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On a rational recursive sequence [X.sub.n+1] = (a - [bx.sup.2.sub.n]/(1 + [x.sup.2.sub.n-1]).

Abstract Our aim in this paper is to investigate the asymptotic stability, and the global attractivity of difference equation of the form

x.sub.n+1 = alpha - [bx.sup.2.sub.n]/1 + [x.sup.2.sub.n-1], n = 0,1, ...,

where a [greater than or equal to] 0, b [greater than or equal to], 0. We obtain sufficient conditions for global attractivity.

Keywords Difference equation, asymptotic stability, global attractivity, positive solution.

[section] 1. Introduction

The behaviour of solutions of the difference equation

[x.sub.n+ = a - [bx.sup.2.sub.n]/1 + [x.sup.2.sub.n-1], n = 0,1, ...,

was investigated, when a, b [member] [0, [infinity]) (see Zhang et. al [1]). Also, Li [5] has studied the asymptotic behaviour of nonlinear delay difference equations of the form [x.sub.n+1] = [x.sub.n] f ([x.sub.n], [x.sub.n] [x.sub.1], [x.sub.n] [x.sub.r]), p [greater than or equal to] 0, n = 0, 1, ..., where [k.sub.1], [k.sub.2], ..., [k.sub.r] are positive integers.

In this paper, our aim is to study the asymptotic stability and global attractivity of the rational recursive sequences

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where a [greater than or equal to] 0, b [greater than or equal to] 0.

Here, we recall some concepts and theorems which will be useful in this paper. Consider the difference equation

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where k is a positive integer and the function F [member of] [C.sup.1][[R.sup.k+1], R].

Assume that a -k, a-, [a.sub.0] [member of] R are arbitrary real numbers, then for initial conditions [x.sub.-i] [a.sub.i] for i = 0, 1, k, the (2) has a unique solution {[x.sub.n]} [inifinity] n = -k.

If n [greater than or equal to] 1, then [x.sub.n] [greater than or equal to] 0. We consider this solution is positive solution of the (2).

Definition 1.1. A point x is called an equilibrium of the (2), if [bar.x] = F ([bar.x], ..., [bar.x]). That is, [x.sub.n] = x, for n [greater than or equal to], 0, is a solution of the (2), or equivalently, is fixed point of F.

Definition 1.2. Let I be an interval of real numbers, the equilibrium point x of the (1) is said to be

(a) locally stable if for every [epsilon] [greater than or equal to] 0, there exists [sigma] [greater than or equal to] 0 such that for all [x.sub.-k], ..., [x.sub.-1], [x.sub.0] [member of] I

with [SIGMA]| [x.sub.i] - [bar.x] < [delta], we have |[x.sub.n], - [bar.x]| < [delta] for all n [greater than or equal to]

(b) locally asymptotically stable if it is locally stable, and if there exists y [greater than or equal to] 0 such that for all [x.sub.-k], ..., [x.sub.-1], x0 [member of] I, with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we have lim [x.sub.n] = [bar.x].

(c) global attractor if for all [x.sub.-k], ..., [x.sub.-1], [x.sub.0] [member of] I, we have lim [x.sub.n] = [bar.x].

(d) global asymptotically stable if x is locally stable and x is also global attractor.

(e) unstable if x is not locally stable. If k = 1, then (2) is in form that

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let [bar.x] is an equilibrium of the (3), f (u, v) is the function associated with the (3).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The linearized equation associated with the (3) about the equilibrium [bar.x] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Its characteristic equation is

(4) [[lambda].sup.2] r[lambda]- s = 0.

Theorem 1.1. Assume that F is a C1 function and let [bar.x] be an equilibrium of the (3).

Then the following statements are true.

(a) If all the roots of the (4) lie in the open unit disk |[lambda]| < 1, then the equilibrium [bar] of the (3) is asymptotically stable.

(b) If at least one of the roots of the (4) has absolute value greater then one, then the equilibrium x of the (3) is unstable.

(c) (4) has all its roots in the open unit disk |[lambda]| < 1 if and only if |r| < 1 - < 2,

then x is a attracting equilibrium.

(d) One root of the (4) has absolute value greater than one and another has absolute value less than one if and only if

[r.sup.2] + 4s > 0 and |r|>|1-s|,

then [bar.x] is a saddle point.

(e) All the roots of the (4) has absolute value greater than one if and only if

|s| > 1 and |r|>|1-s|,

then [bar.x] is a repelling equilibrium or source.

Theorem 1.2. Consider the equation

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [k.sub.i](i = 1, 2, ..., r) are positive integers. Denote by k the maximum of [k.sub.l], ..., [k.sub.r].

Also, assume that the function f satisfies the following hypotheses.

([H.sub.1]) f [member of] C[(0, [infinity]) x (0, [infinity])r, (0, [infinity])], and g [member of] C[(0, [infinity])[r.sup.+1], (0, [infinity])], where g([u.sub.0], [.sub.1], ..., [u.sub.0] f ([u.sub.0], [u.sub.1], ..., [u.sub.r]) for [u.sub.0] [member of] (0, [infinity]) and [u.sub.1], ..., ur [member of] [0, [infinity])

([H.sub.2]) f ([u.sub.0], [u.sub.1], ..., [u.sub.r]) is nonincreasing in [u.sub.1], =, [u.sub.r], respectively.

([H.sub.3]) The equation f (x, x, ..., x) = 1 has a unique positive solution [bar.x].

([H.sub.4]) Either the function f ([u.sub.0], [u.sub.1], ..., [u.sub.r]) does not depend on [u.sub.0] or for every x [greater than or equal to] 0 and u[greater than or equal to]0 [f (x, u, ... [l.sub.u]) ... f ([bar.x] ..., u)] (x - x) [less than or equal to] < O

with

[f (X,[bar.x], ... [bar.x]) = [bar.x] [bar.x] ..., [bar.x])(x - [bar.x) < 0 for x not equal to] [bar.x].

Now define a new function

x [less than or equal to] y [less than or equal to] [bar.x]maxG(x,y) for 0[less than or equal to] x [less than or equal to] Y,

x [less than or equal to] y [less than or equal to]xmaxG(x,y) for 0 [less than or equal to] x [less than or equal to]Y,

F(x) - {T\y\xminG(x,y) for x [greater than or equal to] T,

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then

(i) F [member of] C[(0, [infinity]), (0, [infinity])] and F is nonincreasing in [0, oc).

(ii) Assume that F has no periodic points of prime period 2. Then x is a global attractor of all positive solutions of (5). Main Results Consider the (1) with

a [greater than or equal to] 0 and b [greater than or equal to] 0.

1). The case a = 0.

In this section, we study the asymptotically stable and the global attractivity for the difference equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where b [member of] (0, [infinty]).

If a - 0, the (6) has no positive equilibrium. Theorem 2.1. If b < 2, then the (6) has a unique equilibrium [bar.x] - 0 and [bar.x] is asymptotically stable. If b - 2, (6) has a unique negative equilibrium [bar.x] = -1, and [bar.x] is unstable.

Proof. Let cP(x) - [x.sup.3] + [bx.sup.3] + [bar.x], it is easy to see that [phi] (x) - [3.sup.x2] + 2bx + 1 [greater than or equal to] 0 when b < 2. So, the (6) has a unique equilibrium [bar.x] = 0, and easy to see that [x.sup.3] is asymptotically stable. The case b - 2, is easy to proof and we omit it. The proof is complete.

Theorem 2.2. If b [greater than or equal to] 2, then the (6) has three equilibriums

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and [x.sub.l] is repelling equilibrium, while [x.sub.2] is saddle point. proof. It is easy to see that cP(x) = [x.sub.3] + [bx.sub.3] + [bar.x] = x ([x.sub.2] + [bx.sub.3] + 1) = 0 has three roots [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. So, the (6) has three [x.sub.0] = [x.sub.1], [x.sub.2] = [X.sub.2].

The asymptotic stability of To has discussed in theorem 2.1, we omit it.

Now, we consider the function f (u, v) = - [bu.sup.2] 1 + [v.sup.2] which is assosiate with the (6).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by parts (a) and (e) of Theorem 1.1, we have [x.sub.1] is a repelling equilibrium.

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by parts (a) and (d) of Theorem 1.1, we have [[bar.x].sub.2] is a saddle point. The proof is complete.

2). The case b = 0.

In this section, we study the asymptotic stability and global attractivity for the difference equation

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where a [member of] (0, [infinity]).

Theorem 2.3. The equation (7) has a unique positive equilibrium x and if 0 < a < 2, then [bar.x] is asymptotically stable.

Proof. Let [bar.x] = a/1 + [[bar.x].sup.2], it can be rewritten as follows:

[[bar.x].sup.2] + [bar.x] a = 0.

Consider the function [phi](x) = [x.sup.3] + [bar.x] - a, we have

[phi](0) = - a < 0, and [phi](a) [a.sup.3] > 0.

Since [phi](x) = [3.sup.x2] + 1 [greater than or equal to] 0, so [phi](x) is a strictly monotonically increasing function. Hence, [phi](x) = 0 has only one positive root x; and 0 < x < a.

Therefore (7) has a unique positive equilibrium 7 and 7 = [bar.x]; so 0 < [bar.x] < a.

Let f (u, v) = a / 1 + [v.sup.2] is assosiated with the (7).

r = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

s = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

1 = s [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since 0 < a < 2, so 1 - s < 2. By parts (a) and (c) of Theoreml.l, x is asymptotically stable.

The proof is complete.

Theorem 2.4. Assume that b = 0 and 0 < a < 2, let {[bar.x].sub.n]} [infinity] n = 1 1 be a positive solution of the (7). Then

lim [x.sub.n]/n[right arrow] [infinity] = x, n [right arrow] [infinity]

where x is the unique positive equilibrium of the (7), that is, [bar.x] is a global attractor of all positive solutions of the (7).

Furtherly, if and only if ([x.sub.-1], [x.sub.0]) IL ([bar.x],[bar.x]), then the semicycles of every positive solution of the (7) has length 2.

A detailed proof of the Theorem 2.4. can be found in the monograph of D. C. Zhang, B. Shi and M. J. Gal, we omit it.

3). The case a [greater than or equal to] 0, b [greater than or equal to] 0 [inifnity].

In this section, we consider the (1), when a, b [member of] E (0, [infinty]).

Theorem 2.5. The equation (1) has a unique positive equilibrium x, and if 0 < a [less than or equal to] 1, 0 < b < [square root of]3, then [bar.x] is asymptotically stable.

Proof. Let [bar.x] = a - b[[bar.x].sup.2]/1 +[[bar.x].sup.2], it can be rewritten as follows

[[bar.x].sup.3]+b[[bar.x].sup.2]+ [bar.x] - a = 0.

Consider the function [phi] (x) = [[bar.x].sup.3]+b[[bar.x].sup.2]+ [bar.x] - a = 0 ..., we have ([phi](0) = - a < 0, and [phi](a) = [a.sup.3]+ a[2.sup.b] [greater than or equal to] 0.

Next ([phi] (x) = 3[x.sup.2] + 2bx + 1 [greater than or equal to] 0. Hence, O(x) is a monotonically increasing function in [0, [infinity]) and [phi](x) = 0 has only one positive root x, 0 < x < a.

So, it is easy to see that the (1) has a unique positive equilibrium [bar.x] = [bar.x], and 0 < x < a.

Now, we consider the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since 0 < a [less than or equal to] < 1, 0 < b < [square root of]3-, we have |r|< 1-s<2.

Then, by parts (a) and (c) of Theorem 1.1, [bar.x] is asymptotically stable. The proof is complete.

Theorem 2.6. If ab < 1 and 0 [less than or equal to] [x.sub.0] [less than or equal to] a/b < 2, then x is a global attractor.

Proof. Let ab < 1 such that [x.sub.n] [less than or equal to] a/b {[x.sub.n]=1} [infinity] -1 is positive solution of (1) when [x.sub.0] [member of]

[0, [square root of]a/b]. The (1) can be rewritten as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It is easy to verify that f ([u.sup.0], [u.sup.1]) and g([u.sup.0], [u.sup.2]) satisfy the hypotheses ([H.sub.1]) - ([H.sub.4]) which were divised in the Theoreml.2.

Next, we define a function as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Therefore we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

such that [x.sub.n] [less than or equal to] [square root to] a/b < 2, hence F has no periodic points of prime period 2. Thus, by Theorem 1.2. [bar.x] is global attractor of all positive solutions of the (1). The proof is complete.

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Wansheng He, Dewang Cul, Honggang Dang and Chuan Qin

Department of Mathematics, Tianshui Normal University,

Tianshui, Gansu, 741001, China
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Author:He, Wansheng; Cui, Dewang; Dang, Honggang; Qin, Chuan
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Date:Jan 1, 2008
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