Global asymptotic stability in a rational recursive sequence

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Abstract

In this paper, we study the global stability of the difference equationxn=a+bxn−1+cxn−12d−xn−2,n=1,2,…,where a, b⩾0 and c, d>0. We show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters, and every positive solution of the equation in the basin exponentially converges to the attractor.

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 equationxn=a+bxn−1d+xn−k,n=1,2,…,where 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:xn=a−bxn−1d+xn−k,n=1,2,…,xn=a−bxn−1d−xn−k,n=1,2,…,xn=a+bxn−1d−xn−k,n=1,2,…,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 equationxn=a+cxn−12d+xn−k2,n=1,2,….Zhang 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:xn=a+bxn−1+cxn−12d−xn−2,n=1,2,…,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×II be a continuous function. For every initial condition 〈x−1,x0〉∈I×I, the difference equationxn=f(xn−1,xn−2),n=1,2,…,has a unique solution {xn}n=−1, which is called a recursive sequence. An equilibrium point of Eq. (2.1) is a point αI with f(α,α)=α.

Definition 2.1

Let α be an equilibrium point of Eq. (2.1).

  • (i)

    α is locally stable if for every ε>0, there exists δ>0 such that for each 〈x−1,x0〉∈I×I with |x−1α|+|x0α|<δ, |xnα|<ε holds for n=1,2,….

  • (ii)

    α is a local attractor if

Equilibria and local asymptotic stability

Consider the difference equationxn=f(xn−1,xn−2)=a+bxn−1+cxn−12d−xn−2,n=1,2,…,wherea,b⩾0,c,d>0.The equilibrium points of this equation are the solutions of the quadratic equation(1+c)x2−(d−b)x+a=0.Supposed>b,(d−b)2>4a(1+c).Then Eq. (3.1) has one nonnegative equilibrium α and one positive equilibrium β>α, whereα=(d−b)−(d−b)2−4a(1+c)2(1+c),β=(d−b)+(d−b)2−4a(1+c)2(1+c).Sinced−α>d−β=d+b+2cd−(d−b)2−4a(1+c)2(1+c)d+b+2cd−(d−b)2(1+c)=b+cd1+c>0,we have0⩽α<β<d.

Theorem 3.1

Assume , hold. Then

  • (i)

    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.xn−α=f(xn−1,xn−2)−f(α,α)=[f(xn−1,xn−2)−f(α,xn−2)]+[f(α,xn−2)−f(α,α)]=a+bxn−1+cxn−12d−xn−2a+bα+cα2d−xn−2+a+bα+cα2d−xn−2a+bα+cα2d−α=b+c(xn−1+α)d−xn−2(xn−1−α)+a+bα+cα2(d−xn−2)(d−α)(xn−2−α).

In view of a++2=α(dα), we derivexn−α=b+cα+cxn−1d−xn−2(xn−1−α)+αd−xn−2(xn−2−α).The following lemma follows from Eq. (4.1).

Lemma 4.1

Assume , hold andxn−2,xn−1〉∈[0,β)×[0,β).

  • (i)

    If xn−1α

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:xn+1−xn=f(xn,xn−1)−f(xn−1,xn−2)=[f(xn,xn−1)−f(xn−1,xn−1)]+[f(xn−1,xn−1)−f(xn−1,xn−2)]=b+c(xn+xn+1)d−xn−1(xn−xn−1)+a+bxn−1+cxn−12(d−xn−1)(d−xn−2)(xn−1−xn−2).

From Eq. (5.1) and by induction on n, we obtain

Lemma 5.1

Assume , hold.

  • (i)

    If there exists N such that xNxN+1xN+2, then xnxn+1 for nN.

  • (ii)

    If there exists N such that xNxN+1xN+2, then xnxn+1 for nN.

From Eqs.

References (19)

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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.

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