥
$N$ and, since
, it is also immediate that
$p(t)=¥alpha[¥{¥phi_{n}(¥tau_{n}(t)):n¥geq N¥}]$ for each positive integer N.
Thus, if $N$ is a positive
integer, $t¥in(0, T_{z})$ , and $h¥in(0, t)$ , then
$ t¥in$
$p$
$t$
$¥tau_{n}(t)=t^{n}k$
$n$
$t¥in[t^{n}i,$
$¥{t^{n}i¥}_{0}^{¥infty}$
$t^{n}k$
$t^{n}i+1$
$|¥phi_{n}^{¥prime}(t)-A¥phi_{n}(¥tau_{n}(t))|¥leq¥epsilon_{n}$
$¥phi_{n}(¥tau_{n}(t))¥in S_{z}$
$t$
$| phi_{n}(t)- phi_{n}( tau_{n}(t))| leq M_{z}|t- tau_{n}(t)| leq M_{z} epsilon_{n}$
$p(t-h)=¥alpha[¥{¥phi_{n}(¥tau_{n}(t-h)):n¥geq N¥}]$
$¥geq¥alpha[¥{¥phi_{n}(¥tau_{n}(t))-hA¥phi_{n}(¥tau_{n}(t)):n¥geq N¥}]$
$-¥alpha[¥{¥phi_{n}(¥tau_{n}(t-h))-¥phi_{n}(¥tau_{n}(t))+hA¥phi_{n}(¥tau_{n}(t)):n¥geq N¥}]$
$¥geq¥alpha[¥{x-hAx¥}¥Omega_{N}]-2¥sup_{n¥geq N}¥{|¥phi_{n}(¥tau_{n}(t-h))-¥phi_{n}(¥tau_{n}(t))+hA¥phi_{n}(-¥tau_{n}(t))|¥}$
where
$¥Omega_{N}=$
$¥{¥phi_{n}(¥tau_{n}(t)):n¥geq N¥}$
$¥epsilon(N, h)=¥sup$
.
However, if
{ $|Ax-Ay|:x$ ,
$y¥in S_{z}$
and
$|x-y|¥leq(¥epsilon_{N}+h)M_{z}$
},
, by the uniform continuity of A.
and
as
then
, we have that if $s¥in[t-h,
by (ii) of Lemma 1 and the definition of
$n¥geq N$,
$¥epsilon(N, h)¥rightarrow 0$
$h¥rightarrow 0+$
Also,
and
$ N¥rightarrow¥infty$
$¥tau_{n}$
t]$
$|¥phi_{n}(¥tau_{n}(s))-¥phi_{n}(¥tau_{n}(t))|¥leq|¥tau_{l},(s)-¥tau_{n}(t)|M_{z}¥leq(s-t+¥epsilon_{n})M_{z}¥leq(h+¥epsilon_{N})M_{z}$
and hence
,
$|A¥phi_{n}(¥tau_{n}(s))-A¥phi_{n}(¥tau_{n}(t))|¥leq¥epsilon(N, h)$
.
Consequently, if
$n¥geq N$
then
$|¥phi_{n}(¥tau_{n}(t-h))-¥phi_{n}(¥tau_{n}(t))+hA¥phi_{n}(¥tau_{n}(t))|$
$=|-¥int_{t-h}^{t}[¥phi_{n}^{¥prime}(s)-A¥phi_{n}(¥tau_{n}^{¥prime}(s))]ds+¥int_{t-h}^{t}[A¥phi_{n}(¥tau_{n}(t))-A¥phi_{n}(¥tau_{n}(s))]ds|$
$¥leq h¥epsilon_{n}+h¥epsilon(N, h)¥leq h(¥epsilon_{N}+¥epsilon(N, h))$
.
,
R. H. MARTIN, JR.
204
Thus, for each
$h¥in(0, t)$
and each positive integer
$N$
,
$(-h)^{-1}(p(t-h)-p(t))¥leq(-h)^{-1}(¥alpha[¥{x-hAx¥}¥Omega_{N}]-¥alpha[¥Omega_{N}])$
$+2(¥epsilon_{N}+¥epsilon(N, h))$
Letting
.
we see that
$h¥rightarrow 0+$
$D_{¥_}p(t)¥leq m_{¥_}[¥alpha, ¥Omega_{N}, A]+2¥epsilon_{N}+2¥epsilon(N, 0)$
$¥leq T[A]¥alpha[¥Omega_{N}]+2¥epsilon_{N}+2¥epsilon(N, 0)$
$=¥mathcal{T}[A]p(t)+2¥epsilon_{N}+2¥epsilon(N, 0)$
.
, the
as
Since this is true for each positive integer $N$ and
assertions of Lemma 6 follow.
Proof of Theorem 4. Solving the differential inequality in Lemma 6,
we have that
$¥epsilon_{N}+¥epsilon(N, 0)¥rightarrow 0$
$ N¥rightarrow¥infty$
,
$¥alpha[¥{¥phi_{n}(t):n¥geq 1¥}]=p(t)¥leq.p(0)e^{¥gamma[A]t}=0$
since $p(0)=¥alpha[¥{z¥}]=0$ . As indicated in the paragraph following the statement
of Theorem 4, this shows the existence of a solution to (IVP) on $[0, T_{z}]$ . Thus
local existence of solutions is established. Now suppose that $D$ is closed and
is a noncontinuable solution to (IVP) defined on [0, $T$ ). If $ M=¥sup$ $¥{|Ax|:x¥in D¥}$
{ $u(t)|¥leq M$ for all $t¥in[O,$ $T)$
then $ M<¥infty$ since $A¥in¥alpha-D_{+}^{u}(D)$ , and hence
and it follows that $|u(t)-u(s)|¥leq M|t-s|$ for all , $s¥in[0,$ $T$ ). If $ T<¥infty$
exists and is in $D$ since $D$ is closed. However this is impossible since is noncontinuable and we conclude that $ T=¥infty$ . This completes the proof of Theorem 4.
and $r>0$ , $S(¥Omega, r)=¥{x¥in E:d(x, ¥Omega)<r¥}$ . Now let $B$ be a function
If
from $D$ into the set of subsets of $D$. Then $B$ is said to be upper semicontinuous
such that $Bx¥subset S$
, there is a
if for each $z¥in D$ and
(Bz, ) whenever $x¥in D$ and $|x-z|¥leq¥delta$ . Also, if $C$ is a function from $D$ into
from $D$ into the set of subsets of $D$
the set of subsets of $D$, then define
.
by ¥
¥
¥
Now suppose that $D$ is a closed bounded subset of $E$ and let $A$ be in
denote the set of all noncontinuable solutions
. For each $z¥in D$ let
is
is nonempty and each member of
to (IVP) such that $u(0)=z$ (note
$W_{A}(t)$
define the mapping
defined on [0, ) by Theorem 4.) For each
$D$
$D$
by
into the set of all subsets of
from
$u$
$|u^{¥prime}(t)|=|$
$t$
$¥mathrm{t}¥mathrm{h}¥mathrm{e}¥lim_{t¥rightarrow T-}u(t)$
$u$
$¥Omega¥subset E$
$¥delta=¥delta(z, ¥epsilon)>0$
$¥epsilon>0$
$(¥mathrm{u}.¥mathrm{s}.¥mathrm{c}.)$
$ ¥epsilon$
$B¥cdot C$
$B cdot Cz= bigcup_{x in Cz}Bx$
$¥alpha-D_{+}^{n}(D)$
$G_{z}$
$G_{z}$
$G_{z}$
$u$
$t¥geq 0$
$¥infty$
(4)
$W_{A}(t)z=¥{u(t):u¥in G_{z}¥}$
(i)
(ii)
$z¥in D$
.
is a closed bounded subset of
$W_{A}(t)$ is defined by (4) for each
. Then
$W_{A}(0)z=¥{z¥}$ for each $z¥in D$ ;
$W_{A}(t+s)z=W_{A}(t)¥cdot W_{A}(s)z$ for each ,
and $z¥in D$ ;
Theorem
, and
$¥alpha-D_{+}^{u}(D)$
for each
Suppose that
$D$
$t¥geq 0$
$t$
$s¥geq 0$
$E$
,
$A$
is in
Approximation and Existence
(iii)
and
of Solutions
205
$¥sup¥{d(x;W_{A}(s)z):x¥in W_{A}(t)z¥}¥leq|t-s|¥sup$ $¥{|Ax|:x¥in D¥}$
$z¥in D$
;
(iv) if
$¥Omega¥subset D$
then
$¥alpha[¥bigcup_{z¥in¥Omega}W_{A}(t)z]¥leq¥alpha[¥Omega]e^{¥gamma[A]t}$
for each
$t¥geq 0$
for each ,
$t$
$s¥geq(1$
;
, then
is a compact subset of $D$ and
is a compact
subset of $D$ and, in particular, $W_{A}(t)z$ is compact for each $z¥in D$ ; and
.
(vi) $W_{A}(t)$ is upper semicontinuous for each
Part (i) of Theorem 5 is trivial and part (ii) follows routinely since
is autonomous. Part (iii) is also easy, for if $M=¥sup¥{|Ax|:x¥in D¥}$ , $z¥in D$, and
$u¥in G_{z}$ then
$|u(t)-u(s)|¥leq M|t-s|$ for all ,
. For the proof of (iv) we
following
use the
lemma:
,
Lemma 7. Suppose that
some index set, is a family of noncontinuable solutions to (IVP) and let
.
for each
Then
is continuous,
has both a right and left derivative, and
, $A$ ] where
and
for each $t>0$ .
$|u_{
¥
lambda}(t)-u_{
¥
lambda}(s)|
¥leq M|t-s|$ for
$M=
¥
sup
¥
{|Ax|:x
¥
in
D
¥
}$
Troof. Let
and note that
all ,
and
. It follows easily that $|p(t)-p(s)|¥leq 2M|t-s|$ for ,
,
and so
, $t>0$ , and $h¥neq 0$ , then
is continuous. If
(v) if
$t¥geq 0$
$¥Omega$
$¥bigcup_{z¥in¥Omega}W_{A}(t)z$
$t¥geq 0$
$(¥mathrm{I}¥mathrm{V}¥mathrm{P})¥}$
$t$
$¥{u_{¥lambda}:
$s¥geq 0$
$¥Lambda$
¥lambda¥in¥Lambda¥}$
$p(t)=¥alpha[¥{u_{¥mathit{1}}(t):¥lambda¥in¥Lambda¥}]$
$p$
$p_{¥_}^{¥prime}(t)=$
$p$
$p_{+}^{¥prime}(t)=m_{+}(¥alpha,$
$m_{¥_}[¥alpha, ¥Omega_{t}, A]$
$s¥geq 0$
$t$
$t¥geq 0$
$¥Omega_{t}=¥{u_{¥lambda}(t):¥lambda¥in¥Lambda¥}$
$¥Omega_{t}$
$¥lambda¥in¥Lambda$
$t$
$s¥geq 0$
$¥lambda¥in¥Lambda$
$p$
$|h^{-1}|¥cdot|u_{¥lambda}(t+h)-u_{¥lambda}(t)-hAu_{¥mathit{1}}(t)|¥leq|h^{-1}|¥int_{t}^{t+|h|}|u^{¥prime}r(s)-Au_{¥lambda}(t)|ds$
$=|h^{-1}|¥int_{t}^{t+|h|}|Au_{¥lambda}(s)-Au_{¥lambda}(t)|ds$
$¥leq¥sup$
By the uniform continuity of $A$ , if
as
. Hence, if $t>0h¥neq 0$ ,
$¥epsilon(h)¥rightarrow 0$
$¥{|Ax-Ay|:|x-y|¥leq|h|M¥}$
$¥epsilon(h)=¥sup¥{|Ax-Ay|:|x-y|¥leq|h|M¥}$ ,
then
$h¥rightarrow 0$
$|h^{-1}(p(t+h)-p(t))-h^{-1}(¥alpha[¥{x+hAx¥}¥Omega_{t}]-¥alpha[¥Omega_{t}])|$
$=|h^{-1}|¥cdot|¥alpha[¥{u_{¥lambda}(t+h):¥lambda¥in¥Lambda¥}]-¥alpha[¥{u_{¥lambda}(t)+hAu_{2}(t):¥lambda¥in¥Lambda¥}]|$
$¥leq|h^{-1}|¥cdot¥alpha[¥{u_{¥mathit{1}}(t+h)-u_{¥lambda}(t)-hAu_{¥lambda}(t):¥lambda¥in¥Lambda¥}]$
$¥leq|h^{-1}|¥cdot¥sup$ $¥{|u_{¥lambda}(t+h)-u_{¥lambda}(t)-hAu_{¥lambda}(t)|:¥lambda¥in¥Lambda¥}2$
$¥leqq ¥mathrm{e}(h)2$
Since
$¥epsilon(h)¥rightarrow 0$
$h¥rightarrow 0+$
.
as
$h¥rightarrow 0$
.
, the assertions of this lemma follow by letting
The proof of part (iv) of Theorem 5 now follows readily.
let
, and set
for each
. Note that
$¥Lambda=¥bigcup_{z¥in¥Omega}G_{z}$
Thus, if
$ u_{¥lambda}=¥lambda$
$p(t)=¥alpha[¥{u_{¥lambda}(t):¥lambda¥in¥Lambda¥}]$
For let
and
$¥Omega¥subset D$
$¥lambda¥in¥Lambda$
$¥{u_{¥lambda}(t):¥lambda¥in¥Lambda¥}=¥bigcup_{z¥in¥Omega}W_{A}(t)z$
and
$¥Omega_{t}=¥{u_{¥lambda}(t):¥lambda¥in¥Lambda¥}$
for each
.
$p_{-}^{¥prime}(t)=m_{-}[¥alpha, ¥Omega_{t}, A]¥leq¥gamma[A]¥alpha[¥Omega_{t}]=¥gamma[A]p(t)$
Consequently,
$h¥rightarrow 0-$
$t¥geq 0$
, then
,
.
R. H. MARTIN, JR.
206
,
$¥alpha[¥cup W_{A}(t)z]=p(t)¥leq p(0)e^{¥gamma[A]t}=¥alpha[¥Omega]e^{¥gamma[A]t}$
$ z¥in¥Omega$
and (iv) is established.
Parts (v) and (vi) of Theorem 5 follow in a straightforward manner from
the following observation:
is a family of noncontinuable solutions
Lemma 8. Suppose that
is compact. Let $T>0$ and let $C([0, T], E)$
to (IVP) such that
denote the space of continuous functions $u:[0, T]¥rightarrow E$ with $|u|=¥max$ $¥{|u(t)|:t¥in$
$[0, T]¥}$ .
, then
If is the restriction to $[0, T]$ $o/$ for each
$M$
is the closure in
is a relatively compact subset of $C([0, T], E)$ . Moreover, if
$v
¥
in
M$
, then
is a solution to (IVP) on $[0, T]$
$C([0, T], E)$ of
and
and $v(t)¥in¥cup W_{A}(t)z$ for each $t¥in[0, T]$ .
$¥{u_{¥lambda} :
¥lambda¥in¥Lambda¥}$
$¥Omega=¥{u_{¥lambda}(0):¥lambda¥in¥Lambda¥}$
$¥lambda¥in¥Lambda$
$u_{¥lambda}$
$v_{¥lambda}$
$¥{v_{¥lambda}:
¥lambda¥in¥Lambda¥}$
$¥{v_{¥lambda}:
¥lambda¥in¥Lambda¥}$
$v$
$ z¥in¥Omega$
Indication of proof. We already have that the family
for each
equicontinuous. If $t¥in[0, T]$ then
$v_{¥lambda}(t)¥in¥bigcup_{z¥in¥Omega}W_{A}(t)z$
$¥{v_{¥mathit{1}} :
$¥lambda¥in¥Lambda$
is
and it
¥lambda¥in¥Lambda¥}$
follows from part (iv) of Theorem 5 that
$¥alpha[¥{u_{¥lambda}(t):¥lambda¥in¥Lambda¥}]¥leq¥alpha[¥bigcup_{z¥in¥Omega}W_{A}(t)z]¥leq¥alpha(¥Omega)e^{¥gamma[A]t}=0$
mpact for each $t¥in[0, T]$ and it follows from
is relatively
Thus
is relatively compact in $C([0, T], E)$ . The final
Ascoli’s Theorem that
assertion is easy since the uniform limit of solutions to (IVP) on $[0, T]$ is also
a solution on $[0, T]$ .
We have now indicated the proof of Theorem 5. Note that Theorem 5 asserts
are “generators” of a certain type of “multivalued”
that members of
semigroups on $D$ which satisfy an exponential growth in terms of the measure
of noncompactness. We have the following type of converse to Theorem 5.
Theorem 6. Suppose that $D$ is a closed bounded subset of $E$ and $A$ is
$¥{v_{1}(t):¥lambda¥in¥Lambda¥}$
$¥mathrm{c}¥mathrm{o}$
$¥{v_{¥lambda} :
¥lambda¥in¥Lambda¥}$
$¥alpha-D_{+}^{u}(D)$
$a$
into $E$ which satisfies each of the following properties:
$A$ is uniformly continuous and bounded on $D$ ;
has a solution
For each in
defined on [0, ); and
and ¥ , then ¥ ¥
such that if
there is a real number
function from
(a)
(b)
(c)
$D$
(5)
if
Proof.
and
and
$¥gamma[A]¥leq¥omega$
.
$ alpha[ {u_{z}(t)$
:
Moreover,
$m_{+}[¥alpha, ¥Omega, A]¥leq¥omega¥alpha[¥Omega]$
for
by Theorem 1.
$p(t)=¥alpha[¥{u_{z}(t):z¥in¥Omega¥}]$ , then
$A$
$t geq 0$
.
$A¥in¥alpha-D_{+}^{u}(D)$
$¥Omega¥subset D$
$¥infty$
$v_{z}$
$¥Omega¥subset D$
$¥omega$
$z¥in¥Omega¥}]¥leq¥alpha[¥Omega]e^{¥omega t}$
Then
$E(¥mathrm{I}¥mathrm{V}¥mathrm{P})$
$z$
is in
$A_{+}(D)$
each
$¥Omega¥subset D$
.
Also, using Lemma 7, we see that
$m_{+}[¥alpha, ¥Omega, A]=¥lim_{h¥rightarrow 0+}h^{-1}(p(h)-p(0))$
$¥leq¥lim_{h¥rightarrow 0+}h^{-1}(¥alpha[¥Omega]e^{¥omega h}-¥alpha[¥Omega])$
of Solutions
Approximation and Existence
.
$=¥omega¥alpha[¥Omega]$
Thus (5) holds and $A¥in¥alpha-D_{+}^{u}(D)$ with
Using Theorem 6, it follows that if
¥
¥
¥ ¥
¥
¥
and
$¥gamma[A]¥leq¥omega$
$m_{+}[ alpha,
$A$
.
and
are in
$B$
$¥alpha-D_{+}^{u}(D)$
for each
$m_{+}[¥alpha, ¥Omega, B]¥leq¥gamma[B]¥alpha[¥Omega]$
Omega, A] leq gamma[A] alpha[ Omega]$
207
$¥Omega¥subset D$
.
then
Thus,
$m_{¥_}[¥alpha, ¥Omega, A+B]¥leq m_{+}[¥alpha, ¥Omega, A+B]$
$¥leq m_{+}$
[ , J2, $A$ ] $+m_{+}[¥alpha, ¥Omega, B]$
$¥alpha$
$¥leq(¥mathcal{T}[A]+¥gamma[B])¥alpha[¥Omega]$
Since
$A+B¥in A_{+}(D)$
$¥alpha-D_{+}^{u}(D)$
in
.
by Theorem 3, we have the following algebraic result for
:
Theorem 7. Suppose that $D$ is a closed bounded subset
, and $r¥geq 0$ . Then
(i) $rA¥in¥alpha-D_{+}^{u}(D)$ and $¥gamma[rA]=r¥gamma[A]$ ; and
(ii) $A+B¥in¥alpha-D_{+}^{u}(D)$ and $¥gamma[A+B]¥leq ¥mathcal{T}[A]+¥mathit{7}[B]$ .
of ,
$E$
$A$
and $B$ are
$¥alpha-D_{+}^{u}(D)$
Nonlinear Operator Equations.
In this section we show how the results of §3 can be applied to nonlinear
operator equations and fixed point problems. We begin with the following fixed
point theorem for multivalued mappings. If
is a subset of $E$ , let clco(i?)
denote the closed convex hull of .
Theorem 8. Suppose that $D$ is a closed, bounded, convex subset of $E$ and
$B$ is an upper semicontinuous mapping
from $D$ into the set of subsets of $D$ such
that $Bx$ is relatively compact for each in $D$ and there is a positive number $L<1$
. Then there is a point $z¥in D$ such
such that
for each
§4.
$¥Omega$
$¥Omega$
$x$
$¥Omega¥subset E$
$¥alpha[¥bigcup_{x¥in¥Omega}Bx]¥leq L¥alpha[¥Omega]$
.
then
is compact.
Also, if
The proof is essentially the same as the single valued case, by using the
multivalued extension of Brouwer’s Theorem given by Kakutani [12], and we
only indicate it here. Using the condition involving a we first reduce the theorem
let
. Define ¥
and,
to the case that $D$ is compact. If
that
$¥mathrm{z}¥in ¥mathrm{c}¥mathrm{l}¥mathrm{c}¥mathrm{o}(¥mathrm{B}¥mathrm{z})$
$¥Lambda=$
$¥Omega¥subset D$
inductively, define
of Lemma 3,
$¥Omega_{n+1}=¥mathrm{c}1¥mathrm{c}¥mathrm{o}(¥mathrm{B}(¥Omega_{n}))$
$¥alpha[¥Omega_{n+1}]=¥alpha$
Thus
that
$¥alpha[¥Omega_{n}]¥leq L^{n}¥alpha[¥Omega_{0}]$
$¥Omega_{¥infty}=¥bigcap_{h=0}^{¥infty}¥Omega_{n}$
and
$¥Lambda$
$¥{z¥in D:¥mathrm{z}¥in ¥mathrm{c}¥mathrm{l}¥mathrm{c}¥mathrm{o}(¥mathrm{B}¥mathrm{z})¥}$
[clco(B
for $n=0,0,1$ ,
$(¥Omega_{n}))$
$¥Omega_{n+1}¥subset¥Omega_{n}$
]
for
$ Omega_{0}=D$
$B(¥Omega)=¥bigcup_{x¥in¥Omega}Bx$
$¥cdots$
.
Then, by (ii) and (vi)
$=¥alpha[B(¥Omega_{n})]¥leq L¥alpha[¥Omega_{n}]$
$n=0,1$ ,
$¥cdots$
.
.
By Lemma 4, we have
is nonempty, compact and also convex.
Moreover, by the
construction, it follows that
for each
. Thus, replacing $D$ by
, it follows
we may assume that $D$ is compact and convex. Since $B$ is
$C$
$x
¥
in
D$
Using
compactness
,
that if Cx
.
for each
then
is .
the
of
$Bx¥subset¥Omega_{¥infty}$
$x¥in¥Omega_{¥infty}$
$¥Omega_{¥infty}$
$¥mathrm{u}.¥mathrm{s}.¥mathrm{c}.$
$=¥mathrm{c}¥mathrm{l}¥mathrm{c}¥mathrm{o}(¥mathrm{B}¥mathrm{x})$
$¥mathrm{u}$
$¥mathrm{s}.¥mathrm{c}$
208
R. H. MARTIN, JR.
, it is enough to show that for each
and the fact that $C$ is
there
is a
. Now let
. Then there is a subset
such that
, ,
and a continuous function
such that
$|g(x)-x|¥leq¥epsilon$ for $x¥in D$ (see,
p.
, [20, Lemma 1,
,
93]). Set
and define
$D$
$¥epsilon>0$
$¥mathrm{u}.¥mathrm{s}.¥mathrm{c}.$
$¥acute{¥{}x_{¥mathrm{I}}$
$¥cdots$
$¥epsilon>0$
$ d(z_{¥epsilon}; Cz_{¥epsilon})¥leq¥epsilon$
$¥dot{z}_{¥epsilon}¥in D$
$x_{n}¥}¥subset D$
$g:¥mathrm{D}¥rightarrow ¥mathrm{c}¥mathrm{l}¥mathrm{c}¥mathrm{o}$
$(¥{x_{1^{ }},¥cdots, x_{¥iota},¥})$
$D_{¥epsilon}=¥mathrm{c}1¥mathrm{c}¥mathrm{o}(¥{¥mathrm{x}_{1},$
$¥mathrm{e}.¥mathrm{g}.$
$¥cdots$
$x_{n}¥})$
$C_{¥epsilon}x=$
clco
$(¥bigcup_{y¥in Bx}¥{g(y)¥})$
,
it now follows that
. mapping of
is an
into the set of compact,
convex subsets of ; so by Kakutani’s theorem [12], there is a
such
. It now follows that
that
and we have that if $¥Lambda=¥{z¥in D$ :
is nonempty. It is easy to see that
is also compact and
clco(Bz) , then
the indication of the proof of Theorem 8 is complete.
We now prove our fundamental result in this section.
Theorem 9. Suppose that $D$ is a closed, bounded and convex subset of $E$
and $A$ is in ¥
with $¥gamma[A]<0$ .
Then $Z=¥{z¥in D:Az=¥theta¥}$ is nonempty
and compact.
Proof. Using the uniform continuity and boundedness of $A$ , for each
there is a
such that if
is a solution to (IVP) then
$C_{¥epsilon}$
$D_{¥epsilon}$
$¥mathrm{u}.¥mathrm{s}.¥mathrm{c}$
$z_{¥epsilon}¥in D_{¥epsilon}¥subset D$
$D_{¥epsilon}$
$ d(z_{¥epsilon} ; Cz_{¥epsilon})¥leq¥epsilon$
$z_{¥epsilon}¥in C_{¥epsilon}z_{¥epsilon}$
$ z¥in$
$¥Lambda$
$¥}$
$¥Lambda$
$ alpha-D_{+}^{u}(D)$
$¥epsilon>0$
$¥delta(¥epsilon)>0$
$u$
$|¥frac{u(h)-u(0)}{h}-Au(0)|¥leq¥epsilon$
for
$h¥in(0, ¥delta(¥epsilon))]$
.
Let $W_{A}$ be defined by (4). By (iv), (v) and (vi) of Theorem 5, Theorem 8, and
the fact that
for $t>0$ , we have that for each $t>0$ there is a $z(t)¥in D$
such that $z(t)¥in clco(W_{A}(t)z(t))$ . Let
and let
be as above. Since
), it follows that there are noncontinuable solutions
clco(W
to (IVP) and positive numbers
such that $u_{i}(0)=z(¥delta)$ ,
, and
$e^{¥gamma[A]t}<1$
$¥epsilon>0$
$ z(¥delta)¥epsilon$
$¥delta=¥delta(¥epsilon)$
$A(¥delta)z(¥delta)$
$¥{u_{1^{ }},¥cdots, u_{n}¥}$
$¥{¥beta_{1^{ }},¥cdots, ¥beta_{n}¥}$
$n$
$l¥sum_{=1}¥beta_{i}=1$
$|z(¥delta)-¥sum_{t=1}^{n}¥beta_{i}u_{i}(¥delta)|¥leq¥delta_{¥epsilon}$
.
Thus we have that
$|Az(¥delta)|=|¥sum_{t=1}^{n}¥beta_{i}Az(¥delta)|$
$¥leq|¥sum_{t=1}^{n}¥beta_{i}¥{¥delta^{-1}[u_{i}(¥delta)-u_{i}(0)]-Az(¥delta)¥}|+|¥sum_{¥iota=1}^{n}¥beta_{i}¥delta^{-1}[u_{i}(¥delta)-u_{i}(0)]$
$¥leq¥sum_{¥iota=1}^{n}¥beta_{i}¥epsilon+¥delta^{-1}|¥sum_{¥iota=1}^{n}¥beta_{i}[u_{i}(¥delta)-z(¥delta)]|$
$=¥epsilon+¥delta^{-1}|¥sum_{¥iota=1}^{n}¥beta_{i}u_{i}(¥delta)-z(¥delta)|¥leq 2¥epsilon$
.
Approximation and Existence
Consequently, for each positive integer
Since $¥alpha[¥{Az_{n}: n¥geq 1¥}]=0$ we have that
$¥alpha[¥{z_{n}:
n¥geq 1¥}]=¥alpha[¥{z_{n}-Az_{n}:
$n$
209
of Solutions
there is a
$z_{n}¥in D$
such that
n¥geq 1¥}]¥geq(1-¥gamma[A])¥alpha[¥{z_{n}:
$|Az_{n}|¥leq n^{-1}$
n¥geq 1¥}]$
.
.
Since $¥gamma[A]<0$ it follows that $¥alpha[¥{z_{n}: n¥geq 1¥}]=0$ ; so (by relabeling if necessary)
Thus $ Az=¥lim Az_{n}=¥theta$ and $Z$ is nonempty. $Z$
we may assume that $¥lim z_{n}=z$ .
is closed since $A$ is continuous and $Z$ is relatively compact since
$ n¥rightarrow¥infty$
$ n¥rightarrow¥infty$
,
$¥alpha[Z]=¥alpha[¥{z-Az:z¥in Z¥}]¥geq(1-¥gamma[A])¥alpha[Z]$
which implies that $¥alpha[Z]=0$ . This completes the proof of Theorem 9.
We now indicate how Theorem 9 may be used to obtain information on the
, and also to obtain existence criteria for fixed
resolvent of members of
$¥alpha-D_{+}^{u}(D)$
points.
Theorem 10.
is a closed, bounded and convex subset of $E$
. Then for each $h>0$ such that $h¥gamma[A]<1$ , the range of
and $A$ is in ¥
the function $I-hA$ :
(where $(I-hA)x=x-hAx$) contains $D$.
Then $ B¥in$
Proof. Let $w¥in D$ and define $Bx=-x+w$ for each $x¥in D$.
Suppose that
$D$
$ alpha-D_{+}^{u}(D)$
$D¥rightarrow E$
$¥alpha-D_{+}^{u}(D)$
with
$¥gamma[B]=-1$
.
Thus, by Theorem 7,
$B+hA¥in¥alpha-D_{+}^{u}(D)$
$¥gamma[B+hA]¥leq¥gamma[B]+h¥gamma[A]=-1+h¥mathcal{T}[A]<0$
with
.
By Theorem 9 there is a $z¥in D$ such that $-z+w+hAz=¥theta$ . It follows that $w$ is
in the range of $I-hA$ and the proof is complete.
Theorem 11. Suppose that $D$ is a closed, bounded and convex subset of $E$
and $B$ is a function from $D$ into $E$ satisfying each of the following:
(a) $B$ is uniformly continuous and bounded on $D$ ;
¥
¥
¥
;
(b) there is a number $L<1$ such that ¥ ¥ ¥
for each
and
(c) $¥lim¥inf d(z+h(Bz-z);D)/h=0$ for each $z¥in D$.
$m_{ _}[ alpha,
Omega, B] leq L alpha[ Omega]$
$¥Omega¥subset D$
$h¥rightarrow 0+$
, then
is nonempty and compact.
, then (b) is
Remark 3. Note that if $¥alpha[B(¥Omega)]¥leq L¥alpha[¥Omega]$ for each
$m_{
¥
_}[
¥
alpha,
¥
Omega,
A]
¥
leq
¥
alpha[A(
¥
Omega)]$
. Note also that (c) is fulfilled if $B$ maps
fulfilled since
$D$ into $D$ or, more generally, if $B$ maps
into $D$. For if is in the interior
, then
of $D$ then (c) trivially holds, and if
If
$¥Lambda=¥{z¥in D:Bz=z¥}$
$¥Lambda$
$¥Omega¥subset D$
$¥partial D$
$z$
$z¥in¥partial D$
$z+h(Bz-z)=(1-h)z+hBz¥in D$
whenever $h¥in(0,1)$ , since $D$ is convex.
Proof of Theorem 11. Define $Ax=Bx-x$ for all $x¥in D$. Then $Bz=z$ if and
only if $ Az=¥theta$ . Assumptions (a) and (c) imply that $A¥in A_{+}(D)$ and assumption
(b) along with part (vi) of Lemma 5 implies that
R. H. MARTIN, JR.
210
$m_{¥_}[¥alpha, ¥Omega, A]¥leq(L-1)¥alpha[¥Omega]$
Thus $A¥in¥alpha-D_{+}^{u}(D)$ with
from Theorem 9.
$¥gamma[A]¥leq(L-1)<0$ ,
for each
$¥Omega¥subset D$
.
and this theorem follows directly
References
[1] A. Ambrosetti, Un teorema di esistenza per le equazioni differenziali negli spazi
di Banach, Rend. Sem. Mat. Univ. Padova, 39 (1967), 349-361.
[2] J. M. Bony, Principe du maximum, inegalite de Harnack et unicite du problemes
de Cauchy pour les operateurs elliptiques degeneres, Ann. Inst. Fourier Grenoble,
19 (1969), 277-304.
[3] H. Brezis, On a characterization of flow-invariant sets, Comm. Pure App. Math.,
23 (1970), 261-263.
[4] F. E. Browder, Nonlinear equations of evolution, Ann. of Math., 80 (1964), 485[5]
[6]
[7]
[8]
[9]
[10]
[4]
[12]
[13]
[14]
[15]
[4]
[4]
[18]
[19]
[20]
[21]
523.
A. Cellina, On the existence of solutions of ordinary differential equations in
Banach spaces, Funkcialaj Ekvacioj, 14 (1971), 129-136.
A. Cellina, On the local existence of solutions of ordinary differential equations,
Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys., 20 (1972), 293-296.
C. Corduneanu, Equazioni differenziali negli spazi di Banach Teoremi di esistenza
e prolungabilitz, Rend. Ace. Naz. Lincei, 23 (1957), 226-230.
M. G. Crandall, Differential equations on convex sets, J. Math. Soc. Japan, 22
(1970), 443-455.
M. G. Crandall, A generalization of Peano’s Theorem and flow invariance, Proc.
Amer. Math. Soc., 36 (1972), 151-155.
P. Hartman, Generalized Lyapunov functions and functional equations, Annali
de Mat. Pura Appl., 69 (1965), 305-320.
P. Hartman, On invariant sets and on a theorem of Wazewski, Proc. Amer.
Math. Soc., 32 (1972), 511-520.
S. Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Math. J.,
8 (1941), 457-459.
K. Kuratowski, “Topology”, Vol. I, Academic Press, New York, 1966.
A. Lasota and J. A. Yorke, The generic property of existence of solutions of
differential equations in Banach spaces, J. Diff. Eqns., 13 (1973), 1-12.
T. Y. Li, Existence of solutions for ordinary differential equations in Banach
spaces, (to appear).
R. H. Martin, Jr., Lyapunov functions and autonomous differential equations in
a Banach space, Math. Systems Theory, 7 (1973) 66-72.
R. H. Martin, Jr., Differential equations on closed subsets of a Banach space,
Trans. Amer. Math. Soc., 179 (1973), 399-414.
H. Murakami, On nonlinear ordinary and evolution equations, Funkcialaj Ekvacioj,
9 (1966), 151-162.
M. Nagumo, Uber die Laga der Integralkurven gewohnlicher Differentialgl.eichungen, Proc. Phys.-Math. Soc. Japan, 24 (1942), 551-559.
L. Nirenberg, ”Functional Analysis”, Courant Institute of Mathematical Sciences,
New York University, New York, 1961.
R. M. Redheffer, The theorems of Bony and Brezis on flow invariant sets, Amer.
Approximation and Existence
of Solutions
211
Math. Monthly, 79 (1972), 740-747.
[22] S. Szufla, Some remarks on ordinary differential equations in Banach spaces,
Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys., 16 (1968), 795-800.
[23] J. A. Yorke, Invariance for ordinary differential equations, Math. Systems,
Theory, 1 (1967), 353-372.
[24] J. A. Yorke, Differential inequalities and non-Lipschitz scalar functions, Math.
Systems Theory, 4 (1970), 140-153.
(Ricevita la 9-an de aprilo, 1973)