Global asymptotic stability in a rational recursive sequence☆
Introduction
Kocic et al. [1] examined the periodicity and oscillating properties of the positive solutions as well as the global attractivity of the nonnegative equilibrium of the difference equationwhere a, b, d⩾0, a+b>0, and k∈{2,3,…}. There are yet three other types of recursive sequences that are formally similar to sequence (1.1), which are listed below:where a,b,d⩾0, a+b>0, and k∈{2,3,…}. Aboutaleb et al. [2] studied the global asymptotic stability of Eq. (1.2) with k=2, and Li and Sun [3] extended the results to Eq. (1.2) with k⩾2. Yan and Li [4] investigated the global attractivity of Eq. (1.3) with k=2, and Yan et al. [5] extended the results to Eq. (1.3) with k⩾2. Yan and Li [6] examined the global asymptotic behavior of Eq. (1.4) with k=2. Sequences , , , have the common feature that the numerator and the denominator in the fraction are both linear in xn. For more recursive sequences with this feature, the reader is referred to [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].
Some rational recursive sequences were also investigated in which the numerator or/and the denominator is quadratic in xn. For instances, Li [17] found some sufficient conditions for the global attractivity of the positive equilibrium point of the difference equationZhang et al. [18] investigated the global stability of the sequence (1.5) with k=2 and d=1.
In this paper, we study the global asymptotic behavior of the following recursive sequence:where a,b⩾0 and c,d>0. Eq. (1.6) is similar to Eq. (1.4) with k=2, with the only difference that the former contains a quadratic term in the numerator of the fraction while the numerator of the latter is linear. Also Eq. (1.6) is similar to (1.5) in that they both contain xn2 term in the numerator.
Section snippets
Preliminaries
Let I be a real interval and let f: I×I→I be a continuous function. For every initial condition 〈x−1,x0〉∈I×I, the difference equationhas a unique solution {xn}n=−1∞, which is called a recursive sequence. An equilibrium point of Eq. (2.1) is a point α∈I with f(α,α)=α. Definition 2.1 Let α be an equilibrium point of Eq. (2.1). α is locally stable if for every ε>0, there exists δ>0 such that for each 〈x−1,x0〉∈I×I with |x−1−α|+|x0−α|<δ, |xn−α|<ε holds for n=1,2,…. α is a local attractor if
Equilibria and local asymptotic stability
Consider the difference equationwhereThe equilibrium points of this equation are the solutions of the quadratic equationSupposeThen Eq. (3.1) has one nonnegative equilibrium α and one positive equilibrium β>α, whereSincewe have Theorem 3.1 Assume , hold. Then α is locally
Global asymptotic stability
In this section, we deal with the global attractivity of α. To this end, we need to make an estimation on the gap between xn and α.
In view of a+bα+cα2=α(d−α), we deriveThe following lemma follows from Eq. (4.1). Lemma 4.1 Assume , hold and 〈xn−2,xn−1〉∈[0,β)×[0,β). If xn−1⩾α
Asymptotic behavior of positive solutions
In this section, we investigate the asymptotic behavior of positive solutions of Eq. (3.1). To this end, we need to make the following estimation:
From Eq. (5.1) and by induction on n, we obtain Lemma 5.1 Assume , hold. If there exists N such that xN⩽xN+1⩽xN+2, then xn⩽xn+1 for n⩾N. If there exists N such that xN⩾xN+1⩾xN+2, then xn⩾xn+1 for n⩾N.
From Eqs.
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Cited by (5)
A note for “On the rational recursive sequence [Formula Presented]”
2012, Arab Journal of Mathematical SciencesCitation Excerpt :The study of rational difference equation such as Eq. (1.1) is quite challenging and rewarding due to the fact that some results of rational difference equations offer prototypes for the development of the basic theory of the global behavior of nonlinear difference equations; moreover, the investigations of rational difference equations are still in its infancy so far. To see this, refer to the monographs [5,6] and the papers [7,2,11,9,10,12,4,13,3,14,8 and the references cited therein]. In this section, we will formulate our main results in this note.
On the difference equation x<inf>n + 1</inf> = frac(px<inf>n - s</inf> + x<inf>n - t</inf>, qx<inf>n - s</inf> + x<inf>n - t</inf>)
2008, Applied Mathematics and ComputationOn the recursive sequence x<inf>n</inf> = ax<inf>n-1</inf>+bx <inf>n-2</inf>/c+dx<inf>n-1</inf>x<inf>n-2</inf>
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This research was partly supported by the Visiting Scholar's Funds of National Education Ministry's Key Laboratory of Electro-Optical Technique and System, Chongqing University.