On the Nonlinear Difference Equation

In this paper, we investigate some qualitative behavior of the solutions of the difference equation where the coefficients a, b and c _i are positive real numbers, and where the initial conditions are arbitrary positive real numbers.

• Difference Equation
• Stability
• Periodicity
• Boundedness
• Global Stability

1. Introduction

Our aim in this paper is to study with some properties of the solutions of the difference equation

where the coefficients and are positive real numbers, and where the initial conditions are arbitrary positive real numbers. There is a class of nonlinear difference equations, known as the rational difference equations, each of which consists of the ratio of two polynomials in the sequence terms in the same form. There has been a lot of work concerning the global asymptotics of solutions of rational difference equations [1] -[8] .

Many researchers have investigated the behavior of the solution of difference equation. For example:

Amleh et al. [9] has studied the global stability, boundedness and the periodic character of solutions of the equation

Our aim in this paper is to extend and generalize the work in [9] , [10] and [11] . That is, we will investigate the global behavior of (1.1) including the asymptotical stability of equilibrium points, the existence of bounded solution, the existence of period two solution of the recursive sequence of Equation (1).

Now we recall some well-known results, which will be useful in the investigation of (1.1) and which are given in [12] .

Let I be an interval of real numbers and let

where F is a continuous function. Consider the difference equation

with the initial condition

Definition 1. (Equilibrium Point)

A point is called an equilibrium point of Equation (1.2) if

That is, for, is a solution of Equation (1.2), or equivalently, is a fixed point of f.

Definition 2. (Stability)

Let be in equilibrium point of Equation (1.2) then

1) An equilibrium point of Equation (1.2) is called locally stable if for every there exists such that, if with then for all.

2) An equilibrium point of Equation (1.2) is called locally asymptotically stable if is locally stable and there exists such that, if with

then

3) An equilibrium point of Equation (1.2) is called a global attractor if for all we have

4) An equilibrium point of Equation (1.2) is called globally asymptotically stable if is locally stable and a global attractor.

5) An equilibrium point of Equation (1.2) is called unstable if is not locally stable.

Definition 3. (Permanence)

Equation (1.2) is called permanent if there exists numbers m and M with such that for any initial conditions there exists a positive integer N which depends on the initial conditions such that

Definition 4. (Periodicity)

A sequence is said to be periodic with period p if for all. A sequence is said to be periodic with prime period p if p is the smallest positive integer having this property.

The linearized equation of Equation (1.2) about the equilibrium point is defined by the equation

where

The characteristic equation associated with Equation (1.3) is

Theorem 1.1. [13] Let be an interval of real numbers and assume that

is a continuous function satisfying the following properties:

(a) is non-increasing in the first (k) terms for each in and non-decreasing in the last term for each in for all

(b) If is a solution of the system

implies

Theorem 1.2. [12] Assume that F is a -function and let be an equilibrium point of Equation (1.2). Then the following statements are true:

1) If all roots of Equation (1.4) lie in the open unit disk, then he equilibrium point is locally asymptotically stable.

2) If at least one root of Equation (1.4) has absolute value greater than one, then the equilibrium point is unstable.

3) If all roots of Equation (1.4) have absolute value greater than one, then the equilibrium point is a source.

Theorem 1.3. [14] Assume that Then

is a sufficient condition for the asymptotically stable of Equation (1.5)

2. Local Stability of Equation (1.1)

In this section we investigate the local stability character of the solutions of Equation (1.1). Equation (1.1) has a unique nonzero equilibrium point

Let

Then, we get

Let be a function defined by

Therefore it follows that

and

Then we see that

and

Then the linearized equation of (1.1) about is

Theorem 2.1. Assume that

Then the equilibrium point of Equation (1.1) is locally stable.

Proof. It is follows by Theorem (1.3) that, Equation (2.2) is locally stable if

That is

This implies that

then

Thus

Hence, the proof is completed.

3. Periodic Solutions

In this section we investigate the periodic character of the positive solutions of Equation (1.1).

Theorem 3.1. Equation (1.1) has positive prime period-two solution only if

Proof. Assume that there exists a prime period-two solution

of (1.1). Let Since, we have Thus, from Equation (1.1), we get

and

Let

and

Then

and

Then

and

Subtracting (3.2) from (3.3) gives

Since, we have

Also, since p and q are positive, should be positive. Again, adding (3.2) and (3.3) yields

It follows by (3.4), (3.5) and the relation

that

Assume that p and q are two distinct real roots of the quadratic equation

and so

which is equivalent to

Thus, the proof is completed.

4. Bounded Solution

Our aim in this section we investigate the boundedness of the positive solutions of Equation (1.1).

Theorem 4.1. The solutions of Equation (1.1) are bounded.

Proof. Let be a solution of Equation (1.1). We see from Equation (1.1) that

Then

On the other hand, we see that the change of variables

transforms Equation (1.1) to the following form:

Hence, we obtain

Thus

and so,

It follows that

Thus we obtain

From (4.1) and (4.2) we see that

Therefore every solution of Equation (1.1) is bounded.

5. Global Stability of Equation (1.1)

Our aim in this section we investigate the global asymptotic stability of Equation (1.1).

Theorem 5.1. If then the equilibrium point of Equation (1.1) is global attractor.

Proof. Let be a function defined by

then we can see that the function is decreasing in the rest of arguments and increasing in.

Suppose that is a solution of the system

Then from Equation (2.1), we see that

then

Thus

It follows by Theorem (1.1) that is a global attractor of Equation (1.1) and then the proof is complete.

6. Numerical

For confirming the results of this section, we consider numerical examples which represent different types of solution of Equation (1.1).Example

Examples 6.1. Consider the difference equation

where Figure 1 shows that the equilibrium point of Equation (1.1) has locally stable, with initial data (see Table 1).

Example 6.2. Consider the difference equation

where Figure 2, shows that Equation (1.1) which is periodic with period two. Where the initial data satisfies condition (3.1) of Theorem (3.1) (see Table 2).

Figure 1

Ref. b1.

Table 1

The equilibrium point of Equation (1.1)

n	x(n)	n	x(n)	n	x(n)	n	x(n)
1	2.0000	16	1.2953	31	1.3187	46	1.3182
2	0.5000	17	1.3359	32	1.3178	47	1.3182
3	1.8846	18	1.3044	33	1.3185	48	1.3182
4	0.8259	19	1.3288	34	1.3179	49	1.3182
5	1.6796	20	1.3099	35	1.3184	50	1.3182
6	1.0232	21	1.3246	36	1.3180	51	1.3182
7	1.5399	22	1.3132	37	1.3183	52	1.3182
8	1.1415	23	1.3220	38	1.3181	53	1.3182
9	1.4526	24	1.3152	39	1.3182	44	1.3182
10	1.2123	25	1.3205	40	1.3181	45	1.3182
11	1.3993	26	1.3164	41	1.3182	46	1.3182
12	1.2547	27	1.3196	42	1.3182	57	1.3182
13	1.3671	28	1.3171	43	1.3182	58	1.3182
14	1.2801	29	1.3190	44	1.3182	59	1.3182
15	1.3476	30	1.3175	45	1.3182	60	1.3182

Figure 2

Ref. b4

Table 2

The initial data satisfies condition (3.1) of Theorem (3.1)

n	x(n)	n	x(n)	n	x(n)	n	x(n)	n	x(n)
1	0.1000	17	2.4834	33	3.5629	49	3.6057	65	3.6064
2	0.3000	18	0.6734	34	0.2809	50	0.2688	66	0.2686
3	0.6964	19	2.7184	35	3.5803	51	3.6060	67	3.6064
4	0.8339	20	0.5658	36	0.2760	52	0.2687	68	0.2686
5	1.3033	21	2.9493	37	3.5907	53	3.6061	69	3.6064
6	1.0945	22	0.4751	38	0.2730	54	0.2687	70	0.2686
7	1.6178	23	3.1503	39	3.5970	55	3.6062	71	3.6064
8	1.1360	24	0.4055	40	0.2713	56	0.2687	72	0.2686
9	1.7886	25	3.3061	41	3.6008	57	3.6063	73	3.6064
10	1.0891	26	0.3561	42	0.2702	58	0.2686	74	0.2686
11	1.9286	27	3.4160	43	3.6030	59	3.6063	75	3.6064
12	1.0058	28	0.3232	44	0.2696	60	0.2686	76	0.2686
13	2.0829	29	3.4886	45	3.6044	61	3.6063	77	3.6064
14	0.9029	30	0.3021	46	0.2692	62	0.2686	78	0.2686
15	2.2675	31	3.5345	47	3.6052	63	3.6064	79	3.6064
16	0.7892	32	0.2889	48	0.2690	64	0.2686	80	0.2686

Remark 6.1. Note that the special cases of Equation (1.1) have been studied in [9] when and in [10] when and in [11] when

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