Publicacions
Ma emá iques,
Vol
34
(1990),
225-239
.
(h
o
,
h)-BOUNDEDNESS
OF
THE
SOLUTIONS
OF
DIFFERENTIAL
SYSTEMS
WITH
IMPULSES
Abs ac
G
.K
.
KULEV
AND
D
.D
.
BAINOV
In
he
p esen
pape
he
ques ion
o
boundedness
o
he
solu ions
o
sys-
ems
o
di e en ial
equa ions
wi h impulses
in
e ms
o
wo measu es
is
conside ed
.
In
he
in es iga ions
piecewise
con inuous
auxilia y
unc ions
a e
used
which
a e
an
analogue
o
he
classical
Lyapuno 's
unc ions
.
The
ideas
o
Lyapuno 's
second
me hod
a e
combined
wi h
he
newes
ideas
o
he heo y
o
s abili y
and
boundedness
o
he
solu ions
o
sys ems
o
di e en ial
equa ions
.
1
.
In oduc ion
Sys ems
o
di e en ial
equa ions
wi h
impulses
ep esen
a
na u al
appa a-
us
o
ma hema ical
simula ion
o
eal
p ocesses
and
phenomena
s udied
in
biology, physics,
con ol
heo y, e c
.
Fo
ins ance,
i
he
popula ion
o a
gi en
species
is
egula ed
by some
impulsi e
ac o s
ac ing
a
ce ain
momen s,
hen
we
ha e
no
easons
o
expec
ha
he
p ocess
will
be
simula ed
by
egula
con-
ol
.
On
he
con a y,
he
solu ions
mus
ha e
jumps
a
hese
momen s
and
he
jumps
a e
gi en
be o ehand
.
Mo eo e ,
he
ma hema ical
heo y
o
he
sys ems
o
di e en ial
equa ions
wi h
impulses
is
much
iche
han
he
espec i e
heo y
o
sys ems wi hou
impulses
.
Tha
is
why
in
he
ecen
yea s
his
heo y
is
an
impo an
ield
o
nume ous
in es iga ions
([1]-[7])
.
The
usage
o
classical
Lyapuno 's
unc ions
in
he
s udy
o
he
s abili y
and
boundedness
o
he
solu ions o
sys ems
o
di e en ial
equa ions
wi h
impulses
ia
Lyapuno 's
second
me hod
cons ic s
he
pliabili y
o he
me hod
.
The
ac
ha
he
solu ions o
such
sys ems
a e
piecewise
con inuous
unc ions
shows
ha
i
is
necessa y
o
in oduce
analogues
o
Lyapuno 's
unc ions
which
ha e
discon inui ies o
he
i s
kind
.
The
in oduc ion
o
such
unc ions
malces
he
applica ion
o
Lyapuno 's
second
me hod
o
sys ems
wi h
impulses
much
mo e
e icien ([1]-[6])
.
In
he
p esen
pape
he
boundedness
o
he
solu ions
o
sys ems
o
di e -
en ial
equa ions
wi h
impulses
in
he
e ms
o
wo
measu es
is
s udied
.
In
he
The
p esen
in es iga ion
is
pa ially
suppo ed
by
he
Minis y
o
Cul u e,
Science
and
Educa ion
o
People's
Republic
o
Bulga ia
unde G an
61
.
22
6

G
.K
.
KULEV,
D
.D
.
BAINOV
in es iga ions
piecewise
con inuous
Lyapuno 's
unc ions
a e
used
which
a e
combined by
he
newes
ideas o
he
heo y
o
s abili y
and
boundedness
o
he
solu ions o
sys ems
o
di e en ial
equa ions
.
The
main
esul s
gene alize
heo ems
o
Yoshizawa
[8]
and
Ha a,
Yoneyama,
Sai oh,
Hi ano
[9]
.
Conside
he
ollowing
sys em
o
di e en ial
equa ions
wi h
impulses
whe e
E
.C[ 8+
x
Rn,
R
n
],
TR
E
C[Rn,
R],
IR
E
C[Rn,
Rn]
and
Ox/ = R(x)
_
x( +)
-
x( _)
.
Le
ú
E
R+
and
xo E
R
n
.
Deno e
by
x(
;
ú,
xo) he
solu ion
o
sys em
(1)
which
sa is ies
he
ini ial
condi ion
x( ó
;
ú,
xo)
=
xo
and by
J
+
( o,
xo)
deno e
he
maximal
in e al
o
he
o m
( o,w)
in
which
he
solu ion
x(
; o,xo)
is
de ined
.
The
solu ions
x( )
=
x(
;
ú,
xo)
o
sys em
(1)
a e
piecewise
con inuous
unc-
ions
wi h
poin s
o
discon inui y
o
he
i s
lcind,
Le
.
a
he
momen
R
when
he
in eg al
cu e
o
he
solu ion
mee s
he hype su ace
he
ollowing
ela ions
hold
2
.
P elimina y
.
no es
and
de ini ions
x
=
( ,x),
:~
TR(x)
;
Ox/ = R(z)
=
IR(x),
aR
=
{( ,
x)
E
R+
X
Rn
:
=
TR(X)}
X
( -)
=
x( R),
Ox/ =
R
=
x( R)
-
x( _
)
=
IR(x( R))
.
Hence o h
we
shall
always
assume
ha
o
all
x
E
Rn
he
ollowing
ela ions
a e
alid
0
<
TI(x)-<
T2
(x)
<
.
. .
<
TR(x)
<
...
and
lim
- R(x)
=
00
R-oo
and
he
in eg al
cu e
o
any
solu ion
x( )
=
x(
;
ú,
xo)
o
sys em
(1)
mee s
each
hype su ace
R
a
mos
once
[7]
.
In
he
u he
conside a ions
we
shall
use he
ollowing
classes
o
unc ions
:
K
=
{u
E
C[R+,
R+]
:
o
is
s ic ly
inc easing
and
u(0)
=
0}
CK
=
.
{u
E
C[ 8+,R
+
] :
Q( ,)
E
K
o
any
E
R
+
}
=
{hEC[R+xR
n
,R+
] :
in

h( ,x)=0
o
any ElR
+
}
DIFFERENTIAL
SYSTEMS
WITH
IMPULSES

227
De ini ion
1
.
Le
h
o
,
h
E
1'
.
We
say
ha
he
solu ions o
sys em
(1)
a e
:
a)
(ho,
h)-equibounded
i
(Va
>
0)(V o
E
R+)(3,a
=
p( o,
a)
>
0)(Vxo
E
Rn,
ho( o,
xo)
<
a)
(V
>
o)
:
h( ,
x(
;
o,
xo))
<
/~
.
b)
(h
o
,
h)-uni o mly
bounded
i
he
numbe
0
o
a)
does
no
depend on
o
E
R+
.
c)
h-ul ima ely
bounded
o
bound
B
i
(V( o,xo)
ER
+
x
R
n
)(
3
T
=T( o,xo)
>
0)(V
>
o
+T)
h( ,
x
(
;
o,
xo
))
<
B
.
d)
(ho,
h)-equi-ul ima ely
bounded
o
bound
B
i
(Va >
0)(V o
E
R+)(
3
T
=
T( o, a)
>
0)(Vxo
ER
n
,
ho( o,
xo)
<
a)
(V
>
o
+
T)
:
h( ,
x(
;
o
,
x
o
))
<
B
.
e)
(h
o
,
h)-uni o mly
ul ima ely
bounded
o
bound
B
i
he
numbe
T
o
d)
does
no
depend
on
o
E
R+
.
De ini ion
2
.
Le he
unc ion
A
:
R
+
--->
R+
be
measu able
.
We
say
ha
A( )
is
in e 
ally
posi i e
i
l
A( ) d
=
co
whene e
I
-
U
[ai,
oi],
al
<
/o
;
<
i=1
al+1
and
Ni-a¡>
_8>0
.
We
shall
in oduce
he
class
Vo
o
pieceiwse
con inuous
auxilia y
unc ions
which
a e
an
analogue
o
Lyapuno 's
unc ions
[3]
.
Le
7-0(x)
=
0
o
xE
Rn
.
Conside
he
se s
GR
=
{( ,
x)
E
IR
+
X
Rn
:
7-
R-1(X)
<
<
TR(x)}
and
De ini ion
3
.
We
say
ha
he
unc ion
V
:
R
+
x
Rn
-+
R
+
belongs
o
he
class
Vo
i
V( ,x)
is
con inuous
in
G,
locally
Lipschi z
con inuous
wi h
espec
o x in
any
o
he
se s
GR
and
o
( o,
xo)
E
Q
'R,
R
=
1,2,
. . .
he e
exis
he
limi s
V( ~
,xo)
=

lim

V( ,x)

,

V( ó
,xo)
=

lim

V( ,x)
( ,x)-( o,xo)

( ,x)-i o,xo)
( ,
x)EGR

( ,x)EGR+1
and,
mo eo e ,
he
equali y
V
( o
,
xo)
=V
( o,
xo)
holds
.
Le
V
E
Vo
.
Fo
( ,
x)
E
G
de ine
he
unc ion
00
G=
UGR
R=1
U(1)( ,
x)
=
lim
Sup
1
[V(
+
h,
x
+
h
( ,
x1)
-
V
( ,
x)]
.
h_o+
h
00
22
8

G
.K
.
KULEV,
D
.D
.
BAINOV
o
:~
R
whe e
R
=
TR(x( R))
.
Le
h,
ho
E
I'
and
V,
W
E
Vo
.

Fo he
sake
o
b e i y
o
he
o mula ion
o
he
main
esul s
we
shall
make
a
lis
o
some
condi ions
o
be
used
in
he
o mula ion
o
he
subsequen
heo ems
.
A
.
I
o
he
solu ion
x(
;
o,
xo) o
sys em
(1)
he e
exis s
bo
>
0
such
ha
h( ,
x(
;
o,
xo))
_<
bo
<
oo o
each
E
T+( o,
xo),
hen
x(
;
o,
xo)
is
de ined
in
he
in e al
( o,
oo)
.
B1
.
The
unc ion
V
is
h- adially
unbounded
.
B2
.
V(1)( ,x)
<
0
o
( ,x) E
G
.
B3
.
V(
1 )
( ,
x)
<
-CV( ,
x)
o
( ,
x)
E
G
whe e
C
>
0
is
a
cons an
.
B4
.

V
(l )( ,
x)
<
-A( )C(h( ,
x)) o
( ,
x)
E
G
whe e
A( )
is
in eg ally
posi i e
and
C
E
K_
B5
.
V(
1
)
( ,
x)
<
-C(W( ,
x))
+
A( )O(V( ,
x)) o
( ,
x)
E
G
whe e
C(y)
is
nonnega i e
and
con inuous
in
R
and
(2)

lim
in
C(y)
>
0
A( )
is
nonnega i e
and
con inuous
in
R
+
and
We
shall
no e
ha
i
x
=
x( )
is
a solu ion o
sys em
(1),
hen
V(
1
)( ,
x( ))
=
D+V( ,
x( ))
=
lim
sup
1
[V(
-}-
h,
x(
+
h))
-
V( ,
x( ))]
n-o+
h
De lni ion
4
.
Le
ho,
h
E
I'
.
The
unc ion
V
E
Vo
is
called
:
a) h- adially
unbounded
i
he e
exis s
a
unc ion
a
EK,
a(y) -> oo
as
y
-->
oo
and
such
ha
V
( +,
x)
>
a(h( ,
x))
o
( ,
x)
E
i8
+
x
Rn
.
b)
ho-dec escen
i
he e
exis
b
>
0
and
a
unc ion
b
E
K
such
ha
V( +,
x)
<_
b(ho( ,x))
o
ho( ,x)
<
b
.
c)
weakly
h
o
-dec escen
i
he e
exis
ó
>
0
and a
unc ion
b
E
CK
such
ha
om
ho( ,
x)
<
b i
ollows ha
V( +,
x)
<
b( ,
ho( ,
x))
.
-y-00
100',
A
( )
d
<
co
O(u)
is
posi i e
and
con inuous
in
U8
and
5)
10,>0
du
O(u)
B6
.
The e
exis s
a
cons an
K
such
ha
V( ,x)>K o any( ,x)ER+xR'
B7
.
V( +,
x
+
IR(x))
<
V( ,
x)
o
( ,
x)
E
UR,
R=
1,
2
. .
.
.
DIFFERENTIAL
SYSTEMS
WITH
IMPULSES

229
C1
.
jWl
l
l( ,x)j
<
p( )w(W( ,x»
o
( ,x)
E
G
whe e
p( )
is
nonnega i e
and
con inuous
in
R+
and
(6)

p(T)
d-
<_
m(
-
s)
o
>_ s
>_
0
9
whe e
m(y)
E
K
and
w(u)
is
posi i e
and
con inuous
in
IFB
and
°°
(7F)

du
=
oo
.
w
(u)
C2
.
W
l
l( ,
x)
<
p( )w(W( ,
x))
o
( ,
x)
E
G
whe e
p( )
and
w(u)
a e
he
unc ions
o
condi ion
C1
.
C3
.
The e
exis s
a
unc ion
m
E
K
such
ha
o
>_
s
>_
0 and
o
any
piecewise
con inuous
in
[s,
]
unc ion
u(T)
wi h
poin s
o
discon inui y
o
he
i s
kind R such
ha
R
=
TR(u( R))
a
which
u(T)
is
con inuous
om
he
le ,
he
ollowing inequali y
holds
(8)

J
W
(1)
(- ,
u(
,
))
d- 1
<
m(
-
s)
.
s
Theo em
1
.
Le¡
condi ion
(A)
hold
and
unc ion
V
E
Vo
exisi
o
which
condi ions
Bl,
B2
and
B7
hold
.
Then
he
solu ions
o
sys em
(1) a e
:
P oo
.
Since
V
is
h- adially
unbounded,
hen
he e
exis s
a
unc ion
aE
K,
a(y)
-->
oo
as
y
-->
oo
and
such
ha
C4
.
W( +,
.x
+
IR(x))
=
W( ,
x)
o
( ,
x)
E
oR
.
C5
.
W( ,x)
is
h- adially
unbounded
.
3
.
Main
esul s
1
.
(ho,
h)-equibounded
i
V
is
weakly
ho-dec escen
.
2
.
(ho,
h)-uni o mly
bounded
i
V
is
h
o
-dec escen
.
V( +
,
x)
>
a(h( ,x))
o
( ,
x)
E
R+
x Rn
1
.
I
V
is
weakly
ho-dec escen ,
hen
he e
exis
óo
>
0
and a
unc ion
b
E
CK
such
ha
(10),

V( +,
x)
<
b( ,
ho( ,
x))
o
ho( ,
x)
<
6o
Le
a
>
0
and
o
E
R
+
(a
<
6
o )
be
gi en
.
Choose
3
=
i(
o
,cY)
>
0
so
ha
(11)

a(0)
>
b( o,a)

230

G
.K
.
KULEV,
D
.D
.
BAINOV
Le
xo
E
R
n
,
ho( o,xo)
<_
a and
le
x( )
=
x(
; o,xo)
.
Se
( )
=
V( ,
x( ))
.
Since
V( ,
x)
is
locally
Lipschi z
con inuous
in
any
o
he
se s
GR,
hen
om
B2
i
ollows ha
D
-1-
( )
<_
0
o
EJ+
( o,
xo),
9~
R
whe e
R
=
TR(x( R))
.
F om
B7
i
ollows ha
( R)
<
( R)
.
Tha
is
why
he
unc-
ion
( )
is
dec easing
in
he
in e al
J+(
o
,xo)
.
Then
om
(9),
(10)
and
(11)
we
ge
a(h( ,
x( ))
<
( +)
<
( )
<
( ó)
<
b(
o
,
h
o
(
o
,
xo))
<
b(
o
,
a)
<
a(0)
o
E
J+( o,
xo)
which
implies
ha
h( ,
x( ))
<
,i
.

F om
condi ion (A)
i
ollows
ha
J+( o,
xo)
=
( o,
oo)
.
Thus
1, is
p o ed
.
2
.
I
V
is
ho-dec escen ,
hen
(10)
and
(11)
hold
o
some
unc ion
b
E
K
independen
o
.
Hence
he
numbe
Q
can
be
chosen
independen
o
o
and
so
ha
o
ho( o,
xo)
<_
a
we
ha e
h( ,
x( »
<0
.
This
shows
ha he
solu ions
o
sys em
(1)
a e (ho,
h)-uni o mly
bounded
.
Theo em
1
is
p o ed
.
Co olla y
1
.
Le
condi ion
(A)
hold
and
unc ion
U
E
Vo
exis
which
is
h-
adially
unboundedand
such
ha
ú
(j) ( ,
x)
_<
A( )O(U( ,
x))
o
( ,
x)
E
G
whe e
¡he
unc ion
A( )
is
nonnega i e
and
con inuous
in
R
+
and
0
00
A( )
d
<
oo
and
O(u)
is
posüi e
and
con inuous
in
H
and
oo
du/«u)
=
oo,
U( +,
x
+
IR(x»
<
U( , x)
o
( ,
x)
E oR,
R
=
1,
2
.
.
. .
Then
¡he
solu ions
o
sys em
(1)
a e
:
1
.
(ho,
h)-equibounded
i
U
is
weakly ho-dec escen
.
2
.
(ho,
h)-uni o mly
bounded
i
U
is
ho-dec escen
.
P ooL
.
I
is
immedia ely
e i ied
ha
he
unc ion
V( ,
x)
=
exp
~-
J
A(s) ds
+
~P(U( ,
x))
}
,
( ,
x)
E
R+x
Rn
,
JJJ
0
whe e
$(u)
=
o
du/O(u)
sa is ies
he
condi ions
o
Theo em
1
.
Theo em
2
.
Le
condi ion
(A)
hold
and
a
unc ion
V
E Vo
exis
which
is
weakly
ho-dec escen
and
o
which
condi ions
Bl,
B3
and
B7
hold
.
Then
he
solu ions
o
sys em
(1)
a e
(h
o ,
h)-equi-ul ima¡
ely
bounded
.
P oo
.
F om
Theo em
1
i
ollows
ha
he
solu ions o
sys em
(1)
a e
(ho,
h)-
equibounded
.
Hence
each
solu ion
x( )
=
x(
; o,xo)
o (1)
is
de ined
in
he
in e al ( o,
oo)
.
DIFFERENTIAL
SYSTEMS
WITH
IMPULSES

23
1
Since
V
is
h- adially
unbounded,
hen
he e
exis
B
> 0
and
a
E K,
a(y)
->
oo
as
y
->
oo such
ha
(12)

V( +,
x)
>
a(h( ,x))
o
h( ,x)
>
B
Since
V
is
weakly
h
o
-dec escen ,
hen
he e
exis
b
o
>
0
and
b
E
CK
such
ha (10)
holds
.
Le
a>
0
and
o
E
R+
be
gi en,
x
o
E
Rn
be such
ha
ho( o,xo)
<_
a
and
le
x( )
=
x(
;
o
,
x
o
)
.
F om
B3
and
B7 we
ob ain
(13)

V( ,x( ))
<
V( ó,x
o
)exp[-C( - o)]
o
>
o
.
Se
T
=
T( o,a)
>
c
ln[b( o,a)/a(B)]
.
Then
om
(12)
and
(13)
i
ollows
ha
o
>
o
+T
he
ollowing
inequali ies
hold
Theo em
3
.
Le
condi ion
(A)
hold
and
a
unc iou
V
E
Vo
exis
which
is
h
o
-dec escen
and
o
which
coudi ions
Bl,
B/,
and
B7
hold
.
Then
he solu ions
o
sys em
(1) a e (ho,
h)-uui o mly
ul ima ely
bounded
.
P oo
.
F om Theo em
1
i
ollows
ha
he
solu ions
o
sys em
(1)
a e
(ho,
h)-
uni o mly
bounded
.
Hence
each
solu ion x( )
=
x( ; o,xo)
o
(1)
is
de ined
in
he
in e al
( o
,
oo)
.
Since
V
is
h- adially
unbounded,
hen
he e
exis
R>
0and
a
EK,
a(y)
-~
oo
as
y
--~
oo
such
ha
(14)

V(
+
,
x)
>
a(h( ,x))
ol
.
la( ,x)
>
R
(15)

V(
+
,
x)
<
b(ho( ,
x))
o
ho( ,
x)
<
6o
.
Choose
B
>_
R
so
ha
a(B)
>
b(R)
.
Le
a
>_
R
be
gi en
.
VVe
shall
p o e
ha
he e
exis s
T=
T(a)
>
0
such
ha
o
any
solu ion
x( )
=
x(
;
ú
,
xo)
o
sys em
(1)
o
which
h
o ( o
,
x
o
)
<
a and
o
some
(E
[
o
,
o
+
T]
he
ollowing
inequali y
holds
(16)
a(h( ,x( ))
<
V( +,x( +))
<
V(T,x( ))
<
< V( ó
,x
o
)exp[-C( - o)]
<
b( o,ho( o,xo))exp(
-CT)<
a(B)
Hence
h( ,x( ))
<
B
o
>
o
+T
.
Theo em
2
is
p o ed
.
Since
V
is
ho-dec escen ,
hen
he e
exis
óo
>
0
and
b
E
K
such
ha
ho«,x«))
<
R
232

G
.K
.
KULEV,
D
.D
.
BAIIVOV
Suppose
ha
his
is
no
ue
.
Then
o
any
T
>
0
he e
exis s
a
solu ion
x( )
=
x(
;
o,
xo
)
o
(1) o
which
h
o
( o,
x
o
)
<
a and
such
ha
o
all
E
[
o
,
o
+
T]
we
ha e
(17)

ho( ,x( ))
>
R
F omB4
and
B7
i
ollows
ha
(18)

V( ,
x( ))
-
V( o
,
xo)
<
~

V(1> (s, x
(s))
ds
<
o
Bu
he
unc ion V( ,
x( »
is
mono onely
dec easing
in
Hence
he e
exis s
he
limi
(19)

lim
V( ,
x( ))
=V
o
>
0
Then
om
(15), (17),
(18)
and
(19)
we
ob ain
F om
he
ha
Then
~~
A( )C(h
o ( ,
x( ))
d
<
b(R)
-
Vo
o
o+T

b(R)
-
V
o
+
1
~( ) d
>

C(R)
0
-

A(s)C(ho(s,
x(s»)
ds,
>
o
o
he
in eg al
( o,
oo)
.
in eg al
posi i i y
o
-A( )
i
ollows ha
he e
exis s
T >
0
such
J
00

o+T
b(R)
-
V
o
>
1,
o
A( )C(ho( ,
x( ))
d
>

A( )C(ho( ,
x( ))
d
>
o
o
+T
>_
C(R)

A( ) d
>
b(R)
-
V
o
+
1
.
o
The
con adic ion
ob ained
shows
ha
he e
exis s
T=
T(n)
>
0
such
ha
o
any
solu ion
x( )
=
x(
; o,xo) o
(1)
o
which
ho( o,xo)
<
a,
he e
exis s
(
E
[ o, o
+
T]
such
ha
(16)
holds
.
Then
o
>_
(
(hence
o
any
>
o
+T
oo)
he
ollowing
inequali ies
hold
a(h( ,x( ))
<
V( +,x( +))
<
V( ,x( ))
<
V«
+
,
x«+»
<
<
b(ho«,x«))
<
b(R)
<
a(B)
.
Hence
he
solu ions o
sys em
(1)
a e
(ho,
h)-uni o mly ul ima ely
bounded
o
bound
B
.
(21)
(22)
(23)
DIFFERENTIAL
SYSTEMS
WITH
IMPULSES

23
3
Theo em
4
.
Le¡
condi ion
(A)
hold
and
unc ions
V,
W
E
Vo
exis
o
which
condi ions
B5,
B6, B7,
Cl,
C!,
and
C5
hold
.
Then
:
1
.
V
is
h- adially
unbounded
.
2
.
The
solu ions
o
sys em
(1) a e
h-ul ima ely
bounded
.
3
.
I
V
is
weakly
ho-dec escen ,
hen
he
solu ions
o
sys em
(1) a e
(ho,
h)-
equibounded
.
4
.
I
V
is
h
o
-dec escen ,
hen
¡he
solu ions
o
sys em
(1) a e
(ho,
h)-uni-
o mly
bounded
.
P oo
.
1
.
Assume
ha
he
asse ion
is
no
ue
.
Then
he e
exis s
N
o
>
0
such
ha
o
any y
>
0
he e
exis
T
E
R+
and xE
Rn
o
which
h(7-,
-
7)
>
y
and
such
ha
V(T+,x)
<
No
.
F om
(2)
i
ollows ha
he e
exis
R
l
>
0and
ó
>
0
such
ha
o
any
y>R
l
weha eC(y)>6
.
Le
L=
'O'>
A( )
d
and
M
=
sup{O(u)
:
K
<
u
<
<D
-
1
(oD(No)
+
L)
whe e
<p
(u)
=
o
du/O(u)
.

-
F om
C5
i
ollows
ha
he e
exis s
a
unc ion
aE
K,
a(y)
->
oo
as
y
-->
oo
and
such
ha
(20)

W(
+
,
x)
>
a(h( ,x))
o
( ,
x)
E
R+
xRn
F om
(4)
and
he
condi ion
a(y)
->
oo
as
y
-->
oo
i
ollows
ha
he e
exis s
R
2
>
R
l
such
ha
a(R2)
>
R
l
and
IPa(R2)

dy

N
o
-
K
+
ML
JR,

w(y)
>
m

6
In
he
abo e assump ion
we
eplace
y by
R2
.
As a
esul
we
ob ain
ha
he e
exis
o
E
IR
+
and
x
o
ERn
such
ha
h(
o
,x
o
)
>
R2
and
V( ó
,xo)
<
No
.
F om
condi ion
(A)
and
om
C5,
Cl and
C4
i
ollows ha
he
solu ion
x( )
=
x(
;
o,
xo)
o
sys em
(1)
is
de ined
in
he
in e al
( o,
oo)
.
F om
B5 and
B7
i
ollows
ha
V i)( ,
x( ))
_<
A( )O(V( ,
x( ))
o
7É
R
whe e
R
=TR(x( R))
and
V( +,x( R))
<
V( R,x( R)), whence
by
in eg a ion
we
ob ain
(
~(V( ,
x( ))
-
~(V( ó
,
xo))
<
~`

Vc~>S,
x
(S))
ds
<L
o
«V(s,x(s»)
Hence
K
<_
V( ,x( ))
_<
4
-1
(-¿(N
o
)
- -
L),
whence we
conclude
ha
g5(V( ,
x( )))
<
M
o
>
o
.
Assume
ha
W( ,
x( ))
>R
l
o
any
>
o
.

Then
om
B5
and
B7
i
ollows
ha
V(i)( ,x( ))
<
-ó
+
MA( )
o
>
o
,
:~
R
V( R,
x( R))
C
V( R,
X( R)),