On the difference 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 equationLater, Saleh and Abu-Baha [11] investigated the difference equationMotivated by the previous works, this paper addresses the difference equationwith 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 Let s, t be distinct nonnegative integers. Consider the difference equationSuppose f satisfies the following two conditions: f: [a, b]2 → [a, b] is a continuous function that is nondecreasing in the first argument and is nonincreasing in the second argument. The system[18]
has a unique solution.
Then is the
Main result
This section addresses Eq. (1.3). Theorem 3.1 Let 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: Either p > q ⩾ 1, or 1 ⩾ p > q, or (1 + 3q)/(1 − q) ⩾ p > 1 > q. 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: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 :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:
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 to describe the growth of population model. Later, Milton and Belair [10] described the growth of bobwhite quail population model by the difference equation . 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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