Abstract
The main goal of the paper is to investigate boundedness, invariant intervals, semicycles, and global attractivity of all nonnegative solutions of the equation 
, 
, where the parameters 
, 
is an integer, and the initial conditions 
. It is shown that the unique positive equilibrium of the equation is globally asymptotically stable under the condition 
. The result partially solves the open problem proposed by Kulenović and Ladas in work (2002).
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1. Introduction
The aim of this paper is to study the dynamical behavior of the following rational difference equation:
where 
is an integer, and the initial conditions 
The related case where 
was investigated in [1].
In 2002, Kulenović and Ladas [1] proposed the following open problem.
Open Problem 1.
Assume that 
and 
. Investigate the global behavior of all positive solutions of (1.1).
If we allow the parameters to be satisfied, 
, then (1.1) contains, as special cases, six difference equations with positive parameters:
Equations (1.2) and (1.3) can be reduced to (1.5), which was studied in [2]. By the change of variables 
, equation (1.4) can be reduced to the linear equation
whose global behavior of solutions is easily derived. Equation (1.6) is investigated in [3].
Equation (1.7) is essentially similar to the Riccati equation. In fact, if 
is a positive solution of (1.7), then 
are 
solutions of the following Riccati equations:
respectively.
Therefore, we only pay attention to investigating (1.1) with positive parameters and omit any further discussion of the above (1.2)–(1.7).
For other related results on difference equations, one can refer to [4–18].
Equation (1.1) has a unique equilibrium 
which is positive and is given by
In [19], the authors investigated the global asymptotic stability of the positive equilibrium 
of (1.1). We summarize their results in the following theorem.
Theorem 1.1.
-
(i)
Assume that
. Then the positive equilibrium 
of (1.1) is locally asymptotically stable. -
(ii)
Assume that
and 
. Then for every solution of (1.1) with initial conditions in invariant interval 
, the positive equilibrium point 
is globally asymptotically stable. -
(iii)
Assume that
and 
. Then for every solution of (1.1) with initial conditions in invariant interval 
, the positive equilibrium point 
is globally asymptotically stable.
. Then the positive equilibrium 
of (1.1) is locally asymptotically stable.
and 
. Then for every solution of (1.1) with initial conditions in invariant interval 
, the positive equilibrium point 
is globally asymptotically stable.
and 
. Then for every solution of (1.1) with initial conditions in invariant interval 
, the positive equilibrium point 
is globally asymptotically stable.Reviewing the proof of Theorem 1.1, one can easily find that the positive equilibrium 
of (1.1) is locally asymptotically stable when 
. So Theorem 1.1 (i) can be improved as the following theorem.
Theorem 1.2.
Assume that 
. Then the positive equilibrium 
of (1.1) is locally asymptotically stable.
By the change of variables 
, (1.1) reduces to the difference equation
where 
, and 
Its unique positive equilibrium is
Motivated by the above open problem, the purpose of this paper is to investigate the boundedness, local stability, invariant intervals, semicycles, and global attractivity of all nonnegative solutions of (1.11). We show that the unique positive equilibrium of (1.11) is a global attractor when 
and our result solves the open problem when 
.
The organization of this paper is as follows. In Section 2, some basic definitions and lemmas regarding difference equations are given. The boundedness and invariant intervals of (1.11) are discussed in Section 3. The semicycle analysis of (1.11) is presented in Section 4. The main results are formulated and proved in Section 5, where the global asymptotic stability of (1.11) is investigated.
2. Some Lemmas
In this section, we recall some definitions and lemmas which will be useful in the sequel.
Let 
be some interval of real numbers and let 
be a continuously differentiable function. Then for initial conditions 
, the difference equation
has a unique solution 
.
An internal
is called an invariant interval of (2.1) if
That is, every solution of (2.1) with initial conditions in 
remains in 
.
Definition 2.1.
Let 
be an equilibrium point of (2.1).
-
(i)
The equilibrium point
of (2.1) is called locally stable if for every 
there exists 
such that for all 
with 
, one has 
. -
(ii)
The equilibrium point
of (2.1) is called locally asymptotically stable if it is locally stable and if there exists 
such that for all 
with 
one has 
-
(iii)
The equilibrium point
of (2.1) is called a global attractor if for every 
one has 
-
(iv)
The equilibrium point
of (2.1) is called globally asymptotically stable if it is locally asymptotically stable and is a global attractor. -
(v)
The equilibrium point
of (2.1) is called unstable if it is not locally stable.
of (2.1) is called locally stable if for every 
there exists 
such that for all 
with 
, one has 
.
of (2.1) is called locally asymptotically stable if it is locally stable and if there exists 
such that for all 
with 
one has 
of (2.1) is called a global attractor if for every 
one has 
of (2.1) is called globally asymptotically stable if it is locally asymptotically stable and is a global attractor.
of (2.1) is called unstable if it is not locally stable.The following lemmas can be found in [3, 20], respectively; also see [19, 21–24].
Lemma 2.2 (see [3]).
Let 
be an interval of real numbers and assume that 
is a continuous function satisfying the following properties.
-
(i)
is nondecreasing in each of its arguments. -
(i)
(ii)The equation
(2.3)has a unique positive solution in the interval
.
is nondecreasing in each of its arguments.
.Then (2.1) has a unique positive equilibrium 
and every solution of (2.1) converges to 
.
Lemma 2.3 (see [20]).
Let 
be an interval of real numbers and assume that 
is a continuous function satisfying the following properties.
-
(i)
is a nondecreasing function in 
and a nonincreasing function in 
. -
(ii)
If
is a solution of the following system:
(2.4)then
.
is a nondecreasing function in 
and a nonincreasing function in 
.
is a solution of the following system:
.Then (2.1) has a unique equilibrium point 
and every solution of (2.1) converges to 
.
The following result was proved in [25].
Lemma 2.4 (see [25]).
Assume that 
and 
hold. Then the positive equilibrium 
of the difference equation
is a global attractor of all positive solutions.
3. Boundedness and Invariant Intervals
The following result about boundedness of (1.11) can be found in [19].
Theorem 3.1.
Every solution of (1.11) is bounded from above and from below by positive constants.
Let 
be a nonnegative solution of (1.11). Then the following identities are easily established:
If 
, then the unique equilibrium is 
and (3.1) and (3.4) become
respectively.
If 
, then the unique equilibrium is 
and (3.4) becomes
Set
Then we have the following monotone character for the function 
.
Lemma 3.2.
Let 
be defined by (3.8). Then the following statements hold true.
-
(i)
Assume that
. Then 
is strictly increasing in each of its arguments for 
and it is strictly increasing in 
and decreasing in 
for 
. -
(ii)
Assume that
. Then 
is strictly increasing in 
and decreasing in 
for 
.
. Then 
is strictly increasing in each of its arguments for 
and it is strictly increasing in 
and decreasing in 
for 
.
. Then 
is strictly increasing in 
and decreasing in 
for 
.Proof.
By calculating, the partial derivatives of the function 
are
from which (i) and (ii) easily follow.
3.1. The Case 
Lemma 3.3.
Assume that 
and 
is a nonnegative solution of (1.11). Then the following statements are true.
-
(i)
If for some
, then 
for 
. -
(ii)
If for some
, then 
. -
(iii)
If for some
, then 
. -
(iv)
If for some
, then 
for 
. -
(v)
If for some
, then 
. -
(vi)
If for some
, then 
. -
(vii)
Equation (1.11) possesses an invariant interval
and 
; moreover, the interval 
is also an invariant interval of (1.11) and 
.
, then 
for 
.
, then 
.
, then 
.
, then 
for 
.
, then 
.
, then 
.
and 
; moreover, the interval 
is also an invariant interval of (1.11) and 
.Proof.
-
(i)–(vi)
Clearly, in this case
holds. The statements directly follow by using the identities (3.1), (3.3) and (3.4). -
(vii)
By Lemma 3.2 (i) the function
is strictly increasing in each of its arguments for 
. Given that 
, then we get 
(3.10)which implies that
. By the induction, 
for every 
, as claimed.
holds. The statements directly follow by using the identities (3.1), (3.3) and (3.4).
is strictly increasing in each of its arguments for 
. Given that 
, then we get 
. By the induction, 
for every 
, as claimed.On the other hand, 
implies that
Furthermore, 
is the unique positive root of the quadratic equation
Since
then we have that 
. That is, 
, finishing the proof.
When 
, identities (3.5) and (3.6) imply that the following results hold.
Lemma 3.4.
Assume that 
and 
is a nonnegative solution of (1.11). Then the following statements are true.
-
(i)
If for some
, then 
for 
. -
(ii)
If for some
, then 
for 
. -
(iii)
If for some
, then 
for 
. -
(iv)
If for some
, then 
. -
(v)
If for some
, then 
.
, then 
for 
.
, then 
for 
.
, then 
for 
.
, then 
.
, then 
.Lemma 3.5.
Assume that 
holds and 
is a nonnegative solution of equation (1.11). Then the following statements are true
-
(i)
If for some
, then 
for 
. -
(ii)
If for some
, then 
. -
(iii)
If for some
, then 
. -
(iv)
Equation (1.11) possesses an invariant interval
and 
. Further, if 
, then 
for all 
, and if 
, then the following statements are also true. -
(v)
If for some
, then 
for 
. -
(vi)
If for some
, then 
. -
(vii)
If for some
, then 
.
, then 
for 
.
, then 
.
, then 
.
and 
. Further, if 
, then 
for all 
, and if 
, then the following statements are also true.
, then 
for 
.
, then 
.
, then 
.Proof.
In this case 
holds.
-
(i)–(iii)
Using the identities (3.3) and (3.4), one can see that the results follow.
-
(iv)
By Lemma 3.2 the function
is strictly increasing in 
and decreasing in 
in 
. Then we can get 
(3.14)
is strictly increasing in 
and decreasing in 
in 
. Then we can get 
which implies that 
. By the induction, 
for every 
.
On the other hand, 
implies that
Then as same as the argument in Lemma 3.3 (vii) it can be proved that 
.
Further, if 
, then the identity (3.1) implies that 
for all 
, and if 
, the results (v)–(vii) can also follow from (3.1).
The proof is complete.
3.2. The Case 
Lemma 3.6.
Assume that 
. Then the interval 
is an invariant interval of (1.11) and 
. Further, if 
, then the following statements are also true.
-
(i)
If for some
, then 
for 
. -
(ii)
If for some
, then 
. -
(iii)
If for some
, then 
.
, then 
for 
.
, then 
.
, then 
.Proof.
Notice that in this case the inequality 
holds; then the results can easily be obtained by using the identity (3.1). Further, 
implies that
finishing the proof.
4. Semicycle Analysis
Here, we present some results regarding the semicycle analysis of the solutions of (1.11). Now recall two definitions from [25].
Definition 4.1.
Let 
be a solution of (2.1). A positive semicycle of the solution 
of (2.1) consists of a "string" of terms 
, all greater than or equal to the equilibrium point 
, with 
and 
such that
Definition 4.2.
Let 
be a solution of (2.1). A negative semicycle of the solution 
of (2.1) consists of a "string" of terms 
, all less than the equilibrium point 
, with 
and 
such that
The next two lemmas can be found in [26] and [19], respectively.
Lemma 4.3 (see [26]).
Assume that 
and that 
is increasing in both arguments. Let 
be a positive equilibrium of (2.1). Then except possibly for the first semicycle, every oscillatory solution of (2.1) has semicycles of length at most 
.
Lemma 4.4 (see [19]).
Assume that 
and that 
is increasing in 
for each fixed 
and is decreasing in 
for each fixed 
. Let 
be a positive equilibrium of (2.1). If 
, then every oscillatory solution of (2.1) has semicycles that are either of length at least 
or of length at most 
.
Using the monotonic character of the function 
from Lemma 3.2, in each of the intervals in Lemmas 3.3–3.6, together with Lemmas 4.3 and 4.4, it is easy to obtain the following results concerning semicycle analysis.
Theorem 4.5.
Assume that 
and 
is a nonnegative solution of (1.11). Then the following statements are true.
-
(i)
If
, then, except possibly for the first semicycle, every oscillatory solution of (1.11) which lies in the invariant interval 
has semicycles of length at most 
. -
(ii)
If
, then (1.11) does not have oscillatory solution with 
or 
. -
(iii)
If
, then, except possibly for the first semicycle, every oscillatory solution of (1.11) which lies in the invariant interval 
has semicycles that are either of length at least 
or of length at most 
.
, then, except possibly for the first semicycle, every oscillatory solution of (1.11) which lies in the invariant interval 
has semicycles of length at most 
.
, then (1.11) does not have oscillatory solution with 
or 
.
, then, except possibly for the first semicycle, every oscillatory solution of (1.11) which lies in the invariant interval 
has semicycles that are either of length at least 
or of length at most 
.Theorem 4.6.
Assume that 
and 
is a nonnegative solution of (1.11). Then, except possibly for the first semicycle, every oscillatory solution of (1.11) which lies in the invariant interval 
has semicycles that are either of length at least 
or of length at most 
.
5. Global Asymptotic Stability for the Case 
In this section, we discuss the global attractivity of the positive equilibrium of (1.11). We show that 
is a global attractor of all nonnegative solutions of (1.11) when 
. Further, the unique positive equilibrium 
of (1.1) is globally asymptotically stable when 
.
Theorem 5.1.
Assume that 
. Then the unique positive equilibrium 
of (1.11) is a global attractor.
The proof is finished by considering the following four cases; see Theorems 5.3, 5.5, 5.7, and 5.9.
Theorem 5.2.
Assume that 
holds, and 
is a nonnegative solution of (1.11). If 
, then 
for 
. Furthermore, every nonnegative solution of (1.11) lies eventually in the interval 
.
Proof.
Firstly, note that in this case 
holds.
If 
, then by Lemma 3.3 (i) and (iv), we have that 
for 
, the first assertion follows.
To complete the proof it remains to show that when 
there exists 
such that 
. There are two cases to be considered.
Case 1 (
).
Lemma 3.3 (ii) and (iii) implies that 
. If 
, then the proof follows from the first assertion. Now assume for the sake of contradiction that all terms of 
never enter the interval 
; then 
would lie in the interval 
for 
. Using Lemma 3.3 (vi), we obtain 
for 
, from which it follows that the sequence 
is strictly decreasing in the interval 
. Hence, 
exists and 
, which is a contradiction, because, in view of Lemma 3.3 (vii), (1.11) has no equilibrium points in the interval 
.
Case 2 (
).
If there exists 
such that 
, then the proof follows from the first assertion. If there exists 
such that 
, then the proof also follows from Case 1. Now assume for the sake of contradiction that 
for all 
, then by Lemma 3.3 (v), we have that 
for 
, which means that 
exists and 
; this contradicts Lemma 3.3 (vii).
The proof is complete.
Theorem 5.3.
Assume that 
holds. Then the unique positive equilibrium 
of (1.11) is a global attractor of all nonnegative solutions of (1.11).
Proof.
Theorem 5.2 and Lemma 3.3 (vii) imply that every nonnegative solution of (1.11) eventually enters the interval 
. Furthermore, the function 
is increasing in each of its arguments in 
and the equation
has a unique positive solution on the interval 
. The proof now immediately follows by applying Lemma 2.2.
Theorem 5.4.
Assume that 
, and 
is a nontrivial nonnegative solution of (1.11). Then the sequence 
is monotonic and 
.
Proof.
In this case, the unique positive equilibrium is 
. By Lemma 3.4 it easy to see that if 
then 
is decreasing and bounded below by 1, if 
then 
is increasing and bounded above by 1, and if 
then 
for 
. Hence, in all case the sequence 
converges to 1, as desired.
By Theorem 5.4, it is easy to see that the following result is true.
Theorem 5.5.
Assume that 
. Then the unique positive equilibrium 
is a global attractor of all nonnegative solutions of (1.11).
Theorem 5.6.
Assume that 
holds and 
is a nonnegative solution of (1.11). If 
, then 
for 
. Furthermore, every nonnegative solution of (1.11) lies eventually in the interval 
.
Proof.
Firstly, note that in this case 
holds.
If 
, then by Lemma 3.5 (iv) and (i), we have that 
for 
; the first assertion follows.
To complete the proof it remains to show that when 
there exists 
such that 
. There are two cases to be considered.
Case 1 (
).
Lemma 3.5 (i) implies that 
for 
. If there exists 
such that 
, then the proof follows from the first assertion. Now assume for the sake of contradiction that all terms of 
never enter the interval 
, then 
would lie in the interval 
for 
. Using Lemma 3.5 (iii), we obtain that 
for 
, from which it follows that 
exists and 
, which is a contradiction, because in view of Lemma 3.5 (iv), (1.11) has no equilibrium points in the interval 
.
Case 2 (
).
If there exists 
such that 
, then the proof follows from the first assertion. If there exists 
such that 
, then the proof also follows from Case 1. Now assume for the sake of contradiction that 
for all 
, then by Lemma 3.5 (ii), we have that 
for 
, from which it follows that 
exists and 
. This contradicts Lemma 3.5 (iv).
The proof is complete.
Theorem 5.7.
Assume that 
holds. Then the unique positive equilibrium 
of (1.11) is a global attractor of all nonnegative solutions of (1.11).
Proof.
Theorem 5.6 implies that every nonnegative solution of (1.11) lies eventually in the invariant interval 
.
Further, the function 
is nondecreasing in 
and nonincreasing in 
in 
. Let 
be a solution of the system
then 
, from which we get that 
, and the proof now follows by applying Lemma 2.3.
Theorem 5.8.
Assume that 
holds and that 
is a nonnegative solution of (1.11). Then every nonnegative solution of (1.11) lies eventually in the invariant interval 
.
Proof.
When 
,
Hence the assertion is true for the case 
. There remains to consider the case 
. In this case 
holds.
If 
, then by Lemma 3.6 (i) and (ii), we have that 
for 
, and the assertion follows.
Given that 
, then by Lemma 3.6 (iii), we have that 
. If 
, then the assertion is true from the above proof. Now assume for the sake of contradiction that 
for all 
. Using identity (3.7), we get that 
for 
, from which it follows that the sequence 
is strictly increasing and there is a finite 
; this contradicts the fact that 
.
The proof is complete.
Theorem 5.9.
Assume that 
holds. Then the unique positive equilibrium 
of (1.11) is a global attractor of all nonnegative solutions of (1.11).
Proof.
Theorem 5.8 implies that there exist a positive integer 
such that 
for 
. Therefore the change of variables
transforms (1.11) to the difference equation
Clearly, 
and 
hold, and the remainder proof now is a straightforward consequence of Lemma 2.4.
In view of Theorem 5.1, we know that the unique positive equilibrium 
of (1.1) is a global attractor when 
. From this and Theorem 1.2, we have the following main result, which partially solves Open Problem 1.
Theorem 5.10.
Assume that 
. Then the unique positive equilibrium 
of (1.1) is globally asymptotically stable.
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Jia, XM., Hu, LX. & Li, WT. Dynamics of a Rational Difference Equation. Adv Differ Equ 2010, 970720 (2010). https://doi.org/10.1155/2010/970720
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DOI: https://doi.org/10.1155/2010/970720