ABSTRACT

In this paper, solutions of difference equation for are investigated, where, are both arbitrary nonzero real numbers. The results are applied to the following difference equation for.

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)

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

then we have

(4)

and

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

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

and

b), where

and

where is the solution of equation (3) with the initial,.

Proof: Part a).

Let,. Then by equation (3) we have

we assume that

(7)

Then by induction, we have

(8)

where, and

Change equation (8) into

(9)

or

(10)

where.

From equation (4), we get

(11)

where

Equation (11) can be changed into

Let and. Then we obtain that

(12)

and           

By induction, we have

(13)

Therefore,

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

c) If, then the positive solution of equation (3) has the properties

Theorem 3.2 Assume that,. Then the following statements are true.

a) If and, then the solution

of equation (3) is periodic with period-3 as follows

(14)

b) If and, then the solution

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

c) If and at least one of p and q is less than 0, then of equation (3) has the following properties

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

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

b) If, then the solution

of equation (2) converges to 0.

c) If, then the solution

of equation (2) has the following properties

d) If, then the solution

of equation (2) has the following properties

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

and is less than 0, then of equation (2) converges to a period-3 solution of equation (2) as one of the following:

i)

ii)

iii)

b) If and at least one of

and is less than 0, then every solution of equation (2) has the following properties:

c) If and at least one of

and is less than 0, then every solution of equation (2) has the following properties:

d) If and at least one of

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.

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