Now, we present the following result which belongs to the classical metric fixed point theory.
Now, we present the main result of the paper.
Proof.
Let us consider the operator
H
defined on
BC
(
R
+
)
as follows:
(20)
(
H
x
)
(
t
)
=
f
(
t
,
x
(
t
)
)
+
∫
0
t
g
(
t
,
s
,
x
(
s
)
)
d
s
.
For convenience, we divide the proof into several steps.
Step
1 (
H
maps
BC
(
R
+
)
into itself). In fact, since
f
and
g
are continuous functions, for
x
∈
BC
(
R
+
)
we infer that
H
x
is continuous on
R
+
. Now, we prove that for
x
∈
BC
(
R
+
)
the function
H
x
is bounded. In fact, for arbitrarily fixed
t
∈
R
+
we get
(21)
|
(
H
x
)
(
t
)
|
≤
|
f
(
t
,
x
(
t
)
)
-
f
(
t
,
0
)
|
+
|
f
(
t
,
0
)
|
+
∫
0
t
|
g
(
t
,
s
,
x
(
s
)
)
|
d
s
≤
φ
(
|
x
(
t
)
|
)
+
|
f
(
t
,
0
)
|
+
u
(
t
)
∫
0
t
v
(
s
)
d
s
≤
φ
(
∥
x
∥
)
+
q
.
This proves that
(22)
sup
t
∈
R
+
|
(
H
x
)
(
t
)
|
≤
φ
(
∥
x
∥
)
+
q
<
∞
.
Therefore,
H
maps
BC
(
R
+
)
into itself.
Step
2 (
H
maps
B
r
0
into itself). It follows from assumption
(
d
)
that
H
maps
B
r
0
into itself.
Step
3 (an estimate of
H
with respect to the quantity
ω
0
)
. For fixed
ε
>
0
and
L
>
0
let us take
t
,
s
∈
[
0
,
L
]
with
|
t
-
s
|
≤
ε
. Without loss of generality, we may assume that
s
<
t
. Then for
x
∈
X
we obtain the following estimate:
(23)
|
(
H
x
)
(
t
)
-
(
H
x
)
(
s
)
|
≤
|
f
(
t
,
x
(
t
)
)
-
f
(
s
,
x
(
s
)
)
|
+
|
∫
0
t
g
(
t
,
τ
,
x
(
τ
)
)
d
τ
-
∫
0
s
g
(
s
,
τ
,
x
(
τ
)
)
d
τ
|
≤
|
f
(
t
,
x
(
t
)
)
-
f
(
s
,
x
(
t
)
)
|
+
|
f
(
s
,
x
(
t
)
)
-
f
(
s
,
x
(
s
)
)
|
+
|
∫
0
t
g
(
t
,
τ
,
x
(
τ
)
)
d
τ
-
∫
0
t
g
(
s
,
τ
,
x
(
τ
)
)
d
τ
|
+
|
∫
0
t
g
(
s
,
τ
,
x
(
τ
)
)
d
τ
-
∫
0
s
g
(
s
,
τ
,
x
(
τ
)
)
d
τ
|
≤
ω
1
L
(
f
,
ε
)
+
φ
(
|
x
(
t
)
-
x
(
s
)
|
)
+
∫
0
t
|
g
(
t
,
τ
,
x
(
τ
)
)
-
g
(
s
,
τ
,
x
(
τ
)
)
|
d
τ
+
∫
s
t
|
g
(
s
,
τ
,
x
(
τ
)
)
|
d
τ
≤
ω
1
L
(
f
,
ε
)
+
φ
(
ω
L
(
x
,
ε
)
)
+
∫
0
L
ω
1
L
(
g
,
ε
)
d
τ
+
u
(
s
)
∫
s
t
v
(
τ
)
d
τ
≤
ω
1
L
(
f
,
ε
)
+
φ
(
ω
L
(
x
,
ε
)
)
+
L
ω
1
L
(
g
,
ε
)
+
ε
sup
s
,
t
∈
[
0
,
L
]
{
u
(
s
)
v
(
t
)
}
,
where we denoted
(24)
ω
1
L
(
f
,
ε
)
=
sup
{
|
f
(
t
,
x
)
-
f
(
s
,
x
)
|
:
t
,
s
∈
[
0
,
L
]
,
m
m
m
m
m
m
m
m
m
m
m
x
∈
[
-
r
0
,
r
0
]
,
|
t
-
s
|
≤
ε
}
,
ω
1
L
(
g
,
ε
)
=
sup
{
|
g
(
t
,
τ
,
x
)
-
g
(
s
,
τ
,
x
)
|
:
t
,
τ
,
s
∈
[
0
,
L
]
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
x
∈
[
-
r
0
,
r
0
]
,
|
t
-
s
|
≤
ε
}
.
From the uniform continuity of the functions
f
and
g
on the sets
[
0
,
L
]
×
[
-
r
0
,
r
0
]
and
[
0
,
L
]
×
[
0
,
L
]
×
[
-
r
0
,
r
0
]
, respectively, it follows that
ω
1
L
(
f
,
ε
)
→
0
and
ω
1
L
(
g
,
ε
)
→
0
when
ε
→
0
. Notice that, since
u
and
v
are continuous on
[
0
,
L
]
, we have that
sup
s
,
t
∈
[
0
,
L
]
{
u
(
s
)
v
(
t
)
}
<
∞
. Therefore, we derive the following estimate:
(25)
ω
L
(
H
x
,
ε
)
≤
ω
1
L
(
f
,
ε
)
+
φ
(
ω
L
(
x
,
ε
)
)
+
L
ω
1
L
(
g
,
ε
)
+
ε
sup
s
,
t
∈
[
0
,
L
]
{
u
(
s
)
v
(
t
)
}
.
Since
φ
is nondecreasing, we obtain
(26)
ω
L
(
H
X
,
ε
)
=
sup
x
∈
X
{
ω
L
(
H
x
,
ε
)
}
≤
ω
1
L
(
f
,
ε
)
+
φ
(
sup
x
∈
X
{
ω
L
(
x
,
ε
)
}
)
+
L
ω
1
L
(
g
,
ε
)
+
ε
sup
s
,
t
∈
[
0
,
L
]
{
u
(
s
)
v
(
t
)
}
=
ω
1
L
(
f
,
ε
)
+
φ
(
ω
L
(
X
,
ε
)
)
+
L
ω
1
L
(
g
,
ε
)
+
ε
sup
s
,
t
∈
[
0
,
L
]
{
u
(
s
)
v
(
t
)
}
.
Hence
(27)
ω
0
L
(
H
X
)
=
lim
ε
→
0
ω
L
(
H
X
,
ε
)
≤
lim
ε
→
0
φ
(
ω
L
(
X
,
ε
)
)
=
φ
(
lim
ε
→
0
ω
L
(
X
,
ε
)
)
=
φ
(
ω
0
L
(
X
)
)
.
Finally, we get
(28)
ω
0
(
H
X
)
=
lim
L
→
∞
ω
0
L
(
H
X
)
≤
lim
L
→
∞
φ
(
ω
0
L
(
X
)
)
=
φ
(
lim
L
→
∞
ω
0
L
(
X
)
)
=
φ
(
ω
0
(
X
)
)
.
Step
4 (an estimate of
H
with respect to the diameter). For
x
,
y
∈
X
and
t
∈
R
+
, we have
(29)
|
(
H
x
)
(
t
)
-
(
H
y
)
(
t
)
|
≤
|
f
(
t
,
x
(
t
)
)
-
f
(
t
,
y
(
t
)
)
|
+
∫
0
t
|
g
(
t
,
s
,
x
(
s
)
)
|
d
s
+
∫
0
t
|
g
(
t
,
s
,
y
(
s
)
)
|
d
s
≤
φ
(
|
x
(
t
)
-
y
(
t
)
|
)
+
2
u
(
t
)
∫
0
t
v
(
s
)
d
s
.
Since
φ
is nondecreasing, from the last inequality it follows that
(30)
diam
(
H
X
)
(
t
)
≤
φ
(
diam
X
(
t
)
)
+
2
u
(
t
)
∫
0
t
v
(
s
)
d
s
.
Consequently, from assumption
(
c
1
)
and the continuity of
φ
, we get
(31)
limsup
t
→
∞
diam
(
H
X
)
(
t
)
≤
φ
(
limsup
t
→
∞
diam
X
(
t
)
)
.
Step
5 (
H
satisfies the contractive condition of Proposition 6). In fact, from assumption
(
b
1
)
, (28), (31), and the definition of the measure of noncompactness
μ
, we infer
(32)
μ
(
H
X
)
=
ω
0
(
H
X
)
+
limsup
t
→
∞
diam
(
H
X
)
(
t
)
≤
φ
(
ω
0
(
X
)
)
+
φ
(
limsup
t
→
∞
diam
X
(
t
)
)
=
2
φ
(
ω
0
(
X
)
)
+
φ
(
limsup
t
→
∞
diam
X
(
t
)
)
2
≤
2
φ
(
ω
0
(
X
)
+
limsup
t
→
∞
diam
X
(
t
)
2
)
=
2
φ
(
μ
(
X
)
2
)
.
Now, considering in
BC
(
R
+
)
the measure of noncompactness
μ
1
defined by
μ
1
(
X
)
=
(
1
/
2
)
μ
(
X
)
, the last estimate can be written in the form
(33)
μ
1
(
H
X
)
≤
φ
(
μ
1
(
X
)
)
.
Therefore, if
μ
1
(
X
)
≠
0
, then
(34)
μ
1
(
H
X
)
≤
φ
(
μ
1
(
X
)
)
μ
1
(
X
)
·
μ
1
(
X
)
or equivalently
(35)
μ
1
(
H
X
)
≤
β
(
μ
1
(
X
)
)
·
μ
1
(
X
)
,
where
β
(
t
)
=
φ
(
t
)
/
t
.
In the case
μ
1
(
X
)
=
0
we have that
X
is a relatively compact subset of
BC
(
R
+
)
and, since
H
is continuous,
H
X
is also relatively compact and thus
μ
1
(
H
X
)
=
0
. This proves that (35) is also satisfied when
μ
1
(
X
)
=
0
. Summarizing, for any nonempty subset
X
of
B
r
0
, we have
(36)
μ
1
(
H
X
)
≤
β
(
μ
1
(
X
)
)
·
μ
1
(
X
)
,
where
β
∈
B
0
(assumption
(
b
2
)
) and
μ
1
is a measure of noncompactness in
BC
(
R
+
)
.
In the sequel, let us consider the sequence of sets
(
B
r
0
n
)
, where
B
r
0
1
=
Conv
H
(
B
r
0
)
,
B
r
0
2
=
Conv
H
(
B
r
0
1
)
, and so on. Notice that the sequence is decreasing; that is,
B
r
0
n
+
1
⊂
B
r
0
n
for
n
=
1,2
,
3
,
…
. Moreover,
B
r
0
1
⊂
B
r
0
and the sets in this sequence are closed, convex, and nonempty.
On the other hand, in view of (32), we get
(37)
μ
1
(
B
r
0
1
)
=
μ
1
(
Conv
H
(
B
r
0
)
)
=
μ
1
(
H
(
B
r
0
)
)
≤
φ
(
μ
1
(
B
r
0
)
)
,
μ
1
(
B
r
0
2
)
=
μ
1
(
Conv
H
(
B
r
0
1
)
)
=
μ
1
(
H
(
B
r
0
1
)
)
≤
φ
(
μ
1
(
B
r
0
1
)
)
≤
φ
(
φ
(
μ
1
(
B
r
0
)
)
)
=
φ
2
(
μ
1
(
B
r
0
)
)
and, by using induction,
(38)
μ
1
(
B
r
0
n
)
≤
φ
n
(
μ
1
(
B
r
0
)
)
,
where we have used the nondecreasing character of
φ
and where
φ
(
n
)
denotes the
n
th iteration. Taking into account
(
b
2
)
, since
φ
(
t
)
/
t
∈
B
0
, we have
φ
(
t
)
<
t
for
t
>
0
and as
φ
is continuous, it follows that
φ
(
n
)
(
t
)
→
0
for
t
>
0
[10].
Therefore, we deduce that
(39)
lim
n
→
∞
μ
1
(
B
r
0
n
)
=
lim
n
→
∞
φ
n
(
μ
1
(
B
r
0
)
)
=
0
.
Now, taking into account Definition 1, we deduce that the set
Y
=
⋂
n
=
1
∞
B
r
0
n
is nonempty, bounded, closed, and convex. Moreover, since
μ
1
(
Y
)
≤
μ
1
(
B
r
0
n
)
for any
n
∈
N
,
Y
is a member of the kernel
ker
μ
1
of the measure of noncompactness
μ
1
. Let us also observe that the operator
H
transforms the set
Y
into itself.
Next, we will prove that
H
is continuous on the set
Y
. To do this let us fix a number
ε
>
0
and we take a sequence
(
x
n
)
⊂
Y
and
x
∈
Y
such that
x
n
→
x
. We have to prove that
H
x
n
→
H
x
.
In fact, since
Y
∈
ker
μ
1
, we have
μ
1
(
Y
)
=
0
and, particularly,
limsup
t
→
∞
diam
Y
(
t
)
=
0
. Then, for
ε
>
0
we can find
T
>
0
such that
|
x
(
t
)
-
y
(
t
)
|
<
ε
for any
x
,
y
∈
Y
and
t
≥
T
. Particularly, since
H
:
Y
→
Y
we have
H
x
n
,
H
x
∈
Y
for any
n
∈
N
, and, thus, for
t
≥
T
,
(40)
|
(
H
x
n
)
(
t
)
-
(
H
x
)
(
t
)
|
<
ε
,
for
any
n
∈
N
.
On the other hand, since
g
:
[
0
,
T
]
×
[
0
,
T
]
×
[
-
r
0
,
r
0
]
→
R
is continuous on a compact set, it is uniformly continuous. This means that for
ε
>
0
we can find
δ
>
0
such that if
max
{
|
t
-
t
′
|
,
|
s
-
s
′
|
,
|
u
-
v
|
}
<
δ
for
t
,
t
′
,
s
,
s
′
∈
[
0
,
T
]
and
u
,
v
∈
[
-
r
0
,
r
0
]
, we have
|
g
(
t
,
s
,
u
)
-
g
(
t
′
,
s
′
,
v
)
|
<
ε
/
2
T
.
Taking into account that
x
n
→
x
, we can find
n
0
∈
N
such that, for
n
≥
n
0
,
∥
x
n
-
x
∥
<
min
{
ε
/
2
,
δ
}
.
For
n
≥
n
0
and
t
∈
[
0
,
T
]
, we have
(41)
|
(
H
x
n
)
(
t
)
-
(
H
x
)
(
t
)
|
≤
|
f
(
t
,
x
n
(
t
)
)
-
f
(
t
,
x
(
t
)
)
|
+
∫
0
t
|
g
(
t
,
s
,
x
n
(
s
)
)
-
g
(
t
,
s
,
x
(
s
)
)
|
d
s
≤
φ
(
|
x
n
(
t
)
-
x
(
t
)
|
)
+
∫
0
T
|
g
(
t
,
s
,
x
n
(
s
)
)
-
g
(
t
,
s
,
x
(
s
)
)
|
d
s
≤
φ
(
∥
x
n
-
x
∥
)
+
∫
0
T
ε
2
T
d
s
≤
φ
(
ε
2
)
+
ε
2
<
ε
2
+
ε
2
=
ε
,
where we have used the fact that
φ
(
t
)
<
t
for
t
>
0
and the nondecreasing character of
φ
.
From (40) and (41),
∥
H
x
n
-
H
x
∥
<
ε
for
n
≥
n
0
. This proves our claim.
Finally, taking into account that as
Y
∈
ker
μ
1
and, consequently,
Y
is relatively compact,
H
:
Y
→
Y
is a continuous operator, applying the classical Schauder fixed point theorem, we infer that the operator
H
has at least one fixed point in
Y
.
In order to prove that solutions of (17) are asymptotically stable, we notice that any solution
x
(
t
)
of (17) in
B
r
0
is a fixed point of
H
. Now, taking into account that
H
transforms
B
r
0
into itself, we have
(42)
μ
1
(
H
(
B
r
0
∩
Fix
H
)
)
=
μ
1
(
B
r
0
∩
H
(
Fix
H
)
)
=
μ
1
(
B
r
0
∩
Fix
H
)
.
Since
μ
1
(
H
X
)
≤
β
(
μ
1
(
X
)
)
·
μ
1
(
X
)
for any nonempty subset
X
of
B
r
0
, we have
(43)
μ
1
(
H
(
B
r
0
∩
Fix
H
)
)
=
μ
1
(
B
r
0
∩
Fix
H
)
≤
β
(
μ
1
(
B
r
0
∩
Fix
H
)
)
·
μ
1
(
B
r
0
∩
Fix
H
)
.
Further, we distinguish two cases.
Case
1
(
β
(
μ
1
(
B
r
0
∩
Fix
H
)
)
=
0
)
. In this case, by (43),
μ
1
(
B
r
0
∩
Fix
H
)
=
0
.
Case
2
(
β
(
μ
1
(
B
r
0
∩
Fix
H
)
)
>
0
)
. In this case, by (43) and taking into account that the range of the function
β
is
[
0,1
)
, we infer
(44)
μ
1
(
H
(
B
r
0
∩
Fix
H
)
)
=
μ
1
(
B
r
0
∩
Fix
H
)
≤
β
(
μ
1
(
B
r
0
∩
Fix
H
)
)
·
μ
1
(
B
r
0
∩
Fix
H
)
<
μ
1
(
B
r
0
∩
Fix
H
)
which is a contradiction. Therefore
μ
1
(
B
r
0
∩
Fix
H
)
=
0
.
Since
μ
1
(
X
)
=
(
1
/
2
)
μ
(
X
)
for any nonempty subset
X
, we deduce that
μ
(
B
r
0
∩
Fix
H
)
=
0
. Taking into account the definition of the measure of noncompactness
μ
(see Section 2), we have
(45)
limsup
t
→
∞
diam
[
(
B
r
0
∩
Fix
H
)
(
t
)
]
=
0
.
But this means that for any
ε
>
0
we can find
T
>
0
such that
(46)
diam
[
(
B
r
0
∩
Fix
H
)
(
t
)
]
≤
ε
for
any
t
≥
T
.
As all solutions of (17) in
B
r
0
are in
B
r
0
∩
Fix
H
, by (46) we have that for
ε
>
0
there exists
T
>
0
such that
(47)
|
x
(
t
)
-
y
(
t
)
|
≤
ε
for
any
t
≥
T
,
where
x
,
y
∈
B
r
0
and they are solutions of (17). This means that solutions of (17) are asymptotically stable. The proof is complete.