Theorem 7.
Assume that (
H
1
)–(
H
4
) and (4) hold. Let
:
D
T
≡
{
(
t
,
s
)
:
t
≥
s
≥
t
0
,
t
,
s
∈
[
t
0
,
∞
)
T
}
→
R
be rdcontinuous function, such that
(29)
H
(
t
,
t
)
=
0
,
t
≥
t
0
;
H
(
t
,
s
)
>
0
,
t
>
s
≥
t
0
,
t
,
s
∈
[
t
0
,
∞
)
T
,
and
H
has a nonpositive continuous
Δ
partial derivative
H
Δ
s
(
t
,
s
)
with respect to the second variable and satisfies (31). Let
h
:
D
T
→
R
be a rdcontinuous function and satisfies
(30)

H
Δ
s
(
t
,
s
)
=
h
(
t
,
s
)
(
H
(
t
,
s
)
)
(
γ

1
)
/
γ
,
(
t
,
s
)
∈
D
T
,
(31)
0
<
inf
s
≥
T
0
[
liminf
t
→
∞
H
(
t
,
s
)
H
(
t
,
T
0
)
]
≤
∞
,
T
0
∈
[
t
0
,
∞
)
T
.
If there exist a positive and differentiable function
δ
:
T
→
R
such that
δ
Δ
(
t
)
≥
0
for
t
∈
[
t
0
,
∞
)
T
and a real rdcontinuous function
Ψ
:
[
t
0
,
∞
)
T
→
R
such that
(32)
limsup
t
→
∞
1
H
(
t
,
T
0
)
∫
T
0
t
a
(
τ
(
s
)
)
(
δ
(
s
)
τ
Δ
(
s
)
)
γ

1
G
+
γ
(
t
,
s
)
Δ
s
<
∞
,
(33)
∫
T
0
∞
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
1
/
(
γ

1
)
(
Ψ
+
(
σ
(
s
)
)
δ
(
σ
(
s
)
)
)
γ
/
(
γ

1
)
Δ
s
=
∞
,
(34)
limsup
t
→
∞
1
H
(
t
,
T
)
∫
T
t
[
a
(
τ
(
s
)
)
γ
γ
(
δ
(
s
)
τ
Δ
(
s
)
)
γ

1
L
H
(
t
,
s
)
δ
(
s
)
q
(
s
)
(
1

r
(
τ
(
s
)
)
)
γ

1

a
(
τ
(
s
)
)
γ
γ
(
δ
(
s
)
τ
Δ
(
s
)
)
γ

1
G
+
γ
(
t
,
s
)
]
Δ
s
≥
Ψ
(
T
)
,
where
G
(
t
,
s
)
=
(
δ
Δ
(
s
)

(
p
(
s
)
/
a
(
s
)
)
δ
(
s
)
)
(
H
(
t
,
s
)
)
1
/
γ

δ
(
s
)
h
(
t
,
s
)
,
G
+
(
t
,
s
)
=
max
{
0
,
G
(
t
,
s
)
}
,
Ψ
+
(
t
)
=
max
{
0
,
Ψ
(
t
)
}
, and
T
∈
[
T
0
,
∞
)
T
, then (1) is oscillatory on
[
t
0
,
∞
)
T
.
Proof.
Assume that (1) has a nonoscillatory solution
x
(
t
)
on
[
t
0
,
∞
)
T
. Without loss of generality we may assume that there exists
t
1
∈
[
t
0
,
∞
)
T
, such that
x
(
t
)
>
0
and
x
[
τ
(
t
)
]
>
0
for all
t
∈
[
t
1
,
∞
)
T
. By the definition of
z
(
t
)
, it follows
(35)
x
(
t
)
=
z
(
t
)

r
(
t
)
x
(
τ
(
t
)
)
≥
z
(
t
)

r
(
t
)
z
(
τ
(
t
)
)
≥
(
1

r
(
t
)
)
z
(
t
)
,
t
∈
[
t
1
,
∞
)
T
.
Since it satisfies
lim
t
→
∞
τ
(
t
)
=
∞
, there exists
T
0
∈
[
t
0
,
∞
)
T
such that
τ
(
t
)
≥
t
1
for all
t
∈
[
T
0
,
∞
)
T
. Then if it satisfies
t
∈
[
T
0
,
∞
)
T
, we have
(36)
x
(
τ
(
t
)
)
≥
(
1

r
(
τ
(
t
)
)
)
z
(
τ
(
t
)
)
.
By Lemma 6 and (
H
3
), we obtain that
(37)
1
z
∘
τ
≥
1
z
∘
τ
σ
,
a
(
z
Δ
)
γ

1
≥
a
σ
(
z
Δ
σ
)
γ

1
on
[
T
0
,
∞
)
T
(where
(
z
Δ
)
σ
is short hand for
z
Δ
σ
), and
(38)
z
Δ
∘
τ
≥
(
a
σ
)
1
/
(
γ

1
)
(
a
∘
τ
)
1
/
(
γ

1
)
z
Δ
σ
holds. Moreover, using Lemmas 3 and 6, it follows that
(39)
[
(
z
∘
τ
)
γ

1
]
Δ
=
(
γ

1
)
(
z
∘
τ
)
Δ
×
∫
0
1
[
h
(
z
∘
τ
σ
)
+
(
1

h
)
(
z
∘
τ
)
]
γ

2
d
h
≥
(
γ

1
)
(
z
∘
τ
)
Δ
×
∫
0
1
[
h
(
z
∘
τ
)
+
(
1

h
)
(
z
∘
τ
)
]
γ

2
d
h
=
(
γ

1
)
(
z
∘
τ
)
γ

2
(
z
∘
τ
)
Δ
.
In Lemma 2, let
v
=
τ
,
w
=
z
, and
T
is unbounded above by
(
H
1
)
, so
T
k
=
T
, and
T
~
=
v
(
T
)
=
τ
(
T
)
=
T
by
(
H
3
)
; using Lemma 2, we get
(40)
(
z
∘
τ
)
Δ
=
(
z
Δ
∘
τ
)
τ
Δ
.
Thus
(41)
[
(
z
∘
τ
)
γ

1
]
Δ
≥
(
γ

1
)
(
z
∘
τ
)
γ

2
(
z
Δ
∘
τ
)
τ
Δ
.
By the above inequality and the first inequality in (37), we obtain that
(42)
[
(
z
∘
τ
)
γ

1
]
Δ
(
z
∘
τ
)
γ

1
≥
(
γ

1
)
(
z
Δ
∘
τ
)
τ
Δ
z
∘
τ
σ
holds on
[
T
0
,
∞
)
T
. Now we define the function
W
by
(43)
W
=
δ
a
(
z
Δ
)
γ

1
(
z
∘
τ
)
γ

1
.
Then we have
W
>
0
on
[
T
0
,
∞
)
T
, and
(44)
W
Δ
=
(
10
)
δ
(
z
∘
τ
)
γ

1
[
a
(
z
Δ
)
γ

1
]
Δ
+
a
σ
(
z
Δ
σ
)
γ

1
(
z
∘
τ
)
γ

1
δ
Δ

δ
[
(
z
∘
τ
)
γ

1
]
Δ
(
z
∘
τ
)
γ

1
(
z
∘
τ
σ
)
γ

1
≤
(
1
)
(
H
4
)

L
q
δ
(
x
∘
τ
)
γ

1
(
z
∘
τ
)
γ

1

p
δ
(
z
∘
τ
)
γ

1
(
z
Δ
)
γ

1
+
a
σ
(
z
Δ
σ
)
γ

1
(
z
∘
τ
)
γ

1
δ
Δ

δ
[
(
z
∘
τ
)
γ

1
]
Δ
(
z
∘
τ
)
γ

1
(
z
∘
τ
σ
)
γ

1
≤
(
36
)

L
q
δ
(
1

r
∘
τ
)
γ

1

p
δ
(
z
∘
τ
)
γ

1
(
z
Δ
)
γ

1
+
a
σ
(
z
Δ
σ
)
γ

1
(
z
∘
τ
)
γ

1
δ
Δ

δ
[
(
z
∘
τ
)
γ

1
]
Δ
(
z
∘
τ
)
γ

1
(
z
∘
τ
σ
)
γ

1
≤
(
43
)

L
q
δ
(
1

r
∘
τ
)
γ

1

p
a
W
+
δ
Δ
δ
σ
W
σ

δ
a
σ
(
z
Δ
σ
)
γ

1
[
(
z
∘
τ
)
γ

1
]
Δ
(
z
∘
τ
)
γ

1
(
z
∘
τ
σ
)
γ

1
≤
(
37
)

L
q
δ
(
1

r
∘
τ
)
γ

1

p
δ
a
δ
σ
W
σ
+
δ
Δ
δ
σ
W
σ

δ
a
σ
(
z
Δ
σ
)
γ

1
[
(
z
∘
τ
)
γ

1
]
Δ
(
z
∘
τ
)
γ

1
(
z
∘
τ
σ
)
γ

1
≤
(
42
)

L
q
δ
(
1

r
∘
τ
)
γ

1
+
(
δ
Δ

p
a
δ
)
W
σ
δ
σ

(
γ

1
)
δ
a
σ
(
z
Δ
σ
)
γ

1
(
z
Δ
∘
τ
)
τ
Δ
(
z
∘
τ
σ
)
γ
≤
(
38
)

L
q
δ
(
1

r
∘
τ
)
γ

1
+
(
δ
Δ

p
a
δ
)
W
σ
δ
σ

(
γ

1
)
δ
τ
Δ
(
a
σ
)
γ
/
(
γ

1
)
(
z
Δ
σ
)
γ
(
a
∘
τ
)
1
/
(
γ

1
)
(
z
∘
τ
σ
)
γ
≤
(
43
)

L
q
δ
(
1

r
∘
τ
)
γ

1
+
(
δ
Δ

p
a
δ
)
W
σ
δ
σ

(
γ

1
)
δ
τ
Δ
(
a
∘
τ
)
1
/
(
γ

1
)
(
δ
σ
)
γ
/
(
γ

1
)
(
W
σ
)
γ
/
(
γ

1
)
;
then we obtain
(45)
W
Δ
(
t
)
≤

L
q
(
t
)
δ
(
t
)
(
1

r
(
τ
(
t
)
)
)
γ

1
+
δ
¯
(
t
)
δ
(
σ
(
t
)
)
W
(
σ
(
t
)
)

(
γ

1
)
δ
(
t
)
τ
Δ
(
t
)
(
a
(
τ
(
t
)
)
)
λ

1
(
δ
(
σ
(
t
)
)
)
λ
(
W
(
σ
(
t
)
)
)
λ
on
[
T
0
,
∞
)
T
, where
λ
=
γ
/
(
γ

1
)
,
δ
¯
(
t
)
=
δ
Δ
(
t
)

(
p
(
t
)
/
a
(
t
)
)
δ
(
t
)
. Thus, for every
t
,
T
∈
[
T
0
,
∞
)
T
with
t
≥
T
≥
T
0
, by (13), we get
(46)
∫
T
t
L
H
(
t
,
s
)
δ
(
s
)
q
(
s
)
(
1

r
(
τ
(
s
)
)
)
γ

1
Δ
s
≤
H
(
t
,
T
)
W
(
T
)

∫
T
t
(

H
Δ
s
(
t
,
s
)
)
W
(
σ
(
s
)
)
Δ
s
+
∫
T
t
H
(
t
,
s
)
δ
¯
(
s
)
δ
(
σ
(
s
)
)
W
(
σ
(
s
)
)
Δ
s

∫
T
t
H
(
t
,
s
)
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
(
W
(
σ
(
s
)
)
)
λ
Δ
s
=
(
30
)
H
(
t
,
T
)
W
(
T
)
+
∫
T
t
δ
¯
(
s
)
H
(
t
,
s
)

δ
(
σ
(
s
)
)
h
(
t
,
s
)
H
1
/
λ
(
t
,
s
)
δ
(
σ
(
s
)
)
W
(
σ
(
s
)
)
Δ
s

∫
T
t
H
(
t
,
s
)
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
(
W
(
σ
(
s
)
)
)
λ
Δ
s
≤
H
(
t
,
T
)
W
(
T
)
+
∫
T
t
δ
¯
(
s
)
H
(
λ

1
)
/
λ
(
t
,
s
)

δ
(
s
)
h
(
t
,
s
)
δ
(
σ
(
s
)
)
×
H
1
/
λ
(
t
,
s
)
W
(
σ
(
s
)
)
Δ
s

∫
T
t
H
(
t
,
s
)
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
(
W
(
σ
(
s
)
)
)
λ
Δ
s
≤
H
(
t
,
T
)
W
(
T
)
+
∫
T
t
G
+
(
t
,
s
)
δ
(
σ
(
s
)
)
H
1
/
λ
(
t
,
s
)
W
(
σ
(
s
)
)
Δ
s

∫
T
t
H
(
t
,
s
)
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
(
W
(
σ
(
s
)
)
)
λ
Δ
s
,
where
G
(
t
,
s
)
=
δ
¯
(
s
)
H
(
λ

1
)
/
λ
(
t
,
s
)

δ
(
s
)
h
(
t
,
s
)
=
(
δ
Δ
(
s
)

(
p
(
s
)
/
a
(
s
)
)
δ
(
s
)
)
(
H
(
t
,
s
)
)
1
/
γ

δ
(
s
)
h
(
t
,
s
)
,
G
+
(
t
,
s
)
=
max
{
0
,
G
(
t
,
s
)
}
. So using Lemma 4, let
(47)
X
=
[
H
(
t
,
s
)
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
]
1
/
λ
W
(
σ
(
s
)
)
,
(48)
Y
=
[
G
+
(
t
,
s
)
λ
δ
(
σ
(
s
)
)
(
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
)

1
/
λ
]
1
/
(
λ

1
)
.
Using the inequality (18), we have
(49)
G
+
(
t
,
s
)
δ
(
σ
(
s
)
)
H
1
/
λ
(
t
,
s
)
W
(
σ
(
s
)
)

H
(
t
,
s
)
×
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
(
W
(
σ
(
s
)
)
)
λ
≤
C
(
G
+
(
t
,
s
)
δ
(
σ
(
s
)
)
)
λ
/
(
λ

1
)
(
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
)

1
/
(
λ

1
)
,
where
C
=
(
λ

1
)
λ

λ
/
(
λ

1
)
(
γ

1
)

1
/
(
λ

1
)
=
1
/
γ
γ
. Thus
(50)
G
+
(
t
,
s
)
δ
(
σ
(
s
)
)
H
1
/
λ
(
t
,
s
)
W
(
σ
(
s
)
)

H
(
t
,
s
)
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
(
W
(
σ
(
s
)
)
)
λ
≤
a
(
τ
(
s
)
)
γ
γ
(
δ
(
s
)
τ
Δ
(
s
)
)
γ

1
G
+
γ
(
t
,
s
)
.
From (46) and (50), we obtain
(51)
∫
T
t
[
a
(
τ
(
s
)
)
γ
γ
(
δ
(
s
)
τ
Δ
(
s
)
)
γ

1
L
H
(
t
,
s
)
δ
(
s
)
q
(
s
)
(
1

r
(
τ
(
s
)
)
)
γ

1

a
(
τ
(
s
)
)
γ
γ
(
δ
(
s
)
τ
Δ
(
s
)
)
γ

1
G
+
γ
(
t
,
s
)
]
Δ
s
≤
H
(
t
,
T
)
W
(
T
)
;
that is,
(52)
1
H
(
t
,
T
)
∫
T
t
[
a
(
τ
(
s
)
)
γ
γ
(
δ
(
s
)
τ
Δ
(
s
)
)
γ

1
L
H
(
t
,
s
)
δ
(
s
)
q
(
s
)
(
1

r
(
τ
(
s
)
)
)
γ

1

a
(
τ
(
s
)
)
γ
γ
(
δ
(
s
)
τ
Δ
(
s
)
)
γ

1
G
+
γ
(
t
,
s
)
]
Δ
s
≤
W
(
T
)
.
From condition (34), we have
(53)
Ψ
(
T
)
≤
W
(
T
)
,
T
∈
[
T
0
,
∞
)
T
,
(54)
limsup
t
→
∞
1
H
(
t
,
T
)
∫
T
t
L
H
(
t
,
s
)
δ
(
s
)
q
(
s
)
(
1

r
(
τ
(
s
)
)
)
γ

1
Δ
s
≥
Ψ
(
T
)
.
By (46), we have
(55)
1
H
(
t
,
T
)
∫
T
t
L
H
(
t
,
s
)
δ
(
s
)
q
(
s
)
(
1

r
(
τ
(
s
)
)
)
γ

1
Δ
s
≤
W
(
T
)
+
1
H
(
t
,
T
)
∫
T
t
G
+
(
t
,
s
)
δ
(
σ
(
s
)
)
H
1
/
λ
(
t
,
s
)
W
(
σ
(
s
)
)
Δ
s

1
H
(
t
,
T
)
∫
T
t
H
(
t
,
s
)
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
×
(
W
(
σ
(
s
)
)
)
λ
Δ
s
,
and from the above inequality, let
T
=
T
0
, and denote that
(56)
A
(
t
)
=
1
H
(
t
,
T
0
)
∫
T
0
t
G
+
(
t
,
s
)
δ
(
σ
(
s
)
)
H
1
/
λ
(
t
,
s
)
W
(
σ
(
s
)
)
Δ
s
,
B
(
t
)
=
1
H
(
t
,
T
0
)
∫
T
0
t
H
(
t
,
s
)
(
γ

1
)
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
×
(
W
(
σ
(
s
)
)
)
λ
Δ
s
;
meanwhile noting (54), we obtain
(57)
liminf
t
→
∞
[
B
(
t
)

A
(
t
)
]
≤
W
(
T
0
)

limsup
t
→
∞
1
H
(
t
,
T
0
)
×
∫
T
0
t
L
H
(
t
,
s
)
δ
(
s
)
q
(
s
)
(
1

r
(
τ
(
s
)
)
)
γ

1
Δ
s
≤
W
(
T
0
)

Ψ
(
T
0
)
<
∞
.
Now we assert that
(58)
∫
T
0
∞
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
(
W
(
σ
(
s
)
)
)
λ
Δ
s
<
∞
holds. Suppose to the contrary that
(59)
∫
T
0
∞
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
(
W
(
σ
(
s
)
)
)
λ
Δ
s
=
∞
.
By (31), there exists a constant
ɛ
>
0
such that
(60)
inf
s
≥
T
0
[
liminf
t
→
∞
H
(
t
,
s
)
H
(
t
,
T
0
)
]
>
ɛ
>
0
.
From (59), there exists a
T
∈
[
T
0
,
∞
)
T
for arbitrary real number
M
>
0
such that
(61)
∫
T
0
t
δ
(
s
)
τ
Δ
(
s
)
(
a
(
τ
(
s
)
)
)
λ

1
(
δ
(
σ
(
s
)
)
)
λ
(
W
(
σ
(
s
)
)
)
λ
Δ
s
≥
M
(
γ

1
)
ɛ
,
for
t
∈
[
T
,
∞
)
T
. By (13), we have
(62)
B
(
t
)
=
1
H
(
t
,
T
0
)
∫
T
0
t
{
(
∫
T
0
s
δ
(
u
)
τ
Δ
(
u
)
(
a
(
τ
(
u
)
)
)
λ

1
(
δ
(
σ
(
u
)
)
)
λ
(
W
(
σ
(
u
)
)
)
λ
Δ
u
)
Δ
s
(
γ

1
)
H
(
t
,
s
)
×
(
∫
T
0
s
δ
(
u
)
τ
Δ
(
u
)
(
a
(
τ
(
u
)
)
)
λ

1
(
δ
(
σ
(
u
)
)
)
λ
×
(
W
(
σ
(
u
)
)
)
λ
Δ
u
∫
T
0
s
δ
(
u
)
τ
Δ
(
u
)
(
a
(
τ
(
u
)
)
)
λ

1
(
δ
(
σ
(
u
)
)
)
λ
)
Δ
s
}
Δ
s
=
1
H
(
t
,
T
0
)
∫
T
0
t
{
∫
T
0
σ
(
s
)
δ
(
u
)
τ
Δ
(
u
)
(
a
(
τ
(
u
)
)
)
λ

1
(
δ
(
σ
(
u
)
)
)
λ
(
W
(
σ
(
u
)
)
)
λ
Δ
u
[

(
γ

1
)
H
Δ
s
(
t
,
s
)
]
×
∫
T
0
σ
(
s
)
δ
(
u
)
τ
Δ
(
u
)
(
a
(
τ
(
u
)
)
)
λ

1
(
δ
(
σ
(
u
)
)
)
λ
×
(
W
(
σ
(
u
)
)
)
λ
Δ
u
δ
(
u
)
τ
Δ
(
u
)
(
a
(
τ
(
u
)
)
)
λ

1
(
δ
(
σ
(
u
)
)
)
λ
}
Δ
s
≥
1
H
(
t
,
T
0
)
∫
T
t
{
∫
T
0
s
δ
(
u
)
τ
Δ
(
u
)
(
a
(
τ
(
u
)
)
)
λ

1
(
δ
(
σ
(
u
)
)
)
λ
(
W
(
σ
(
u
)
)
)
λ
Δ
u
[

(
γ

1
)
H
Δ
s
(
t
,
s
)
]
×
∫
T
0
s
δ
(
u
)
τ
Δ
(
u
)
(
a
(
τ
(
u
)
)
)
λ

1
(
δ
(
σ
(
u
)
)
)
λ
×
(
W
(
σ
(
u
)
)
)
λ
Δ
u
δ
(
u
)
τ
Δ
(
u
)
(
a
(
τ
(
u
)
)
)
λ

1
(
δ
(
σ
(
u
)
)
)
λ
}
Δ
s
≥
1
H
(
t
,
T
0
)
∫
T
t
[

(
γ

1
)
H
Δ
s
(
t
,
s
)
]
M
(
γ

1
)
ɛ
Δ
s
=
M
ɛ
H
(
t
,
T
)
H
(
t
,
T
0
)
.
From (60), there exists
t
2
∈
[
T
,
∞
)
T
such that
H
(
t
,
T
)
/
H
(
t
,
T
0
)
≥
ɛ
for
t
∈
[
t
2
,
∞
)
T
, so
B
(
t
)
≥
M
. Since
M
is arbitrary, we have
(63)
lim
t
→
∞
B
(
t
)
=
∞
.
Selecting a sequence
{
T
n
}
n
=
1
∞
:
T
n
∈
[
T
0
,
∞
)
T
with
lim
n
→
∞
T
n
=
∞
satisfying
(64)
lim
n
→
∞
[
B
(
T
n
)

A
(
T
n
)
]
=
liminf
t
→
∞
[
B
(
t
)

A
(
t
)
]
<
∞
,
then there exists a constant
M
0
>
0
such that
(65)
B
(
T
n
)

A
(
T
n
)
≤
M
0
for sufficiently large positive integer
n
. From (63), we can easily see
(66)
lim
n
→
∞
B
(
T
n
)
=
∞
,
and (65) implies that
(67)
lim
n
→
∞
A
(
T
n
)
=
∞
.
From (65) and (66), we have
(68)
A
(
T
n
)
B
(
T
n
)

1
≥

M
0
B
(
T
n
)
>

M
0
2
M
0
=

1
2
;
that is,
(69)
A
(
T
n
)
B
(
T
n
)
>
1
2
for sufficiently large positive integer
n
, which together with (67) implies
(70)
lim
n
→
∞
[
A
(
T
n
)
]
γ
[
B
(
T
n
)
]
γ

1
=
lim
n
→
∞
[
A
(
T
n
)
B
(
T
n
)
]
γ

1
A
(
T
n
)
=
∞
.
On the other hand, by Lemma 5, we obtain
(71)
A
(
T
n
)
=
1
H
(
T
n
,
T
0
)
∫
T
0
T
n
G
+
(
T
n
,
s
)
δ
(
σ
(
s
)
)
H
1
/
λ
(
T
n
,
s
)
W
(
σ
(
s
)
)
Δ
s
=
∫
T
0
T
n
{
[
(
γ

1
)
H
(
T
n
,
s
)
δ
(
s
)
τ
Δ
(
s
)
H
(
T
n
,
T
0
)
]
(
γ

1
)
/
γ
×
W
(
σ
(