# Positive Solutions and Mann Iterative Algorithms for a Nonlinear Three-Dimensional Difference System

Authors: Zeqing Liu; Yan Lu; Shin Min Kang; Young Chel Kwun
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The existence of uncountably many positive solutions and Mann iterative approximations for a nonlinear three-dimensional difference system are proved by using the Banach fixed point theorem. Four illustrative examples are also provided.


---

## Body

## 1. Introduction

In recent years, the oscillation, nonoscillation, asymptotic behavior, existence and multiplicity of solutions, bounded solutions, unbounded solutions, positive solutions, and nonoscillatory solutions for some two- and three-dimensional difference systems have been studied by many authors and a significant number of important results have been found; see [1–12] and the references therein.In order to solve the problem that the Picard iteration fails to converge under some conditions, Mann [13] introduced a modified iteration, which is now called Mann iterative scheme. It is well known that the Mann iterative schemes are often used in the fields of nonlinear differential equations, nonlinear equations, nonlinear mappings, optimization, variational inequalities, nonlinear analysis, and so forth.Note that the difference systems in [1–12] are as follows:
(1)
Δ
x
n
=
b
n
g
(
y
n
)
,
n
≥
n
0
,
Δ
y
n
=
-
a
n
f
(
x
n
)
,
n
≥
n
0
,
Δ
x
n
=
a
n
f
(
y
n
)
,
n
≥
n
0
,
Δ
y
n
=
b
n
g
(
x
n
)
,
n
≥
n
0
,
Δ
x
n
=
b
n
g
(
y
n
)
,
n
≥
n
0
,
Δ
y
n
-
1
=
-
a
n
f
(
x
n
)
,
n
≥
n
0
,
Δ
x
n
=
b
n
g
(
y
n
)
,
n
≥
n
0
,
Δ
y
n
=
-
a
n
f
(
x
n
)
+
r
n
,
n
≥
n
0
,
where n
0
∈
ℕ
0,  f
,
g
∈
C
(
ℝ
,
ℝ
),  {
a
n
}
n
∈
ℕ
n
0 and {
b
n
}
n
∈
ℕ
n
0 are nonnegative sequences, and {
r
n
}
n
∈
ℕ
n
0 is a real sequence with ∑
i
=
n
0
∞
‍
|
r
i
|
<
∞:
(2)
Δ
x
n
=
b
n
g
(
y
n
)
,
n
≥
n
0
,
Δ
y
n
-
1
=
-
a
n
f
(
x
n
)
+
r
n
,
n
≥
n
0
,
where n
0
∈
ℕ
0, f
,
g
∈
C
(
ℝ
,
ℝ
),  {
a
n
}
n
∈
ℕ
n
0 and {
r
n
}
n
∈
ℕ
n
0 are real sequences with ∑
i
=
n
0
∞
‍
|
r
i
|
<
∞, and   {
b
n
}
n
∈
ℕ
n
0 is a nonnegative sequence:
(3)
Δ
(
x
n
+
p
1
n
x
n
-
τ
1
)
+
f
1
(
n
,
x
a
1
n
,
x
a
h
n
,
y
b
1
n
,
y
b
k
n
)
=
q
1
n
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
h
h
h
h
h
l
n
≥
n
0
,
Δ
(
y
n
+
p
2
n
y
n
-
τ
2
)
+
f
2
(
n
,
x
c
1
n
,
x
c
h
n
,
y
d
1
n
,
y
d
k
n
)
=
q
2
n
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
h
h
h
h
h
h
h
l
n
≥
n
0
,
where h
,
τ
i
,
k
,
n
0
∈
ℕ,  {
p
i
n
}
n
∈
ℕ
n
0, {
q
i
n
}
n
∈
ℕ
n
0
⊂  ℝ, f
i
∈
C
(
ℕ
n
0
×
ℝ
h
+
k
,
ℝ
), {
h
l
n
}
n
∈
ℕ
n
0, {
f
l
n
}
n
∈
ℕ
n
0
⊆
ℕ for i
∈
Λ
2, and {
a
l
n
}
n
∈
ℕ
n
0,  {
b
l
n
}
n
∈
ℕ
n
0,  {
c
l
n
}
n
∈
ℕ
n
0,  {
d
l
n
}
n
∈
ℕ
n
0
⊂  ℤ with
(4)
lim
n
→
∞
a
l
n
=
lim
n
→
∞
b
l
n
=
lim
n
→
∞
c
l
n
=
lim
n
→
∞
d
l
n
=
+
∞
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
l
h
l
l
∈
Λ
k
,
Δ
x
n
=
a
n
f
(
y
n
-
l
)
,
n
≥
n
0
,
Δ
y
n
=
b
n
g
(
z
n
-
m
)
,
n
≥
n
0
,
Δ
z
n
=
δ
c
n
h
(
x
n
-
k
)
,
n
≥
n
0
.
However, to the best of our knowledge, there exists no result in the literature dealing with the following nonlinear three-dimensional difference system:
(5)
Δ
2
(
x
1
n
+
b
1
n
x
1
(
n
-
τ
1
)
)
hh
+
f
1
(
n
,
x
1
r
11
n
,
…
,
x
1
r
1
k
n
,
x
2
s
11
n
,
…
,
x
2
s
1
k
n
,
hhhhhhh
x
3
t
11
n
,
…
,
x
3
t
1
k
n
)
=
c
1
n
,
n
∈
ℕ
n
0
,
Δ
2
(
x
2
n
+
b
2
n
x
2
(
n
-
τ
2
)
)
hh
+
f
2
(
n
,
x
1
r
21
n
,
…
,
x
1
r
2
k
n
,
x
2
s
21
n
,
…
,
x
2
s
2
k
n
,
hhhhhhh
x
3
t
21
n
,
…
,
x
3
t
2
k
n
)
=
c
2
n
,
n
∈
ℕ
n
0
,
Δ
2
(
x
3
n
+
b
3
n
x
3
(
n
-
τ
3
)
)
hh
+
f
3
(
n
,
x
1
r
31
n
,
…
,
x
1
r
3
k
n
,
x
2
s
31
n
,
…
,
x
2
s
3
k
n
,
hhhhhhh
x
3
t
31
n
,
…
,
x
3
t
3
k
n
)
=
c
3
n
,
n
∈
ℕ
n
0
,
which is abbreviated as, for convenience,
(6)
Δ
2
(
x
a
n
+
b
a
n
x
a
(
n
-
τ
a
)
)
hh
+
f
a
(
n
,
x
1
r
a
1
n
,
…
,
x
1
r
a
k
n
,
x
2
s
a
1
n
,
…
,
x
2
s
a
k
n
,
hhhhhhh
x
3
t
a
1
n
,
…
,
x
3
t
a
k
n
)
=
c
a
n
,
(
n
,
a
)
∈
ℕ
n
0
×
Λ
3
,
where n
0
∈
ℕ
0,  τ
a
,
k
∈
ℕ,  {
b
a
n
}
n
∈
ℕ
n
0,  {
c
a
n
}
n
∈
ℕ
n
0
⊂
ℝ, f
a
∈
C
(
ℕ
n
0
×
ℝ
3
k
,
ℝ
), and {
r
a
l
n
,
s
a
l
n
,
t
a
l
n
:
n
∈
ℕ
n
0, l
∈
Λ
k
}
⊂
ℕ with
(7)
lim
n
→
∞
r
a
l
n
=
lim
n
→
∞
s
a
l
n
=
lim
n
→
∞
t
a
l
n
=
+
∞
,
(
l
,
a
)
∈
Λ
k
×
Λ
3
.The main purpose of this paper is to study solvability and convergence of the Mann iterative schemes for the system (6). Sufficient conditions for the existence of uncountably many positive solutions of the system (6) and convergence of the Mann iterative schemes relative to these positive solutions are provided by utilizing the Banach fixed point theorem. Four illustrative examples are given.


## 2. Preliminaries

Throughout this paper, we assume thatΔ is the forward difference operator defined by Δ
x
n
=
x
n
+
1
-
x
n, Δ
2
x
n
=
Δ
(
Δ
x
n
), ℝ
=
(
-
∞
,
+
∞
), ℝ
+
=
[
0
,
+
∞
), ℝ
-
=
(
-
∞
,
0
], and ℤ,  ℕ, and ℕ
0 stand for the sets of all integers, positive integers, and nonnegative integers, respectively, as
(8)
Λ
n
0
=
{
1,2
,
…
,
n
0
}
,
ℕ
n
0
=
{
n
:
n
∈
ℕ
0
with
n
≥
n
0
}
,
n
0
∈
ℕ
0
,
β
=
min
{
n
0
-
τ
a
,
inf
{
r
a
l
n
,
s
a
l
n
,
t
a
l
n
:
n
∈
ℕ
n
0
,
l
∈
Λ
k
}
:
hhhhh
a
∈
Λ
3
{
r
a
l
n
,
s
a
l
n
,
t
a
l
n
:
n
∈
ℕ
n
0
,
l
∈
Λ
k
}
}
∈
ℕ
,
P
a
n
=
max
{
r
a
l
n
,
s
a
l
n
,
t
a
l
n
:
l
∈
Λ
k
}
,
(
n
,
a
)
∈
ℕ
n
0
×
Λ
3
.
Let l
β
∞ denote the Banach space of all sequences in ℕ
β with norm
(9)
∥
x
∥
=
sup
n
∈
ℕ
β
|
x
n
n
|
<
+
∞
for
each
x
=
{
x
n
}
n
∈
ℕ
β

∈
l
β
∞
and put
(10)
d
^
=
{
n
d
}
n
∈
ℕ
β
∈
l
β
∞
for
each
d
∈
ℝ
,
Ω
(
d
^
,
D
)
=
{
x
=
{
x
n
}
n
∈
ℕ
β

∈
l
β
∞
:
∥
x
-
d
^
∥
≤
D
}
hhhhhhhhl
for
each
(
D
,
d
^
)
∈
(
ℝ
+
∖
{
0
}
)
×
l
β
∞
.
It is easy to see that for each (
D
1
,
D
2
,
D
3
,
d
^
1
,
d
^
2
,
d
^
3
)
∈
(
ℝ
+
∖
{
0
}
)
3
×
(
l
β
∞
)
3, Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) is a nonempty closed convex subset of the Banach space (
l
β
∞
)
3 with norm ∥
(
x
,
y
,
z
)
∥
1
=
max
{
∥
x
∥
,
∥
y
∥
,
∥
z
∥
} for each (
x
,
y
,
z
)
∈
(
l
β
∞
)
3.By a solution of the system (6), we mean a three-dimensional sequence {
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β
∈
(
l
β
∞
)
3 with a positive integer T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β such that (6) holds for all n
≥
T.LetK be a nonempty convex subset of a Banach space X and let S
:
K
→
K be a mapping. For any given x
0
∈
K, the sequence {
x
m
}
m
∈
ℕ
0 defined by
(11)
x
m
+
1
=
(
1
-
α
m
)
x
m
+
α
m
S
x
m
,
m
∈
ℕ
0
,
where {
α
m
}
m
∈
ℕ
0 is a sequence in [
0,1
] with certain condition, is called the Mann iterative scheme.Lemma 1.
Let{
p
t
}
t
∈
ℕ be a nonnegative sequence and n
,
τ
∈
ℕ.(i)
If∑
t
=
n
∞
‍
t
p
t
<
+
∞, then
(12)
∑
s
=
n
∞
‍
∑
t
=
s
∞
‍
p
t
=
∑
t
=
n
∞
‍
(
t
-
n
+
1
)
p
t
≤
∑
t
=
n
∞
‍
t
p
t
.
(ii)
If∑
t
=
n
+
τ
∞
‍
t
2
p
t
<
+
∞, then
(13)
∑
i
=
1
∞
‍
∑
s
=
n
+
i
τ
∞
‍
∑
t
=
s
∞
‍
p
t
≤
∑
t
=
n
+
τ
∞
‍
t
2
p
t
.Proof.
It is easy to see that(14)
∑
s
=
n
∞
‍
∑
t
=
s
∞
‍
p
t
=
∑
t
=
n
∞
‍
p
t
+
∑
t
=
n
+
1
∞
‍
p
t
+
∑
t
=
n
+
2
∞
‍
p
t
+
⋯
=
∑
t
=
n
∞
‍
(
1
+
t
-
n
)
p
t
≤
∑
t
=
n
∞
‍
t
p
t
,
which yields (12). Note that
(15)
∑
i
=
1
∞
‍
∑
s
=
n
+
i
τ
∞
‍
∑
t
=
s
∞
‍
p
t
=
∑
i
=
1
∞
‍
(
∑
t
=
n
+
i
τ
∞
‍
p
t
+
∑
t
=
n
+
1
+
i
τ
∞
‍
p
t
+
∑
t
=
n
+
2
+
i
τ
∞
‍
p
t
+
⋯
)
=
∑
i
=
1
∞
‍
∑
t
=
n
+
i
τ
∞
‍
(
1
+
t
-
n
-
i
τ
)
p
t
≤
∑
i
=
1
∞
‍
∑
t
=
n
+
i
τ
∞
‍
t
p
t
=
∑
t
=
n
+
τ
∞
‍
t
p
t
+
∑
t
=
n
+
2
τ
∞
‍
t
p
t
+
∑
t
=
n
+
3
τ
∞
‍
t
p
t
+
⋯
=
∑
t
=
n
+
τ
∞
‍
(
1
+
[
t
-
n
-
τ
τ
]
)
t
p
t
≤
∑
t
=
n
+
τ
∞
‍
(
1
+
t
-
n
-
τ
τ
)
t
p
t
≤
∑
t
=
n
+
τ
∞
‍
t
-
n
τ
t
p
t
≤
∑
t
=
n
+
τ
∞
‍
t
2
p
t
,
where [
(
t
-
n
-
τ
)
/
τ
] denotes the largest integral number not exceeding (
t
-
n
-
τ
)
/
τ. Hence (13) holds. This completes the proof.


## 3. Uncountably Many Positive Solutions and Mann Iterative Schemes

Our main results are as follows.Theorem 2.
Assume that there exist constantsn
1
∈
ℕ
n
0, d
a
,
D
a
∈
ℝ
+
∖
{
0
} and nonnegative sequences {
F
a
n
}
n
∈
ℕ
n
0 and {
U
a
n
}
n
∈
ℕ
n
0 satisfying
(16)
d
a
>
D
a
,
b
a
n
=
-
1
,
(
n
,
a
)
∈
ℕ
n
1
×
Λ
3
;
(17)
|
f
a
(
n
,
u
1
,
…
,
u
k
,
v
1
,
…
,
v
k
,
w
1
,
…
,
w
k
)
-
f
a
(
n
,
u
¯
1
,
…
,
u
¯
k
,
v
¯
1
,
…
,
v
¯
k
,
w
¯
1
,
…
,
w
¯
k
)
|
≤
U
a
n
max
{
|
u
l
-
u
¯
l
|
,
|
v
l
-
v
¯
l
|
,
|
w
l
-
w
¯
l
|
:
l
∈
Λ
k
}
,
(
n
,
u
l
,
u
¯
l
,
v
l
,
v
¯
l
,
w
l
,
w
¯
l
)
∈
ℕ
n
0
×
Π
w
=
1
3
[
d
w
-
D
w
,
d
w
+
D
w
]
2
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
(
l
,
a
)
∈
Λ
k
×
Λ
3
;
(18)
h
h
h
l
|
f
a
(
n
,
u
1
,
…
,
u
k
,
v
1
,
…
,
v
k
,
w
1
,
…
,
w
k
)
|
≤
F
a
n
,
(
n
,
u
l
,
v
l
,
w
l
)
∈
ℕ
n
0
×
Π
w
=
1
3
[
d
w
-
D
w
,
d
w
+
D
w
]
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
h
(
l
,
a
)
∈
Λ
k
×
Λ
3
;
(19)
lim
n
→
∞
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
max
{
F
a
j
,
|
c
a
j
|
,
U
a
j
P
a
j
}
=
0
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
a
∈
Λ
3
.
Then one has the following.
(a) For any(
L
1
,
L
2
,
L
3
)
∈
Π
w
=
1
3
(
d
w
-
D
w
,
d
w
+
D
w
), there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1 + τ
2
+
τ
3
+
β such that, for each (
x
1
0
,
x
2
0
,
x
3
0
)
=
{
(
x
1
n
0
,
x
2
n
0
,
x
3
n
0
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), the Mann iterative sequence {
(
x
1
m
,
x
2
m
,
x
3
m
)
}
m
∈
ℕ
0
=
{
{
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
}
n
∈
ℕ
β
}
m
∈
ℕ
0
⊂
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) generated by the scheme(20)
x
a
n
m
+
1
=
{
(
1
-
α
m
)
x
a
n
m
+
α
m
{
n
L
a
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
}
,
α
m
{
n
L
a
+
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
}
n
≥
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
,
(
1
-
α
m
)
x
a
n
m
+
α
m
{
n
L
a
+
n
T
∑
q
=
1
∞
‍
∑
i
=
T
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
}
,
α
m
{
n
L
a
+
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
}
β
≤
n
<
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
converges to a positive solution(
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) of the system (6) with lim
n
→
∞
(
z
a
n
/
n
)
=
L
a for each a
∈
Λ
3 and has the following error estimate:(21)
∥
(
x
1
m
+
1
,
x
2
m
+
1
,
x
3
m
+
1
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
≤
e
-
(
1
-
θ
)
∑
n
=
0
m
α
n
∥
(
x
1
0
,
x
2
0
,
x
3
0
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
m
∈
ℕ
0
,
where {
α
m
}
m
∈
ℕ
0 is an arbitrary sequence in [
0,1
] such that
(22)
∑
m
=
0
∞
‍
α
m
=
+
∞
.
(b) The system (6) has uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Proof.
Firstly, we prove that (a) holds. Put(
L
1
,
L
2
,
L
3
)
∈
Π
w
=
1
3
(
d
w
-
D
w
,
d
w
+
D
w
). It follows from (19) that there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β satisfying
(23)
θ
=
max
{
θ
a
:
a
∈
Λ
3
}
,
where
(24)
θ
a
=
1
T
∑
q
=
1
∞
‍
∑
i
=
T
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
U
a
j
P
a
j
,
a
∈
Λ
3
;
(25)
1
T
∑
q
=
1
∞
‍
∑
i
=
T
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
<
D
a
-
|
L
a
-
d
a
|
,
hlhhhhhhhhhhhhhhhhhhhhhhhhhlh
a
∈
Λ
3
.
Define mappings A
L
1
,
A
L
2
,
A
L
3
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
l
β
∞ and S
L
1
,
L
2
,
L
3
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
(
l
β
∞
)
3 by(26)
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
=
{
n
L
a
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
]
,
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
]
n
≥
T
,
a
∈
Λ
3
,
n
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
,
β
≤
n
<
T
,
a
∈
Λ
3
,
(27)
S
L
1
,
L
2
,
L
3
(
x
1
,
x
2
,
x
3
)
=
(
A
L
1
(
x
1
,
x
2
,
x
3
)
,
A
L
2
(
x
1
,
x
2
,
x
3
)
,
A
L
3
(
x
1
,
x
2
,
x
3
)
)
,for each (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).
Now we assert that(28)
A
L
a
(
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
)
⊆
Ω
(
d
^
a
,
D
a
)
,
a
∈
Λ
3
,
(29)
S
L
1
,
L
2
,
L
3
(
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
)
⊆
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
.
Using (18), (25), and (26), we get that, for any (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) and a
∈
Λ
3,
(30)
∥
A
L
a
(
x
1
,
x
2
,
x
3
)
-
d
^
a
∥
=
sup
n
∈
ℕ
β
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
-
n
d
a
n
|
=
max
{
sup
β
≤
n
<
T
|
n
T
·
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
n
-
d
a
|
sup
n
≥
T
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
n
-
d
a
|
,
hhhhhhl
sup
β
≤
n
<
T
|
n
T
·
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
n
-
d
a
|
}
=
sup
n
≥
T
|
∑
i
=
n
+
q
τ
a
∞
‍
L
a
-
d
a
hhhhhl
+
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
hhhhhhhhhl
+
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
]
∑
i
=
n
+
q
τ
a
∞
‍
|
≤
sup
n
≥
T
(
|
L
a
-
d
a
|
+
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
)
≤
|
L
a
-
d
a
|
+
1
T
∑
q
=
1
∞
‍
∑
i
=
T
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
≤
|
L
a
-
d
a
|
+
D
a
-
|
L
a
-
d
a
|
=
D
a
,
which implies (28). Thus (29) follows from (27) and (28).
Next we claim that(31)
∥
A
L
a
(
x
1
,
x
2
,
x
3
)
-
A
L
a
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
≤
θ
a
max
{
∥
x
w
-
x
¯
w
∥
:
w
∈
Λ
3
}
,
a
∈
Λ
3
,
(32)
∥
S
L
1
,
L
2
,
L
3
(
x
1
,
x
2
,
x
3
)
-
S
L
1
,
L
2
,
L
3
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
≤
θ
∥
(
x
1
,
x
2
,
x
3
)
-
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
,
for all (
x
1
,
x
2
,
x
3
), (
x
¯
1
,
x
¯
2
,
x
¯
3
)
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
). In view of (17), (23), (26), and (27), we deduce that, for any (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β, (
x
¯
1
,
x
¯
2
,
x
¯
3
)
=
{
(
x
¯
1
n
,
x
¯
2
n
,
x
¯
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), and a
∈
Λ
3,(33)
∥
A
L
a
(
x
1
,
x
2
,
x
3
)
-
A
L
a
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
=
sup
n
∈
ℕ
β
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
-
A
L
a
(
x
¯
1
n
,
x
¯
2
n
,
x
¯
3
n
)
n
|
=
max
{
sup
n
≥
T
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
-
A
L
a
(
x
¯
1
n
,
x
¯
2
n
,
x
¯
3
n
)
n
|
,
sup
β
≤
n
<
T
|
n
T
·
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
-
A
L
a
(
x
¯
1
T
,
x
¯
2
T
,
x
¯
3
T
)
n
|
}
=
sup
n
≥
T
1
n
|
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
-
f
a
(
j
,
x
¯
1
r
a
1
j
,
…
,
x
¯
1
r
a
k
j
,
x
¯
2
s
a
1
j
,
…
,
x
¯
2
s
a
k
j
,
x
¯
3
t
a
1
j
,
…
,
x
¯
3
t
a
k
j
)
]
∑
q
=
1
∞
‍
|
≤
sup
n
≥
T
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
(
U
a
j
max
{
|
x
1
r
a
l
j
-
x
¯
1
r
a
l
j
|
,
|
x
2
s
a
l
j
-
x
¯
2
s
a
l
j
|
,
|
x
3
t
a
l
j
-
x
¯
3
t
a
l
j
|
:
l
∈
Λ
k
}
)
≤
1
T
∑
q
=
1
∞
‍
∑
i
=
T
+
q
τ
a
∞
‍
∑
j
=
i
∞
(
U
a
j
P
a
j
max
{
|
x
1
r
a
l
j
-
x
¯
1
r
a
l
j
|
r
a
l
j
,
|
x
2
s
a
l
j
-
x
¯
2
s
a
l
j
|
s
a
l
j
,
|
x
3
t
a
l
j
-
x
¯
3
t
a
l
j
|
t
a
l
j
:
l
∈
Λ
k
}
)
‍
≤
θ
a
max
{
∥
x
w
-
x
¯
w
∥
:
w
∈
Λ
3
}
=
θ
a
∥
(
x
1
,
x
2
,
x
3
)
-
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
,
∥
S
L
1
,
L
2
,
L
3
(
x
1
,
x
2
,
x
3
)
-
S
L
1
,
L
2
,
L
3
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
=
∥
(
A
L
1
(
x
1
,
x
2
,
x
3
)
,
A
L
2
(
x
1
,
x
2
,
x
3
)
,
A
L
3
(
x
1
,
x
2
,
x
3
)
)
-
(
A
L
1
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
,
A
L
2
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
,
A
L
3
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
)
∥
1
=
max
{
∥
A
L
a
(
x
1
,
x
2
,
x
3
)
-
A
L
a
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
:
a
∈
Λ
3
}
≤
max
{
θ
a
∥
(
x
1
,
x
2
,
x
3
)
-
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
:
a
∈
Λ
3
}
=
θ
∥
(
x
1
,
x
2
,
x
3
)
-
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
,which yield (31) and (32). Clearly, (29) and (32) ensure that S
L
1
,
L
2
,
L
3 is a contraction mapping in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) and it has a unique fixed point (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), that is,
(34)
z
a
n
=
n
L
a
hhhl
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
h
z
2
s
a
k
j
,
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
]
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
(
n
,
a
)
∈
ℕ
T
×
Λ
3
,
z
a
(
n
-
τ
a
)
=
(
n
-
τ
a
)
L
a
hhhl
+
∑
q
=
1
∞
‍
∑
i
=
n
+
(
q
-
1
)
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
hhhhhhhhhhhhhhhhhhhhhl
z
2
s
a
k
j
,
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
]
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhlh
(
n
,
a
)
∈
ℕ
T
+
τ
a
×
Λ
3
,
which guarantee that
(35)
Δ
2
(
z
a
n
-
z
a
(
n
-
τ
a
)
)
=
-
f
a
(
n
,
z
1
r
a
1
n
,
…
,
z
1
r
a
k
n
,
z
2
s
a
1
n
,
…
,
z
2
s
a
k
n
,
z
3
t
a
1
n
,
…
,
z
3
t
a
k
n
)
+
c
a
n
,
(
n
,
a
)
∈
ℕ
T
+
τ
1
+
τ
2
+
τ
3
×
Λ
3
,
which together with (16) means that (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β is a positive solution of the system (6) in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
). Observe that (18) and (19) give that, for each a
∈
Λ
3,
(36)
|
z
a
n
n
-
L
a
|
hhh
=
|
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
hhhhhhhhhhhhhhhhhhhh
z
2
s
a
k
j
,
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
]
∑
j
=
i
∞
‍
|
hhh
≤
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
⟶
0
as
n
⟶
∞
,
which implies that
(37)
lim
n
→
∞
z
a
n
n
=
L
a
,
a
∈
Λ
3
.
By means of (20), (26), and (31), we infer that
(38)
|
x
a
n
m
+
1
-
z
a
n
|
n
=
|
-
c
a
j
]
-
z
a
n
∑
q
=
1
∞
‍
)
x
a
n
m
-
z
a
n
n
(
1
-
α
m
)
x
a
n
m
-
z
a
n
n
+
α
m
n
hhh
×
(
n
L
a
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
‍
L
a
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
[
f
a
(
j
,
x
2
m
,
…
,
‍
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
L
a
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
[
f
a
(
j
,
x
2
m
,
,
‍
-
c
a
j
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
]
-
z
a
n
∑
q
=
1
∞
‍
)
x
a
n
m
-
z
a
n
n
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
h
h
+
α
m
·
1
n
|
A
L
a
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
-
A
L
a
(
z
1
n
,
z
2
n
,
z
3
n
)
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
hh
+
α
m
∥
A
L
a
(
x
1
m
,
x
2
m
,
x
3
m
)
-
A
L
a
(
z
1
,
z
2
,
z
3
)
∥
≤
(
1
-
(
1
-
θ
a
)
α
m
)
∥
x
a
m
-
z
a
∥
,
n
≥
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
,
|
x
a
n
m
+
1
-
z
a
n
|
n
=
|
-
c
a
j
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
]
-
z
a
n
∑
i
=
T
+
q
τ
a
∞
‍
)
x
a
n
m
-
z
a
n
n
(
1
-
α
m
)
x
a
n
m
-
z
a
n
n
+
α
m
n
h
h
h
×
(
∑
i
=
T
+
q
τ
a
∞
‍
n
L
a
+
n
T
∑
q
=
1
∞
∑
i
=
T
+
q
τ
a
∞
∑
j
=
i
∞
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
L
a
h
h
h
h
h
h
h
h
h
h
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
h
h
h
h
n
L
a
h
h
l
n
T
n
T
n
T
n
T
n
T
n
T
+
n
T
-
c
a
j
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
]
-
z
a
n
∑
i
=
T
+
q
τ
a
∞
‍
)
x
a
n
m
-
z
a
n
n
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
n
·
n
T
h
h
×
|
∑
q
=
1
∞
‍
T
L
a
+
∑
q
=
1
∞
∑
i
=
T
+
q
τ
a
∞
∑
j
=
i
∞
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
L
a
+
∑
q
=
1
∞
∑
i
=
T
+
q
τ
a
∞
∑
j
=
i
∞
[
f
a
(
j
,
x
1
r
a
1
j
m
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
L
a
+
∑
q
=
1
∞
∑
i
=
T
+
q
τ
a
∞
∑
j
=
i
∞
[
(
x
1
r
a
1
j
m
,
-
c
a
j
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
]
-
A
L
a
(
z
1
T
,
z
2
T
,
z
3
T
)
∑
q
=
1
∞
‍
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
·
1
T
|
A
L
a
(
x
1
T
m
,
x
2
T
m
,
x
3
T
m
)
-
A
L
a
(
z
1
T
,
z
2
T
,
z
3
T
)
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
∥
A
L
a
(
x
1
m
,
x
2
m
,
x
3
m
)
-
A
L
a
(
z
1
,
z
2
,
z
3
)
∥
≤
(
1
-
(
1
-
θ
a
)
α
m
)
∥
x
a
m
-
z
a
∥
,
h
h
h
h
h
h
h
h
h
h
h
h
h
l
h
h
h
h
β
≤
n
<
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
,
which imply that
(39)
∥
(
x
1
m
+
1
,
x
2
m
+
1
,
x
3
m
+
1
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
=
max
{
sup
n
∈
ℕ
β
|
x
a
n
m
+
1
-
z
a
n
n
|
:
a
∈
Λ
3
}
=
max
{
sup
n
≥
T
|
x
a
n
m
+
1
-
z
a
n
n
|
,
sup
β
≤
n
<
T
|
x
a
n
m
+
1
-
z
a
n
n
|
:
a
∈
Λ
3
}
≤
max
{
(
1
-
(
1
-
θ
a
)
α
m
)
∥
x
a
m
-
z
a
∥
:
a
∈
Λ
3
}
≤
(
1
-
(
1
-
θ
)
α
m
)
∥
(
x
1
m
,
x
2
m
,
x
3
m
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
≤
e
-
(
1
-
θ
)
α
m
∥
(
x
1
m
,
x
2
m
,
x
3
m
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
≤
e
-
(
1
-
θ
)
∑
n
=
0
m
α
n
∥
(
x
1
0
,
x
2
0
,
x
3
0
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
,
m
∈
ℕ
0
;
that is, (21) holds. It follows from (21) and (22) that lim
m
→
∞
∥
(
x
1
m
,
x
2
m
,
x
3
m
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
=
0.
Secondly, we show that(
b
) holds. Let (
L
11
,
L
21
,
L
31
), (
L
12
,
L
22
,
L
32
)
∈
Π
w
=
1
3
(
d
w
-
D
w
,
d
w
+
D
w
) with max
{
|
L
a
1
-
L
a
2
|
:
a
∈
Λ
3
}
>
0. As in the proof of (
a
), we infer similarly that, for each l
∈
Λ
2, there exist constants θ
l
,
θ
1
l
,
θ
2
l
,
θ
3
l
∈
(
0,1
), T
l
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β and mappings A
L
1
l
,
A
L
2
l
,
A
L
3
l
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
l
β
∞ and S
L
1
l
,
L
2
l
,
L
3
l
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
(
l
β
∞
)
3 satisfying (23)~(32), where θ, θ
1, θ
2, θ
3, T, L
1, L
2, L
3, A
L
1, A
L
2, A
L
3, and S
L
1
,
L
2
,
L
3 are replaced by θ
l, θ
1
l, θ
2
l, θ
3
l,  T
l, L
1
l, L
2
l, L
3
l, A
L
1
l, A
L
2
l, A
L
3
l, and S
L
1
l
,
L
2
l
,
L
3
l, respectively, and the contraction mapping S
L
1
l
,
L
2
l
,
L
3
l has a unique fixed point (
z
1
l
,
z
2
l
,
z
3
l
)
=
{
(
z
1
n
l
,
z
2
n
l
,
z
3
n
l
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), which is a positive solution of the system (6); that is,
(40)
z
a
n
l
=
n
L
a
l
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
l
,
…
,
z
1
r
a
k
j
l
,
z
2
s
a
1
j
l
,
…
,
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
hhhhh
‍
z
2
s
a
k
j
l
,
z
3
t
a
1
j
l
,
…
,
z
3
t
a
k
j
l
)
-
c
a
j
]
,
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
hhhhh
5,55
‍
(
n
,
l
,
a
)
∈
ℕ
T
1
×
Λ
2
×
Λ
3
.
On account of (17), (23), and (40), we conclude that(41)
|
z
a
n
1
-
z
a
n
2
n
|
=
|
∑
q
=
1
∞
‍
L
a
1
-
L
a
2
+
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
1
,
…
,
z
1
r
a
k
j
1
,
z
2
s
a
1
j
1
,
…
,
z
2
s
a
k
j
1
,
z
3
t
a
1
j
1
,
…
,
z
3
t
a
k
j
1
)
L
a
1
L
a
2
+
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
∑
j
=
i
∞
‍
∑
j
=
i
∞
‍
5
-
f
a
(
j
,
z
1
r
a
1
j
2
,
…
,
z
1
r
a
k
j
2
,
z
2
s
a
1
j
2
,
…
,
z
2
s
a
k
j
2
,
z
3
t
a
1
j
2
,
…
,
z
3
t
a
k
j
2
)
]
∑
i
=
n
+
q
τ
a
∞
‍
|
≥
|
L
a
1
-
L
a
2
|
-
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
(
U
a
j
max
{
|
z
1
r
a
l
j
1
-
z
1
r
a
l
j
2
|
,
|
z
2
s
a
l
j
1
-
z
2
s
a
l
j
2
|
,
|
z
3
t
a
l
j
1
-
z
3
t
a
l
j
2
|
:
l
∈
Λ
k
}
)
≥
|
L
a
1
-
L
a
2
|
-
1
max
{
T
1
,
T
2
}
×
∑
q
=
1
∞
‍
∑
i
=
max
{
T
1
,
T
2
}
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
(
U
a
j
P
a
j
max
{
|
z
1
r
a
l
j
1
-
z
1
r
a
l
j
2
|
r
a
l
j
,
|
z
2
s
a
l
j
1
-
z
2
s
a
l
j
2
|
s
a
l
j
,
|
z
3
t
a
l
j
1
-
z
3
t
a
l
j
2
|
t
a
l
j
:
l
∈
Λ
k
}
)
≥
|
L
a
1
-
L
a
2
|
-
(
1
max
{
T
1
,
T
2
}
∑
q
=
1
∞
‍
∑
i
=
max
{
T
1
,
T
2
}
+
q
τ
a
∞
‍
∑
j
=
i
∞
‍
U
a
j
P
a
j
)
max
{
∥
z
1
1
-
z
1
2
∥
,
∥
z
2
1
-
z
2
2
∥
,
∥
z
3
1
-
z
3
2
∥
}
≥
|
L
a
1
-
L
a
2
|
-
max
{
θ
1
,
θ
2
}
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
,
(
n
,
a
)
∈
ℕ
max
{
T
1
,
T
2
}
×
Λ
3
,which means that
(42)
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
=
max
{
sup
n
∈
ℕ
β
|
z
a
n
1
-
z
a
n
2
n
|
:
a
∈
Λ
3
}
≥
max
{
sup
n
∈
ℕ
max
{
T
1
,
T
2
}
|
z
a
n
1
-
z
a
n
2
n
|
:
a
∈
Λ
3
}
≥
max
{
|
L
a
1
-
L
a
2
|
-
max
{
θ
1
,
θ
2
}
hhhhhhhll
×
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
:
a
∈
Λ
3
}
=
max
{
|
L
a
1
-
L
a
2
|
:
a
∈
Λ
3
}
hhhl
-
max
{
θ
1
,
θ
2
}
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
,
which gives that
(43)
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
≥
max
{
|
L
a
1
-
L
a
2
|
:
a
∈
Λ
3
}
1
+
max
{
θ
1
,
θ
2
}
>
0
;
that is, (
z
1
1
,
z
2
1
,
z
3
1
)
≠
(
z
1
2
,
z
2
2
,
z
3
2
). Hence the system (6) possesses uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
). This completes the proof.Theorem 3.
Assume that there exist constantsn
1
∈
ℕ
n
0, d
a
,
D
a
∈
ℝ
+
∖
{
0
} and nonnegative sequences {
α
m
}
m
∈
ℕ
0
⊂
[
0,1
], {
F
a
n
}
n
∈
ℕ
n
0, and {
U
a
n
}
n
∈
ℕ
n
0 satisfying (17), (18), and (22) as follows:
(44)
d
a
>
D
a
,
b
a
n
=
1
,
(
n
,
a
)
∈
ℕ
n
1
×
Λ
3
,
(45)
lim
n
→
∞
1
n
∑
i
=
n
+
τ
a
∞
‍
∑
j
=
i
∞
‍
max
{
F
a
j
,
|
c
a
j
|
,
U
a
j
P
a
j
}
=
0
,
a
∈
Λ
3
.
Then one has the following.
(a) For any(
L
1
,
L
2
,
L
3
)
∈
Π
w
=
1
3
(
d
w
-
D
w
,
d
w
+
D
w
), there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β such that, for each (
x
1
0
,
x
2
0
,
x
3
0
)
=
{
(
x
1
n
0
,
x
2
n
0
,
x
3
n
0
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), the Mann iterative sequence {
(
x
1
m
,
x
2
m
,
x
3
m
)
}
m
∈
ℕ
0
=
{
{
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
}
n
∈
ℕ
β
}
m
∈
ℕ
0
⊂
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) generated by the scheme(46)
x
a
n
m
+
1
=
{
(
1
-
α
m
)
x
a
n
m
+
α
m
{
n
L
a
-
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
}
,
{
n
L
a
l
∑
q
=
1
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
l
-
c
a
j
]
}
n
≥
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
,
(
1
-
α
m
)
x
a
n
m
+
α
m
{
n
L
a
-
n
T
∑
q
=
1
∞
‍
∑
i
=
T
+
(
2
q
-
1
)
τ
a
T
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
‍
}
,
{
n
L
a
l
∑
q
=
1
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
l
-
c
a
j
]
}
β
≤
n
<
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3converges to a positive solution (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) of the system (6) with lim
n
→
∞
(
z
a
n
/
n
)
=
L
a  for each a
∈
Λ
3 and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Proof.
Firstly, we prove that (a) holds. Set(
L
1
,
L
2
,
L
3
)
∈
Π
w
=
1
3
(
d
w
-
D
w
,
d
w
+
D
w
). It follows from (45) that there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β satisfying
(47)
θ
=
max
{
θ
a
:
a
∈
Λ
3
}
,
where
(48)
θ
a
=
1
T
∑
i
=
T
+
τ
a
∞
‍
∑
j
=
i
∞
‍
U
a
j
P
a
j
,
a
∈
Λ
3
,
(49)
1
T
∑
i
=
T
+
τ
a
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
<
D
a
-
|
L
a
-
d
a
|
,
a
∈
Λ
3
.
Define mappings A
L
1
,
A
L
2
,
A
L
3
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
l
β
∞ and S
L
1
,
L
2
,
L
3
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
(
l
β
∞
)
3 by (27) and(50)
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
=
{
n
L
a
-
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
]
,
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
,
c
a
j
]
n
≥
T
,
a
∈
Λ
3
,
n
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
,
β
≤
n
<
T
,
a
∈
Λ
3
,for each (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
). It follows from (18), (49), and (50) that, for any (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) and a
∈
Λ
3,
(51)
∥
A
L
a
(
x
1
,
x
2
,
x
3
)
-
d
^
a
∥
=
sup
n
∈
ℕ
β
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
-
n
d
a
n
|
=
max
{
sup
n
≥
T
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
n
-
d
a
|
,
h
h
h
h
h
h
l
sup
β
≤
n
<
T
|
n
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
n
-
d
a
|
}
=
sup
n
≥
T
|
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
L
a
-
d
a
hhhhhll
-
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
-
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
-
1
n
∑
q
=
1
∞
‍
x
=
-
1
n
h
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
-
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
-
1
n
∑
q
=
1
∞
‍
=
-
1
n
h
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
]
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
|
≤
sup
n
≥
T
(
+
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
|
L
a
-
d
a
|
hhhhhhh
+
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
)
≤
|
L
a
-
d
a
|
+
1
T
∑
i
=
T
+
τ
a
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
≤
|
L
a
-
d
a
|
+
D
a
-
|
L
a
-
d
a
|
=
D
a
,
which means (28). It is easy to see that (27) and (28) yield (29). By virtue of (17), (47), and (50), we know that for any (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β
,
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
=
{
(
x
¯
1
n
,
x
¯
2
n
,
x
¯
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), and a
∈
Λ
3,(52)
∥
A
L
a
(
x
1
,
x
2
,
x
3
)
-
A
L
a
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
=
sup
n
∈
ℕ
β
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
-
A
L
a
(
x
¯
1
n
,
x
¯
2
n
,
x
¯
3
n
)
n
|
=
max
{
sup
n
≥
T
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
-
A
L
a
(
x
¯
1
n
,
x
¯
2
n
,
x
¯
3
n
)
n
|
,
sup
β
≤
n
<
T
|
n
T
·
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
-
A
L
a
(
x
¯
1
T
,
x
¯
2
T
,
x
¯
3
T
)
n
|
}
=
sup
n
≥
T
1
n
|
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
h
-
1
n
∑
q
=
1
∞
‍
-
f
a
(
j
,
x
¯
1
r
a
1
j
,
…
,
x
¯
1
r
a
k
j
,
x
¯
2
s
a
1
j
,
…
,
x
¯
2
s
a
k
j
,
x
¯
3
t
a
1
j
,
…
,
x
¯
3
t
a
k
j
)
]
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
|
≤
sup
n
≥
T
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
(
U
a
j
max
{
|
x
1
r
a
l
j
-
x
¯
1
r
a
l
j
|
,
|
x
2
s
a
l
j
-
x
¯
2
s
a
l
j
|
,
|
x
3
t
a
l
j
-
x
¯
3
t
a
l
j
|
:
l
∈
Λ
k
}
)
≤
1
T
∑
i
=
T
+
τ
a
∞
‍
∑
j
=
i
∞
‍
(
U
a
j
P
a
j
max
{
|
x
1
r
a
l
j
-
x
¯
1
r
a
l
j
|
r
a
l
j
,
|
x
2
s
a
l
j
-
x
¯
2
s
a
l
j
|
s
a
l
j
,
|
x
3
t
a
l
j
-
x
¯
3
t
a
l
j
|
t
a
l
j
:
l
∈
Λ
k
}
)
≤
θ
a
max
{
∥
x
w
-
x
¯
w
∥
:
w
∈
Λ
3
}
=
θ
a
∥
(
x
1
,
x
2
,
x
3
)
-
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
,
∥
S
L
1
,
L
2
,
L
3
(
x
1
,
x
2
,
x
3
)
-
S
L
1
,
L
2
,
L
3
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
=
∥
(
A
L
1
(
x
1
,
x
2
,
x
3
)
,
A
L
2
(
x
1
,
x
2
,
x
3
)
,
A
L
3
(
x
1
,
x
2
,
x
3
)
)
-
(
A
L
1
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
,
A
L
2
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
,
A
L
3
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
)
∥
1
=
max
{
∥
A
L
a
(
x
1
,
x
2
,
x
3
)
-
A
L
a
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
:
a
∈
Λ
3
}
≤
θ
∥
(
x
1
,
x
2
,
x
3
)
-
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
,which yield (31) and (32). Thus (29) and (32) guarantee that S
L
1
,
L
2
,
L
3 is a contraction mapping in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) and it has a unique fixed point (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), that is,
(53)
z
a
n
=
n
L
a
hhhhl
-
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
hhhhh
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
-
1
n
∑
q
=
1
∞
‍
z
2
s
a
k
j
,
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
]
,
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
1
n
∑
q
=
1
∞
‍
1
n
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
h
h
l
h
5
∑
q
=
1
∞
‍
(
n
,
a
)
∈
ℕ
T
×
Λ
3
,
z
a
(
n
-
τ
a
)
hh
=
(
n
-
τ
a
)
L
a
h
h
h
h
l
-
∑
q
=
1
∞
‍
∑
i
=
n
+
2
(
q
-
1
)
τ
a
n
+
(
2
q
-
1
)
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
hh
-
1
n
∑
q
=
1
∞
‍
∑
q
=
1
∞
‍
z
2
s
a
k
j
,
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
]
,
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
∑
q
=
1
∞
‍
1
n
-
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
ll
‍
(
n
,
a
)
∈
ℕ
T
+
τ
a
×
Λ
3
,
which mean that
(54)
Δ
(
z
a
n
+
z
a
(
n
-
τ
a
)
)
h
h
=
2
L
a
+
∑
j
=
n
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
z
2
s
a
k
j
,
hhhhhhhhhhhhhhh
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
]
,
hhhhhhhhhhhlhh
(
n
,
a
)
∈
ℕ
T
+
τ
1
+
τ
2
+
τ
3
×
Λ
3
,
which implies that
(55)
Δ
2
(
z
a
n
+
z
a
(
n
-
τ
a
)
)
=
-
f
a
(
n
,
z
1
r
a
1
n
,
…
,
z
1
r
a
k
n
,
z
2
s
a
1
n
,
…
,
z
2
s
a
k
n
,
z
3
t
a
1
n
,
…
,
z
3
t
a
k
n
)
h
l
+
c
a
n
,
(
n
,
a
)
∈
ℕ
T
+
τ
1
+
τ
2
+
τ
3
×
Λ
3
,
which together with (44) ensures that (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β is a positive solution of the system (6) in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
). Obviously, (18) and (45) yield that
(56)
|
z
a
n
n
-
L
a
|
=
1
n
|
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
ghgg
-
1
n
∑
q
=
1
∞
‍
z
2
s
a
k
j
,
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
]
∑
q
=
1
∞
‍
|
≤
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
≤
1
n
∑
i
=
n
+
τ
a
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
⟶
0
as
n
⟶
∞
,
which means that
(57)
lim
n
→
∞
z
a
n
n
=
L
a
,
a
∈
Λ
3
.
In light of (31), (46), and (50), we get that
(58)
|
x
a
n
m
+
1
-
z
a
n
|
n
=
|
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
(
1
-
α
m
)
x
a
n
m
-
z
a
n
n
+
α
m
n
hhhhhhhh
×
(
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
n
L
a
+
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
‍
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhl
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
hhhhhhhhhhhhhhhhhhhhhhhhhhhhl
-
c
a
j
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
]
-
z
a
n
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
)
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
·
1
n
|
A
L
a
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
-
A
L
a
(
z
1
n
,
z
2
n
,
z
3
n
)
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
∥
A
L
a
(
x
1
m
,
x
2
m
,
x
3
m
)
-
A
L
a
(
z
1
,
z
2
,
z
3
)
∥
≤
(
1
-
(
1
-
θ
a
)
α
m
)
∥
x
a
m
-
z
a
∥
,
[
f
a
(
,
,
…
,
β
≤
n
<
T
,
n
≥
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
,
|
x
a
n
m
+
1
-
z
a
n
|
n
=
|
∑
i
=
T
+
(
2
q
-
1
)
τ
a
T
+
2
q
τ
a
-
1
‍
(
1
-
α
m
)
x
a
n
m
-
z
a
n
n
+
α
m
n
hh
×
(
∑
i
=
T
+
(
2
q
-
1
)
τ
a
T
+
2
q
τ
a
-
1
‍
n
L
a
+
n
T
∑
q
=
1
∞
‍
∑
i
=
T
+
(
2
q
-
1
)
τ
a
T
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
hhhhhhhhhhhhhhhhhhhhhhhhlhlhh
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
hhhhhhhhhhhhhhhhhhhhhhhhhlhh
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
hhhhhhhhhhhhhhhhhhhhhhhh
-
c
a
j
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
]
-
z
a
n
∑
i
=
T
+
(
2
q
-
1
)
τ
a
T
+
2
q
τ
a
-
1
‍
)
∑
i
=
T
+
(
2
q
-
1
)
τ
a
T
+
2
q
τ
a
-
1
‍
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
n
·
n
T
×
|
∑
i
=
T
+
(
2
q
-
1
)
τ
a
T
+
2
q
τ
a
-
1
‍
T
L
a
hhhh
+
∑
q
=
1
∞
‍
∑
i
=
T
+
(
2
q
-
1
)
τ
a
T
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
hhhhhhhhhhhhhhhhhhhhlhh
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
hhhhhhhhhhhhhhhhlhhhhhh
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
]
hhhh
-
A
L
a
(
z
1
T
,
z
2
T
,
z
3
T
)
∑
i
=
T
+
(
2
q
-
1
)
τ
a
T
+
2
q
τ
a
-
1
‍
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
·
1
T
|
A
L
a
(
x
1
T
m
,
x
2
T
m
,
x
3
T
m
)
-
A
L
a
(
z
1
T
,
z
2
T
,
z
3
T
)
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
∥
A
L
a
(
x
1
m
,
x
2
m
,
x
3
m
)
-
A
L
a
(
z
1
,
z
2
,
z
3
)
∥
≤
(
1
-
(
1
-
θ
a
)
α
m
)
∥
x
a
m
-
z
a
∥
,
[
f
a
(
,
…
,
β
≤
<
T
,
β
≤
n
<
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
,
which yield that
(59)
∥
(
x
1
m
+
1
,
x
2
m
+
1
,
x
3
m
+
1
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
=
max
{
sup
n
∈
ℕ
β
|
x
a
n
m
+
1
-
z
a
n
n
|
:
a
∈
Λ
3
}
=
max
{
sup
n
≥
T
|
x
a
n
m
+
1
-
z
a
n
n
|
,
sup
β
≤
n
<
T
|
x
a
n
m
+
1
-
z
a
n
n
|
:
a
∈
Λ
3
}
≤
max
{
(
1
-
(
1
-
θ
a
)
α
m
)
∥
x
a
m
-
z
a
∥
:
a
∈
Λ
3
}
≤
(
1
-
(
1
-
θ
)
α
m
)
∥
(
x
1
m
,
x
2
m
,
x
3
m
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
≤
e
-
(
1
-
θ
)
α
m
∥
(
x
1
m
,
x
2
m
,
x
3
m
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
≤
e
-
(
1
-
θ
)
∑
n
=
0
m
α
n
∥
(
x
1
0
,
x
2
0
,
x
3
0
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
,
m
∈
ℕ
0
;
that is, (21) holds. It follows from (21) and (22) that lim
m
→
∞
∥
(
x
1
m
,
x
2
m
,
x
3
m
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
=
0.
Secondly, we show that (b) holds. Put(
L
11
,
L
21
,
L
31
), (
L
12
,
L
22
,
L
32
)
∈
Π
w
=
1
3
(
d
w
-
D
w
,
d
w
+
D
w
) with max
{
|
L
a
1
-
L
a
2
|
:
a
∈
Λ
3
}
>
0. Similar to the proof of (
a
), we conclude that, for each l
∈
Λ
2, there exist constants θ
l, θ
1
l, θ
2
l, θ
3
l
∈
(
0,1
),  T
l
≥
1
+
n
0+n
1
+
τ
1
+
τ
2
+
τ
3
+
β and mappings A
L
1
l
,
A
L
2
l
,
A
L
3
l
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
l
β
∞ and S
L
1
l
,
L
2
l
,
L
3
l
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
(
l
β
∞
)
3 satisfying (27)~(32) and (47)~(50), where θ, θ
1, θ
2, θ
3,   T,   L
1, L
2, L
3,   A
L
1, A
L
2, A
L
3, and S
L
1
,
L
2
,
L
3 are replaced by θ
l, θ
1
l, θ
2
l, θ
3
l,  T
l,L
1
l, L
2
l, L
3
l, A
L
1
l, A
L
2
l, A
L
3
l, and S
L
1
l
,
L
2
l
,
L
3
l, respectively, and the contraction mapping S
L
1
l
,
L
2
l
,
L
3
l has a unique fixed point (
z
1
l
,
z
2
l
,
z
3
l
)
=
{
(
z
1
n
l
,
z
2
n
l
,
z
3
n
l
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), which is a positive solution of the system (6); that is,(60)
z
a
n
l
=
n
L
a
l
-
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
l
,
…
,
z
1
r
a
k
j
l
,
z
2
s
a
1
j
l
,
…
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
z
2
s
a
k
j
l
,
z
3
t
a
1
j
l
,
…
,
z
3
t
a
k
j
l
)
-
c
a
j
]
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
h
(
n
,
l
,
a
)
∈
ℕ
T
l
×
Λ
2
×
Λ
3
.
Using (17), (47), and (60), we infer that, for each n
≥
max
{
T
1
,
T
2
} and a
∈
Λ
3,(61)
|
z
a
n
1
-
z
a
n
2
n
|
=
|
z
2
s
a
k
j
2
,
z
3
t
a
1
j
2
,
…
,
z
3
t
a
k
j
2
)
]
L
a
1
-
L
a
2
-
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
1
,
…
,
z
1
r
a
k
j
1
,
z
2
s
a
1
j
1
,
…
,
z
2
s
a
k
j
1
,
z
3
t
a
1
j
1
,
…
,
z
3
t
a
k
j
1
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
-
f
a
(
j
,
z
1
r
a
1
j
2
,
…
,
z
1
r
a
k
j
2
,
z
2
s
a
1
j
2
,
…
,
z
2
s
a
k
j
2
,
z
3
t
a
1
j
2
,
…
,
z
3
t
a
k
j
2
)
]
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
|
≥
|
L
a
1
-
L
a
2
|
-
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
(
U
a
j
max
{
|
z
1
r
a
l
j
1
-
z
1
r
a
l
j
2
|
,
|
z
2
s
a
l
j
1
-
z
2
s
a
l
j
2
|
,
|
z
3
t
a
l
j
1
-
z
3
t
a
l
j
2
|
:
l
∈
Λ
k
}
)
≥
|
L
a
1
-
L
a
2
|
-
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
a
n
+
2
q
τ
a
-
1
‍
∑
j
=
i
∞
‍
(
U
a
j
P
a
j
max
{
|
z
1
r
a
l
j
1
-
z
1
r
a
l
j
2
|
r
a
l
j
,
|
z
2
s
a
l
j
1
-
z
2
s
a
l
j
2
|
s
a
l
j
,
|
z
3
t
a
l
j
1
-
z
3
t
a
l
j
2
|
t
a
l
j
:
l
∈
Λ
k
}
)
≥
|
L
a
1
-
L
a
2
|
-
(
1
max
{
T
1
,
T
2
}
∑
i
=
max
{
T
1
,
T
2
}
+
τ
a
∞
‍
∑
j
=
i
∞
‍
U
a
j
P
a
j
)
max
{
∥
z
1
1
-
z
1
2
∥
,
∥
z
2
1
-
z
2
2
∥
,
∥
z
3
1
-
z
3
2
∥
}
≥
|
L
a
1
-
L
a
2
|
-
max
{
θ
1
,
θ
2
}
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
,which implies that
(62)
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
=
max
{
sup
n
∈
ℕ
β
|
z
a
n
1
-
z
a
n
2
n
|
:
a
∈
Λ
3
}
≥
max
{
sup
n
∈
ℕ
max
{
T
1
,
T
2
}
|
z
a
n
1
-
z
a
n
2
n
|
:
a
∈
Λ
3
}
≥
max
{
|
L
a
1
-
L
a
2
|
-
max
{
θ
1
,
θ
2
}
hhhhhhhh
×
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
:
a
∈
Λ
3
|
L
a
1
-
L
a
2
|
}
=
max
{
|
L
a
1
-
L
a
2
|
:
a
∈
Λ
3
}
-
max
{
θ
1
,
θ
2
}
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
,
which gives that
(63)
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
≥
max
{
|
L
a
1
-
L
a
2
|
:
a
∈
Λ
3
}
1
+
max
{
θ
1
,
θ
2
}
>
0
;
that is, (
z
1
1
,
z
2
1
,
z
3
1
)
≠
(
z
1
2
,
z
2
2
,
z
3
2
). Hence the system (6) possesses uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
). This completes the proof.Theorem 4.
Assume that there exist constantsn
1
∈
ℕ
n
0,  d
a
,
D
a
,
b
a
∈
ℝ
+
∖
{
0
} and nonnegative sequences {
α
m
}
m
∈
ℕ
0
⊂
[
0,1
], {
U
a
n
}
n
∈
ℕ
n
0, and {
F
a
n
}
n
∈
ℕ
n
0 satisfying (17), (18), and (22) as follows:
(64)
1
<
d
a
D
a
<
2
-
b
a
b
a
,
0
≤
b
a
n
≤
b
a
<
1
,
(
n
,
a
)
∈
ℕ
n
1
×
Λ
3
,
(65)
lim
n
→
∞
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
max
{
F
a
j
,
|
c
a
j
|
,
U
a
j
P
a
j
}
=
0
,
a
∈
Λ
3
.
Then one has the following.
(a) For any(
L
1
,
L
2
,
L
3
)
∈
Π
w
=
1
3
(
b
w
(
d
w
+
D
w
)
+
d
w
-
D
w
,
d
w
+
D
w
), there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β such that, for each (
x
1
0
,
x
2
0
,
x
3
0
)
=
{
(
x
1
n
0
,
x
2
n
0
,
x
3
n
0
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), the Mann iterative sequence {
(
x
1
m
,
x
2
m
,
x
3
m
)
}
m
∈
ℕ
0
=
{
{
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
}
n
∈
ℕ
β
}
m
∈
ℕ
0
⊂
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) generated by the scheme(66)
x
a
n
m
+
1
=
{
(
1
-
α
m
)
x
a
n
m
+
α
m
{
n
L
a
-
b
a
n
x
a
(
n
-
τ
a
)
m
-
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
}
,
{
n
L
a
-
b
a
n
x
a
(
n
-
τ
a
)
m
-
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
}
n
≥
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
,
(
1
-
α
m
)
x
a
n
m
+
α
m

{
n
L
a
-
n
T
b
a
T
x
a
(
T
-
τ
a
)
m
-
n
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍

[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
}
,
{
n
L
a
-
b
a
n
x
a
(
n
-
τ
a
)
m
h
-
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
}
β
≤
n
<
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3converges to a positive solution (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) of the system (6) with lim
n
→
∞
(
z
a
n
+
b
a
n
z
a
(
n
-
τ
a
)
)
/
n
=
L
afor each a
∈
Λ
3 and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Proof.
Firstly, we show that (a) holds. Let(
L
1
,
L
2
,
L
3
)
∈
Π
w
=
1
3
(
b
w
(
d
w
+
D
w
)
+
d
w
-
D
w
,
d
w
+
D
w
). Observe that (65) implies that there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β satisfying
(67)
θ
=
max
{
θ
a
:
a
∈
Λ
3
}
,
where(68)
θ
a
=
|
b
a
|
+
1
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
U
a
j
P
a
j
,
a
∈
Λ
3
,
(69)
1
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
hh
<
min
{
d
a
+
D
a
-
L
a
,
L
a
-
b
a
(
d
a
+
D
a
)
-
(
d
a
-
D
a
)
}
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhllhhhh
a
∈
Λ
3
.
Define mappings A
L
1
,
A
L
2
,
A
L
3
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
l
β
∞ and S
L
1
,
L
2
,
L
3
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
(
l
β
∞
)
3 by (27) and(70)
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
=
{
n
L
a
-
b
a
n
x
a
(
n
-
τ
a
)
-
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
]
,
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
h
-
c
a
j
]
n
≥
T
,
a
∈
Λ
3
,
n
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
,
β
≤
n
<
T
,
a
∈
Λ
3
,for each (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
). It follows from (18), (69), and (70) that, for any (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
),(71)
1
n
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
=
L
a
-
b
a
n
x
a
(
n
-
τ
a
)
n
hh
-
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
hhhhhhhhhhhhh
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
]
≤
L
a
+
1
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
<
L
a
+
min
{
d
a
+
D
a
-
L
a
,
L
a
-
b
a
(
d
a
+
D
a
)
-
(
d
a
-
D
a
)
}
≤
d
a
+
D
a
,
(
n
,
a
)
∈
ℕ
T
×
Λ
3
,
1
n
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
=
1
n
·
n
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
hhhhhhhhhhhhhhl
=
1
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
<
d
a
+
D
a
,
hhhhhhhhhhhhhhhhhhhhhhhl
1
h
β
≤
n
<
T
,
a
∈
Λ
3
,
1
n
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
hh
=
L
a
-
b
a
n
x
a
(
n
-
τ
a
)
n
hhhl
-
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
hhhhhhhhhhhlhhh
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
]
≥
L
a
-
b
a
(
d
a
+
D
a
)
-
1
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
>
L
a
-
b
a
(
d
a
+
D
a
)
-
min
{
d
a
+
D
a
-
L
a
,
L
a
-
b
a
(
d
a
+
D
a
)
-
(
d
a
-
D
a
)
}
≥
d
a
-
D
a
,
(
n
,
a
)
∈
ℕ
T
×
Λ
3
,
1
n
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
=
1
n
·
n
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
hhhhhhhhhhhhhhl
=
1
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
>
d
a
-
D
a
,
hhhhhhhhhhhhhhhhhhhhhhhlhl
β
≤
n
<
T
,
a
∈
Λ
3
,
which mean (28). Consequently, (29) follows from (27) and (28). By virtue of (17), (67), and (70), we infer that, for any (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β, (
x
¯
1
,
x
¯
2
,
x
¯
3
) hh {
(
x
¯
1
n
,
x
¯
2
n
,
x
¯
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), and a
∈
Λ
3,(72)
∥
A
L
a
(
x
1
,
x
2
,
x
3
)
-
A
L
a
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
=
sup
n
∈
ℕ
β
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
-
A
L
a
(
x
¯
1
n
,
x
¯
2
n
,
x
¯
3
n
)
n
|
=
max
{
sup
n
≥
T
|
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
-
A
L
a
(
x
¯
1
n
,
x
¯
2
n
,
x
¯
3
n
)
n
|
,
h
h
h
h
h
h
sup
β
≤
n
<
T
|
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
-
A
L
a
(
x
¯
1
T
,
x
¯
2
T
,
x
¯
3
T
)
n
n
T
h
h
h
h
h
h
h
h
h
h
h
h
h
·
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
-
A
L
a
(
x
¯
1
T
,
x
¯
2
T
,
x
¯
3
T
)
n
|
}
=
sup
n
∈
ℕ
β
(
∑
i
=
n
∞
‍
1
n
|
∑
i
=
n
∞
‍
b
a
n
(
x
a
(
n
-
τ
a
)
-
x
¯
a
(
n
-
τ
a
)
)
hhhhhhhlh
+
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
hhhhhhhhhhhhhhhhhhl
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
hhhhhhhhhhhhhhhh
-
f
a
(
j
,
x
¯
1
r
a
1
j
,
…
,
x
¯
1
r
a
k
j
,
x
¯
2
s
a
1
j
,
…
,
hhhhhhhhhhhhhhhhhhhhll
x
¯
2
s
a
k
j
,
x
¯
3
t
a
1
j
,
…
,
x
¯
3
t
a
k
j
)
]
∑
i
=
n
∞
‍
|
∑
i
=
n
∞
‍
)
≤
b
a
∥
x
a
-
x
¯
a
∥
hh
+
1
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
(
U
a
j
max
{
|
x
1
r
a
l
j
-
x
¯
1
r
a
l
j
|
,
|
x
2
s
a
l
j
-
x
¯
2
s
a
l
j
|
,
hhhhhhhhhhhhhhhhhhl
|
x
3
t
a
l
j
-
x
¯
3
t
a
l
j
|
:
l
∈
Λ
k
}
)
≤
b
a
∥
x
a
-
x
¯
a
∥
hh
+
1
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
(
U
a
j
P
a
j
max
{
|
x
1
r
a
l
j
-
x
¯
1
r
a
l
j
|
r
a
l
j
,
|
x
2
s
a
l
j
-
x
¯
2
s
a
l
j
|
s
a
l
j
,
hhhhhhhhhhhhhhhhhhhhhhl
|
x
3
t
a
l
j
-
x
¯
3
t
a
l
j
|
t
a
l
j
:
l
∈
Λ
k
}
)
≤
θ
a
max
{
∥
x
w
-
x
¯
w
∥
:
w
∈
Λ
3
}
=
θ
a
∥
(
x
1
,
x
2
,
x
3
)
-
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
,
∥
S
L
1
,
L
2
,
L
3
(
x
1
,
x
2
,
x
3
)
-
S
L
1
,
L
2
,
L
3
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
=
∥
(
A
L
1
(
x
1
,
x
2
,
x
3
)
,
A
L
2
(
x
1
,
x
2
,
x
3
)
,
A
L
3
(
x
1
,
x
2
,
x
3
)
)
hhh
-
(
A
L
1
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
,
A
L
2
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
,
A
L
3
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
)
∥
1
=
max
{
∥
A
L
a
(
x
1
,
x
2
,
x
3
)
-
A
L
a
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
:
a
∈
Λ
3
}
≤
θ
∥
(
x
1
,
x
2
,
x
3
)
-
(
x
¯
1
,
x
¯
2
,
x
¯
3
)
∥
1
,
which yield (31) and (32). Thus (29) and (32) ensure that S
L
1
,
L
2
,
L
3 is a contraction mapping in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) and it has a unique fixed point (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), that is,
(73)
z
a
n
=
n
L
a
-
b
a
n
z
a
(
n
-
τ
a
)
-
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
z
2
s
a
k
j
,
hhhhhhhhhh
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
]
,
hhhhhhhhhhhhhhl
(
n
,
a
)
∈
ℕ
T
×
Λ
3
,
which yields that
(74)
Δ
(
z
a
n
+
b
a
n
z
a
(
n
-
τ
a
)
)
hh
=
L
a
+
∑
j
=
n
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
z
2
s
a
k
j
,
hhhhhhhhhhhhhhl
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
]
,
hhhhhhhhhhhhhl
(
n
,
a
)
∈
ℕ
T
+
τ
1
+
τ
2
+
τ
3
×
Λ
3
,
which implies that
(75)
Δ
2
(
z
a
n
+
b
a
n
z
a
(
n
-
τ
a
)
)
=
-
f
a
(
n
,
z
1
r
a
1
n
,
…
,
z
1
r
a
k
n
,
z
2
s
a
1
n
,
…
,
z
2
s
a
k
n
,
z
3
t
a
1
n
,
…
,
z
3
t
a
k
n
)
hl
+
c
a
n
,
(
n
,
a
)
∈
ℕ
T
+
τ
1
+
τ
2
+
τ
3
×
Λ
3
,
which together with (64) means that (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β is a positive solution of the system (6) in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
). Note that (18) and (65) give that, for each a
∈
Λ
3,
(76)
|
z
a
n
+
b
a
n
z
a
(
n
-
τ
a
)
n
-
L
a
|
hl
=
1
n
|
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
,
…
,
z
1
r
a
k
j
,
z
2
s
a
1
j
,
…
,
z
2
s
a
k
j
,
hhhhhhhhhhhhhh
z
3
t
a
1
j
,
…
,
z
3
t
a
k
j
)
-
c
a
j
]
∑
i
=
n
∞
‍
|
hl
≤
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
⟶
0
as
n
⟶
∞
,
which means that
(77)
lim
n
→
∞
z
a
n
+
b
a
n
z
a
(
n
-
τ
a
)
n
=
L
a
,
a
∈
Λ
3
.
Making use of (31), (66), and (70), we get that
(78)
|
x
a
n
m
+
1
-
z
a
n
|
n
=
|
∑
i
=
n
∞
‍
(
1
-
α
m
)
x
a
n
m
-
z
a
n
n
hhh
+
α
m
n
(
∑
i
=
n
∞
‍
n
L
a
-
b
a
n
x
a
(
n
-
τ
a
)
hhhhhhhhhl
-
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
hhhhhhhhhhhhhhhlh
-
c
a
j
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
]
-
z
a
n
∑
i
=
n
∞
‍
)
∑
i
=
n
∞
‍
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
·
1
n
|
A
L
a
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
-
A
L
a
(
z
1
n
,
z
2
n
,
z
3
n
)
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
∥
A
L
a
(
x
1
m
,
x
2
m
,
x
3
m
)
-
A
L
a
(
z
1
,
z
2
,
z
3
)
∥
≤
(
1
-
(
1
-
θ
a
)
α
m
)
∥
x
a
m
-
z
a
∥
,
n
≥
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
,
|
x
a
n
m
+
1
-
z
a
n
|
n
=
|
∑
i
=
T
∞
‍
(
1
-
α
m
)
x
a
n
m
-
z
a
n
n
hhl
+
α
m
n
(
∑
i
=
T
∞
‍
n
L
a
-
n
T
b
a
T
x
a
(
T
-
τ
a
)
hhhhhhhhl
-
n
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
hhhhhhhhhhhhhhhhhhhh
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
hhhhhhhhhhhhhhhhh
-
c
a
j
x
2
s
a
k
j
,
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
]
-
z
a
n
∑
i
=
n
∞
‍
)
∑
i
=
T
∞
‍
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
hh
+
α
m
n
·
n
T
|
∑
i
=
T
∞
‍
T
L
a
-
b
a
T
x
a
(
T
-
τ
a
)
hhhhhhhhhhl
-
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
m
,
…
,
x
1
r
a
k
j
m
,
x
2
s
a
1
j
m
,
…
,
hhhhhhhhhhhhhhhhhhhhhl
x
2
s
a
k
j
m
,
x
3
t
a
1
j
m
,
…
,
x
3
t
a
k
j
m
)
-
c
a
j
]
hhhhhhhhhhh
-
A
L
a
(
z
1
T
,
z
2
T
,
z
3
T
)
∑
i
=
T
∞
‍
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
·
1
T
|
A
L
a
(
x
1
T
m
,
x
2
T
m
,
x
3
T
m
)
-
A
L
a
(
z
1
T
,
z
2
T
,
z
3
T
)
|
≤
(
1
-
α
m
)
∥
x
a
m
-
z
a
∥
+
α
m
∥
A
L
a
(
x
1
m
,
x
2
m
,
x
3
m
)
-
A
L
a
(
z
1
,
z
2
,
z
3
)
∥
≤
(
1
-
(
1
-
θ
a
)
α
m
)
∥
x
a
m
-
z
a
∥
,
&
[
f
a
1111
β
≤
<
T
,
β
≤
n
<
T
,
(
m
,
a
)
∈
ℕ
0
×
Λ
3
,
which yield that
(79)
∥
(
x
1
m
+
1
,
x
2
m
+
1
,
x
3
m
+
1
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
=
max
{
sup
n
∈
ℕ
β
|
x
a
n
m
+
1
-
z
a
n
n
|
:
a
∈
Λ
3
}
=
max
{
sup
n
≥
T
|
x
a
n
m
+
1
-
z
a
n
n
|
,
sup
β
≤
n
<
T
|
x
a
n
m
+
1
-
z
a
n
n
|
:
a
∈
Λ
3
}
≤
max
{
(
1
-
(
1
-
θ
a
)
α
m
)
∥
x
a
m
-
z
a
∥
:
a
∈
Λ
3
}
≤
(
1
-
(
1
-
θ
)
α
m
)
∥
(
x
1
m
,
x
2
m
,
x
3
m
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
≤
e
-
(
1
-
θ
)
α
m
∥
(
x
1
m
,
x
2
m
,
x
3
m
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
≤
e
-
(
1
-
θ
)
∑
n
=
0
m
α
n
∥
(
x
1
0
,
x
2
0
,
x
3
0
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
,
m
∈
ℕ
0
;
that is, (21) holds. It follows from (21) and (22) that lim
m
→
∞
∥
(
x
1
m
,
x
2
m
,
x
3
m
)
-
(
z
1
,
z
2
,
z
3
)
∥
1
=
0.
Secondly, we show that (b) holds. Let(
L
11
,
L
21
,
L
31
), (
L
12
,
L
22
,
L
32
) ∈ Π
w
=
1
3
(
b
w
(
d
w
+
D
w
) +
d
w
-
D
w
,
d
w
+
D
w
) with max
{
|
L
a
1
-
L
a
2
|
:
a
∈
Λ
3
}
>
0. Similarly we infer that, for each l
∈
Λ
2, there exist constants θ
l
,
θ
1
l
,
θ
2
l
,
θ
3
l
∈
(
0,1
),  T
l
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β and mappings A
L
1
l
,
A
L
2
l
,
A
L
3
l
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
l
β
∞ and S
L
1
l
,
L
2
l
,
L
3
l
:
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
)
→
(
l
β
∞
)
3 satisfying (27)~(32) and (67)~(70), where θ, θ
1, θ
2, θ
3, T, L
1, L
2, L
3,  A
L
1, A
L
2, A
L
3, and S
L
1
,
L
2
,
L
3 are replaced by θ
l, θ
1
l, θ
2
l, θ
3
l, T
l, L
1
l, L
2
l, L
3
l, A
L
1
l, A
L
2
l, A
L
3
l, and S
L
1
l
,
L
2
l
,
L
3
l, respectively, and the contraction mapping S
L
1
l
,
L
2
l
,
L
3
l has a unique fixed point (
z
1
l
,
z
2
l
,
z
3
l
)
=
{
(
z
1
n
l
,
z
2
n
l
,
z
3
n
l
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), which is a positive solution of the system (6); that is,
(80)
z
a
n
l
=
n
L
a
l
-
b
a
n
z
a
(
n
-
τ
a
)
l
-
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
l
,
…
,
z
1
r
a
k
j
l
,
z
2
s
a
1
j
l
,
…
,
z
2
s
a
k
j
l
,
h
h
h
h
h
h
h
h
h
h
h
l
z
3
t
a
1
j
l
,
…
,
z
3
t
a
k
j
l
)
-
c
a
j
]
,
h
h
h
h
h
h
h
h
h
h
l
(
n
,
l
,
a
)
∈
ℕ
T
l
×
Λ
2
×
Λ
3
.
Using (17), (67), (70), and (80), we know that, for each (
n
,
a
)
∈
ℕ
max
{
T
l
,
T
2
}
×
Λ
3,
(81)
|
z
a
n
1
-
z
a
n
2
n
|
hh
=
|
∑
i
=
n
∞
‍
L
a
1
-
L
a
2
-
1
n
b
a
n
(
z
a
(
n
-
τ
a
)
1
-
z
a
(
n
-
τ
a
)
2
)
hhhhh
-
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
z
1
r
a
1
j
1
,
…
,
z
1
r
a
k
j
1
,
z
2
s
a
1
j
1
,
…
,
z
2
s
a
k
j
1
,
hhhhhhhhhhhhhhhhl
z
3
t
a
1
j
1
,
…
,
z
3
t
a
k
j
1
)
hhhhhhhhhhhhh
-
f
a
(
j
,
z
1
r
a
1
j
2
,
…
,
z
1
r
a
k
j
2
,
z
2
s
a
1
j
2
,
…
,
z
2
s
a
k
j
2
,
hhhhhhhhhhhhhhhhhh
z
3
t
a
1
j
2
,
…
,
z
3
t
a
k
j
2
)
]
∑
i
=
n
∞
‍
|
hh
≥
|
L
a
1
-
L
a
2
|
-
b
a
∥
z
a
1
-
z
a
2
∥
hhhh
-
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
(
U
a
j
max
{
|
z
1
r
a
l
j
1
-
z
1
r
a
l
j
2
|
,
|
z
2
s
a
l
j
1
-
z
2
s
a
l
j
2
|
,
hhhhhhhhhhhhhhhhhhhl
|
z
3
t
a
l
j
1
-
z
3
t
a
l
j
2
|
:
l
∈
Λ
k
}
)
hh
≥
|
L
a
1
-
L
a
2
|
-
b
a
∥
z
a
1
-
z
a
2
∥
hhhh
-
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
(
U
a
j
P
a
j
max
{
|
z
1
r
a
l
j
1
-
z
1
r
a
l
j
2
|
r
a
l
j
,
|
z
2
s
a
l
j
1
-
z
2
s
a
l
j
2
|
s
a
l
j
,
hhhhhhhhhhhhhhhhhhhhhhll
|
z
3
t
a
l
j
1
-
z
3
t
a
l
j
2
|
t
a
l
j
:
l
∈
Λ
k
}
)
hh
≥
|
L
a
1
-
L
a
2
|
-
(
b
a
+
1
max
{
T
1
,
T
2
}
∑
i
=
max
{
T
1
,
T
2
}
∞
‍
∑
j
=
i
∞
‍
U
a
j
P
a
j
)
h
h
h
h
×
max
{
∥
z
1
1
-
z
1
2
∥
,
∥
z
2
1
-
z
2
2
∥
,
∥
z
3
1
-
z
3
2
∥
}
hh
≥
|
L
a
1
-
L
a
2
|
-
max
{
θ
1
,
θ
2
}
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
,
which implies that
(82)
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
=
max
{
sup
n
∈
ℕ
β
|
z
a
n
1
-
z
a
n
2
n
|
:
a
∈
Λ
3
}
≥
max
{
sup
n
∈
ℕ
max
{
T
1
,
T
2
}
|
z
a
n
1
-
z
a
n
2
n
|
:
a
∈
Λ
3
}
≥
max
{
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
|
L
a
1
-
L
a
2
|
-
max
{
θ
1
,
θ
2
}
hhhhhhh
×
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
:
a
∈
Λ
3
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
}
=
max
{
|
L
a
1
-
L
a
2
|
:
a
∈
Λ
3
}
-
max
{
θ
1
,
θ
2
}
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
,
which gives that
(83)
∥
(
z
1
1
,
z
2
1
,
z
3
1
)
-
(
z
1
2
,
z
2
2
,
z
3
2
)
∥
1
≥
max
{
|
L
a
1
-
L
a
2
|
:
a
∈
Λ
3
}
1
+
max
{
θ
1
,
θ
2
}
>
0
;
that is, (
z
1
1
,
z
2
1
,
z
3
1
)
≠
(
z
1
2
,
z
2
2
,
z
3
2
). Hence the system (6) possesses uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
). This completes the proof.Theorem 5.
Assume that there exist constantsn
1
∈
ℕ
n
0, d
a
,
D
a
∈
ℝ
+
∖
{
0
}, b
a
∈
ℝ
-
∖
{
0
} and nonnegative sequences {
α
m
}
m
∈
ℕ
0
⊂
[
0,1
], {
U
a
n
}
n
∈
ℕ
n
0, and {
F
a
n
}
n
∈
ℕ
n
0 satisfying (17), (18), (22), (65), and
(84)
1
<
d
a
D
a
<
b
a
+
2
-
b
a
,
-
1
<
b
a
≤
b
a
n
≤
0
,
(
n
,
a
)
∈
ℕ
n
1
×
Λ
3
.
Then one has the following.
(a) For any(
L
1
,
L
2
,
L
3
)
∈
Π
w
=
1
3
(
d
w
-
D
w
,
(
1
+
b
w
)
(
d
w
+
D
w
)
), there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β such that, for each (
x
1
0
,
x
2
0
,
x
3
0
)
=
{
(
x
1
n
0
,
x
2
n
0
,
x
3
n
0
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), the Mann iterative sequence {
(
x
1
m
,
x
2
m
,
x
3
m
)
}
m
∈
ℕ
0
=
{
{
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
}
n
∈
ℕ
n
0
}
m
∈
ℕ
0
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) generated by (66) converges to a positive solution (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) of the system (6) with lim
n
→
∞
(
z
a
n
+
b
a
n
z
a
(
n
-
τ
a
)
)
/
n
=
L
a for each a
∈
Λ
3 and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Proof.
Let(
L
1
,
L
2
,
L
3
)
∈
Π
w
=
1
3
(
d
w
-
D
w
,
(
1
+
b
w
)
(
d
w
+
D
w
)
). It follows from (65) that there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β satisfying (67) and
(85)
1
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
hh
<
min
{
(
1
+
b
a
)
(
d
a
+
D
a
)
-
L
a
,
L
a
-
(
d
a
-
D
a
)
}
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhlll
a
∈
Λ
3
.
Let A
L
1, A
L
2, A
L
3, and S
L
1
,
L
2
,
L
3 be defined by (27) and (70), respectively. It follows from (18), (70), and (85) that, for any (
x
1
,
x
2
,
x
3
)
=
{
(
x
1
n
,
x
2
n
,
x
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
),
(86)
1
n
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
hh
=
L
a
-
1
n
b
a
n
x
a
(
n
-
τ
a
)
hhhh
-
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
hhhhhhhhhhhhhhh
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
]
hh
≥
L
a
-
1
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
hh
>
L
a
-
min
{
(
1
+
b
a
)
(
d
a
+
D
a
)
-
L
a
,
L
a
-
(
d
a
-
D
a
)
}
hh
≥
d
a
-
D
a
,
(
n
,
a
)
∈
ℕ
T
×
Λ
3
,
1
n
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
=
1
n
·
n
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
hhhhhhhhhhhhhhll
=
1
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
>
d
a
-
D
a
,
hhhhhhhhhhhhhhhhhhhlhhhhhh
β
≤
n
<
T
,
a
∈
Λ
3
,
1
n
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
=
L
a
-
1
n
b
a
n
x
a
(
n
-
τ
a
)
hh
-
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
a
(
j
,
x
1
r
a
1
j
,
…
,
x
1
r
a
k
j
,
x
2
s
a
1
j
,
…
,
x
2
s
a
k
j
,
hhhhhhhhhhhhhl
x
3
t
a
1
j
,
…
,
x
3
t
a
k
j
)
-
c
a
j
]
≤
L
a
-
b
a
(
d
a
+
D
a
)
+
1
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
(
F
a
j
+
|
c
a
j
|
)
<
L
a
-
b
a
(
d
a
+
D
a
)
+
min
{
(
1
+
b
a
)
(
d
a
+
D
a
)
-
L
a
,
L
a
-
(
d
a
-
D
a
)
}
≤
d
a
+
D
a
,
(
n
,
a
)
∈
ℕ
T
×
Λ
3
,
1
n
A
L
a
(
x
1
n
,
x
2
n
,
x
3
n
)
=
1
n
·
n
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
hhhhhhhhhhhhhhl
=
1
T
A
L
a
(
x
1
T
,
x
2
T
,
x
3
T
)
<
d
a
+
D
a
,
hhhhhhhhhhhhhhhlhhhhhhhlhl
β
≤
n
<
T
,
a
∈
Λ
3
,
which lead to (28). The rest of the proof is similar to that of Theorem 4 and is omitted. This completes the proof.Similar to the proofs of Theorems2
~
5, we have the following results and omit their proofs.Theorem 6.
Assume that there exist constantsn
1
∈
ℕ
n
0, d
a
,
D
a
,
b
3
∈
ℝ
+
∖
{
0
} and nonnegative sequences {
α
m
}
m
∈
ℕ
0
⊂
[
0,1
], {
F
a
n
}
n
∈
ℕ
n
0, and  {
U
a
n
}
n
∈
ℕ
n
0 for each a
∈
Λ
3 satisfying (17), (18), and (22) as follows: (87)
d
1
>
D
1
,
d
2
>
D
2
,
1
<
d
3
D
3
<
2
-
b
3
b
3
,
b
1
n
=
-
1
,
b
2
n
=
1
,
0
≤
b
3
n
≤
b
3
<
1
,
n
∈
ℕ
n
1
(88)
lim
n
→
∞
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
max
{
F
1
j
,
|
c
1
j
|
,
U
1
j
P
1
j
}
=
0
,
lim
n
→
∞
1
n
∑
i
=
n
+
τ
2
∞
‍
∑
j
=
i
∞
‍
max
{
F
2
j
,
|
c
2
j
|
,
U
2
j
P
2
j
}
=
0
,
lim
n
→
∞
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
max
{
F
3
j
,
|
c
3
j
|
,
U
3
j
P
3
j
}
=
0
.
Then one has the following.
(a)  For any(
L
1
,
L
2
,
L
3
)
∈
(
d
1
-
D
1
,
d
1
+
D
1
)
×
(
d
2
-
D
2
,
d
2
+
D
2
)   ×  (
b
3
(
d
3
+
D
3
)
+
d
3
-
D
3
,
d
3
+
D
3
), there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β such that, for each (
x
1
0
,
x
2
0
,
x
3
0
)
=
{
(
x
1
n
0
,
x
2
n
0
,
x
3
n
0
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), the Mann iterative sequence {
(
x
1
m
,
x
2
m
,
x
3
m
)
}
m
∈
ℕ
0
=
{
{
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
}
n
∈
ℕ
β
}
m
∈
ℕ
0
⊂
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) generated by the scheme(89)
x
1
n
m
+
1
=
{
(
1
-
α
m
)
x
1
n
m
+
α
m
{
n
L
1
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
,
{
n
L
1
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
n
≥
T
,
m
∈
ℕ
0
,
(
1
-
α
m
)
x
1
n
m
+
α
m
{
n
L
1
+
n
T
∑
q
=
1
∞
‍
∑
i
=
T
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
,
{
n
L
1
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
β
≤
n
<
T
,
m
∈
ℕ
0
,
(90)
x
2
n
m
+
1
=
{
(
1
-
α
m
)
x
2
n
m
+
α
m
{
n
L
2
-
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
2
n
+
2
q
τ
2
-
1
‍
∑
j
=
i
∞
‍
[
f
2
(
j
,
x
1
r
21
j
m
,
…
,
x
1
r
2
k
j
m
,
x
2
s
21
j
m
,
…
,
x
2
s
2
k
j
m
,
x
3
t
21
j
m
,
…
,
x
3
t
2
k
j
m
)
-
c
2
j
]
}
,
{
n
L
1
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
n
≥
T
,
m
∈
ℕ
0
,
(
1
-
α
m
)
x
2
n
m
+
α
m
{
n
L
2
-
n
T
∑
q
=
1
∞
‍
∑
i
=
T
+
(
2
q
-
1
)
τ
2
T
+
2
q
τ
2
-
1
‍
∑
j
=
i
∞
‍
[
f
2
(
j
,
x
1
r
21
j
m
,
…
,
x
1
r
2
k
j
m
,
x
2
s
21
j
m
,
…
,
x
2
s
2
k
j
m
,
x
3
t
21
j
m
,
…
,
x
3
t
2
k
j
m
)
]
-
c
2
j
}
,
{
n
L
1
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
β
≤
n
<
T
,
m
∈
ℕ
0
,
(91)
x
3
n
m
+
1
=
{
(
1
-
α
m
)
x
3
n
m
+
α
m
{
n
L
3
-
b
3
n
x
3
(
n
-
τ
3
)
m
-
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
3
(
j
,
x
1
r
31
j
m
,
…
,
x
1
r
3
k
j
m
,
x
2
s
31
j
m
,
…
,
x
2
s
3
k
j
m
,
x
3
t
31
j
m
,
…
,
x
3
t
3
k
j
m
)
-
c
3
j
]
}
,
{
n
L
1
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
n
≥
T
,
m
∈
ℕ
0
,
(
1
-
α
m
)
x
3
n
m
+
α
m

{
n
L
3
-
n
T
b
3
T
x
3
(
T
-
τ
3
)
m
-
n
T
∑
i
=
T
∞
‍
∑
j
=
i
∞

‍
[
f
3
(
j
,
x
1
r
31
j
m
,
…
,
x
1
r
3
k
j
m
,
x
2
s
31
j
m
,
…
,
x
2
s
3
k
j
m
,
x
3
t
31
j
m
,
…
,
x
3
t
3
k
j
m
)
-
c
3
j
]
}
,
{
n
L
1
+
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
[
f
1
h
.
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
β
≤
n
<
T
,
m
∈
ℕ
0converges to a positive solution (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) of the system (6) with(92)
lim
n
→
∞
z
1
n
n
=
L
1
,
lim
n
→
∞
z
2
n
n
=
L
2
,
lim
n
→
∞
z
3
n
+
b
3
n
z
3
(
n
-
τ
3
)
n
=
L
3
and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Theorem 7.
Assume that there exist constantsn
1
∈
ℕ
n
0, d
a
,
D
a
∈
ℝ
+
∖
{
0
},  b
3
∈
ℝ
-
∖
{
0
} and nonnegative sequences {
α
m
}
m
∈
ℕ
0
⊂
[
0,1
], {
F
a
n
}
n
∈
ℕ
n
0, and {
U
a
n
}
n
∈
ℕ
n
0 for each a
∈
Λ
3 satisfying (17), (18), (22), (88), and
(93)
d
1
>
D
1
,
d
2
>
D
2
,
1
<
d
3
D
3
<
b
3
+
2
-
b
3
,
b
1
n
=
-
1
,
b
2
n
=
1
,
-
1
<
b
3
≤
b
3
n
≤
0
,
n
∈
ℕ
n
1
.
Then one has the following.
(a)  For any(
L
1
,
L
2
,
L
3
)
∈
(
d
1
-
D
1
,
d
1
+
D
1
)
×
(
d
2
-
D
2
,
d
2
+
D
2
)  ×  (
d
3
-
D
3
,
(
1
+
b
3
)
(
d
3
+
D
3
)
), there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1
+
τ
2
+
τ
3
+
β such that, for each (
x
1
0
,
x
2
0
,
x
3
0
)
=
{
(
x
1
n
0
,
x
2
n
0
,
x
3
n
0
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), the Mann iterative sequence {
(
x
1
m
,
x
2
m
,
x
3
m
)
}
m
∈
ℕ
0
=
{
{
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
}
n
∈
ℕ
β
}
m
∈
ℕ
0
⊂
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) generated by the scheme (89)~(91) converges to a positive solution (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) of the system (6) with
(94)
lim
n
→
∞
z
1
n
n
=
L
1
,
lim
n
→
∞
z
2
n
n
=
L
2
,
lim
n
→
∞
z
3
n
+
b
3
n
z
3
(
n
-
τ
3
)
n
=
L
3
and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Theorem 8.
Assume that there exist constantsn
1
∈
ℕ
n
0, d
a
,
D
a
,
b
3
∈
ℝ
+
∖
{
0
},  b
2
∈
ℝ
-
∖
{
0
} and nonnegative sequences {
α
m
}
m
∈
ℕ
0
⊂
[
0,1
], {
F
a
n
}
n
∈
ℕ
n
0, and {
U
a
n
}
n
∈
ℕ
n
0 for each a
∈
Λ
3 satisfying (17), (18), and (22) as follows:
(95)
d
1
>
D
1
,
1
<
d
2
D
2
<
b
2
+
2
-
b
2
,
1
<
d
3
D
3
<
2
-
b
3
b
3
,
b
1
n
=
-
1
,
-
1
<
b
2
≤
b
2
n
≤
0
,
0
≤
b
3
n
≤
b
3
<
1
,
n
∈
ℕ
n
1
,
(96)
lim
n
→
∞
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
max
{
F
1
j
,
|
c
1
j
|
,
U
1
j
P
1
j
}
=
0
,
lim
n
→
∞
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
max
{
F
2
j
,
|
c
2
j
|
,
U
2
j
P
2
j
,
F
3
j
,
|
c
3
j
|
,
U
3
j
P
3
j
}
=
0
.
Then one has the following.
(a) For any(
L
1
,
L
2
,
L
3
)
∈
(
d
1
-
D
1
,
d
1
+
D
1
)
×
(
d
2
-
D
2
,
(
1
+
b
2
)
(
d
2
+
D
2
)
)
×
(
b
3
(
d
3
+
D
3
)
+
d
3
-
D
3, d
3
+
D
3
), there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1 +
τ
1
+
τ
2
+
τ
3 +
β such that, for each (
x
1
0
,
x
2
0
,
x
3
0
)
=
{
(
x
1
n
0
,
x
2
n
0
,
x
3
n
0
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), the Mann iterative sequence {
(
x
1
m
,
x
2
m
,
x
3
m
)
}
m
∈
ℕ
0
=
{
{
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
}
n
∈
ℕ
β
}
m
∈
ℕ
0
⊂
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) generated by the scheme (89), (91), and(97)
x
2
n
m
+
1
=
{
(
1
-
α
m
)
x
2
n
m
+
α
m
{
n
L
2
-
b
2
n
x
2
(
n
-
τ
2
)
m
-
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
2
(
j
,
x
1
r
21
j
m
,
…
,
x
1
r
2
k
j
m
,
x
2
s
21
j
m
,
…
,
x
2
s
2
k
j
m
,
x
3
t
21
j
m
,
…
,
x
3
t
2
k
j
m
)
-
c
2
j
]
}
,
{
n
L
2
-
b
2
n
x
2
(
n
-
τ
2
)
m
-
,
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
2
(
j
,
x
1
r
21
j
m
,
…
,
x
1
r
2
k
j
m
,
x
2
s
21
j
m
,
…
,
x
2
s
2
k
j
m
,
x
3
t
21
j
m
,
…
,
x
3
t
2
k
j
m
)
-
c
2
j
]
}
n
≥
T
,
m
∈
ℕ
0
,
(
1
-
α
m
)
x
2
n
m
+
α
m
{
n
L
2
-
n
T
b
2
T
x
2
(
T
-
τ
2
)
m
-
n
T
∑
i
=
T
∞
‍
∑
j
=
i
∞
‍
[
f
2
(
j
,
x
1
r
21
j
m
,
…
,
x
1
r
2
k
j
m
,
x
2
s
21
j
m
,
…
,
x
2
s
2
k
j
m
,
x
3
t
21
j
m
,
…
,
x
3
t
2
k
j
m
)
-
c
2
j
x
2
s
2
k
j
m
,
x
3
t
21
j
m
,
…
,
x
3
t
2
k
j
m
)
]
}
,
{
n
L
2
-
b
2
n
x
2
(
n
-
τ
2
)
m
h
-
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
[
f
2
(
j
,
x
1
r
21
j
m
,
…
,
x
1
r
2
k
j
m
,
x
2
s
21
j
m
,
…
,
x
2
s
2
k
j
m
,
x
3
t
21
j
m
,
…
,
x
3
t
2
k
j
m
)
-
c
2
j
]
}
β
≤
n
<
T
,
m
∈
ℕ
0converges to a positive solution (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) of the system (6) with
(98)
lim
n
→
∞
z
1
n
n
=
L
1
,
lim
n
→
∞
z
2
n
+
b
2
n
z
2
(
n
-
τ
2
)
n
=
L
2
,
lim
n
→
∞
z
3
n
+
b
3
n
z
3
(
n
-
τ
3
)
n
=
L
3
and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Theorem 9.
Assume that there exist constantsn
1
∈
ℕ
n
0, d
a
,
D
a
,
b
3
∈
ℝ
+
∖
{
0
},  b
2
∈
ℝ
-
∖
{
0
} and nonnegative sequences {
α
m
}
m
∈
ℕ
0
⊂
[
0,1
], {
F
a
n
}
n
∈
ℕ
n
0, and {
U
a
n
}
n
∈
ℕ
n
0 for each a
∈
Λ
3 satisfying (17), (18), and (22) as follows:
(99)
d
1
>
D
1
,
1
<
d
2
D
2
<
b
2
+
2
-
b
2
,
1
<
d
3
D
3
<
2
-
b
3
b
3
,
b
1
n
=
1
,
-
1
<
b
2
≤
b
2
n
≤
0
,
0
≤
b
3
n
≤
b
3
<
1
,
n
∈
ℕ
n
1
,
lim
n
→
∞
1
n
∑
i
=
n
+
q
τ
1
∞
‍
∑
j
=
i
∞
‍
max
{
F
1
j
,
|
c
1
j
|
,
U
1
j
P
1
j
}
=
0
,
lim
n
→
∞
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
max
{
F
2
j
,
|
c
2
j
|
,
U
2
j
P
2
j
,
F
3
j
,
|
c
3
j
|
,
U
3
j
P
3
j
}
=
0
.
Then one has the following.
(a) For any(
L
1
,
L
2
,
L
3
)
∈
(
d
1
-
D
1
,
d
1
+
D
1
)
×
(
d
2
-
D
2
,
(
1
+
b
2
)
(
d
2
+
D
2
)
)
×
(
b
3
(
d
3
+
D
3
)
+
d
3
-
D
3, d
3
+
D
3
), there exist θ
∈
(
0,1
) and T
≥
1
+
n
0
+
n
1
+
τ
1 +
τ
2
+
τ
3
+
β such that, for each (
x
1
0
,
x
2
0
,
x
3
0
)
= {
(
x
1
n
0
,
x
2
n
0
,
x
3
n
0
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
), the Mann iterative sequence {
(
x
1
m
,
x
2
m
,
x
3
m
)
}
m
∈
ℕ
0
=
{
{
(
x
1
n
m
,
x
2
n
m
,
x
3
n
m
)
}
n
∈
ℕ
β
}
m
∈
ℕ
0
⊂
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) generated by the scheme (91), (96), and(100)
x
1
n
m
+
1
=
{
(
1
-
α
m
)
x
1
n
m
+
α
m
{
n
L
1
-
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
1
n
+
2
q
τ
1
-
1
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
,
{
n
L
1
-
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
1
n
+
2
q
τ
1
-
1
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
n
≥
T
,
m
∈
ℕ
0
,
(
1
-
α
m
)
x
1
n
m
+
α
m
{
n
L
1
-
n
T
∑
q
=
1
∞
‍
∑
i
=
T
+
(
2
q
-
1
)
τ
1
T
+
2
q
τ
1
-
1
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
,
…
,
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
,
{
n
L
1
-
∑
q
=
1
∞
‍
∑
i
=
n
+
(
2
q
-
1
)
τ
1
n
+
2
q
τ
1
-
1
‍
∑
j
=
i
∞
‍
[
f
1
(
j
,
x
1
r
11
j
m
,
…
,
x
1
r
1
k
j
m
,
x
2
s
11
j
m
…
x
2
s
1
k
j
m
,
x
3
t
11
j
m
,
…
,
x
3
t
1
k
j
m
)
-
c
1
j
]
}
β
≤
n
<
T
,
m
∈
ℕ
0converges to a positive solution (
z
1
,
z
2
,
z
3
)
=
{
(
z
1
n
,
z
2
n
,
z
3
n
)
}
n
∈
ℕ
β
∈
Π
w
=
1
3
Ω
(
d
^
w
,
D
w
) of the system (6) with
(101)
lim
n
→
∞
z
1
n
n
=
L
1
,
lim
n
→
∞
z
2
n
+
b
2
n
z
2
(
n
-
τ
2
)
n
=
L
2
,
lim
n
→
∞
z
3
n
+
b
3
n
z
3
(
n
-
τ
3
)
n
=
L
3
and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).


## 4. Examples

Now we suggest four examples to explain the main results in Section3.Example 1.
Consider the nonlinear three-dimensional difference system(102)
Δ
2
(
x
1
n
-
x
1
(
n
-
τ
1
)
)
+
4
n
x
1
(
2
n
+
6
)
3
-
x
2
(
n
+
1
)
2
+
2
x
3
(
n
-
2
)
n
6
+
1
=
n
2
-
3
n
+
1
n
7
+
n
4
+
1
,
n
≥
8
,
Δ
2
(
x
2
n
-
x
2
(
n
-
τ
2
)
)
+
(
n
+
3
)
x
1
(
3
n
-
10
)
2
+
n
2
x
2
(
n
2
-
1
)
+
x
3
(
4
n
2
-
1
)
3
n
8
+
ln
n
=
2
n
+
(
-
1
)
n
ln
n
n
6
+
3
n
2
+
5
n
+
1
,
n
≥
8
,
Δ
2
(
x
3
n
-
x
3
(
n
-
τ
3
)
)
+
sin
2
(
n
2
x
2
(
n
-
2
)
-
n
3
x
3
(
n
-
7
)
3
)
(
n
+
2
)
8
+
n
|
x
1
(
2
n
-
1
)
|
=
n
-
sin
n
n
8
+
3
n
2
+
2
n
+
5
,
n
≥
8
,
where n
0
=
8 and τ
1
,
τ
2
,
τ
3
∈
ℕ are fixed. Let n
1
=
3, k
=
1, d
1
=
6, D
1
=
2, d
2
=
10, D
2
=
6, d
3
=
14, D
3
=
10, β
=
min
{
8
-
τ
1
,
8
-
τ
2
,
8
-
τ
3
,
1
}
∈
ℕ, and
(103)
b
1
n
=
b
2
n
=
b
3
n
=
-
1
,
r
11
n
=
2
n
+
6
,
s
11
n
=
n
+
1
,
t
11
n
=
n
-
2
,
r
21
n
=
3
n
-
10
,
s
21
n
=
n
2
-
1
,
t
21
n
=
4
n
2
-
1
,
r
31
n
=
2
n
-
1
,
s
31
n
=
n
-
2
,
t
31
n
=
n
-
7
,
c
1
n
=
n
2
-
3
n
+
1
n
7
+
n
4
+
1
,
c
2
n
=
2
n
+
(
-
1
)
n
ln
n
n
6
+
3
n
2
+
5
n
+
1
,
c
3
n
=
n
-
sin
n
n
8
+
3
n
2
+
2
n
+
5
,
f
1
(
n
,
u
,
v
,
w
)
=
4
n
u
3
-
v
2
+
2
w
n
6
+
1
,
f
2
(
n
,
u
,
v
,
w
)
=
(
n
+
3
)
u
2
+
n
2
v
+
w
3
n
8
+
ln
n
,
f
3
(
n
,
u
,
v
,
w
)
=
sin
2
(
n
2
v
-
n
3
w
)
(
n
+
2
)
8
+
n
|
u
|
,
U
1
n
=
800
n
5
,
U
2
n
=
40
n
6
,
U
3
n
=
3
n
5
,
F
1
n
=
2500
n
5
,
F
2
n
=
15000
n
6
,
F
3
n
=
1
n
8
,
P
1
n
=
2
n
+
6
,
P
2
n
=
4
n
2
-
1
,
P
3
n
=
2
n
-
1
,
hhhhhhhhhhhhhhhhhhll
(
n
,
u
,
v
,
w
)
∈
ℕ
n
0
×
ℝ
3
.
It is clear that (16)~(18) are satisfied. Note that Lemma 1 implies that
(104)
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
∑
j
=
i
∞
‍
max
{
F
a
j
,
|
c
a
j
|
,
U
a
j
P
a
j
:
a
∈
Λ
3
}
≤
1
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
∑
j
=
i
∞
‍
max
{
2500
j
5
,
15000
j
6
,
1
j
8
,
2
j
5
,
3
j
5
,
hhhhhhhhhhhhhhhhhlhhhhhhh
1
j
7
,
800
j
5
·
3
j
,
hhhhhhhhhhhhhhhhhlhhhhhhh
1300
j
6
·
4
j
2
,
3
j
5
·
2
j
}
≤
15000
n
∑
q
=
1
∞
‍
∑
i
=
n
+
q
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
∑
j
=
i
∞
‍
1
j
4
≤
15000
n
∑
j
=
n
+
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
j
2
·
1
j
4
≤
15000
n
∑
j
=
n
+
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
1
j
2
⟶
0
as
n
⟶
∞
,
which yield that (19) holds. Thus Theorem 2 implies that the system (102) possesses uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Example 2.
Consider the nonlinear three-dimensional difference system(105)
Δ
2
(
x
1
n
+
x
1
(
n
-
τ
1
)
)
+
n
2
x
2
(
n
+
1
)
+
x
3
(
n
2
-
3
)
ln
n
n
8
+
|
x
1
(
2
n
-
1
)
|
=
(
-
1
)
n
n
-
5
n
9
+
3
n
-
1
,
n
≥
40
,
Δ
2
(
x
2
n
+
x
2
(
n
-
τ
2
)
)
+
x
1
(
n
2
-
1
)
+
(
-
1
)
n
x
3
(
2
n
+
1
)
n
5
+
x
2
(
n
-
1
)
2
=
n
3
-
sin
(
n
-
1
)
n
12
+
3
n
2
+
11
,
n
≥
40
,
Δ
2
(
x
3
n
+
x
3
(
n
-
τ
3
)
)
+
n
2
x
1
(
n
-
30
)
+
n
x
2
(
2
n
-
2
)
+
x
3
(
3
n
+
1
)
3
n
7
+
sin
(
n
3
+
1
)
=
n
cos
(
n
3
-
2
n
)
n
7
+
2
n
3
+
1
,
n
≥
40
,
where n
0
=
40 and τ
1
,
τ
2
,
τ
3
∈
ℕ are fixed. Let n
1
=
40,  k
=
1, d
1
=
3, D
1
=
1, d
2
=
4, D
2
=
2, d
3
=
5, D
3
=
3, β
=
min
{
40
-
τ
1
,
40
-
τ
2
,
40
-
τ
3
,
10
}
∈
ℕ, and
(106)
b
1
n
=
b
2
n
=
b
3
n
=
1
,
r
11
n
=
2
n
-
1
,
s
11
n
=
n
+
1
,
t
11
n
=
n
2
-
3
,
r
21
n
=
n
2
-
1
,
s
21
n
=
n
-
1
,
t
21
n
=
2
n
+
1
,
r
31
n
=
n
-
30
,
s
31
n
=
2
n
-
2
,
t
31
n
=
3
n
+
1
,
c
1
n
=
(
-
1
)
n
n
-
5
n
9
+
3
n
-
1
,
c
2
n
=
n
3
-
sin
(
n
-
1
)
n
12
+
3
n
2
+
11
,
c
3
n
=
n
cos
(
n
3
-
2
n
)
n
7
+
2
n
3
+
1
,
f
1
(
n
,
u
,
v
,
w
)
=
n
2
v
+
w
ln
n
n
8
+
|
u
|
,
f
2
(
n
,
u
,
v
,
w
)
=
u
+
(
-
1
)
n
w
n
5
+
v
2
,
f
3
(
n
,
u
,
v
,
w
)
=
n
2
u
+
n
v
+
w
3
n
7
+
sin
(
n
3
+
1
)
,
U
1
n
=
24
n
6
,
U
2
n
=
3
n
5
,
U
3
n
=
400
n
5
,
F
1
n
=
7
n
6
,
F
2
n
=
12
n
5
,
F
3
n
=
600
n
5
,
P
1
n
=
n
2
-
3
,
P
2
n
=
n
2
-
1
,
P
3
n
=
3
n
+
1
,
hhhhhhhhhhhhhhhhhlh
(
n
,
u
,
v
,
w
)
∈
ℕ
n
0
×
ℝ
3
.
Obviously, (17) and (18) are satisfied. Note that Lemma 1 yields that
(107)
1
n
∑
i
=
n
+
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
∑
j
=
i
∞
‍
max
{
F
a
j
,
|
c
a
j
|
,
U
a
j
P
a
j
:
a
∈
Λ
3
}
≤
1
n
∑
i
=
n
+
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
∑
j
=
i
∞
‍
max
{
7
j
6
,
12
j
5
,
600
j
5
,
1
j
8
,
2
j
9
,
1
j
6
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
l
h
h
h
l
24
j
6
·
j
2
,
3
j
5
·
j
2
,
400
j
5
·
4
j
}
≤
600
n
∑
i
=
n
+
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
∑
j
=
i
∞
‍
1
j
3
≤
600
n
∑
j
=
n
+
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
j
·
1
j
3
=
600
n
∑
j
=
n
+
min
{
τ
a
:
a
∈
Λ
3
}
∞
‍
1
j
2
⟶
0
as
n
⟶
∞
,
which implies that (45) holds. Thus Theorem 3 implies that the system (105) possesses uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Example 3.
Consider the nonlinear three-dimensional difference system(108)
Δ
2
(
x
1
n
+
3
n
cos
2
n
5
n
+
3
x
1
(
n
-
τ
1
)
)
h
h
h
h
+
(
n
x
1
(
n
2
-
1
)
x
2
(
n
+
5
)
+
sin
(
x
1
(
n
-
2
)
x
2
(
n
+
7
)
)
h
h
h
h
h
h
h
+
cos
(
x
3
(
2
n
+
1
)
x
3
(
2
n
-
1
)
)
)
(
n
15
+
ln
n
)
-
1
hh
=
(
-
1
)
n
n
3
n
7
+
1
,
n
≥
5
,
Δ
2
(
x
2
n
+
3
n
3
sin
4
n
6
n
3
+
1
x
2
(
n
-
τ
2
)
)
h
h
h
h
+
(
n
-
2
x
2
(
n
2
-
2
)
2
x
1
(
n
+
2
)
x
2
(
3
n
+
1
)
+
n
x
1
(
2
n
+
2
)
x
3
(
3
n
+
5
)
h
h
h
h
h
h
+
n
-
2
x
2
(
n
2
-
2
)
2
x
3
(
4
n
+
1
)
2
)
(
n
6
+
n
2
+
1
)
-
1
hh
=
(
-
1
)
n
(
n
2
-
1
)
n
8
+
n
+
1
,
n
≥
5
,
Δ
2
(
x
3
n
+
4
n
21
5
n
21
+
ln
n
x
3
(
n
-
τ
3
)
)
h
h
h
h
+
x
1
(
n
+
3
)
x
2
(
n
+
1
)
+
x
2
(
n
+
4
)
x
3
(
n
2
-
2
)
+
x
1
(
2
n
+
3
)
x
3
(
n
2
-
3
)
n
6
+
n
x
1
(
n
+
3
)
2
+
x
3
(
n
2
-
2
)
2
hh
=
n
sin
n
n
9
+
1
,
n
≥
5
,
where n
0
=
5 and τ
1
,
τ
2
,
τ
3
∈
ℕ are fixed. Let n
1
=
8, k
=
2, d
1
=
2, D
1
=
1, d
2
=
4, D
2
=
2, d
3
=
5, D
3
=
4, β
=
min
{
5
-
τ
1
,
5
-
τ
2
,
5
-
τ
3
,
3
}
∈
ℕ, b
1
=
3
/
5, b
2
=
1
/
2, b
3
=
4
/
5, and
(109)
b
1
n
=
3
n
cos
2
n
5
n
+
3
,
b
2
n
=
3
n
3
sin
4
n
6
n
3
+
1
,
b
3
n
=
4
n
21
5
n
21
+
ln
n
,
c
1
n
=
(
-
1
)
n
n
3
n
7
+
1
,
c
2
n
=
(
-
1
)
n
(
n
2
-
1
)
n
8
+
n
+
1
,
c
3
n
=
n
sin
n
n
9
+
1
,
r
11
n
=
n
2
-
1
,
r
12
n
=
n
-
2
,
s
11
n
=
n
+
5
,
s
12
n
=
n
+
7
,
t
11
n
=
2
n
+
1
,
t
12
n
=
2
n
-
1
,
r
21
n
=
n
+
2
,
r
22
n
=
2
n
+
2
,
s
21
n
=
3
n
+
1
,
s
22
n
=
n
2
-
2
,
t
21
n
=
3
n
+
5
,
t
22
n
=
4
n
+
1
,
r
31
n
=
n
+
3
,
r
32
n
=
2
n
+
3
,
s
31
n
=
n
+
1
,
s
32
n
=
n
+
4
,
t
31
n
=
n
2
-
2
,
t
32
n
=
n
2
-
3
,
f
1
(
n
,
u
1
,
u
2
,
v
1
,
v
2
,
w
1
,
w
2
)
=
n
u
1
v
1
+
sin
(
u
2
v
2
)
+
cos
(
w
1
w
2
)
n
15
+
ln
n
,
f
2
(
n
,
u
1
,
u
2
,
v
1
,
v
2
,
w
1
,
w
2
)
=
u
1
v
1
+
n
u
2
w
1
+
n
-
2
v
2
2
w
2
2
n
6
+
n
2
+
1
,
f
3
(
n
,
u
1
,
u
2
,
v
1
,
v
2
,
w
1
,
w
2
)
=
u
1
v
1
+
v
2
w
1
+
u
2
w
2
n
6
+
n
u
1
2
+
w
1
2
,
U
1
n
=
40
n
14
,
U
2
n
=
1700
n
5
,
U
3
n
=
5000
n
6
,
F
1
n
=
20
n
14
,
F
2
n
=
3000
n
5
,
F
3
n
=
100
n
6
,
P
1
n
=
n
2
-
1
,
P
2
n
=
n
2
-
2
,
P
3
n
=
n
2
-
2
,
h
h
h
h
h
h
h
h
l
(
n
,
u
1
,
u
2
,
v
1
,
v
2
,
w
1
,
w
2
)
∈
ℕ
n
0
×
ℝ
6
.
It is easy to see that (17), (18), and (64) are satisfied. Note that Lemma 1 yields that
(110)
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
max
{
F
a
j
,
|
c
a
j
|
,
U
a
j
P
a
j
:
a
∈
Λ
3
}
≤
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
max
{
20
j
14
,
3000
j
5
,
100
j
6
,
1
j
4
,
1
j
6
,
2
j
8
,
40
j
14
·
j
2
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
1700
j
5
·
j
2
,
5000
j
6
·
j
2
}
≤
5000
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
1
j
3
≤
5000
n
∑
j
=
n
∞
‍
j
·
1
j
3
=
5000
n
∑
j
=
n
∞
‍
1
j
2
⟶
0
as
n
⟶
∞
,
which implies that (65) holds. Thus Theorem 4 implies that the system (108) possesses uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).Example 4.
Consider the nonlinear three-dimensional difference system(111)
Δ
2
(
x
1
n
-
4
n
3
+
2
5
n
3
+
3
x
1
(
n
-
τ
1
)
)
+
n
2
-
x
1
(
n
-
1
)
x
2
(
n
2
-
1
)
+
x
2
(
n
2
-
1
)
x
3
(
n
3
-
2
)
n
7
+
1
=
(
-
1
)
n
n
2
-
1
n
8
+
4
n
3
+
5
,
n
≥
4
,
Δ
2
(
x
2
n
-
3
n
cos
6
(
n
2
-
1
)
5
n
+
1
x
2
(
n
-
τ
2
)
)
+
n
3
/
2
x
1
(
n
+
3
)
+
x
2
(
n
2
-
3
)
x
3
(
3
n
-
1
)
ln
n
n
8
+
1
=
sin
(
n
2
-
1
)
n
5
+
1
,
n
≥
4
,
Δ
2
(
x
3
n
-
2
n
2
sin
2
(
n
2
+
3
)
4
n
2
+
3
x
3
(
n
-
τ
3
)
)
+
n
x
2
(
n
-
3
)
+
x
3
(
4
n
-
3
)
n
6
+
x
1
(
n
2
-
2
)
2
=
(
-
1
)
n
+
1
n
n
6
+
n
4
+
n
2
+
1
,
n
≥
4
,
where n
0
=
4 and τ
1
,
τ
2
,
τ
3
∈
ℕ are fixed. Let n
1
=
10, k
=
1, d
1
=
4, D
1
=
3, d
2
=
2, D
2
=
1, d
3
=
5, D
3
=
4, β
=
min
{
4
-
τ
1
,
4
-
τ
2
,
4
-
τ
3
,
1
}
∈
ℕ, b
1
=
-
4
/
5, b
2
=
-
3
/
5, b
3
=
-
1
/
2, and
(112)
b
1
n
=
-
4
n
3
+
2
5
n
3
+
3
,
b
2
n
=
-
3
n
cos
6
(
n
2
-
1
)
5
n
+
1
,
b
3
n
=
-
2
n
2
sin
2
(
n
2
+
3
)
4
n
2
+
3
,
c
1
n
=
(
-
1
)
n
n
2
-
1
n
8
+
4
n
3
+
5
,
c
2
n
=
sin
(
n
2
-
1
)
n
5
+
1
,
c
3
n
=
(
-
1
)
n
+
1
n
n
6
+
n
4
+
n
2
+
1
,
r
11
n
=
n
-
1
,
s
11
n
=
n
2
-
1
,
t
11
n
=
n
3
-
2
,
r
21
n
=
n
+
3
,
s
21
n
=
n
2
-
3
,
t
21
n
=
3
n
-
1
,
r
31
n
=
n
2
-
2
,
s
31
n
=
n
-
3
,
t
31
n
=
4
n
-
3
,
f
1
(
n
,
u
,
v
,
w
)
=
n
2
-
u
v
+
v
w
n
7
+
1
,
f
2
(
n
,
u
,
v
,
w
)
=
n
3
/
2
u
2
+
v
w
ln
n
n
8
+
1
,
f
3
(
n
,
u
,
v
,
w
)
=
n
v
+
w
n
6
+
u
2
,
U
1
n
=
22
n
7
,
U
2
n
=
26
n
6
,
U
3
n
=
300
n
5
,
F
1
n
=
47
n
5
,
F
2
n
=
10
n
6
,
F
3
n
=
6
n
5
,
P
1
n
=
n
3
-
2
,
P
2
n
=
n
2
-
3
,
P
3
n
=
n
2
-
4
,
hhhhhhhhhhhhhhhhhh
(
n
,
u
,
v
,
w
)
∈
ℕ
n
0
×
ℝ
3
.
Clearly, (17), (18), and (84) hold. Observe that Lemma 1 means that
(113)
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
max
{
F
a
j
,
|
c
a
j
|
,
U
a
j
P
a
j
:
a
∈
Λ
3
}
≤
1
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
max
{
4
j
5
,
10
j
6
,
6
j
5
,
2
j
6
,
1
j
5
,
1
j
5
,
22
j
7
·
j
3
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
26
j
6
·
j
2
,
300
j
5
·
j
2
}
≤
300
n
∑
i
=
n
∞
‍
∑
j
=
i
∞
‍
1
j
3
≤
300
n
∑
j
=
n
∞
‍
j
·
1
j
3
=
300
n
∑
j
=
n
∞
‍
1
j
2
⟶
0
as
n
⟶
∞
;
that is, (65) holds. Thus Theorem 5 implies that the system (111) possesses uncountably many positive solutions in Π
w
=
1
3
Ω
(
d
^
w
,
D
w
).



---

*Source: 954182-2014-01-08.xml*