On the difference equation xn+1=pxn-s+xn-tqxn-s+xn-t

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Abstract

In this paper, we address the difference equationxn+1=pxn-s+xn-tqxn-s+xn-tn=0,1,with positive initial conditions. We establish a sufficient condition for the global asymptotic stability of the positive equilibrium of this equation.

Introduction

Qualitative analysis of nonlinear difference equations is a fertile research area and increasingly attracts many concerns. Recently there has been a lot of work concerning the asymptotic behavior of rational difference equations [1], [2], [3], [4], [5], [6], [7], [8], [9], [11], [13], [14], [15], [16], [17], [18]. In particular, Kulenovic and Ladas [7] studied the difference equationxn+1=pxn+xn-1qxn+xn-1n=0,1,Later, Saleh and Abu-Baha [11] investigated the difference equationxn+1=pxn+xn-tqxn+xn-tn=0,1,Motivated by the previous works, this paper addresses the difference equationxn+1=pxn-s+xn-tqxn-s+xn-tn=0,1,with positive initial conditions where s, t are distinct nonnegative integers, p, q > 0, p  q. Clearly, Eq. (1.3) subsumes Eqs. (1.1), (1.2). We establish a sufficient condition for the global asymptotic stability of the positive equilibrium of Eq. (1.3).

The remaining materials are organized in this fashion: Section 2 provides preliminary knowledge necessary for our study. Section 3 establishes the main result of this paper. Section 4 gives some computational results.

Section snippets

Preliminary knowledge

For fundamental terminologies concerned with difference equations, the reader is referred to Refs. [4], [7].

Theorem 2.1

[18]

Let s, t be distinct nonnegative integers. Consider the difference equationxn+1=f(xn-s,xn-t),n=0,1,x-max{s,t},x1-max{s,t},,x0[a,b].Suppose f satisfies the following two conditions:

  • (C1)

    f: [a, b]2  [a, b] is a continuous function that is nondecreasing in the first argument and is nonincreasing in the second argument.

  • (C2)

    The system

x=f(x,y),y=f(y,x)x,y[a,b]has a unique solution.

Then x˜ is the

Main result

This section addresses Eq. (1.3).

Theorem 3.1

Let {xn}n=-k+ be a positive solution of Eq. (1.3). Then we have min{1, p/q} < xn < max{1, p/q} for all n  1.

Proof

By applying Theorem 2.2. 

Theorem 3.2

The positive equilibrium of Eq. (1.3) is globally asymptotically stable if one of the following two conditions is satisfied:

  • (H1)

    Either p > q  1, or 1  p > q, or (1 + 3q)/(1  q)  p > 1 > q.

  • (H2)

    Either q > p  1, or 1  q > p, or (1 + 3p)/(1  p)  q > 1 > p.

Proof

We prove only the global asymptotic stability of the positive equilibrium under condition (H1); the assertion under

Numerical examples

To illustrate the theoretical results established in the previous sections, this section presents several numerical examples.

Example 4.1

Consider the following difference equation for p > q > 1:xn+1=2.1xn-1+xn-21.3xn-1+xn-2n=0,1,with the initial conditions x−2 = 0.1, x−1 = 0.2, x0 = 0.3 (Fig. 1).

Example 4.2

Consider the following difference equation for 1+3q1-qp>1>q:xn+1=1.2xn+xn-20.7xn+xn-2n=0,1,with the initial conditions x−2 = 0.1, x−1 = 0.2, x0 = 0.3 (Fig. 2).

Example 4.3

Consider the following difference equation for 1  q > p:xn+1=0.5xn-1+xn-

Conclusion

Nonlinear difference equation has received much attention recently and is applied universally in population biology, economic, signal processing and so on. In particular, Utida [12] proposed a difference equation xt+1=xt(1b+cxt-d) to describe the growth of population model. Later, Milton and Belair [10] described the growth of bobwhite quail population model by the difference equation xt+1=αxt+βxt1+(xt)r. Recently, their results were generalized by Graef et al. [3] to the higher-order

Acknowledgements

We are grateful to the anonymous reviewers for their valuable comments. The work of this paper was supported by Program for New Century Excellent Talent of Educational Ministry of China (NCET-05-0759), Doctorate Foundation of Educational Ministry of China (20050611001) and National Natural Science Foundation of China (10771227).

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