Global Behavior of a Nonlinear Difference Equation with Applications ()
1. Introduction
Amleh, Grove and Ladas [1] studied the global stability boundedness character and periodic nature of positive solutions of difference equation
(1)
where
and initial conditions
and
are both arbitrary positive real numbers.
Amleh, Grove and Ladas [1] obtain the following theorem.
Theorem A (Amleh, Grove and Ladas [1]) Let
and
be a solution of equation (1)
with initial conditions
and
.
Then the following statements are true.
1) ![](https://www.scirp.org/html/7-1200070\c0e409fa-5274-4358-9580-076a31b32415.jpg)
2)
.
Now, we can see that if
and
, then
. So, the theorem A does not hold for
.
Kulenovic and Glass in their monograph [2] give an open problem as follows.
Open Problem 6.10.7. For the following difference equation determine the “good” set
of the initial conditions
throng with the equation is well defined for all
. Then for every
, investigate the long-term behavior of the solution
of
(2)
Let
. Then equation (2) can be rewritten as follows
(3)
where
and
are arbitrary nonzero real numbers. To this end, we study equation (3) and use the results of equation (3) to equation (2).
2. Some Lemmas
It is easy for one to see that if
![](https://www.scirp.org/html/7-1200070\3ce889e7-4557-4873-97ce-9fc30206dfbe.jpg)
then we have
(4)
and
![](https://www.scirp.org/html/7-1200070\7139b769-54e3-4383-b1d8-0e3d2265459b.jpg)
Lemma 2.1 (Kocic and Ladas [3]) Consider the difference equation
(5)
Assume that
is a
function and
is an equilibrium of equation (5).
Then the linearized equation associated with equation (5) about the equilibrium
is
![](https://www.scirp.org/html/7-1200070\450af75f-a950-4e5b-b794-930d81df6ed5.jpg)
and the following statements are true.
a) If all roots of the polynomial equation
(6)
lie in the open unit disk
, then the equilibrium
of equation (5) is asymptotically stable;
b) If at least one root of equation (6) has absolute value greater than one, then equilibrium
of equation (5) is unstable.
One can refer to Kocic and Ladas [3, Corallary 1.3.2, p14 ].
Lemma 2.2 Equation (3) has two equilibriums
and
.
It is easy to see that
has two roots and the proof is complete.
3. Main Results
Theorem 3.1 Let
and
. Then the following statements are true.
a)
, where
![](https://www.scirp.org/html/7-1200070\472639ea-3a57-492b-afac-39461558219f.jpg)
and
![](https://www.scirp.org/html/7-1200070\74239fa0-4b94-4177-8804-ab23c1817bbd.jpg)
b)
, where
![](https://www.scirp.org/html/7-1200070\85c17f72-c942-485f-a700-f0cbf4b8654b.jpg)
and
![](https://www.scirp.org/html/7-1200070\1218a4cd-4cc6-4f60-8cc0-819425a338f6.jpg)
where
is the solution of equation (3) with the initial
,
.
Proof: Part a).
Let
,
. Then by equation (3) we have
![](https://www.scirp.org/html/7-1200070\96093c28-154a-4655-b0f9-d11609120da3.jpg)
we assume that
(7)
Then by induction, we have
(8)
where
,
and ![](https://www.scirp.org/html/7-1200070\3e19d23a-9876-4a93-8319-9eff65dedbac.jpg)
Change equation (8) into
(9)
or
(10)
where
.
From equation (4), we get
(11)
where
![](https://www.scirp.org/html/7-1200070\ea186216-9a2a-404e-9e3f-e565361c6f98.jpg)
Equation (11) can be changed into
![](https://www.scirp.org/html/7-1200070\d7c7a6f1-8abd-47a8-a501-967e2a53692b.jpg)
Let
and
. Then we obtain that
(12)
and ![](https://www.scirp.org/html/7-1200070\0b7b631c-a724-41b8-816e-5670d9c18958.jpg)
By induction, we have
(13)
Therefore,
![](https://www.scirp.org/html/7-1200070\83595a82-0a48-4644-be8b-36a4d78d1386.jpg)
Hence, the proof of part (a) is complete.
The proof of part (b) can be similarly given, so we omit it. This can complete the proof of theorem 3.1.
By theorem 3.1, we get the following corollary.
Corollary 3.1 Assume that
,
. Then the following statements are true.
a) If
, then the positive solution
of equation (3) converges to 1, i.e,
.
b) If
, then the positive solution
of equation (3) has the properties
![](https://www.scirp.org/html/7-1200070\11a923af-9415-400d-871b-e55e22d54190.jpg)
c) If
, then the positive solution
of equation (3) has the properties
![](https://www.scirp.org/html/7-1200070\e8cfb0ae-c300-4e08-a2e6-8f18fdce0c86.jpg)
Theorem 3.2 Assume that
,
. Then the following statements are true.
a) If
and
, then the solution ![](https://www.scirp.org/html/7-1200070\8cf97e60-7bc5-472e-a239-b0c63d82f058.jpg)
of equation (3) is periodic with period-3 as follows
(14)
b) If
and
, then the solution ![](https://www.scirp.org/html/7-1200070\3fa9ccde-52d7-4ece-9019-e614c6fe5b9b.jpg)
of equation (3) is periodic with period-3 as follows
(15)
c) If
and
, then the solution
of equation (3) is periodic with period-3 as follows
(16)
The proof of theorem 3.2 is very easy, so we will omit it.
By theorems 3.1 and 3.2, we can obtain the following corollary.
Corollary 3.2 Assume that
. Then the following statements are true.
a) If
and at least one of p and q is less than 0, then
of equation (3) converges to a period-3 solution of equation (3) as one of (10)-(12).
b) If
and at least one of p and q is less than 0, then
of equation (3) has the following properties
![](https://www.scirp.org/html/7-1200070\2005d4e1-3fa6-4981-8522-4c44909cd376.jpg)
c) If
and at least one of p and q is less than 0, then
of equation (3) has the following properties
![](https://www.scirp.org/html/7-1200070\c0cba647-c296-46e7-9030-38e33cc1a728.jpg)
d) If at least one of p and q is less than 0, then every solution of equation (3) strictly oscillates about the equilibrium
.
e) If
and at least one of p and q is less than 0, then every solution of equation (3) strictly oscillates about the equilibrium
.
Theorem 3.3 The equilibrium
of equation (3) is unstable.
Proof: The linearize equation associated with equation (3) about the equilibrium
is
(17)
The characteristic equation of (17) is
![](https://www.scirp.org/html/7-1200070\78da967a-cf4f-4203-beb0-5c5d5e811b0d.jpg)
Thus, we obtain two roots
. Noting that
. Therefore, by lemma 3.1, we know that the equilibrium
of equation (3) is unstable. The proof of theorem 3.3 is complete.
4. Application
By theorem 3.1, we have the following theorem.
Theorem 4.1 Assume that
and
. Then the following statements are true.
a) Every solution
of equation (2) satisfies
for ![](https://www.scirp.org/html/7-1200070\de8b1292-82f6-4835-b234-c5f200d826b4.jpg)
b) If
, then the solution ![](https://www.scirp.org/html/7-1200070\e85167e3-6cbd-4792-b475-335cd3882adf.jpg)
of equation (2) converges to 0.
c) If
, then the solution ![](https://www.scirp.org/html/7-1200070\7be73746-cfb2-4790-81e7-56299c99bc1d.jpg)
of equation (2) has the following properties
![](https://www.scirp.org/html/7-1200070\3ad7e9c2-bc4d-4adf-9341-651f20d9f424.jpg)
d) If
, then the solution ![](https://www.scirp.org/html/7-1200070\1afaed52-c65a-4785-b56e-0bb32d71df05.jpg)
of equation (2) has the following properties
![](https://www.scirp.org/html/7-1200070\07513c38-23f6-4d49-811e-822651ec2425.jpg)
By corollary 3.2, we get the following theorem.
Theorem 4.2 Assume that
. Then the following statements are true.
a) If
and at least one of ![](https://www.scirp.org/html/7-1200070\267fe783-557c-4fc0-be43-c3f5043b5081.jpg)
and
is less than 0, then
of equation (2) converges to a period-3 solution of equation (2) as one of the following:
i) ![](https://www.scirp.org/html/7-1200070\175a54ea-afc2-4cdd-980d-8a09b52cd45c.jpg)
ii) ![](https://www.scirp.org/html/7-1200070\98f597bb-0c8a-42ef-b843-1fb7dd7bbd5f.jpg)
iii) ![](https://www.scirp.org/html/7-1200070\947fa472-4fe9-45d8-a201-8fdfa83bb800.jpg)
b) If
and at least one of ![](https://www.scirp.org/html/7-1200070\9e2bf20d-ac9d-4192-8f23-512934e6378d.jpg)
and
is less than 0, then every solution
of equation (2) has the following properties:
![](https://www.scirp.org/html/7-1200070\b35e2f58-2f25-4bfe-b341-51da438e0d22.jpg)
c) If
and at least one of ![](https://www.scirp.org/html/7-1200070\cc1d169f-c29e-4b7c-b3e0-de2a2562a236.jpg)
and
is less than 0, then every solution
of equation (2) has the following properties:
![](https://www.scirp.org/html/7-1200070\016ddf7c-703e-4085-b156-44c15a1996cc.jpg)
d) If
and at least one of ![](https://www.scirp.org/html/7-1200070\3a96849b-bf7f-46f5-8184-84f159d1ec91.jpg)
and
is less than 0, then every solution
of equation (2) strictly oscillates about the equilibrium
of equation (2).
5. Acknowledgements
Research supported by Distinguished Expert Foundation and Youth Science Foundation of Naval Aeronautical and Astronautical University.