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Global attractivity of a higher-order nonlinear difference equation.

1 Introduction

Recently, many researchers are interested in the boundedness, invariant intervals, periodic character and global asymptotic stability of positive solutions for nonlinear difference equations, for example, [1,3-6,10-12]. Here, we would like to mention the results of Li et al. [8], Yan et al. [13] and He et al. [2]. They investigated the global asymptotic stability of positive solutions for the difference equations

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respectively, where the coefficients [alpha], [beta], [gamma], A, a and b are nonnegative real numbers, k [member of] (1, 2,...}. They established that every positive equilibrium of these equations is a global attractor with a basin under certain conditions. In addition, they obtained some sufficient conditions for global asymptotic stability of positive equilibria of these equations.

In 2005, Li et al. [9] studied the difference equation

[x.sub.n+1] = [a + b[x.sub.n]/A + [x.sub.n-k]], n = 0, 1,..., (1.1)

where a, b, A [member of] (0, [infinity]), k is a positive integer and the initial conditions [x.sub.-k],..., [x.sub.0] are arbitrary positive numbers. They obtained the global attractivity of the unique positive equilibrium for Eq. (1.1).

It is also noted that the above mentioned references [1,3-6,10-12] only considered the global attractivity of positive solutions of the difference equation. However, they did not further provide the global attractivity of negative solutions for the difference equation. Hence, in this paper, we deal with the global attractivity ofnegative solutions for the difference equation

[x.sub.n+1] = [1 - [x.sub.n-k]/A + [x.sub.n]], n = 0, 1,..., (1.2)

where A [member of] (-[infinity], 0) is a real number, k is a positive integer and the initial conditions [x.sub.-k],..., [x.sub.0] are arbitrary real numbers.

Our aim is to investigate the periodic character, invariant intervals and the global attractivity of negative solutions of Eq. (1.2). It is shown that the unique negative equilibrium of Eq. (1.2) is a global attractor with a basin that depends on certain conditions of the coefficient A.

Now, we present some definitions and results which will be useful in the sequel. Let I be some interval of real numbers and F be a continuous function defined on [I.sup.k+1]. Then, for initial conditions [x.sub.-k],..., [x.sub.0] [member of] I, it is easy to see that the difference equation

[x.sub.n+1] = F([x.sub.n], [x.sub.n-1],..., [x.sub.n-k]), n = 0, 1,... (1.3)

has a unique solution [{[x.sub.n]}.sup.[infinity].sub.n=-k]. A point [bar.x] is called an equilibrium of Eq. (1.3) if

[bar.x] = F([bar.x],..., [bar.x]).

That is, [x.sub.n] = [bar.x] for n [greater than or equal to] 0 is a solution of Eq. (1.3), or equivalently, [bar.x] is a fixed point of F. An interval J [subset or equal to] I is called an invariant interval for Eq. (1.3) if

[x.sub.-k],..., [x.sub.0] [member of] J [right arrow] [x.sub.n] [member of] J for all n > 0.

That is, every solution of Eq. (1.3) with initial conditions in J remains in J. The linearized equation associated with Eq. (1.3) about the equilibrium [bar.x] is

[y.sub.n+1] = [k.summation over (i=0)] [[partial derivative]F/[partial derivative][u.sub.i]] ([bar.x],..., [bar.x])[y.sub.n-i], n = 0, 1,...,

and its characteristic equation is

[[lambda].sup.k+1] = [k.summation over (i=0)] [[partial derivative]F/[partial derivative][u.sub.i]] ([bar.x],..., [bar.x]) [[lambda].sub.k-i].

Definition 1.1. Let [bar.x] be an equilibrium point of Eq. (1.3).

(i) The equilibrium point [bar.x] of Eq. (1.3) is called locally stable if for every [epsilon] > 0, there exists [delta] > 0 such that for all [x.sub.-k], ..., [x.sub.0] [member of] I with [0.summation over (i=-k)] [absolute value of [x.sub.i] - [bar.x]] < [delta], we have [absolute value of [x.sub.n] - [bar.x]] < [epsilon] for all n [greater than or equalto] -k.

(ii) The equilibrium point [bar.x] of Eq. (1.3) is called locally asymptotically stable if it is locally stable, and if there exist [gamma] > 0 such that for all [x.sub.-k],..., [x.sub.0] [member of] I with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii) The equilibrium point [bar.x] of Eq. (1.3) is called a global attractor if [x.sub.-k],..., [x.sub.0] [member of] I always implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iv) The equilibrium point [bar.x] of Eq. (1.3) is called global asymptotically stable if it is locally asymptotically stable and a global attractor.

(v) The equilibrium point [bar.x] of Eq. (1.3) is called unstable if it is not locally stable.

Theorem 1.2 (See [5]). Assume that p, q [member of] R and k [member of] {0,1,...}. Then [absolute value of p] + [absolute value of q] < 1

is a sufficient condition for asymptotic stability of the difference equation

[x.sup.n+1] + p[x.sub.n] + q[x.sub.n-k] = 0, n = 0, 1,....

Theorem 1.3 (See [7]). Consider the difference equation

[x.sub.n+1] = f([x.sub.n],[x.sub.n-k]), n = 0, 1,..., (1.4)

where k [greater than or equal to] 1 is an integer. Let I = [a, b] be some interval of real numbers, and assume that f : [a, b] x [a, b] [right arrow] [a, b] is a continuous function satisfying the following properties:

(a) f (u, v) is a nonincreasing function in u, and a nondecreasing function in v;

(b) If (m, M) [member of] [a, b] x [a, b] is a solution of

m = f (M,m) and M = f (m, M),

then m = M.

Then Eq. (1.4) has a unique equilibrium x and every solution of Eq. (1.4) converges to [bar.x].

2 Main Results

In this section, we are concerned with the global attractivity of Eq. (1.2) and obtain that the unique negative equilibrium of equation Eq. (1.2) is a global attractor with a basin under certain conditions. The unique negative equilibrium of Eq. (1.2) is

[bar.x] = [-(A +1) - [square root of [(A + 1).sup.2] + 4]/2].

The linearized equation associated with Eq. (1.2) about the equilibrium [bar.x] is

[y.sub.n+1] + [[bar.x]/A + [bar.x]][y.sub.n] + [1/A + [bar.x]][y.sub.n-k] = 0, n = 0, 1,...,

and its characteristic equation is

[[lambda].sub.k+1] + [[bar.x]/A + [bar.x]][[lambda].sup.k] + [1/A + [bar.x]] = 0.

By Theorem 1.2, it is easy to obtain the following result.

Theorem 2.1. Assume that A < -1. Then the unique negative equilibrium [bar.x] of Eq. (1.2) is locally asymptotically stable.

Theorem 2.2. Assume that A < -1. Then Eq. (1.2) has no negative prime period two solution.

Proof. For the sake of contradiction, assume that there exist distinct negative real numbers [phi] and [psi], such that

..., [phi], [psi], [phi], [psi],...,

is a prime period two solution of Eq. (1.2). Then, there are two cases to be considered. Case (i): If k is odd, then [x.sub.n+1] = [x.sub.n-k], and [phi], [psi] satisfy the system

[phi] = [1 - [phi]/A + [psi]], [psi] = [1 - [psi]/A + [phi]].

Thus

([phi] - [psi])(A + 1) = 0.

In view of A + 1 < 0, we get [phi] = [psi], which contradicts the hypothesis that [phi] [not equal to] [psi].

Case (ii): If k is even, then [phi], [psi] satisfy the system

[phi] = [1 - [psi]/A + [psi]], [psi] = [1 - [phi]/A + [phi]].

Thus, we have ([phi] - [psi])(A - 1) = 0. Clearly, A - 1 < 0, so [phi] = [psi], which is also a contradiction. The proof is complete. []

Lemma 2.3. If A < -1 and f (u, v) = [1 - v/A + u], then the following statements are true.

(i) -1 < [bar.x] = [-(1 + A) - [square root of [(1 + A).sup.2] + 4]/2] < 0.

(ii) If u, v [member of] (-[infinity], 0], then f (u, v) isa strictly decreasing function in u, and a strictly increasing function in v.

Proof. The proof is simple and omitted. []

Theorem 2.4 Assume that A [member of] (-[infinity], [-1 - [square root of 5]/2]]. Then [A, 0] is an invariant interval of Eq. (1.2).

Proof. Let ([x.sub.n]} be a solution of Eq. (1.2) with initial conditions [x.sub.k],..., [x.sub.-1], [x.sub.0] [member of] [A, 0]. By Lemma 2.3, the function f(u, v) is strictly decreasing in u, and strictly increasing in v for each fixed u, v [member of] (-[infinity], 0], we obtain that

[x.sub.1] = f([x.sub.0], [x.sub.-k]) < f(A, 0) = [1/2A] < 0

and

[x.sub.1] = f ([x.sub.0], [x.sub.-k]) > f(0, A) = [1 - A/A] [greater than or equal to] A,

which implies that [x.sub.1] [member of] [A, 0]. By induction, it follows that [x.sub.n] [member of] [A, 0] for n [greater than or equal to] 1. The proof is complete. []

Theorem 2.5 Assume that A [member of] (-[infinity], [-1 - [square root of 5]/2]]. Then the unique negative equilibrium [bar.x] of Eq. (1.2) is a global attractor with a basin

S = [[A, 0].sup.k+1].

Proof. Set

f(u, v) = [1 - v/A+u].

Then f : [A, 0] x [A, 0] [right arrow] [A, 0] is a continuous function and nonincreasing in u and nondecreasing in v. Let {[x.sub.n]} be a solution of Eq. (1.2) with initial conditions [x.sub.k],..., [x.sub.-1], [x.sub.0] [member of] S and m, M [member of] [A, 0] be a solution of the system

m = f (M, m), M = f (m, M ).

Then (m - M)(A + 1) = 0. Since A + 1 < 0, we get m = M. Applying Theorem 1.3, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The proof is complete. []

Acknowledgments

We would like to express our sincere thanks to the referees for valued comments which improved the presentation of the paper.

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Xiu-Mei Jia

Hexi University

Department of Mathematics

Zhangye, Gansu 734000, P. R. China

jiaxium07@lzu.cn

Guo-Mei Tang

Northwest University for Nationalities

School of Mathematics and Computer Science

Lanzhou, Gansu 730000, P. R. China

Received March 11, 2009; Accepted May 8, 2009 Communicated by Wan-Tong Li
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Author:Jia, Xiu-Mei; Tang, Guo-Mei
Publication:International Journal of Difference Equations
Article Type:Report
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Date:Jun 1, 2010
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