# Existence and Multiplicity of Positive Solutions for a System of Fourth-Order Boundary Value Problems

Authors: Shoucheng Yu; Zhilin Yang
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

We study the existence and multiplicity of positive solutions for the system of fourth-order boundary value problemsx
(
4
)
=
f
t
,
x
,
x
′
,
-
x
′
′
,
-
x
′
′
′
,
y
,
y
′
,
-
y
′
′
,
-
y
′
′
′
,
y
(
4
)
=
g
t
,
x
,
x
′
,
-
x
′
′
,
-
x
′
′
′
,
y
,
y
′
,
-
y
′
′
,
-
y
′
′
′,x
(
0
)
=
x
′
(
1
)
=
x
′
′
(
0
)
=
x
′
′
′
(
1
)
=
0
, and y
(
0
)
=
y
′
(
1
)
=
y
′
′
(
0
)
=
y
′
′
′
(
1
)
=
0
, where f
,
g
∈
C
(
[
0,1
]
×
R
+
8
,
R
+
)
(
R
+
:
=
[
0
,
∞
)
). We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some integral identities and inequalities and R
+
2-monotone matrices.


---

## Body

## 1. Introduction

In this paper we study the existence and multiplicity of positive solutions for the system of fourth-order boundary value problems:(1)
x
(
4
)
=
f
(
t
,
x
,
x
′
,
-
x
′′
,
-
x
′′′
,
y
,
y
′
,
-
y
′′
,
-
y
′′′
)
,
y
(
4
)
=
g
(
t
,
x
,
x
′
,
-
x
′′
,
-
x
′′′
,
y
,
y
′
,
-
y
′′
,
-
y
′′′
)
,
x
(
0
)
=
x
′
(
1
)
=
x
′′
(
0
)
=
x
′′′
(
1
)
=
0
,
y
(
0
)
=
y
′
(
1
)
=
y
′′
(
0
)
=
y
′′′
(
1
)
=
0
,
where f
,
g
∈
C
(
[
0,1
]
×
R
+
8
,
R
+
)
(
R
+
∶
=
[
0
,
∞
)
). It is well known that the deflection of elastic beams can be described by some fourth-order boundary value problems; for example, see [1, 2]. Consequently, fourth-order boundary value problems play a very important role in both theory and applications. Boundary value problems for systems of nonlinear ordinary differential equations have been also studied by several authors; for example, see [3–11] and the references therein. In [3], Li et al. use fixed point theorems on cones to establish the existence of positive solutions for a system of third-order boundary value problems:
(2)
-
u
′′′
=
a
(
t
)
f
(
t
,
v
)
,
-
v
′′′
=
b
(
t
)
h
(
t
,
u
)
,
u
(
0
)
=
u
′
(
0
)
=
0
,
u
′
(
1
)
=
α
u
′
(
η
)
,
v
(
0
)
=
v
′
(
0
)
=
0
,
v
′
(
1
)
=
α
v
′
(
η
)
.
In [4], Lü et al. study the existence of positive solutions for a system of boundary value problems:
(3)
u
(
4
)
=
f
(
t
,
v
)
,
-
v
′′
=
g
(
t
,
u
)
,
t
∈
(
0,1
)
,
u
(
0
)
=
u
(
1
)
=
u
′′
(
0
)
=
u
′′
(
1
)
=
v
(
0
)
=
v
(
1
)
=
0
,
where f
,
g
∈
C
(
[
0,1
]
×
R
+
,
R
+
). In [11], the authors investigate the existence and multiplicity of positive solutions for the system
(4)
(
-
1
)
m
w
(
2
m
)
=
f
(
t
,
w
,
w
′
,
-
w
′′′
,
…
,
(
-
1
)
m
-
1
w
(
2
m
-
1
)
,
W
W
W
W
W
W
W
z
,
z
′
,
-
z
′′′
,
…
,
(
-
1
)
n
-
1
z
(
2
n
-
1
)
)
,
(
-
1
)
n
z
(
2
n
)
=
g
(
t
,
w
,
w
′
,
-
w
′′′
,
…
,
(
-
1
)
m
-
1
w
(
2
m
-
1
)
,
W
W
W
W
W
W
z
,
z
′
,
-
z
′′′
,
…
,
(
-
1
)
n
-
1
z
(
2
n
-
1
)
)
,
w
(
2
i
)
(
0
)
=
w
(
2
i
+
1
)
(
1
)
=
0
(
i
=
0,1
,
…
,
m
-
1
)
,
z
(
2
j
)
(
0
)
=
z
(
2
j
+
1
)
(
1
)
=
0
(
j
=
0,1
,
…
,
n
-
1
)
.
The hypotheses imposed on the nonlinearities f and g are formulated in terms of two linear functions h
1
(
x
) and h
2
(
y
). The main results in [11] are established by using fixed point index theory based on a priori estimates of positive solutions achieved by utilizing new integral inequalities and nonnegative matrices.This paper is organized as follows. In Section2, we use the method of order reduction to transform (1) into a system of boundary value problems for second-order integrodifferential equations. Also, in this section, we develop some basic integral identities and inequalities that are useful in deriving a priori estimates in Section 3. Our main results, namely, Theorems 11–14, are stated and proved in Section 3. Finally, three examples that illustrate our main results are presented in Section 4.


## 2. Preliminaries

LetE
∶
=
C
1
(
[
0,1
]
,
R
), ∥
u
∥
∶
=
max
⁡
{
∥
u
∥
0
,
∥
u
′
∥
0
}
(
∥
u
∥
0
∶
=
max
⁡
{
|
u
(
t
)
|
:
t
∈
[
0,1
]
}
), P
∶
=
{
u
∈
E
:
u
(
t
)
⩾
0
,
u
′
(
t
)
⩾
0
,
∀
t
∈
[
0,1
]
}. Then (
E
,
∥
·
∥
) is a real Banach space and P is a solid cone in E. Let
(5)
k
(
t
,
s
)
∶
=
min
⁡
{
t
,
s
}
,
0
⩽
t
,
s
⩽
1
.
Define the linear integral operator B by
(6)
(
B
ω
)
(
t
)
∶
=
∫
0
1
‍
k
(
t
,
s
)
ω
(
s
)
d
s
.
It is well known that B
:
E
→
E is a completely continuous, positive, linear operator and
(7)
(
B
ψ
)
(
s
)
=
∫
0
1
‍
k
(
t
,
s
)
ψ
(
t
)
d
t
=
r
(
B
)
ψ
(
s
)
,
where ψ
(
t
)
∶
=
sin
⁡
(
π
/
2
)
t. Substituting u
(
t
)
∶
=
-
x
′′
(
t
), v
(
t
)
∶
=
-
y
′′
(
t
) into (1), we transform (1) into the following system of seconed-order boundary value problems for the integrodifferential equations:
(8)
-
u
′′
=
f
(
t
,
B
u
,
(
B
u
)
′
,
u
,
u
′
,
B
v
,
(
B
v
)
′
,
v
,
v
′
)
,
-
v
′′
=
g
(
t
,
B
u
,
(
B
u
)
′
,
u
,
u
′
,
B
v
,
(
B
v
)
′
,
v
,
v
′
)
,
u
(
0
)
=
u
′
(
1
)
=
0
,
v
(
0
)
=
v
′
(
1
)
=
0
,
which is equivalent to the system of nonlinear integral equations:
(9)
u
(
t
)
=
∫
0
1
k
(
t
,
s
)
f
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
WWWWWWWW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
,
v
(
t
)
=
∫
0
1
k
(
t
,
s
)
g
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
WWWWWWWW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
.
Define the operators A
i
(
i
=
1,2
)
:
P
2
→
P and A
:
P
2
→
P
2 by
(10)
A
1
(
u
,
v
)
(
t
)

=

∫
0
1
‍
k
(
t
,
s
)
f

(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
WWWWWWWWWiWwWvWiWi
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
,
A
2
(
u
,
v
)
(
t
)

=

∫
0
1
k
(
t
,
s
)
g

(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
W
W
W
W
W
W
W
WWiWwv
W
W
i
W
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
,
A
(
u
,
v
)
(
t
)
=
(
A
1
(
u
,
v
)
(
t
)
,
A
2
(
u
,
v
)
(
t
)
)
.
If f
,
g
∈
C
(
[
0,1
]
×
R
+
8
,
R
+
), then A
i and A are completely continuous operators. Clearly, the existence of positive solutions for (1) is equivalent to that of positive fixed points of A
:
P
2
→
P
2.Lemmas1–4 below are cited from [12].Lemma 1.
Ifω
∈
P
∩
C
2
[
0,1
], ω
(
0
)
=
ω
′
(
1
)
=
0, then
(11)
∫
0
1
‍
(
-
ω
′′
(
t
)
)
t
e
t
d
t
=
∫
0
1
‍
(
ω
(
t
)
+
2
ω
′
(
t
)
)
t
e
t
d
t
.Lemma 2.
Ifω
∈
P, then
(12)
∫
0
1
‍
(
(
B
ω
)
(
t
)
+
2
(
B
ω
)
′
(
t
)
)
t
e
t
d
t
=
∫
0
1
‍
ω
(
t
)
t
e
t
d
t
.Lemma 3.
Ifω
∈
P, ω
(
0
)
=
0, and ω
′ is decreasing on [
0,1
], then
(13)
∫
0
1
‍
ω
(
t
)
t
e
t
d
t
⩾
(
e
-
2
)
∫
0
1
‍
ω
′
(
t
)
t
e
t
d
t
.Lemma 4.
Ifω
∈
P, ω
(
0
)
=
0, then
(14)
ω
(
1
)
⩽
∫
0
1
‍
(
ω
(
t
)
+
2
ω
′
(
t
)
)
t
e
t
d
t
.Lemma 5.
Supposew
∈
C
(
[
0,1
]
,
R
+
) is not identically vanishing on [
0,1
], and ω
∈
C
(
[
0,1
]
,
R
+
) is a concave function. Let ϱ
(
w
)
∶
=
∫
0
1
‍
t
w
(
t
)
d
t
>
0. Then
(15)
∫
0
1
‍
w
(
t
)
ω
(
t
)
d
t
⩾
ω
(
1
)
ϱ
(
w
)
.Proof.
By the concavity ofv and the nonnegativity of w, we have
(16)
∫
0
1
‍
w
(
t
)
ω
(
t
)
d
t
=
∫
0
1
‍
w
(
t
)
ω
(
t
·
1
+
(
1
-
t
)
·
0
)
d
t
⩾
ω
(
1
)
∫
0
1
‍
t
w
(
t
)
d
t
=
ω
(
1
)
ϱ
(
w
)
.
This completes the proof.Lemma 6 (see [13]).
LetE be a real Banach space and P a cone in E. Suppose that Ω
⊂
E is a bounded open set and that T
:
Ω
¯
∩
P
→
P is a completely continuous operator. If there exists ω
0
∈
P
∖
{
0
} such that
(17)
ω
-
T
ω
≠
λ
ω
0
,
∀
λ
⩾
0
,
ω
∈
∂
Ω
∩
P
,
then i
(
T
,
Ω
∩
P
,
P
)
=
0, where i indicates the fixed point index on P.Lemma 7 (see [13]).
LetE be a real Banach space and P a cone in E. Suppose that Ω
⊂
E is a bounded open set with 0
∈
Ω and that T
:
Ω
¯
∩
P
→
P is a completely continuous operator. If
(18)
ω
-
λ
T
ω
≠
0
,
∀
λ
∈
[
0,1
]
,
ω
∈
∂
Ω
∩
P
,
then i
(
T
,
Ω
∩
P
,
P
)
=
1.Lemma 8 (see [14, Lemma 2.4]).
Ifp is concave on [
d
,
∞
), with lim
⁡
y
→
∞
p
(
y
)
/
y
≥
0, then p is increasing on [
d
,
∞
) and
(19)
p
(
y
+
z
-
d
)
⩽
p
(
y
)
+
p
(
z
)
-
p
(
d
)
for all y
,
z
∈
[
d
,
∞
).


## 3. Main Results

Letx
∶
=
(
x
1
,
x
2
,
x
3
,
x
4
)
∈
R
+
4
,
y
∶
=
(
y
1
,
y
2
,
y
3
,
y
4
)
∈
R
+
4. Let I
ρ
∶
=
[
0
,
ρ
] for ρ
>
0. We list our hypotheses as follows.(H1)
f
,
g
∈
C
(
[
0,1
]
×
R
+
8
,
R
+
).
(H2)
There exist a nonnegative matrix(
a
i
j
)
2
×
4 and a constant c
>
0 that
(20)
(
f
(
t
,
x
,
y
)
g
(
t
,
x
,
y
)
)
⩾
(
a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
)
(
x
1
+
2
x
2
x
3
+
2
x
4
y
1
+
2
y
4
y
3
+
2
y
4
)
-
(
c
c
)
holds for all (
t
,
x
,
y
)
∈
[
0,1
]
×
R
+
8, with the matrix
(21)
B
1
∶
=
(
(
1
-
2
e
-
1
)
a
11
+
a
12
-
1
(
1
-
2
e
-
1
)
a
13
+
a
14
(
1
-
2
e
-
1
)
a
21
+
a
22
(
1
-
2
e
-
1
)
a
23
+
a
24
-
1
)
being an R
+
2-monotone matrix.
(H3)
There existp
∈
C
(
R
+
,
R
+
) and q
∈
C
(
R
+
,
R
+
) such that (1) p is concave; (2) there are two constants c
>
0 and μ
1
>
1 such that
(22)
f
(
t
,
x
,
y
)
⩾
p
(
y
3
)
-
c
,
g
(
t
,
x
,
y
)
⩾
q
(
x
3
)
-
c
,
WWWWW
∀
(
t
,
x
,
y
)
∈
[
0,1
]
×
R
+
8
,
(23)
p
(
q
(
z
)
)
⩾
π
4
μ
1
16
z
-
c
,
∀
z
∈
R
+
.
(H4)
There exist two functionsΦ
M
,
Ψ
M
∈
C
(
R
+
,
R
+
), such that
(24)
f
(
t
,
x
,
y
)
⩽
Φ
M
(
x
4
+
y
4
)
,
g
(
t
,
x
,
y
)
⩽
Ψ
M
(
x
4
+
y
4
)
,
for all (
t
,
x
,
y
)
∈
[
0,1
]
×
I
M
3
×
R
+
×
I
M
3
×
R
+, and
(25)
∫
0
∞
‍
τ
d
τ
Φ
N
(
τ
)
+
Ψ
N
(
τ
)
+
4
μ
0
>
2
M
,
where M
∶
=
max
⁡
⁡
{
c
1
,
c
2
}, with c
1
>
0
,
c
2
>
0
,
μ
0
>
0 being specified in Theorem 11.
(H5)
There exist a nonnegative matrix(
a
i
j
)
2
×
4 and a constant r
>
0 that
(26)
(
f
(
t
,
x
,
y
)
g
(
t
,
x
,
y
)
)
⩽
(
a
31
a
32
a
33
a
34
a
41
a
42
a
43
a
44
)
(
x
1
+
2
x
2
x
3
+
2
x
4
y
1
+
2
y
4
y
3
+
2
y
4
)
holds for all (
t
,
x
,
y
)
∈
[
0,1
]
×
I
r
8, with the matrix
(27)
B
2
∶
=
(
1
-
a
11
-
a
12
,
-
a
13
-
a
14
-
a
21
-
a
22
,
1
-
a
23
-
a
24
)
being an R
+
2-monotone matrix.
(H6)
There exist a nonnegative matrix(
b
i
j
)
2
×
4 and a constant c
>
0 such that
(28)
(
f
(
t
,
x
,
y
)
g
(
t
,
x
,
y
)
)
⩽
(
b
11
b
12
b
13
b
14
b
21
b
22
b
23
b
24
)
(
x
1
+
2
x
2
x
3
+
2
x
4
y
1
+
2
y
4
y
3
+
2
y
4
)
+
(
c
c
)
holds for all (
t
,
x
,
y
)
∈
[
0,1
]
×
R
+
8, with the matrix
(29)
B
3
∶
=
(
1
-
b
11
-
b
12
,
-
b
13
-
b
14
-
b
21
-
b
22
,
1
-
b
23
-
b
24
)
being an R
+
2-monotone matrix.
(H7)
There exist a nonnegative matrix(
b
i
j
)
2
×
4 and a constant r
>
0 that
(30)
(
f
(
t
,
x
,
y
)
g
(
t
,
x
,
y
)
)
⩾
(
b
31
b
32
b
33
b
34
b
41
b
42
b
43
b
44
)
(
x
1
+
2
x
2
x
3
+
2
x
4
y
1
+
2
y
4
y
3
+
2
y
4
)
holds for all (
t
,
x
,
y
)
∈
[
0,1
]
×
I
r
8, with the matrix
(31)
B
4
∶
=
(
(
1
-
2
e
-
1
)
b
11
+
b
12
-
1
(
1
-
2
e
-
1
)
b
13
+
b
14
(
1
-
2
e
-
1
)
b
21
+
b
22
(
1
-
2
e
-
1
)
b
23
+
b
24
-
1
)
being anR
+
2-monotone matrix.
(H8)
f
(
t
,
x
,
y
) and g
(
t
,
x
,
y
) are increasing in (
x
,
y
)
∈
R
+
8, and there is a constant ω
>
0 such that
(32)
∫
0
1
‍
f
(
s
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
)
d
s
<
ω
,
(33)
∫
0
1
‍
g
(
s
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
)
d
s
<
ω
.Remark 9.
A real matrixB is said to be nonnegative if all elements of B are nonnegative.Remark 10 (see [15, page 112]).
A real square matrixM
=
(
m
i
j
)
2
×
2 is called R
+
2-monotone if, for any column vector x
∈
R
2, M
x
∈
R
+
2
⇒
x
∈
R
+
2.Theorem 11.
If (H1), (H2), (H4), and (H5) hold, then (1) has at least one positive solution.Proof.
Let(34)
M
1
∶
=
{
(
u
,
v
)
∈
P
2
:
(
u
,
v
)
=
A
(
u
,
v
)
+
μ
(
φ
,
φ
)
,
μ
⩾
0
}
,
where φ
(
t
)
∶
=
t
e
-
t. We will prove that M
1 is bounded. Indeed, if (
u
,
v
)
∈
M
1, then there exists μ
⩾
0 such that (
u
,
v
)
=
A
(
u
,
v
)
+
μ
(
φ
,
φ
), which can be written in the form
(35)
u
(
t
)
=
∫
0
1
k
(
t
,
s
)
f
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWWWW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWW
+
μ
φ
(
t
)
,
v
(
t
)
=
∫
0
1
k
(
t
,
s
)
g
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWWWW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWW
+
μ
φ
(
t
)
.
Taking differentiation of the preceding equations twice, we obtain
(36)
-
u
′′
(
t
)
=
f
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
WWWWWW
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
+
μ
(
2
-
t
)
e
-
t
,
-
v
′′
(
t
)
=
g
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
WWWWWW
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
+
μ
(
2
-
t
)
e
-
t
,
and thus we have, by (H2),
(37)
(
-
u
′′
(
t
)
-
v
′′
(
t
)
)
⩾
(
a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
)
(
(
B
u
)
(
t
)
+
2
(
B
u
)
′
(
t
)
u
(
t
)
+
2
u
′
(
t
)
(
B
v
)
(
t
)
+
2
(
B
v
)
′
(
t
)
v
(
t
)
+
2
v
′
(
t
)
)
-
(
c
c
)
.
Multiply the above by φ
(
t
)
∶
=
t
e
-
t and integrate over [
0,1
] and use (11) and to obtain (12)
(38)
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
⩾
(
a
11
a
12
a
13
a
14
a
21
a
22
a
23
a
24
)
(
∫
0
1
‍
u
(
t
)
t
e
t
d
t
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
v
(
t
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
-
(
c
c
)
.
It is readily seen that, for any u
,
v
∈
M
1,u
′ and v
′ are decreasing on [
0,1
]. By (13), we obtain
(39)
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
⩾
(
(
1
-
2
e
-
1
)
a
11
+
a
12
(
1
-
2
e
-
1
)
a
13
+
a
14
(
1
-
2
e
-
1
)
a
21
+
a
22
(
1
-
2
e
-
1
)
a
23
+
a
24
)
×
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
-
(
c
c
)
so that
(40)
B
1
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
⩽
(
c
c
)
.
Since B
1 is an R
+
2-monotone matrix, we have
(41)
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
⩽
B
1
-
1
(
c
c
)
∶
=
(
c
1
c
2
)
.
Now Lemma 4 implies
(42)
∥
u
∥
0
=
u
(
1
)
⩽
c
1
,
∥
v
∥
0
=
v
(
1
)
⩽
c
2
,
∀
(
u
,
v
)
∈
M
1
.
Furthermore, these estimates lead to
(43)
∥
(
B
u
)
(
t
)
∥
0
=
(
B
u
)
(
1
)
=
∫
0
1
‍
s
u
(
s
)
d
s
⩽
c
1
,
∥
(
B
v
)
(
t
)
∥
0
=
(
B
v
)
(
1
)
=
∫
0
1
‍
s
v
(
s
)
d
s
⩽
c
2
,
∥
(
B
u
)
′
(
t
)
∥
0
=
(
B
u
)
′
(
0
)
=
∫
0
1
‍
u
(
s
)
d
s
⩽
c
1
,
∥
(
B
v
)
′
(
t
)
∥
0
=
(
B
v
)
′
(
0
)
=
∫
0
1
‍
v
(
s
)
d
s
⩽
c
2
,
for all (
u
,
v
)
∈
M
1. Let
(44)
Π
∶
=
{
μ
⩾
0
:
there exists some
(
u
,
v
)
∈
P
2
,
W
W
I
W
such
that
(
u
,
v
)
=
A
(
u
,
v
)
+
μ
(
φ
,
φ
)
}
.
Equation (42) implies that μ
0
∶
=
sup
⁡
Π
<
∞. Let M
∶
=
max
⁡
{
c
1
,
c
2
}. By (H4) again, there are two functions Φ
M
,
Ψ
M
∈
C
(
R
+
,
R
+
), for all (
u
,
v
)
∈
M
1, such that
(45)
f
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
⩽
Φ
M
(
u
′
(
t
)
+
v
′
(
t
)
)
,
g
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
⩽
Ψ
M
(
u
′
(
t
)
+
v
′
(
t
)
)
.
Thus we have
(46)
-
u
′′
(
t
)
=
f
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
WWWWiiW
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
+
μ
(
2
-
t
)
e
-
t
WWiW
⩽
Φ
M
(
u
′
(
t
)
+
v
′
(
t
)
)
+
μ
(
2
-
t
)
e
-
t
WWiW
⩽
Φ
M
(
u
′
(
t
)
+
v
′
(
t
)
)
+
2
μ
0
,
-
v
′′
(
t
)
=
g
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
WWWWiiW
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
+
μ
(
2
-
t
)
e
-
t
WWiW
⩽
Ψ
M
(
u
′
(
t
)
+
v
′
(
t
)
)
+
μ
(
2
-
t
)
e
-
t
WWiW
⩽
Ψ
M
(
u
′
(
t
)
+
v
′
(
t
)
)
+
2
μ
0
.
This implies, for all (
u
,
v
)
∈
M
1
(47)
-
(
u
′′
(
t
)
+
v
′′
(
t
)
)
(
u
′
(
t
)
+
v
′
(
t
)
)
WW
⩽
(
u
′
(
t
)
+
v
′
(
t
)
)
(
Φ
N
(
u
′
(
t
)
+
v
′
(
t
)
)
WWWWWWWWiiW
+
Ψ
N
(
u
′
(
t
)
+
v
′
(
t
)
)
+
4
μ
0
)
,
and, in turn,
(48)
∫
0
u
′
(
0
)
+
v
′
(
0
)
τ
d
τ
Φ
N
(
τ
)
+
Ψ
N
(
τ
)
+
4
μ
0
‍
WW
⩽
∫
0
1
‍
(
u
′
(
t
)
+
v
′
(
t
)
)
d
t
=
u
(
1
)
+
v
(
1
)
⩽
2
M
.
By (H4) again, there exists a constant N
1
>
0 such that
(49)
∥
u
′
+
v
′
∥
0
=
u
′
(
0
)
+
v
′
(
0
)
⩽
N
1
,
∀
(
u
,
v
)
∈
M
1
.
This means that M
1 is bounded. Taking R
>
sup
⁡
{
∥
(
u
,
v
)
∥
:
(
u
,
v
)
∈
M
1
}, we have
(50)
(
u
,
v
)
≠
A
(
u
,
v
)
+
μ
(
φ
,
φ
)
,
∀
(
u
,
v
)
∈
∂
Ω
R
∩
P
2
,
μ
⩾
0
.
Now Lemma 6 yields
(51)
i
(
A
,
Ω
R
∩
P
2
,
P
2
)
=
0
.
Let
(52)
M
2
∶
=
{
(
u
,
v
)
∈
Ω
¯
r
∩
P
2
:
(
u
,
v
)
=
λ
A
(
u
,
v
)
,
λ
∈
[
0,1
]
}
,
where r
>
0 is given in (H5). We are in the position to prove M
2
=
{
0
}. Indeed, if (
u
,
v
)
∈
M
2, then (
u
,
v
)
=
λ
A
(
u
,
v
) for some λ
∈
[
0,1
], which can be written componentwise as
(53)
u
(
t
)
=
λ
∫
0
1
k
(
t
,
s
)
f
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
‍
W
W
W
W
W
W
W
W
W
u
(
s
)
,
u
′
(
s
)
,
(
B
v
)
(
s
)
,
W
W
W
W
W
W
W
W
W
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
,
v
(
t
)
=
λ
∫
0
1
k
(
t
,
s
)
g
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
‍
W
W
W
W
W
W
W
W
W
u
(
s
)
,
u
′
(
s
)
,
(
B
v
)
(
s
)
,
W
W
W
W
W
W
W
W
W
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
.
Taking differentiation of the preceding equations twice, we obtain
(54)
-
u
′′
(
t
)
=
λ
f
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
W
W
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
,
-
v
′′
(
t
)
=
λ
g
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
W
W
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
.
By (H5), we have
(55)
(
-
u
′′
(
t
)
-
v
′′
(
t
)
)
⩽
(
a
31
a
32
a
33
a
34
a
41
a
42
a
43
a
44
)
(
(
B
u
)
(
t
)
+
2
(
B
u
)
′
(
t
)
u
(
t
)
+
2
u
′
(
t
)
(
B
v
)
(
t
)
+
2
(
B
v
)
′
(
t
)
v
(
t
)
+
2
v
′
(
t
)
)
.
Multiply the above by φ
(
t
)
∶
=
t
e
-
t and use (11) and (12) to obtain
(56)
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
WW
⩽
(
a
31
a
32
a
33
a
34
a
41
a
42
a
43
a
44
)
(
∫
0
1
‍
u
(
t
)
t
e
t
d
t
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
v
(
t
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
WW
⩽
(
a
31
+
a
32
a
33
+
a
34
a
41
+
a
42
a
43
+
a
44
)
(
∫
0
1
‍
(
u
(
t
)

+

2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)

+

2
v
′
(
t
)
)
t
e
t
d
t
)
,
so that
(57)
B
2
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
⩽
(
0
0
)
.
Since B
2 is an R
+
2-monotone matrix, it follows that ∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
=
0 and ∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
=
0, so that u
=
v
≡
0. This proves M
2
=
{
0
}, as required. Consequently,
(58)
(
u
,
v
)
≠
λ
A
(
u
,
v
)
,
∀
(
u
,
v
)
∈
∂
Ω
r
∩
P
2
,
λ
∈
[
0,1
]
.
Now Lemma 7 yields
(59)
i
(
A
,
Ω
r
∩
P
2
,
P
2
)
=
1
.
Combining (51) and (59) we arrive at
(60)
i
(
A
,
(
Ω
R
∖
Ω
¯
r
)
∩
P
2
,
P
2
)
=
0
-
1
=
-
1
.
Therefore A has at least one fixed point on (
Ω
R
∖
Ω
¯
r
)
∩
P
2. Thus (1) has at least one positive solution, which completes the proof.Theorem 12.
If (H1), (H3), (H4), and (H5) are satisfied, then (1) has at least one positive solution.Proof.
By (H3), we obtain(61)
A
1
(
u
,
v
)
(
t
)
⩾
∫
0
1
‍
k
(
t
,
s
)
p
(
v
(
s
)
)
d
s
-
c
^
1
,
A
2
(
u
,
v
)
(
t
)
⩾
∫
0
1
‍
k
(
t
,
s
)
q
(
u
(
s
)
)
d
s
-
c
^
1
for all (
u
,
v
)
∈
P
2, t
∈
[
0,1
]. We claim that the set
(62)
M
3
∶
=
{
(
u
,
v
)
∈
P
2
:
(
u
,
v
)
=
A
(
u
,
v
)
+
λ
(
φ
,
φ
)
,
λ
⩾
0
}
is bounded, where φ
(
t
)
∶
=
t
e
-
t. Indeed, if (
u
,
v
)
∈
M
3, then there exists a constant λ
⩾
0 such that (
u
,
v
)
=
A
(
u
,
v
)
+
λ
(
φ
,
φ
), which can be written in the form
(63)
u
(
t
)
=
∫
0
1
k
(
t
,
s
)
f
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWWWW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWW
+
λ
φ
(
t
)
,
v
(
t
)
=
∫
0
1
k
(
t
,
s
)
g
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWWWW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWW
+
λ
φ
(
t
)
.
By (H3) and (61), we have
(64)
u
(
t
)
⩾
∫
0
1
‍
k
(
t
,
s
)
p
(
v
(
s
)
)
d
s
-
c
^
1
,
v
(
t
)
⩾
∫
0
1
‍
k
(
t
,
s
)
q
(
u
(
s
)
)
d
s
-
c
^
1
for all t
∈
[
0,1
]. The nonnegativity and concavity of p imply lim
⁡
y
→
∞
p
(
y
)
/
y
⩾
0. Now Lemma 8 and Jensen’s inequality imply
(65)
p
(
v
(
t
)
)
⩾
p
(
v
(
t
)
+
c
^
1
)
-
p
(
c
^
1
)
⩾
∫
0
1
‍
k
(
t
,
s
)
p
(
q
(
u
(
s
)
)
)
d
s
-
p
(
c
^
1
)
.
This, together with (64) and (H3), implies
(66)
u
(
t
)
⩾
∫
0
1
‍
k
(
t
,
s
)
[
∫
0
1
‍
k
(
s
,
τ
)
p
(
q
(
u
(
τ
)
)
)
d
τ
-
p
(
c
^
1
)
]
d
s
-
c
^
1
⩾
∫
0
1
‍
∫
0
1
‍
k
(
t
,
s
)
k
(
s
,
τ
)
p
(
q
(
u
(
τ
)
)
)
d
s
d
τ
-
c
^
2
⩾
∫
0
1
‍
∫
0
1
‍
k
(
t
,
s
)
k
(
s
,
τ
)
[
π
4
μ
1
16
u
(
τ
)
-
c
]
d
s
d
τ
-
c
^
2
⩾
π
4
μ
1
16
∫
0
1
‍
∫
0
1
‍
k
(
t
,
s
)
k
(
s
,
τ
)
u
(
τ
)
d
s
d
τ
-
c
^
3
.
Multiply the last inequality by φ
(
t
)
∶
=
t
e
-
t and integrate over [
0,1
] and use (7) twice to obtain
(67)
∫
0
1
‍
φ
(
t
)
u
(
t
)
d
t
WW
⩾
π
4
μ
1
16
∫
0
1
‍
∫
0
1
‍
∫
0
1
‍
φ
(
t
)
k
(
t
,
s
)
k
(
s
,
τ
)
u
(
τ
)
d
t
d
s
d
τ
WWW
-
2
c
^
3
π
WW
=
μ
1
∫
0
1
‍
φ
(
t
)
u
(
t
)
d
t
-
2
c
^
3
π
,
so that
(68)
∫
0
1
‍
φ
(
t
)
u
(
t
)
d
t
⩽
2
c
^
3
π
(
μ
1
-
1
)
.
By Lemma 5, we have
(69)
∥
u
∥
0
=
u
(
1
)
⩽
2
c
^
3
π
ϱ
(
φ
)
(
μ
1
-
1
)
=
c
^
3
π
2
(
μ
1
-
1
)
.
Multiply the first inequality of (64) by φ
(
t
), integrate over [
0,1
], and use (7) to obtain
(70)
∫
0
1
‍
u
(
t
)
φ
(
t
)
d
t
⩾
4
π
2
∫
0
1
‍
p
(
v
(
t
)
)
φ
(
t
)
d
t
-
2
π
c
^
1
.
This, along with (68), implies
(71)
∫
0
1
‍
p
(
v
(
t
)
)
φ
(
t
)
d
t
⩽
π
2
4
(
2
c
^
1
π
+
∫
0
1
‍
u
(
t
)
φ
(
t
)
d
t
)
⩽
c
^
1
π
2
+
c
^
3
π
2
(
μ
1
-
1
)
.
By Lemma 5, we have
(72)
∥
v
∥
0
=
v
(
1
)
⩽
1
ϱ
(
φ
)
∫
0
1
‍
v
(
t
)
φ
(
t
)
d
t
=
∥
v
∥
0
ϱ
(
φ
)
p
(
∥
v
∥
0
)
∫
0
1
‍
φ
(
t
)
v
(
t
)
∥
v
∥
0
p
(
∥
v
∥
0
)
d
t
⩽
∥
v
∥
0
ϱ
(
φ
)
p
(
∥
v
∥
0
)
∫
0
1
‍
φ
(
t
)
p
(
v
(
t
)
)
d
t
,
so that
(73)
p
(
∥
v
∥
0
)
⩽
1
ϱ
(
φ
)
∫
0
1
‍
φ
(
t
)
p
(
v
(
t
)
)
d
t
⩽
c
^
1
π
3
8
+
c
^
3
π
4
16
(
μ
1
-
1
)
.
(H3) implies that p is strictly increasing and lim
⁡
x
→
∞
p
(
x
)
=
∞ (see Lemma 8). Consequently, there exists c
^
4
>
0 such that
(74)
∥
v
∥
0
⩽
c
^
4
.
Let N
∶
=
max
⁡
{
c
^
3
π
/
2
(
μ
1
-
1
)
,
c
^
4
}. Then
(75)
∥
u
∥
0
⩽
N
,
∥
v
∥
0
⩽
N
,
∀
(
u
,
v
)
∈
M
3
.
This establishes the a priori bound of M
3 for ∥
(
u
,
v
)
∥
0. Now it remains to derive the a priori bound of M
3 for ∥
(
u
′
,
v
′
)
∥
0, which is similar to the derivation of the a priori bound of M
1 for ∥
(
u
′
,
v
′
)
∥
0 in Theorem 11. This means that M
3 is bounded. Taking R
>
sup
⁡
{
∥
(
u
,
v
)
∥
:
(
u
,
v
)
∈
M
3
}, we have
(76)
(
u
,
v
)
≠
A
(
u
,
v
)
+
μ
(
φ
,
φ
)
,
∀
(
u
,
v
)
∈
∂
Ω
R
∩
P
2
,
μ
⩾
0
.
Now Lemma 6 yields
(77)
i
(
A
,
Ω
R
∩
P
2
,
P
2
)
=
0
.
Note that (H1) and (H5) imply (59) holds (see the proof of Theorem 11). Combining (77) and (59) we arrive at
(78)
i
(
A
,
(
Ω
R
∖
Ω
¯
r
)
∩
P
2
,
P
2
)
=
0
-
1
=
-
1
.
Therefore A has at least one fixed point on (
Ω
R
∖
Ω
¯
r
)
∩
P
2. Thus (1) has at least one positive solution, which completes the proof.Theorem 13.
If (H1), (H6), and (H7) are satisfied, then (1) has at least one positive solution.Proof.
Let(79)
M
4
∶
=
{
(
u
,
v
)
∈
P
2
:
(
u
,
v
)
=
λ
A
(
u
,
v
)
,
λ
∈
[
0,1
]
}
.
We will prove that M
4 is bounded. Indeed, if (
u
,
v
)
∈
M
4, then (
u
,
v
)
∈
P
2 and (
u
,
v
)
=
λ
A
(
u
,
v
), for some λ
∈
[
0,1
], which can be written componentwise as
(80)
u
(
t
)
=
λ
∫
0
1
k
(
t
,
s
)
f
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWWWWW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
,
v
(
t
)
=
λ
∫
0
1
k
(
t
,
s
)
g
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
W
W
W
W
W
W
W
W
W
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
.
Taking differentiation of the preceding equations twice, we obtain
(81)
-
u
′′
(
t
)
=
λ
f
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
(
B
v
)
(
t
)
,
W
W
W
W
i
W
W
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
,
-
v
′′
(
t
)
=
λ
g
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
(
B
v
)
(
t
)
,
W
W
W
W
i
W
W
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
.
By (H6), we have
(82)
(
-
u
′′
(
t
)
-
v
′′
(
t
)
)
WW
⩽
(
b
11
b
12
b
13
b
14
b
21
b
22
b
23
b
24
)
(
(
B
u
)
(
t
)
+
2
(
B
u
)
′
(
t
)
u
(
t
)
+
2
u
′
(
t
)
(
B
v
)
(
t
)
+
2
(
B
v
)
′
(
t
)
v
(
t
)
+
2
v
′
(
t
)
)
+
(
c
c
)
.
Multiply the above by φ
(
t
)
∶
=
t
e
-
t and use (11) and (12) to obtain:
(83)
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
WW
⩽
(
b
11
b
12
b
13
b
14
b
21
b
22
b
23
b
24
)
(
∫
0
1
‍
u
(
t
)
t
e
t
d
t
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
v
(
t
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
WWW
+
(
c
c
)
WW
⩽
(
b
11
+
b
12
b
13
+
b
14
b
21
+
b
22
b
23
+
b
24
)
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
WWW
+
(
c
c
)
,
so that
(84)
B
3
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
⩽
(
c
c
)
.
Since B
3 is an R
+
2-monotone matrix, we have
(85)
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
⩽
B
3
-
1
(
c
c
)
∶
=
(
c
3
c
4
)
.
Now Lemma 4 implies
(86)
∥
u
∥
0
=
u
(
1
)
⩽
c
3
,
∥
v
∥
0
=
v
(
1
)
⩽
c
4
,
∀
(
u
,
v
)
∈
M
1
.
Furthermore, these estimates lead to
(87)
∥
(
B
u
)
(
t
)
∥
0
=
(
B
u
)
(
1
)
=
∫
0
1
‍
s
u
(
s
)
d
s
⩽
c
3
,
∥
(
B
v
)
(
t
)
∥
0
=
(
B
v
)
(
1
)
=
∫
0
1
‍
s
v
(
s
)
d
s
⩽
c
4
,
∥
(
B
u
)
′
(
t
)
∥
0
=
(
B
u
)
′
(
0
)
=
∫
0
1
‍
u
(
s
)
d
s
⩽
c
3
,
∥
(
B
v
)
′
(
t
)
∥
0
=
(
B
v
)
′
(
0
)
=
∫
0
1
‍
v
(
s
)
d
s
⩽
c
4
.
Therefore, for all (
u
,
v
)
∈
M
4, we have
(88)
(
-
u
′′
(
t
)
-
v
′′
(
t
)
)
WW
⩽
(
b
11
b
12
b
13
b
14
b
21
b
22
b
23
b
24
)
(
(
B
u
)
(
t
)
+
2
(
B
u
)
′
(
t
)
u
(
t
)
+
2
u
′
(
t
)
(
B
v
)
(
t
)
+
2
(
B
v
)
′
(
t
)
v
(
t
)
+
2
v
′
(
t
)
)
WWW
+
(
c
c
)
WW
⩽
(
3
b
11
+
b
12
3
b
13
+
b
14
2
b
12
2
b
14
3
b
21
+
b
22
3
b
23
+
b
24
2
b
22
2
b
24
)
(
c
3
c
4
u
′
(
t
)
v
′
(
t
)
)
WWW
+
(
c
c
)
.
Applying Lemma 3 in [16], we may establish the boundedness of u
′
,
v
′ of M
4. Taking R
>
sup
⁡
{
∥
(
u
,
v
)
∥
:
(
u
,
v
)
∈
M
4
}, we have
(89)
(
u
,
v
)
≠
λ
A
(
u
,
v
)
,
∀
(
u
,
v
)
∈
∂
Ω
R
∩
(
P
×
P
)
,
λ
∈
[
0,1
]
.
Lemma 7 yields
(90)
i
(
A
,
Ω
R
∩
P
2
,
P
2
)
=
1
.
Let
(91)
M
5
∶
=
{
(
u
,
v
)
∈
Ω
¯
r
2
∩
P
2
:
W
(
u
,
v
)
=
A
(
u
,
v
)
+
λ
(
φ
,
φ
)
,
λ
⩾
0
}
,
where φ
(
t
)
∶
=
t
e
-
t and r
>
0 is given in (H7). Next we will prove M
5
⊂
{
0
}. Indeed, if (
u
,
v
)
∈
M
5, then there exists λ
⩾
0 such that (
u
,
v
)
=
A
(
u
,
v
)
+
λ
(
φ
,
φ
), which can be written in the form
(92)
u
(
t
)
=
∫
0
1
k
(
t
,
s
)
f
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWiWWW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWW
+
λ
φ
(
t
)
,
v
(
t
)
=
∫
0
1
k
(
t
,
s
)
g
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWWWiW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWW
+
λ
φ
(
t
)
.
Taking differentiation of the preceding equations twice, we obtain
(93)
-
u
′′
(
t
)
=
f
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
WWWWWW
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
WWWW
+
λ
(
2
-
t
)
e
-
t
,
-
v
′′
(
t
)
=
g
(
t
,
(
B
u
)
(
t
)
,
(
B
u
)
′
(
t
)
,
u
(
t
)
,
u
′
(
t
)
,
W
W
W
W
W
W
(
B
v
)
(
t
)
,
(
B
v
)
′
(
t
)
,
v
(
t
)
,
v
′
(
t
)
)
WWWW
+
λ
(
2
-
t
)
e
-
t
.
By (H7), we have
(94)
(
-
u
′′
(
t
)
-
v
′′
(
t
)
)
WW
⩾
(
b
31
b
32
b
33
b
34
b
41
b
42
b
43
b
44
)
(
(
B
u
)
(
t
)
+
2
(
B
u
)
′
(
t
)
u
(
t
)
+
2
u
′
(
t
)
(
B
v
)
(
t
)
+
2
(
B
v
)
′
(
t
)
v
(
t
)
+
2
v
′
(
t
)
)
.
Multiply the above by φ
(
t
)
∶
=
t
e
-
t and integrate over [
0,1
] and use (11) and (13). Notice that u
′
,
v
′ are decreasing on [
0,1
]; we obtain
(95)
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
WW
⩾
(
b
31
b
32
b
33
b
34
b
41
b
42
b
43
b
44
)
(
∫
0
1
‍
u
(
t
)
t
e
t
d
t
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
v
(
t
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
WW
⩾
(
(
1
-
2
e
-
1
)
b
31
+
b
32
(
1
-
2
e
-
1
)
b
33
+
b
34
(
1
-
2
e
-
1
)
b
41
+
b
42
(
1
-
2
e
-
1
)
b
43
+
b
44
)
W
W
i
W
×
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
,
so that
(96)
B
4
(
∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
)
⩽
(
0
0
)
.
Since B
4 is an R
+
2-monotone matrix, it follows that ∫
0
1
‍
(
u
(
t
)
+
2
u
′
(
t
)
)
t
e
t
d
t
=
0 and ∫
0
1
‍
(
v
(
t
)
+
2
v
′
(
t
)
)
t
e
t
d
t
=
0, so that u
=
v
≡
0. This proves M
5
⊂
{
0
}, as required. Consequently,
(97)
(
u
,
v
)
≠
A
(
u
,
v
)
+
λ
(
φ
,
φ
)
,
∀
(
u
,
v
)
∈
∂
Ω
r
2
∩
P
2
,
λ
⩾
0
.
Now Lemma 6 yields
(98)
i
(
A
,
Ω
r
2
∩
P
2
,
P
2
)
=
0
.
Combining (90) and (98), we arrive at
(99)
i
(
A
,
(
Ω
R
∖
Ω
¯
r
2
)
∩
P
2
,
P
2
)
=
1
.
Therefore A has at least one fixed point on (
Ω
R
∖
Ω
¯
r
2
)
∩
P
2. Hence (1) has at least one positive solution, which completes the proof.Theorem 14.
If (H1), (H2), (H4), (H7), and (H8) hold, then (1) has at least two positive solutions.Proof.
By (H8), we have(100)
∫
0
1
‍
f
(
s
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
)
d
s
<
ω
,
∫
0
1
‍
g
(
s
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
)
d
s
<
ω
,
for all t
∈
[
0,1
] and all (
x
,
y
)
∈
I
ω
8. Consequently, we have, for all (
u
,
v
)
∈
∂
Ω
ω
∩
P
2,
(101)
∥
A
1
(
u
,
v
)
∥
0
=
A
1
(
u
,
v
)
(
1
)
WWWWW
=
∫
0
1
‍
s
f
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
W
W
W
W
W
W
W
W
W
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWWWW
⩽
∫
0
1
f
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
W
W
W
W
W
W
W
i
W
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWWWW
⩽
∫
0
1
‍
f
(
s
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
)
d
s
WWWWW
<
ω
=
∥
(
u
,
v
)
∥
,
∥
A
2
(
u
,
v
)
∥
0
=
A
2
(
u
,
v
)
(
1
)
WWWWW
=
∫
0
1
s
g
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWWWWW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWWWW
⩽
∫
0
1
‍
g
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
WWWWWWWiW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWWWW
⩽
∫
0
1
g
(
s
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
)
d
s
‍
WWWWW
<
ω
=
∥
(
u
,
v
)
∥
,
∥
A
1
(
u
,
v
)
′
∥
0
=
A
1
(
u
,
v
)
′
(
0
)
WWWWW
=
∫
0
1
f
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWWWiW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWWWW
⩽
∫
0
1
f
(
s
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
)
d
s
‍
WWWWW
<
ω
=
∥
(
u
,
v
)
∥
,
∥
A
2
(
u
,
v
)
′
∥
0
=
A
2
(
u
,
v
)
′
(
0
)
WWWWiW
=
∫
0
1
g
(
s
,
(
B
u
)
(
s
)
,
(
B
u
)
′
(
s
)
,
u
(
s
)
,
u
′
(
s
)
,
‍
WWWWWWWiiW
(
B
v
)
(
s
)
,
(
B
v
)
′
(
s
)
,
v
(
s
)
,
v
′
(
s
)
)
d
s
WWWWW
⩽
∫
0
1
g
(
s
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
,
ω
)
d
s
‍
WWWWW
<
ω
=
∥
(
u
,
v
)
∥
.
The preceding inequalities imply
(102)
∥
A
(
u
,
v
)
∥
=
∥
(
A
1
(
u
,
v
)
,
A
2
(
u
,
v
)
)
∥
<
ω
=
∥
(
u
,
v
)
∥
;
thus
(103)
(
u
,
v
)
≠
λ
A
(
u
,
v
)
,
∀
(
u
,
v
)
∈
∂
Ω
ω
∩
P
2
,
0
⩽
λ
⩽
1
.
Now Lemma 7 yields
(104)
i
(
A
,
Ω
ω
∩
P
2
,
P
2
)
=
1
.
By (H2), (H4), and (H7), we find that (51) and (98) hold. Note that we can choose R
>
ω
>
r
2 in (51) and (98) (see the proofs of Theorems 11 and 13). Combining (51), (98), and (104), we obtain
(105)
i
(
A
,
(
Ω
R
∖
Ω
¯
ω
)
∩
P
2
,
P
2
)
=
0
-
1
=
-
1
,
i
(
A
,
(
Ω
ω
∖
Ω
¯
r
2
)
∩
P
2
,
P
2
)
=
1
-
0
=
1
.
Therefore, A has at least two fixed points, with one on (
Ω
R
∖
Ω
¯
ω
)
∩
P
2 and the other on (
Ω
ω
∖
Ω
¯
r
2
)
∩
P
2. Hence (1) has at least two positive solutions. This completes the proof.


## 4. Examples

Example 15 .
Let(106)
f
(
t
,
x
,
y
)
∶
=
(
∑
i
=
1
4
a
i
x
i
+
∑
i
=
1
4
a
i
+
4
y
i
)
p
,
g
(
t
,
x
,
y
)
∶
=
(
∑
i
=
1
4
b
i
x
i
‍
+
∑
i
=
1
4
‍
b
i
+
4
y
i
)
q
,
where a
i
,
b
i
>
0, 1
<
p
<
2, and 1
<
q
<
2. Then (H1), (H2), (H4), and (H5) hold. By Theorem 11, (1) has at least one positive solution.Example 16.
Let(107)
f
(
t
,
x
,
y
)
∶
=
(
∑
i
=
1
4
‍
c
i
x
i
+
∑
i
=
1
4
‍
c
i
+
4
y
i
)
p
,
g
(
t
,
x
,
y
)
∶
=
(
∑
i
=
1
4
‍
d
i
x
i
+
∑
i
=
1
4
‍
d
i
+
4
y
i
)
q
,
where c
i
,
d
i
>
0, 0
<
p
<
1, and 0
<
q
<
1. Then (H1), (H6), and (H7) hold. By Theorem 13, (1) has at least one positive solution.Example 17.
Let(108)
f
(
t
,
x
,
y
)
∶
=
λ
(
∑
i
=
1
4
‍
a
i
x
i
+
∑
i
=
1
4
a
i
+
4
y
i
)
p
1
+
λ
(
∑
i
=
1
4
c
i
x
i
+
∑
i
=
1
4
c
i
+
4
y
i
)
p
2
,
g
(
t
,
x
,
y
)
∶
=
λ
(
∑
i
=
1
4
b
i
x
i
+
∑
i
=
1
4
b
i
+
4
y
i
)
q
1
+
λ
(
∑
i
=
1
4
d
i
x
i
+
∑
i
=
1
4
d
i
+
4
y
i
)
q
2
,
where a
i
,
b
i
,
c
i
,
d
i
>
0, 0
<
p
1
<
1, p
2
>
1, 0
<
q
1
<
1, and q
2
>
1; λ
>
0 is sufficiently small. Then (H1), (H2), (H4), (H7), and (H8) hold. By Theorem 14, (1) has at least two positive solutions.



---

*Source: 717290-2014-06-18.xml*