This paper aims to investigate the global stability of negative solutions of the difference equation xn+1=(α+βxn-k)/(γ+xn), n=0,1,2,…, where the initial conditions x-k,…,x0∈-∞,0, k is a positive integer, and the parameters β, γ<0, α>0. By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end.

1. Introduction

The study of nonlinear difference equations has always attracted a considerable attention (see, e.g., [1–30] and the references cited therein). In particular, some references investigated the dynamical behavior of positive solutions of difference equations (see, e.g., [3–5, 11]), and some references examined the dynamical behavior of negative solutions of some difference equations (see, e.g., [6, 7]).

Gibbons et al. [4] studied the behavior of nonnegative solutions to the recursive sequence
(1)xn+1=α+βxn-1γ+xn,n=0,1,…,
with α,β,γ≥0, and also presented an open problem, which had been solved by Stevic´ in [8]. Kulenovic´ and Ladas, in addition, considered (1) in their book [9].

Stevic´ [10] considered the behavior of nonnegative solutions of the following second-order difference equation
(2)xn+1=α+βxn-11+g(xn),n=0,1,…,
where g:ℝ+⋃{0}→ℝ is a nonnegative increasing mapping.

Douraki et al. [11] studied the qualitative behavior of positive solutions of the difference equation
(3)xn+1=p+qxn-k1+xn,n=0,1,…,
where the initial values x-k,…,x-1,x0∈(0,+∞), k is a positive integer, and p,q≥0. Moreover, (3) is a special case of the following open problem (see also in [11]), which was proposed by Kulenovic´ and Ladas in [9].

Open Problem (equation <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M18"><mml:mo stretchy="false">(</mml:mo><mml:mn>6.97</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> in [<xref ref-type="bibr" rid="B9">9</xref>]).

Assume that p,q∈[0,∞). Investigate the global behavior of positive solutions of (3).

Stevic´ [12] considered the boundedness, oscillatory behavior, and global stability of nonnegative solutions of the difference equation
(4)xn+1=α+βxn-kf(xn,…,xn-k+1),n=0,1,…,
where k∈ℕ,α,β≥0, and f:ℝ+k→ℝ+ is a continuous function nondecreasing in each variable such that f(0,…,0)>0.

It is worthwhile to note that the above mentioned references ([4, 10–12]), especially [4, 11], only discussed the dynamical behavior of positive solutions of difference equation. Furthermore, inspired by the above work and [6, 7], the main goal of this paper is to study the global behavior of negative solutions of the difference equation
(5)xn+1=α+βxn-kγ+xn,n=0,1,…,
where k is a positive integer, α>0,β,γ<0, and the initial conditions x-k,…,x0∈(-∞,0).

In fact, it is easy to see that (5) is an extension of an open problem introduced by Kulenovic´ and Ladas in [9] and also is a special case of (4) by a simple change. However, here we establish some results regarding the global stability, invariant interval, and periodic character of negative solutions of (5).

2. Linearized Stability and Period 2 Solutions

The aim of this section is to discuss the local stability of the unique negative equilibrium of (5). The period 2 solutions of (5), in addition, will be verified.

In this section, we need the following lemma.

Lemma 1 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Assume that a,b∈ℝ and k∈{1,2,…}. Then
(6)|a|+|b|<1
is a sufficient condition for the asymptotic stability of the difference equation
(7)xn+1-axn+bxn-k=0,n=0,1,….

Suppose in addition that one of the following two cases holds.

k is odd and b<0.

k is even and ab<0.

Then (6) is also a necessary condition for the asymptotic stability of (7).

Assume that x¯ is an equilibrium of (5). Then it satisfies the equation x¯2+(γ-β)x¯-α=0, which implies that the unique negative equilibrium of (5) is
(8)x¯=β-γ-(β-γ)2+4α2.

Let
(9)xn+1=f(xn,xn-1,…,xn-k)=α+βxn-kγ+xn.
Then the linearized equation of (5) at x¯ is
(10)un+1=p0un+p1un-1+⋯+pkun-k,
where
(11)pi=∂f∂xn-i(x¯,x¯,…,x¯).
Straightforward calculations yield
(12)un+1+x¯γ+x¯un-βγ+x¯un-k=0.

From here and Lemma 1, we obtain the following result.

Theorem 2.

The unique negative equilibrium x¯ of (5) is locally asymptotically stable if γ<β.

Proof.

If γ<β, then
(13)|x¯γ+x¯|+|-βγ+x¯|=β+x¯γ+x¯<1.
It follows from Lemma 1 that x¯ is locally asymptotically stable. Thus the proof is complete.

Now, we will examine the existence of period 2 solutions of (5).

Theorem 3.

Let {xn}n=-k∞ be a negative solution of (5). Then the following statements hold.

Assume that k is odd. Then,

equation (5) has a negative prime period 2 solution …,w,r,w,r,… if and only if β=γ;

if β=γ, then the values w and r of all negative prime period 2 solutions are given by {w,r∈(-∞,0):wr=α};

if β≠γ, then (5) has no negative solutions with prime period 2.

Assume that k is even. Then (5) has no negative solutions with prime period 2.

Proof.

Let {xn}n=-k∞ be a negative solution of (5).

Assume that k is odd. Then,

suppose that …,w,r,w,r,… is a negative prime period 2 solution. Then from (5), we have
(14)w=α+βwγ+r,r=α+βrγ+w.
From the above relations, we derive that
(15)(w-r)(β-γ)=0.
Since w≠r, then β=γ.The reverse part is clear by a simple computation. So it is omitted.

Applying the relations (14), we get that w(γ+r)=α+βw and r(γ+w)=α+βr; namely, w(γ-β)+wr=α and r(γ-β)+wr=α. As β=γ, then wr=α.

If β≠γ, then it follows from (15) that w=r.

Assume that k is even. Let …,w,r,w,r,…(w≠r) be a negative prime period 2 solution; then
(16)w=α+βrγ+r,r=α+βwγ+w.
It follows from the above equations that (w-r)(β+γ)=0. As β+γ<0, then w=r, which contradicts the hypothesis that w≠r.

3. Invariant Interval

In this section, we will consider the invariant interval of negative solutions of (5).

Let
(17)f(u,v)=α+βvγ+u.

Lemma 4.

The following statements are true.

Assume that γ∈(-∞,(β-β2+4α)/2]. Then,

γ<β<0;

if β<-α/2, then βγ>α and x¯∈(β,0)⊂[γ,0].

Assume that f(u,v) is defined by (17) and u,v∈(-∞,0]. Then f(u,v) is nonincreasing in u and nondecreasing in v.

Proof.

The proofs of (a) and (b) are as follows:

if γ∈(-∞,(β-β2+4α)/2], then

γ-β≤(β-β2+4α)/2-β=-(β+β2+4α)/2<0, clearly, γ<β<0;

the condition β<-α/2 implies that α/β2<2. Then
(18)|(β-β2+4α)/2||α/β|=β(β-β2+4α)2α=-2ββ+β2+4α=1((-1/2)+(1/4)+(α/β2))>1(-1/2+1/4+2)=1,
which leads to γ≤(β-β2+4α)/2<α/β. Note that γ<β<0; thus βγ>α. Also
(19)x¯-β=2(βγ-α)(β-γ)2+4α-(β+γ)>0.
Clearly, x¯∈(β,0)⊂[γ,0].

Note that
(20)∂f∂u=-α+βv(γ+u)2<0,∂f∂v=βγ+u>0.
So the result holds.

Theorem 5.

Assume that γ∈(-∞,(β-β2+4α)/2]. Then [γ,0] is an invariant interval of (5).

Proof.

Suppose that {xn}n=-k∞ is a solution to (5) with initial conditions x-k,…,x0∈[γ,0]. Since γ∈(-∞,(β-β2+4α)/2], it follows by a direct computation that α+βγ-γ2≤0. By Lemma 4(b), we immediately get that the function f(u,v) is nonincreasing in u and nondecreasing in v with u,v∈(-∞,0). Then
(21)x1=α+βx-kγ+x0=f(x0,x-k)<f(x0,0)<f(γ,0)=α2γ<0,x1=α+βx-kγ+x0=f(x0,x-k)>f(0,x-k)>f(0,γ)=α+βγγ≥γ,
which implies that x1∈[γ,0]. It follows by induction that xn∈[γ,0] for all n≥1. Thus, the proof is complete.

4. Global Stability

Recall that (5) has a unique negative equilibrium x¯, which is locally asymptotically stable if γ<β by Theorem 2. In this section, we will show that (a) x¯ is globally asymptotically stable under the conditions γ∈(-∞,(β-β2+4α)/2] and β<-α/2; (b) every negative solution converges to x¯ when βγ=α.

Lemma 6 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Consider the difference equation
(22)yn+1=f(yn,yn-k),n=0,1,…,
where k∈{1,2,…}. Let I=[a,b] be some interval of real numbers and assume that
(23)f:[a,b]×[a,b]⟶[a,b]
is a continuous function satisfying the following properties:

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

if (m,M)∈[a,b] is a solution of the system
(24)m=f(M,m),M=f(m,M),

then m=M.

Then (22) has a unique equilibrium y¯ and every solution of (22) converges to y¯.
Theorem 7.

Assume that γ∈(-∞,(β-β2+4α)/2] and β<-α/2. Then the following statements are true:

the unique negative equilibrium x¯=(β-γ-(β-γ)2+4α)/2 of (5) is a global attractor with a basin S=[γ,0]k+1;

the unique negative equilibrium x¯=(β-γ-(β-γ)2+4α)/2 of (5) is globally asymptotically stable.

Proof.

Suppose that {xn}n=-k∞ is a solution to (5) with initial values x-k,…,x0∈S. By Lemma 4(b), we immediately get that the function f(u,v) is continuous and nonincreasing in u and nondecreasing in v on the invariant interval [γ,0]. Furthermore, let m,M∈I be a solution of the system m=f(m,M), M=f(M,m); then by Lemma 4(a)(i) and (ii), we have γ<β<0 and x¯∈(β,0)⊂[γ,0], and by Theorem 3, we obtain m=M. Finally, it follows by Lemma 6 that limn→∞xn=x¯. Hence, the proof is complete.

Lemma 8.

Let βγ=α. Then (5) has no nontrivial negative periodic solutions of (not necessarily prime) period k.

Proof.

If βγ=α, then substituting (8), it becomes that x¯=β. Suppose that {xn}n=-k∞ is a negative solution to (5) satisfying xn=xn-k for all n≥0; then
(25)xn+1=α+βxnγ+xn.
Simplifying the above equation, it follows that (γ+xn)(xn+1-β)=0. Clearly, xn+1=β=x¯ for all n≥0. The proof is complete.

Theorem 9.

Assume that βγ=α. Then every negative solution to (5) converges to the unique negative equilibrium x¯.

Proof.

Assume that βγ=α. It follows from Lemma 8 that x¯=β. Then
(26)xn+1-x¯=α+βxn-kγ+xn-β=β(xn-k-xn)γ+xn.

Case 1. If xn-k-xn<0, then xn+1<x¯, so xn-k<xn<x¯.

Case 2. If xn-k-xn≥0, then xn+1≥x¯, so x¯≤xn≤xn-k.

Next, we consider Case 1 (Case 2 is similar and thus it is omitted). If xn-k<xn<x¯ for all n≥0, then it is clear that for i∈{0,1,…,k-1} there exists αi such that
(27)limm→∞xmk+i=αi.
But then α0,α1,…,αk-1 is a periodic solution of (not necessarily prime) period k. By Lemma 8 the result holds.

5. Examples

To illustrate the main results in Section 4, here we present two examples.

Example 1.

Consider (1) with α=8,β=-4,γ=-6, and k=3. Then x¯=-2. As γ∈(-∞,(β-β2+4α)/2] and β<-α/2, it follows from Theorem 7(ii) that the unique negative equilibrium is globally asymptotically stable. For the initial conditions x-i=-10,i=0,1,2,3, Figure 1 exhibits how xn+1 evolves with n.

Evolution of xn+1 for (1) with parameters and initial condition given in Example 1.

Example 2.

Consider (1) with α=12,β=-3,γ=-4, and k=3. Then x¯=-3. As βγ=α, it follows from Theorem 9 that the unique negative equilibrium is globally attractive. For the initial conditions x-i=-4,i=0,1,2,3, Figure 2 exhibits how xn+1 evolves with n.

Evolution of xn+1 for (1) with parameters and initial condition given in Example 2.

Acknowledgments

The authors are greatly indebted to the referees for their valuable suggestions. This work is financially supported by the National Natural Science Foundation of China (no. 10771227) and the Fundamental Research Funds for the Central Universities (no. CDJXS10181130).

AboutalebM. T.El-SayedM. A.HamzaA. E.Stability of the recursive sequence xn+1=(α-βxn)/(γ+xn-1)YangX.YangY.LuoJ.On the difference equation xn=(p+xn-s)/(qxn-1+xn-s)DeVaultR.KosmalaW.LadasG.SchultzS. W.Global behavior of yn+1=(p+yn-k)/(qyn+yn-k)GibbonsC. H.KulenovićM. R. S.LadasG.On the recursive sequence xn+1=(α+βxn-1)/(γ+xn)SunF.YangX.ZhangC.On the recursive sequence xn=A+xn-kp/xn-1rJiaX.-M.HuL.-X.Global attractivity of a higher-order nonlinear difference equationSalehM.AloqeiliM.On the difference equation yn+1=A+yn/yn-k with A<0StevićS.On the recursive sequence xn+1=xn-1/g(xn)KulenovićM. R. S.LadasG.StevićS.On the recursive sequence xn+1=(α+βxn-1)/(1+g(xn))DourakiM. J.DehghanM.RazzaghiM.The qualitative behavior of solutions of a nonlinear difference equationStevićS.On the recursive sequence xn+1=(α+βxn-k)/(f(xn,…,xn-k+1))CunninghamK. C.KulenovićM. R. S.LadasG.ValicentiS. V.On the recursive sequence xn+1=(α+βxn)/(Bxn+Cxn-1)HuL.-X.LiW.-T.StevićS.Global asymptotic stability of a second order rational difference equationKocićV. L.LadasG.KulenovićM. R. S.MerinoO.Global attractivity of the equilibrium of xn+1=(pxn+xn-1)/(qxn+xn-1) for q<pLiW.-T.SunH.-R.Global attractivity in a rational recursive sequenceLiW.ZhangY.SuY.Global attractivity in a class of higher-order nonlinear difference equationLiuW.YangX.Quantitative bounds for positive solutions of a Stević difference equationLiuW.YangX.CaoJ.On global attractivity of a class of nonautonomous difference equationsStevićS.On the difference equation xn+1=α+xn-1/xnStevićS.Global stability of a max-type difference equationStevićS.On a generalized max-type difference equation from automatic control theorySuY.-H.LiW.-T.StevićS.Dynamics of a higher order nonlinear rational difference equationSuY.-H.LiW.-T.Global attractivity of a higher order nonlinear difference equationYanX.-X.LiW.-T.SunH.-R.Global attractivity in a higher order nonlinear difference equationYanX.-X.LiW.-T.Global attractivity in the recursive sequence xn+1=(α-βxn)/(γ-xn-1)YangX.LiuW.LiuJ.Global attractivity of a family of max-type difference equationsYangX.TangY. Y.CaoJ.Global asymptotic stability of a family of difference equationsYangY.YangX.On the difference equation xn=(pxn-s+xn-t)/(qxn-s+xn-t)