# Global Asymptotic Stability of a Rational System

Authors: Lin-Xia Hu; Xiu-Mei Jia
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The main goal of this paper is to investigate the global asymptotic behavior of the difference equationx
n
+
1
=
β
1
x
n
/
A
1
+
y
n, y
n
+
1
=
β
2
x
n
+
γ
2
y
n
/
x
n
+
y
n, n
=
0,1
,
2
,
… with β
1
,
β
2
,
γ
2
,
A
1
∈
(
0
,
∞
) and the initial value (
x
0
,
y
0
)
∈
[
0
,
∞
)
×
[
0
,
∞
) such that x
0
+
y
0
≠
0. The major conclusion shows that, in the case where γ
2
<
β
2, if the unique positive equilibrium (
x
-
,
y
-
) exists, then it is globally asymptotically stable.


---

## Body

## 1. Introduction and Preliminaries

Our aim in this paper is to investigate the dynamics of the following difference equation:(1)
x
n
+
1
=
β
1
x
n
A
1
+
y
n
,
y
n
+
1
=
β
2
x
n
+
γ
2
y
n
x
n
+
y
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
n
=
0,1
,
2
,
…
with β
1
,
β
2
,
γ
2
,
A
1
∈
(
0
,
∞
) and the initial value (
x
0
,
y
0
)
∈
[
0
,
∞
)
×
[
0
,
∞
) such that x
0
+
y
0
≠
0.System (1) is a special case of the rational system
(2)
x
n
+
1
=
α
1
+
β
1
x
n
+
γ
1
y
n
A
1
+
B
1
x
n
+
C
1
y
n
,
y
n
+
1
=
α
2
+
β
2
x
n
+
γ
2
y
n
A
2
+
B
2
x
n
+
C
2
y
n
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
n
=
0
,
1
,
2
,
…
,
where all parameters and the initial value (
x
0
,
y
0
) are nonnegative such that denominators are always positive. There is some interest in systems of rational and related difference equations, for example, see [1–9]. In this paper, we will determine the global convergence properties of the system (1) under certain conditions.Whenβ
2
=
γ
2, the first component {
x
n
} of the solution (
x
n
,
y
n
) of the system (1) satisfies the first-order linear difference equation
(3)
x
n
+
1
=
β
1
A
1
+
β
2
x
n
,
n
=
0,1
,
…
and the second component {
y
n
} is constant and equal to γ
2 for n
≥
1.If the initial value is given byx
0
>
0, then by simple iteration, it is easy to find that
(4)
x
n
=
(
β
1
A
1
+
β
2
)
n
x
0
is the solution of (3). If β
1
>
A
1
+
β
2, then lim
⁡
n
→
∞
x
n
=
∞. If β
1
=
A
1
+
β
2, then x
n
=
x
0 for all n
>
0, and for β
1
<
A
1
+
β
2, we have lim
⁡
n
→
∞
x
n
=
0.Therefore, in the remaining part, we will assume thatβ
2
≠
γ
2.Clearly,(
0
,
γ
2
) is always an equilibrium, and when
(5)
A
1
+
min
⁡
⁡
{
β
2
,
γ
2
}
<
β
1
<
A
1
+
max
⁡
⁡
{
β
2
,
γ
2
}
,(1) also has a unique positive equilibrium
(6)
(
x
-
,
y
-
)
=
(
(
β
1
-
A
1
)
(
β
1
-
A
1
-
γ
2
)
A
1
+
β
2
-
β
1
,
β
1
-
A
1
)
.Equation (1) was investigated in [10] and the main result they obtained is the following.Theorem 1.
(i) Assume thatγ
2
>
β
2. Then every solution of the system (1) converges to (
0
,
γ
2
) if and only if β
1
≤
A
1
+
β
2, and when β
1
>
A
1
+
β
2, the system (1) has unbounded solutions.
(ii) Assume thatβ
2
>
γ
2. Then every positive solution of the system (1) is bounded if and only if β
1
<
A
1
+
β
2. In particular, when β
1
≤
A
1
+
γ
2, the equilibrium (
0
,
γ
2
) is a global attractor of all solutions of the system (1).In [10], the author proposed the following conjecture.Conjecture 2.
Assume that(7)
γ
2
<
β
1
-
A
1
<
β
2
.
Show that the unique positive equilibrium (
x
-
,
y
-
) of the system (1) is globally asymptotically stable.Inspired by Conjecture2, we investigate the global behavior of the system (1). To start our discussion, some basic results should be presented which will be useful in the sequel.Consider the system(8)
x
n
+
1
=
f
(
x
n
,
y
n
)
,
y
n
+
1
=
g
(
x
n
,
y
n
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
n
=
0,1
,
2
,
…
,
where F
=
(
f
,
g
)
:
D
→
R
2 is continuous and D
⊂
R
2.A vital tool for dealing with the linearized stability of (8) is the following well-known result which we incorporate in the following lemma (see, e.g., [11, 12]).Lemma 3.
LetF
=
(
f
,
g
) be a continuously differentiable function defined on an open set D
⊂
R
2. (a)
If the eigenvalues of the Jacobian matrixJ
F
(
(
x
-
,
y
-
)
) have modulus less than one, then the equilibrium of (8) is locally asymptotically stable.
(b)
If at least one of the eigenvalues of the Jacobian matrixJ
F
(
(
x
-
,
y
-
)
) has modulus greater than one, then the equilibrium of (8) is unstable.
(c)
The equilibrium(
x
-
,
y
-
) of (8) is locally asymptotically stable if every solution of the characteristic equation of the Jacobian matrix
(9)
λ
2
-
p
λ
+
q
=
0
lies inside the unit circle, that is, if
(10)
|
p
|
<
1
+
q
<
2
.
In this case, (
x
-
,
y
-
) is also called a sink.
(d)
The equilibrium(
x
-
,
y
-
) of (8) is a repeller if every solution of characteristic equation (9) lies outside the unit circle, which is equivalent to the following condition:
(11)
|
q
|
>
1
,
|
p
|
<
|
1
+
q
|
.
(e)
The equilibrium(
x
-
,
y
-
) of (8) is a saddle point if the Jacobian matrix J
F
(
(
x
-
,
y
-
)
) has one eigenvalue that lies inside the unit circle and if the other one lies outside the unit circle, that is, if and only if
(12)
|
p
|
>
|
1
+
q
|
,
p
2
-
4
q
>
0
.The following well-known comparison result will be used in estimating the value of a solution of the system (1).Lemma 4 (a comparison result).
Assume thatα
∈
(
0
,
∞
) and β
∈
R. Let {
u
n
}
n
=
0
∞ and {
v
n
}
n
=
0
∞ be sequences of real numbers such that u
0
≤
v
0 and
(13)
u
n
+
1
≤
α
u
n
+
β
,
v
n
+
1
=
α
v
n
+
β
,
n
=
0,1
,
2
,
…
.
Then u
n
≤
v
n for n
≥
0.Consider the following difference equation:(14)
u
n
+
1
=
f
(
u
n
)
,
n
=
0,1
,
2
,
…
.The following result of Hautus and Bolis [13] (see also [11, 12]) deals with the global attractivity of (14).Lemma 5.
LetI
⊆
[
0
,
∞
) be some interval and assume that f
∈
C
[
I
,
(
0
,
∞
)
] satisfies the following conditions: (i)
f
(
u
) is nondecreasing in u;
(ii)
equation (14) has a unique positive equilibrium u
-
∈
I and the function f
(
u
) satisfies the negative feedback condition:
(15)
(
u
-
u
-
)
(
f
(
u
)
-
u
)
<
0
f
o
r
e
v
e
r
y
u
∈
I
∖
{
u
-
}
.Then every positive solution of (14) with initial conditions in I converges to u
-.To prepare for our major investigation, we consider the following equation:(16)
u
n
+
1
=
g
(
u
n
,
u
n
-
1
)
,
n
=
0,1
,
2
,
…
,
and the following lemma should be mentioned which is from [12].Lemma 6.
Let[
a
,
b
] be an interval of real numbers and assume that g
:
[
a
,
b
]
2
→
[
a
,
b
] is a continuous function satisfying the following properties: (i)
g
(
x
,
y
) is nondecreasing in each of its arguments;
(ii)
the functiong
(
m
,
m
)
=
m has a unique positive solution.Then (16) has a unique equilibrium u
-
∈
[
a
,
b
] and every solution of (16) converges to u
-.


## 2. Linearized Stability

In this section, we will make some conclusions about linearized stability. Consider the mapT on R
2 associated with the system (1), that is,
(17)
T
(
x
,
y
)
=
(
f
1
(
x
,
y
)
f
2
(
x
,
y
)
)
=
(
β
1
x
A
1
+
y
β
2
x
+
γ
2
y
x
+
y
)
.
Calculating the partial derivatives of the functions f
1
(
x
,
y
) and f
2
(
x
,
y
) shows that
(18)
∂
f
1
∂
x
=
β
1
A
1
+
y
,
∂
f
1
∂
y
=
-
β
1
x
(
A
1
+
y
)
2
,
∂
f
2
∂
x
=
(
β
2
-
γ
2
)
y
(
x
+
y
)
2
,
∂
f
2
∂
y
=
(
γ
2
-
β
2
)
x
(
x
+
y
)
2
.The Jacobian matrix ofT evaluated at (
0
,
γ
2
) is
(19)
J
F
(
(
0
,
γ
2
)
)
=
(
β
1
A
1
+
γ
2
0
β
2
-
γ
2
γ
2
0
)
,
and its eigenvalues are λ
1
=
0 and λ
2
=
β
1
/
(
A
1
+
γ
2
).Another equilibrium(
x
-
,
y
-
), namely, (6), exists if and only if (5) holds. Using the equality A
1
+
y
-
=
β
1, the Jacobian matrix of T evaluated at (
x
-
,
y
-
) is
(20)
J
F
(
(
x
-
,
y
-
)
)
=
(
1
-
x
-
β
1
(
β
2
-
γ
2
)
y
-
(
x
-
+
y
-
)
2
(
γ
2
-
β
2
)
x
-
(
x
-
+
y
-
)
2
)
,
and its characteristic equation associated with (
x
-
,
y
-
) is given by
(21)
λ
2
-
p
λ
+
q
=
0
,
where
(22)
p
=
1
+
(
γ
2
-
β
2
)
x
-
(
x
-
+
y
-
)
2
=
(
β
1
-
A
1
)
(
β
2
-
γ
2
)
-
(
β
1
-
A
1
-
γ
2
)
(
A
1
+
β
2
-
β
1
)
(
β
1
-
A
1
)
(
β
2
-
γ
2
)
,
q
=
(
β
2
-
γ
2
)
x
-
(
y
-
-
β
1
)
β
1
(
x
-
+
y
-
)
2
=
-
A
1
(
β
1
-
A
1
-
γ
2
)
(
A
1
+
β
2
-
β
1
)
β
1
(
β
1
-
A
1
)
(
β
2
-
γ
2
)
.Whenβ
2
>
γ
2, we find that γ
2
<
β
1
-
A
1
<
β
2 and A
1
/
β
1
<
1. Thus p
>
0 and
(23)
1
+
q
=
1
-
A
1
(
β
1
-
A
1
-
γ
2
)
(
A
1
+
β
2
-
β
1
)
β
1
(
β
1
-
A
1
)
(
β
2
-
γ
2
)
>
1
-
(
β
1
-
A
1
-
γ
2
)
(
A
1
+
β
2
-
β
1
)
(
β
1
-
A
1
)
(
β
2
-
γ
2
)
=
|
p
|
,
(24)
0
>
q
=
-
A
1
(
β
1
-
A
1
-
γ
2
)
(
A
1
+
β
2
-
β
1
)
β
1
(
β
1
-
A
1
)
(
β
2
-
γ
2
)
=
-
A
1
β
1
β
1
-
A
1
-
γ
2
β
1
-
A
1
·
β
2
-
(
β
1
-
A
1
)
β
2
-
γ
2
>
-
1
.Whenγ
2
>
β
2, A
1
/
β
1
<
1 holds and by simple computation, we have
(25)
p
2
-
4
q
=
[
1
+
(
γ
2
-
β
2
)
x
-
(
x
-
+
y
-
)
2
]
2
-
4
(
β
2
-
γ
2
)
x
-
(
y
-
-
β
1
)
β
1
(
x
-
+
y
-
)
2
>
[
1
+
(
γ
2
-
β
2
)
x
-
(
x
-
+
y
-
)
2
]
2
-
4
(
γ
2
-
β
2
)
x
-
(
x
-
+
y
-
)
2
=
[
1
-
(
γ
2
-
β
2
)
x
-
(
x
-
+
y
-
)
2
]
2
≥
0
.Furthermore,p
,
q
>
0 and
(26)
|
1
+
q
|
=
1
+
A
1
(
γ
2
-
β
2
)
x
-
β
1
(
x
-
+
y
-
)
2
<
1
+
(
γ
2
-
β
2
)
x
-
(
x
-
+
y
-
)
2
=
|
p
|
.Employing Lemma3, we formulate the results in the following.Theorem 7.
(i) The equilibrium(
0
,
γ
2
) of the system (1) is locally asymptotically stable when β
1
-
A
1
<
γ
2, and it is unstable (a saddle point) when β
1
-
A
1
>
γ
2, and it is nonhyperbolic when β
1
-
A
1
=
γ
2.
(ii) Assume thatγ
2
<
β
2 and (7) holds. Then the unique positive equilibrium (
x
-
,
y
-
) of the system (1) is locally asymptotically stable.
(iii) Assume thatβ
2
<
γ
2 and β
2
<
β
1
-
A
1
<
γ
2. Then the unique positive equilibrium (
x
-
,
y
-
) of the system (1) is unstable; further, it is a saddle point.


## 3. Global Attractivity

In this section, we will commence global asymptotic stability analysis. Let(
x
n
,
y
n
) be a solution of the system (1), then it is easy to obtain the following result from the second equation of the system (1).Theorem 8.
(i) Assume thatβ
2
>
γ
2. Then every solution (
x
n
,
y
n
) of the system (1) satisfies γ
2
≤
y
n
≤
β
2 for n
≥
1.
(ii) Assume thatγ
2
>
β
2. Then every solution (
x
n
,
y
n
) of the system (1) satisfies β
2
≤
y
n
≤
γ
2 for n
≥
1.Theorem 9.
Every solution of the system (1) with x
0
=
0 converges to (
0
,
γ
2
).Proof.
Sincex
0
=
0 implies that x
n
=
0 for n
≥
1, thus lim
⁡
n
→
∞
y
n
=
γ
2, finishing the proof.Theorem 10.
Assume thatγ
2
<
β
2
≤
β
1
-
A
1. Then every solution of the system (1) with x
0
>
0 satisfies lim
⁡
n
→
∞
x
n
=
∞, lim
⁡
n
→
∞
y
n
=
β
2.Proof.
Using Theorem8, we get that when γ
2
<
β
2
<
β
1
-
A
1,
(27)
x
n
+
1
=
β
1
A
1
+
y
n
x
n
>
β
1
A
1
+
β
2
x
n
⟶
∞
,
and when γ
2
<
β
2
=
β
1
-
A
1,
(28)
x
n
+
1
=
β
1
A
1
+
y
n
x
n
>
β
1
A
1
+
β
2
x
n
=
x
n
⟶
∞
,
since the only equilibrium of the system (1) is (
0
,
γ
2
) when β
1
=
A
1
+
β
2.
Further, using the boundedness ofy
n, we have
(29)
y
n
+
1
=
β
2
x
n
+
γ
2
y
n
x
n
+
y
n
=
β
2
+
γ
2
(
y
n
/
x
n
)
1
+
(
y
n
/
x
n
)
⟶
β
2
.
The proof is complete.For the case whereβ
1
≤
A
1
+
γ
2, the authors had obtained that the unique positive equilibrium (
0
,
γ
2
) is a global attractor of all solutions of the system (1) in [10], see Theorem 1 (ii). Moreover, in view of Theorem 7 (i), we may formulate the result in the following theorem.Theorem 11.
Assume thatβ
1
-
A
1
<
γ
2
<
β
2. Then the unique equilibrium (
0
,
γ
2
) of the system (1) is globally asymptotically stable.Now, we pay attention to dealing with the global attractivity of the unique positive equilibrium(
x
-
,
y
-
), namely, (6), under the condition that γ
2
<
β
2. In this case, (
x
-
,
y
-
) exists if and only if (7) holds. To obtain the global attractivity of (
x
-
,
y
-
), the following useful lemma should first be established.Consider the following difference equation:(30)
u
n
+
1
=
a
u
n
(
u
n
+
1
)
u
n
+
b
,
n
=
1,2
,
…
,
where 0
<
b
<
a
<
1 and the initial value u
1
=
x
1
/
y
1
=
β
1
x
0
(
x
0
+
y
0
)
/
(
A
1
+
y
0
)
(
β
2
x
0
+
γ
2
y
0
). Equation (30) possesses two equilibria, namely, zero and u
-
=
(
a
-
b
)
/
(
1
-
a
).Lemma 12.
Every positive solution of (30) converges to the unique positive equilibrium u
-.Proof.
Clearly,u
0
>
0 implies that u
n
>
0 for n
≥
1. Let f
(
u
)
=
a
u
(
u
+
1
)
/
(
u
+
b
), then f
(
u
) is increasing in u for u
>
0 and
(31)
(
u
-
u
-
)
[
a
u
(
u
+
1
)
u
+
b
-
u
]
=
u
(
u
-
u
-
)
[
a
-
b
-
(
1
-
a
)
u
u
+
b
]
=
-
(
1
-
a
)
u
(
u
-
u
-
)
2
u
+
b
<
0
.
Thus lim
⁡
n
→
∞
u
n
=
u
- for u
0
>
0 by applying Lemma 5.
The proof is complete.Theorem 13.
Assume that (7) holds. Then the unique positive equilibrium (
x
-
,
y
-
) of the system (1) is globally asymptotically stable.Proof.
In view of Theorem7, it is sufficient to show that (
x
-
,
y
-
) is a global attractor of all positive solutions of the system (1).
In this case,y
n
≥
γ
2
>
0 holds for n
≥
1 and thus the system (1) yields
(32)
x
n
+
1
y
n
+
1
=
β
1
(
x
n
/
y
n
)
1
+
(
A
1
/
y
n
)
(
x
n
/
y
n
)
+
1
β
2
(
x
n
/
y
n
)
+
γ
2
=
β
1
β
2
(
1
+
(
A
1
/
y
n
)
)
(
x
n
/
y
n
)
(
(
x
n
/
y
n
)
+
1
)
(
x
n
/
y
n
)
+
(
γ
2
/
β
2
)
,
y
n
+
1
=
β
2
(
x
n
/
y
n
)
+
γ
2
(
x
n
/
y
n
)
+
1
,
for n
≥
1. Let u
n
=
x
n
/
y
n, v
n
=
y
n, then the system (1) becomes
(33)
u
n
+
1
=
β
1
β
2
(
1
+
(
A
1
/
v
n
)
)
u
n
(
u
n
+
1
)
u
n
+
(
γ
2
/
β
2
)
,
v
n
+
1
=
β
2
u
n
+
γ
2
u
n
+
1
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
n
=
1,2
,
…
.
Further, the system (33) may reduce to the following second-order difference equation:
(34)
u
n
+
1
=
β
1
u
n
(
u
n
+
1
)
β
2
u
n
+
γ
2
β
2
u
n
-
1
+
γ
2
β
2
u
n
-
1
+
γ
2
+
A
1
(
u
n
-
1
+
1
)
,
n
=
2,3
,
…
.
Clearly, zero is always the equilibrium of (34) and when (7) holds, (34) also possesses a unique positive equilibrium
(35)
u
-
=
β
1
-
A
1
-
γ
2
A
1
+
β
2
-
β
1
.
Notice that γ
2
≤
v
n
=
y
n
≤
β
2 for n
≥
1, and we get
(36)
β
1
γ
2
β
2
(
A
1
+
γ
2
)
≤
β
1
β
2
(
1
+
(
A
1
/
v
n
)
)
≤
β
1
A
1
+
β
2
,
and thus
(37)
β
1
γ
2
β
2
(
A
1
+
γ
2
)
u
n
(
u
n
+
1
)
u
n
+
(
γ
2
/
β
2
)
≤
u
n
+
1
=
β
1
β
2
(
1
+
(
A
1
/
v
n
)
)
u
n
(
u
n
+
1
)
u
n
+
(
γ
2
/
β
2
)
≤
β
1
A
1
+
β
2
u
n
(
u
n
+
1
)
u
n
+
(
γ
2
/
β
2
)
,
n
≥
1
.
Leta
=
β
1
γ
2
/
β
2
(
A
1
+
γ
2
), b
=
γ
2
/
β
2, then a
<
(
β
1
/
β
2
)
(
β
2
/
(
A
1
+
β
2
)
)
=
β
1
/
(
A
1
+
β
2
)
<
1 and b
<
a
<
1. Hence by Lemma 12, we get that every positive solution of the following difference equation
(38)
u
˘
n
+
1
=
β
1
γ
2
β
2
(
A
1
+
γ
2
)
u
˘
n
(
u
˘
n
+
1
)
u
˘
n
+
(
γ
2
/
β
2
)
,
n
=
1,2
,
…
converges to its unique positive equilibrium u
˘
=
γ
2
(
β
1
-
(
A
1
+
γ
2
)
)
/
(
β
2
A
1
+
γ
2
(
β
2
-
β
1
)
).
Leta
=
β
1
/
(
A
1
+
β
2
), b
=
γ
2
/
β
2, then γ
2
/
β
2
<
(
A
1
+
γ
2
)
/
(
A
1
+
β
2
)
<
β
1
/
(
A
1
+
β
2
)
<
1, which means that b
<
a
<
1. Similarly, by Lemma 12, we know that every positive solution of the following difference equation
(39)
u
^
n
+
1
=
β
1
A
1
+
β
2
u
^
n
(
u
^
n
+
1
)
u
^
n
+
(
γ
2
/
β
2
)
,
n
=
1,2
,
…
converges to its unique positive equilibrium u
^
=
(
β
1
β
2
-
γ
2
(
A
1
+
β
2
)
)
/
β
2
(
A
1
+
β
2
-
β
1
).
Applying Lemma4 and (37), we find that every solution of (34) with initial value u
˘
1
=
u
^
1
=
u
1
=
(
x
1
/
y
1
)
>
0 satisfies
(40)
u
˘
n
≤
u
n
≤
u
^
n
,
for
n
≥
1
.
Hence for 0
<
ϵ
<
u
˘
/
2, there exists an integer N such that for n
>
N,
(41)
u
˘
2
<
u
˘
-
ϵ
<
u
˘
n
≤
u
n
≤
u
^
n
<
u
^
+
ϵ
<
u
^
+
u
˘
2
.
Moreover,
(42)
u
˘
2
≤
liminf
⁡
n
→
∞
u
n
≤
limsup
⁡
n
→
∞
u
n
≤
u
^
+
u
˘
2
.
LetI
=
u
˘
/
2, S
=
u
^
+
(
u
˘
/
2
), then S
>
I
>
0 and every solution of (34) eventually enters the invariant interval [
I
,
S
].
Denote the function(43)
g
(
u
,
v
)
=
β
1
u
(
u
+
1
)
β
2
u
+
γ
2
β
2
v
+
γ
2
β
2
v
+
γ
2
+
A
1
(
v
+
1
)
,
and simple computation shows that
(44)
∂
g
∂
u
=
β
1
(
β
2
u
2
+
2
γ
2
u
+
γ
2
)
(
β
2
u
+
γ
2
)
2
β
2
v
+
γ
2
β
2
v
+
γ
2
+
A
1
(
v
+
1
)
>
0
,
(45)
∂
g
∂
v
=
β
1
u
(
u
+
1
)
β
2
u
+
γ
2
A
1
(
β
2
-
γ
2
)
[
β
2
v
+
γ
2
+
A
1
(
v
+
1
)
]
2
>
0
.
Applying Lemma6, to establish the global attractivity of the equilibrium u
- of (34), it is sufficient to confirm that the following equation
(46)
g
(
m
,
m
)
=
β
1
m
(
m
+
1
)
β
2
m
+
γ
2
β
2
m
+
γ
2
β
2
m
+
γ
2
+
A
1
(
m
+
1
)
=
m
has a unique positive solution.
Solving (46), we get
(47)
β
1
(
m
+
1
)
=
β
2
m
+
γ
2
+
A
1
(
m
+
1
)
,
from which it follows that
(48)
m
=
β
1
-
A
1
-
γ
2
A
1
+
β
2
-
β
1
=
u
-
.
Therefore,lim
⁡
n
→
∞
u
n
=
u
-, and hence,
(49)
lim
⁡
n
→
∞
y
n
=
lim
⁡
n
→
∞
v
n
=
β
2
u
-
+
γ
2
u
-
+
1
=
β
1
-
A
1
=
y
-
.
Furthermore,
(50)
lim
⁡
n
→
∞
x
n
=
lim
⁡
n
→
∞
u
n
·
lim
⁡
n
→
∞
y
n
=
u
-
·
y
-
=
(
β
1
-
A
1
)
(
β
1
-
A
1
-
γ
2
)
A
1
+
β
2
-
β
1
=
x
-
,
and thus the result follows.
The proof is complete.



---

*Source: 286375-2014-08-07.xml*