Publicacions
Ma emá iques,
Vol
37
(1993),
209-223
.
A
bs ac
ON
THE
PERTURBATION
PROPAGATION
IN
THE
INITIAL-BOUNDARY
VALUE
PROBLEM
FOR
QUASILINEAR
FIRST
ORDER
EQUATIONS
in
he
domain
wi h
he
condi ions
Yu
.
G
.
RYKOV
The
pape
deals
wi h
ini ial-bounda y
alue
p oblem
o
gene -
alized
solu ions
o single
quasilinea
nonau onomous
conse a ion
law
.
Fo
he
case
so-called
"p ocesses
wi h
agg a a ion"
he
local-
iza ion
p ope y
and
inne
boundedness
a e
s udied
.
Also
in
case
when
bounda y
unc ion
ends
o
ze o
as
=>
+oo
he
localiza ion
e ec
is
ega ded
.
1
.
In oduc ion
This
pape
s udies
gene alized
solu ions
o
he
equa ions
in
he
o m
(1
.1)

Lu
-
u
+
[A( ,
x,
u)],,
+
B( ,
x,
u)
=
H( , x)
Q={( ,x)
: e(0,T),
0<T<+oc,
xER
+
}
u(0, x)
=
0,

u( ,
0)
=
ul
( )
.
such
ha
He e

A( ,
x,
u)

and

B( ,
x,
u)

a e

con inuous

unc ions
A( ,
x,
0)

=

B( ,
x, 0)

=

0
;
B( ,
x,
u)
is
mono onically
inc easing
in
u
;
A( ,
x,
u)
is
con inuosly
di e en iable
wi h
espec
o u,
x
;
A
u

>
0
;
A( ,
0,
u)

#
0
;
A
x
( ,
x,
u)
+
B( ,
x, u)

>_
0
;
H( ,
x)
is
a
measu able
unc ion
bounded
o
bounded
;
ul
E Cl
([0,
T»,
u,
>
0
.
The
de ini ion
o
gene alized
solu ion
and
p oo s
o
he
exis en e
and
uniqueness
heo ems
can
be
ound
in
[3], [4], [7],
[8]
o
[10]
.
210

Y
.
G
.
RYKOV
In Sec ion
2
he
de ini ion o
gene alized
solu ion
and
compa ison
he-
o em
a e
gi en
.
In Sec ion
3
we
deal
wi h
he
case
when
he e
exis s
T<
+oo
such
ha
ui
(T
-
0)
=
+oo
.
Acco ding
o
he
e minology
o
[1],
[5]
i
co esponds
o
he
so-called
"p ocesses
wi h
agg a a ion"
.
De ini ion
1
.1
.
One
says
ha
localiza ion
in
he
p oblem
(1
.1),
(1
.2)
occu s
i
he e
exis s
X
>
0
such
ha
u( ,
x)
-
0
o
x
>_
X,
0
<_
<_
T
.
One
says
ha
localiza ion
does
no
occu
i
o
e e y
su icien ly
la ge
x,
k
>
0
he e
exis s
*
>
0
such ha
u(
*
,
x,,)
:,~
0
.
In
he
pape
[1]
au onomous
equa ions
wi h
powe
nonlinea i ies
and
ze o
lowe
o de
e m
we e
s udied
.
The e
necessa y
and
su icien
cón-
di ions
o
he
occu ence
o
localiza ion
and
o
inne
boundedness
o
solu ions
we e ob ained
.
In Sec ion
3
we
shall
s udy
such
ques ions
o
a bi a y
nonlinea i ies
and
in
he
nonau onomous
case
.
Sec ion
4
is
de o ed
o
localiza ion
in
he
case
when
ui
( )
is
de ined
o
e e y
E
[0,
+oo)
and
may
end
o ze o
as
=>
+oo
.
Some
supplemen a y
esul s
on
he
localiza ion
a e
gi en
in
Sec ion
5
o
he
equa ion
(1
.3)

u
+
(T
-
)p(u'''),,
+
(T
-
)qu
,
=
0,

( ,
x)
E
(0,
T)
x
R
+
.
The e
a e
ce ain
peculia i ies
o he
on
beha io
in
his
case
.
2
.
The
de ini ion
o
gene alized
solu ion
.
A
compa ison
heo em
Now,
le
us in oduce
he
no ion o
gene alized
solu ion
.
De ini ion
2
.1
.
A
measu able
unc ion
u( ,
x)
bounded
o
bounded
is
called
a
gene alized
solu ion
(abb e ia ion
:
g
.s .)
o
he
p oblem
(1
.1),
(1
.2)
in
Q
i
:
1)
o
e e y
w
( ,
x)
>
0,
w
E
Có
(Q) he
inequali y
A
{lu( ,
x)
-
sjw
+
sign(u( ,
x)
-
s)
[A( ,
x,
u( ,
x))
-
A( ,
x,
s)]w
x
-
-
sign(u( ,
x)
-
s)
[A
x
( ,
x, s)
+
B( ,
x,
u( ,
x))
-
H( , x)]w}
d
dx
>
0
holds,
whe e
s
=
cons
is
a bi a y
;
2)
he e
exis s
a
se
El
C
[0,
T],
mes
El
=
0,
such
ha
o
E
[0,
T)
E,u( ,
x)
is
de ined
o
almos
e e y
x
E
l[8
+
and
o
e e y
R>
0
R
lim

ju( ,
x)
1
dx
=
0
;
0
E
[0,T)
El
PERTURBATIONS
FOR
QUASILINEAR
EQUATIONS

211
3)
he e
exis s
a
se
E2
C
[0,
+oo),
mes
E2
=
0,
such
ha
o
x
E
[0,
+oo) E2u( ,
x)
is
de ined
o
almos
e e y
E
[0,
T)
and
o
e e y
T
i
,
0<T,
<T,
and
he
s abili y
condi ion
l
limo o

I
u( ,
x)
-
uI
( )
I
d
=
0
.
c
xE
[O,+oo)
E2
Rema k
2
.1
.
I
u( ,
x)
is
a
piecewise
con inuous
g
.s
.
o
he
p oblem
(1
.1),
(1
.2)
hen
De ini ion
2
.1
implies
(see
[6])
a he
line
o
discon inui y
x
=
y( )
o
u( ,
x)
he
Hugonio
condi ion
(2
.1)

y
=
[A( , y( ),
u+
)
-
A( ,
y( ),
u-)]/(u+
-
u-)
(2
.2)

sign(u+
-u
-
)
[A( , y( ),
pu
-
+
(1
-
p)u+)-
-
pA( ,
y( ),
u)
-
(1-
p)A( ,
y( ),
u+)1
>-
0
o
e e y
,
E
(0,1)
;
he e
u-
=
u( ,
y
-
0),
u+
=
u( ,
y
+
0)
.
The
exis ence
o
g
.s
.
e
he
p oblem
(1
.1),
(1
.2)
unde
a ious
es ic-
ions
en
bounda y
condi ions
and
ini ial
da a
was
p o ed,
o ins ance,
in
[3],
[4],
[10]
.
Theo em

2
.1
.
Suppose
h( ,
x),
g( ,
x)

a e

measu able
unc ions
bounded
o
_<
Ti,
whe e
TI
<
T
is
a bi a y
.

Suppose
w( ,
x)
is
a
g
.s
.

o
he
equa ion
Lw
=
h( ,
x) in
Q
wi h da a w(0,
x)
=
0,
w( ,
0)
=
wl( )
E
L-
([0,T)),
and
( ,x)
is
a g
.s
.

o he
equa ion
L
=
g( ,x)
in
Q
wi h da a (0,x)
=
0,
( ,0)
=
i
( )

E
LOO
([0,
T))
.

Suppose
wl
( ) <_
i
( )
almos
e e ywhe e
in
[0,
T) and
h( ,
x)
<_ g( ,
x)
almos
e e ywhe e
in
Q
.
Then
w( ,
x)
<
( ,
x)
almos
e e ywhe e
in
Q
.
Fo
he
p oo
o his
heo em
simila
me hods
o hose o
pape s
[2],
[10]
a e
used
.
The
uniqueness
o
he
g
.s
.
o
p oblem
(1
.1),
(1
.2)
ollows
om
Theo em 2
.1
.
One
deno es
below
by
u( ,
x)
he
g
.s
.
o
he
p oblem
(1
.1),
(1
.2)
wi h
II
( ,
x)
-
0
.
3
.
P ocess
wi h
agg a a ion
(The
case
T
<
+oo)
Theo em
3
.1
.
Suppose
he
ollowing condi ions
hold
1)
A( ,
x,
)/
<
A( , x,
w)/w,
0
<
<
w,
w
E
I[8
+
;
212

Y
.
C
.
RYKOV
2)
A( ,
x,
)/
<_
ao(T
-
)a( ),
E
1[8
+
,
aE
C'
(R+
) 1
C(R
+
),
ao
E
C([0, T)),
ao
>
0,
a(0)
=
0,
a
is
inc easing
;
3)
u,
( )
<
W(1/(T
-
)),
cp
E
C([1/T,+oo)),
W(11T)
=
0,
cp
is
in~
c easing
;
4)
o
ao(s)a
o
W(1/s)
ds
<
+oo
.
Then
localiza ion
in
he
p oblem
(1
.1),
(1
.2)
occu s
and
u( ,
x)
=
0 o
x
>
T
ao(s)a
o
cp(1/s)
ds
.
P oo
..
Suppose
he
line
x
=
y( )
is
de ined
by
he
equa ions
A( ,
x,
ul ( ))/u
l ( ),

i
ul
( )
:?É0,
y( )
_
{
Au( ,x,0),

i
u,
( )
=0
;
wi h
he
ini ial
da um
y(0)
=
0
.

Le
us
se
Al( ,x)
=
u,
( )
o
0
<_
x
<
y( )
and
Al
( ,
x)
=
0
o
x
>
y( )
.

I
is
easy
o
see
ha
LA,
>_
0
when
x
:,A
y( )
and
a
he
line
o
discon inui y
x
=
y( )
(2
.1),
(2
.2)
hold
.
Flz he ,
hence
y
<-
ao(T
-
)a °
W(
1
/(T
-
)),

y(
0
)
=
0,
T
y
( )
<

ao(s)a
o
cp(1/s)
ds
.
T-
Wi h
he aid
o
assump ion
4),
he
applica ion
o
Theo em
2
.1
gi es
he
equi ed
esul
.
Rema k
3
.1
.
Suppose
(1
.1)
has he
o m
.(3
.1)

u
+A
l
(T
-
)P(u'') x
=
0,
whe e Al
=
cons
>
0,
p E
R,
m
>
1
and
ul
( )
=
(T- )`-T',
ca
>
0
.
Then
Theo em
3
.1
asse s
he
p esence
o
localiza ion
when
p
-
a(m
-
1)
>
-1
.
Theo em
3
.2
.
Suppose
he
ollowing
condi ions
hold
1)
A( ,
x,
)/
<_
A( ,
x,
w)/w, 0
<
<_
w,
w
E
R
+
;
2)

A,
;
( ,
x,
)
+B
( ,
x,
)
>
bo(T
-
)
,
E
l[8
+
,
bo
E
C([0,T)),
bo
>_
0
;
3)
A

( ,x, )
<
ao(T
-
)a( ),
ao
E
C([O,T),
ao
>
0,
a(0)
=
0,
a
E
C'(R+
)
n
C(R+
),
a
inc eases,
4)
a(a3)
<
x(a)a(a),
a
E
[0,1],
a
ER+, x E
C,
x(O)
=
0,
x
in-
c eases
;
5) ul( )
<
~o(1/(T
-
)),
cp
E
C([1/T,+oo)),
w(11T)
=
0,
cp
in-
c eases
;
PERTURBATIONS
FOR
QUASILINEAR
EQUATIONS

213
6)
o
g(s)
ds

<

+oo,
a( )
o
'( )
g(s)
ds

<

C

<

+oo,

E

1[8
+
,
C
=
cons
>
0,
g(s)
=
ao(s)X
(exp
(- Tbo(o,)do,»
,
,
(s)
,
p(s)
exp
(
~s
bo(a)
da)
1
WI( )=
1/ i
1(w)
.
Then
localiza ion
in he
p oblem
(1
.1),
(1
.2)
occu s
.
P oo
..
Le
us
conside
he
unc ion
T
w0
( ,
x)
-
( ,
x)
exp
(-

bo(s)
ds)
,
T-
whe e
( ,
x)
is
de ined
by
he
ela ion
T-
(3
.2)

0
=
x
+
a( )

g(s)
ds
=-
x
+
G( ,
)
.
wl
( )
The
equa ion
G( ,
)
=
0
wi h
espec
o
has
wo
oo s
:
=
0,
=
l(1/(T
-
))
.
When
x
a ies
he
solu ion
o (3
.2)
may
s op
o exis
i
G

( ,
)
=
0
.
Consequen ly
he
se
o
( ,
x)
whe e
he
solu ion
o
(3
.2)
does
no
exis
can
be
desc ibed
by
he
sys em
(3
.3)

x
+
G( ,
)
=
0,

G
( ,
)
=
0
.
Now,
le
us
conside
he
unc ion
y( )
de ined
in
he
ollowing
way
y
=
A
( ,
y,
wo( ,
y))/wo( ,
y),
y(0)
=
0
.
Then
y
:~
A ( ,
y,
wo)
:
:~
ao(T
-
)a(wo)
<
g(T
-
)a( )
.
F om
he
sys em
(3
.3)
o
i s
solu ion
x
=
z( )
one
has
:
z
=
-
G
-
G
=
-
G
=
g(T
-
)a( ),
so
y
<
z
and
lines
x
=
y( )
and x
=
z( )
do
no
in e sec
.
Suppose
A2( ,x)
=
wo( ,x)
o
x
<
y( )
and
2( ,x)
=
0
o
x
>
y( )
.
I is
easy
o
see
ha
T
exp
~-

bo(s)
ds
i
Lwo
>
[a( )g(T
-
)
-
a(wo)ao(T
-
)]I
G
.
T-
Hence
wi h
he
aid
o
assump ion
4)
and
G
>_
0 o
x
<
z( )
one
ob ains
L, 2
>_
0
o
x
<
y( )
.
Besides,
a
he
line
x
=
y( )
(2
.1),
(2
.2)
hold
.
Since
u(0, x)
<
A2
(0,
x)
we
ha e
u( ,
x)
<
A2
( ,
x)
in
Q
.

214

Y
.
G
.
RYKOV
Le
us
ew i e
(3
.2)
:
Hence
¡
T-

¡
wl
( )
x
+
a( )

0

0
J

g(s)
ds
-
a( )
J

g(s)
ds
=
0
.
by
i ue
o
assump ion
6)
.
When
x
is
su icien ly
la ge
he e
is
no
solu ion
o (3
.2)
and
z(T
-
0)
<
+oo
.
This ends
he
p oo
.
Co olla y
3
.
l
.
I
in
addi ion
o
assump ions
o
Theo em
3
.2
he
ol-
lowing
inequali y
holds
o
ixed
x
7~
0 and
=>
T
-
0
.
o
T-
0
x
+
a( )

g(s)
ds
<_
C=
cons
wl( )
a( )

g(s)
ds
<
l( ),

E
R
+
,
0
whe e
77( )
dec eases,
l(+oo)
=
0 hen
u( ,
x)
is
bounded
as
=>
T-
0
o e e y
ixed
x
:y~
0
.
P oo
.
Indeed,
om
(3
.2)
we
ha e
¡
wl
( )

¡
T-
l( )
>
a( )
J

g(s)
ds
=
x
+
a( )
J

g(s)
ds
>
x,
0

o
o
<
l
-1
(x)
.
Since
u( ,
x)
<
A2( ,
x) one
ge s
he
boundedness
u( ,
x)
Rema k
3
.2
.
Fo he
equa ion
(3
.1)
Theo em
3
.2
gi es
he
localiza-
ion
p esence
when
p
-
a(m
-
1)
>_
-1,
while Co olla y
3
.1
gi es
he
boundedness
o
g
.s
.
o
x
7~
0
and
=~>
T
-
0
when
p
-
a(m
-
1)
>
-1
.
Theo em
3
.3
.
Suppose
he
ollowing
condi ions
hold
:
1)
A( ,
x,
)/
<
A( ,
x,
w)/w,
0
<
<_
w,
w
E
R
+
;
2)
A,,
( ,
x,
)

>_

ao(T
-
)a( ),

E

l[8
+
,
ao

E

C([O,
T)),
a
0

>_
0,
a(0)
=
0,
a
E
C
l
(I[8
+
)
n
C(&
+
),
a
inc eases
;
3) 5
2ao
(T
-
)a( )
_<
A( ,
x,
)/
<_
b
l
a
o
(T
-
)a( ),
E
I[8
+
,
0
<
62
<5
1
<1
;
4)
px(a)a(a)
?
a(a l)
>_
X(n)a(0),
w
>_
1,
61p
<
1,
aE
[0,1],
a
E
R+,
X
E
C([0,1]),
X(0)
=
0,
X
inc eases
;
5)
B( ,x, )+A
.,( ,x,
)
<_
b
o
(T- ) ,
E
I[8
+
,
b o
E
C([O,T]),
b
o
>_
0
;
6) u,
( )
>
cp(1/(T
-
)),
cp
E
C([1/T,+oo)),
w(11T)
=
0,
cp
in-
c eases
PERTURBATIONS
FOR
QUASILINEAR
EQUATIONS

215
7)
g(s)s
<
01(s)
o g(o
,
)
da,
0
<
s
<
T
;
cp'(s)s
>
02(s)
;p(s),
s
>
1/T
;
sa'(s)
>
03(s)a(s),
s
>
0;'04(s)
=
bo(1/s)/s
+
02(s),
whe e
o
¡
(s)
(i
=
1,
2,
3)
a e
mono onic
(in
pa icula
may
be
cons an e)
and
1
-
Y'1
0
wl( )/[03( )04(1/wl( ))]
>
M( ),
E
R
+
,
pE
C(R
+
),
>
0,
l
#
0,
M,
does
no
inc ease
;
8
)
o
H(s)ds=+oo,
H(s)-ao(s)a(exp(
T
bo(a)da)
-1
( o
g(a)lo))
V(V)
-
m( )
o
1( )
g(s) ds, e
=
cons
>
0,
o
g(s)
ds
<
+oo
.
Then
he e
is
no
localiza ion
in
he
p oblem
(1
.1),
(1
.2)
and
u( ,
x)
>
0
o
0
<
x
<
52
T
H(a)
da
.
P oo
.
Le
us
conside
he
unc ion
wo( ,
x)
in oduced
in
he
p oo
o
Theo em
3
.2
.
Suppose
y( )
is
de ined
by
he
equa ion
y
=
A( ,
y,
wo( , y))/wo( ,
y)
wi h
he
ini ial
da um
y(0)
=
0
.
By
analogy
wi h
he
p oo
o
Theo em
3
.2
one
s a es
ha
he
cu e
x
=
y( )
is
con ained
in
he
domain
o
exis en e
o
he
solu ion
o
equa ion
(3
.2)
.
Le
us ega d
he
same
compa ison
unc ion
2( ,x)
as
in
he
p oo
o
Theo em
3
.2
.
As
G,>
0
o
x
<
z( )
one has
Lwo
<
0 o
x
<
y( )
and
U
( ,
x)
>
>12
( ,
x) in
Q
.
Now
he
equa ion
G( ,
)
=
0
has
wo
oo s
and
he
oo
o
he
equa-
ion
G
=
0
lies
be ween
hem
by
i ue
o
Rolle's
Theo em
.
Conse-
quen ly
he
solu ion
>
0 o
he
equa ion
(3
.2)
wi h
ixed
x,
always
exceeds
he
solu ion
o
he
equa ion
G
=
0
wi h
he
same
ixed
.
Hence
T-
wi( )
0
=G

=á
( )
(1

g(s) ds
-

g(s)
ds)
+
0

0
o
+
a( )g
o
wl( )( 1
1)
I
( )wl( )
2
,
T-

wi
( )
g(s)
ds
=
g(s)
ds
-
a( )
g °
wl( )( i
1y(V)W1(V)2
.
o

o

a
'( )
Using
condi ions
7)
one
es ima es
:
T-
g(s)
ds
>
wi
( )
9(s)
ds
[
1
-
a'( )
i
o (
1)
( )
1

wl( )]
,
T
s i(s)
=
exp
(£

bo(o
,
)
do
,
)
[s
-1
bo( 1
/s)cp(s)
+
sw
(s)]
?
VI(S)04(8)
;
lis
l
T-

wl( )

a( )

P1
ow,
( )
g(s)
ds
>
~

g(s)
ds
[
1
-
a(V)
04(
1
/W1( ))]
wi( )

'~~

wi( )

g(s)
ds
~1
-

Y51
0
wl( )

g
(s)
dsp( )= ( )
.
o

03( )V)4(
1
/wl( ))
o
21
6

Y
.
G
.
RYKOV
Since
p(s)
and
w1(s)
do
no
inc ease (s)
does
no
inc ease
oo
.
Consequen ly
( ,
x)
>
-1
( o
-
g(s)
ds)
,
hence
T

T-
wo( , x)
>
exp
~-

bo(s)
ds

-1

g(s)
ds

-
H1
(T
-
)
.
T-

)

(
0
Fu he ,
y
=
A( ,
y,
w
0
)/w
0
?
62ao(T
-
)a(wo)
T

I'-
>_
S2ao(T- )a
(exp
(~

bo(s)
d)
s
-1
(0
g(s)
ds

=6
2
H(T- )
.
T-
This
inequali y
implies
y( )
>
52
T
H(u)
_

du and we
ob ain he
e-
qui ed
esul
wi h
he aid
o
assump ion
8)
.
Co olla y
3
.2
.
Suppose
condi ions
1)-7) o
he
Theo em
3
.3
hold,
bu
ins ead
o
condi ion
8)
assume
lim
H
1
(T
-
)
=
+oo
.
Suppose
u( ,
xo)
>
=>T
0
o
some
xo and
close
o
T
.
Then
u( ,
x0)
unbounded
as
=>
T
-
0
.
P oo
..
In he
p oo
o
Theo em
3
.3
we had
he es ima e
w
0
( ,
x)
>_
H
1
(T
-
)
.
Since
u( , x)
>_
wo( ,
x)
o
x
<
y( ),
he
asse ion
o he
co olla y
is
ue
.
Rema k
3
.3
.
Fo
he
equa ion
(3
.1)
Theo em 3
.3
asse s
he
local-
iza ion
absence
when
p-
ca(m
-
1)
<
-1,
p
>
-1
.

Indeed,
in his
case ao(s)
=
A1sP,
bo(s)

0,
X(s)
=
s
m-1
,
a(s)
=
msm
-1
,
g(s)
_
A
l
sP,
01
(s)

p
+
1,
02
(s)

a,
Y'4(8)
=
02(s),
03
(s)
=
m
-
1,
P(s)
1
-
(p
+
1)/(a(m
-
1)),
(s)
=
A1
[a(m
-
1)
-
p
-
1]
(s
+
T-«)-(P+1)/«,
a(m
-
1)
(p
+
1)
H(s)
=
mA1sP
I
s-«
(

a(m
-
1)

)
-

-
T-«
a(m-1)-p-1
I
ollows
om
Co olla y
3
.2
ha
u( ,
x)
is
unbounded
as
=>
T
-
0
and
x
ixed,
since
-«

n
(
m
-
1)

1
-
«/(P+1)
H
l
(S)
-
s

la(m

1)

1
Suppose
p-
a(m
-
1)
=
-1
.
Then
Theo em
3
.3
is
in alid
because
o
assump ion
8)
.
Bu
one
can
choose
02(8)
=
as«/(s«
-T
-
«),
(s)
=
A1(p+
1)-1T-«(s+T-«)-m,
H1(s)
=
T-«[(T/s)«(m-1)/m
-
l]
.
The
unboundedness
o
u( ,
x)
as
=>
T
-
0
and
x
is
no oo
la ge
ollows
om
Co olla y
3
.2
.
PERTURBATIONS
FOR
QUASILINEAR
EQUATIONS

217
Theo em
3
.4
.
Suppose
condi ions 1)-6)
o
Theo em
3
.3
hold
and
o
g(s)
ds
=
+oo
.
Then
he e
is
no
localiza ion
in
he
p oblem
(1
.1),
(1
.2)
and
u
( ,
x)
>
0
o
0
<
x
<
cons
( T
g(s)
ds)62
.
P oo
.
Le
us
conside
he
unc ion
A2( ,
x)
de ined
in
he
p oo
o
Theo em
3
.2
.
We
ha e
y
=
A( ,
y,
w
0
)/w0,
y(0)
=
0
.
The
solu ion
o
his
Cauchy
p oblem
is
no
iden ically
ze o
since
A( ,
0,
wo)
#
0
by
assump ion
.
Hence,
he e
exis
such x*
>
0, *
>
0
ha
y( *)
=
x*
.
F
u he ,
o
>
*
one
ob ains
y
>
62ao(T
-
)a(wo)
>
62g(T
-
)a( )
.
I is
ob ious
ha
wl
( )
<
T
by
he
de ini ion
o
unc ion
w,
( )
.
So
we
ha e om
(3
.2)
Now,
y
=
a( )
-1
g(s)
ds
<
a( )
T

g
(s)
ds
T

T-
7,

1
-
1
y
>
62yg(T
-
)
~

g(s)
ds
J
,
T-
Y( )
>
x*

¡T

g(s)
ds1
-bz
~
J
¡7
.

g(s)
ds
16z
~
J

=>
+oo
T- *

T-
J
as
=>
T
-
0
.
This
ends
he
p oo
.
y( *)
=
Rema k
3
.4
.
In he
case
o equa ion
(3
.1)
Theo em
3
.4
s a es
he
absence
o
localiza ion
o
p
<
-1
.
Theo em
3
.5
.
Suppose
assump ions
1)-6)
o
Theo em
3
.3
hold
and
o
g(s)
ds
=
+oo
.
Suppose
he
ollowing
condi ions
hold
:
1)
,
T
g(T)
dT
<
sg(s)O1(s),
0
<
s
<
T
;
W'(s)s
>
cp(s)02(S),
s
>
1/T
;
sa'(s)
>
a(s)03(s),
s
>
0
;
04(s)
-
bo(1/s)/s+02(s),whe e
o¡
(s)
(i
=
1, 2, 3)
a e
mono onic
unc ions
(in
pa icula
may
be
cons an e)
and
0
1
o
wl( )
+

3( )
4(l/wl( ))
<
M( ),
E
1I8
+
,
M
E
lPlP
C(R
+
),
,
>
0,
l
9~
0,
i
inc eases
;
2)
( )
-
g(wi( ))w,( )p( )
=*
+oo
as
=
:>
+oo
;
3)
H2
(s)
-
exp
(-
s
bo(o-)
do)
-i
( s
g
(u)
do
,
==>
+oo
as
s
+0
.