# On the Dynamics of the Recursive Sequencexn+1=α+xn−kp/xnq

Authors: Mehmet Gümüş; Özkan Öcalan; Nilüfer B. Felah
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equationxn+1=α+xn-kp/xnq,n=0,1,…, where k∈{2,3…}, α,p,q∈(0,∞) and the initial conditions x-k,…,x0 are arbitrary positive numbers. We investigate the boundedness character for p∈(0,∞). Also, we investigate the existence of a prime two periodic solution for k is odd. Moreover, when k is even, we prove that there are no prime two periodic solutions of the equation above.


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## Body

## 1. Introduction

Our aim in this paper is to study the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation(1.1)xn+1=α+xn-kpxnq,n=0,1,…,
where k∈{2,3,…}, α is a positive, p,q∈(0,∞) and the initial conditions x-k,…,x0 are arbitrary positive numbers. Equation (1.1) was studied by many authors for different cases of k,p,q.In [1] the authors studied the global stability, the boundedness character, and the periodic nature of the positive solutions of the difference equation
(1.2)xn+1=α+xn-1xn,n=0,1,…,
where α is positive, and the initial values x-1,x0 are positive numbers (see also [2–4] for more results on this equation).In [5] the authors studied the boundedness, the global attractivity, the oscillatory behaviour, and the periodicity of the positive solutions of the difference equation
(1.3)xn+1=α+xn-1pxnp,n=0,1,…,
where α, p are positive, and the initial values x-1,x0 are positive numbers (see also [6–8] for more results on this equation).In [9] the authors studied general properties, the boundedness, the global stability, and the periodic character of the solutions of the difference equation
(1.4)xn+1=α+xn-kxn,n=0,1,…,
where α,p are positive, k∈{2,3,…}, and the initial values x-k,x-k+1,…,x-1,x0 are positive numbers.In [10, 11] the authors studied the boundedness, the persistence, the attractivity, the stability, and the periodic character of the positive solutions of the difference equation
(1.5)xn+1=α+xn-1pxnq,n=0,1,…,
where α, p, q are positive, and the initial values x-1, x0 are positive numbers.Finally in [12, 13] the authors studied the oscillatory, the behaviour of semicycle, and the periodic character of the positive solution of the difference equation
(1.6)xn+1=α+xn-kpxnp,n=0,1,…,
where k∈{2,3,…} and α, p>0 under the initial conditions x-k,x-k+1,…,x-1,x0 are positive numbers.There exist many other papers related with (1.1) and on its extensions (see [14–16]).Motivated by the above papers, we study of the boundedness character, the oscillatory, and the periodic character of positive solutions of (1.1).In this paper, also we investigate the casep=1, k=1 and q∈(0,∞) of (1.1) and we give a correction for [2, Theorem  2.5].We say that the equilibrium pointx¯ of the equation
(1.7)xn+1=F(xn,xn-1,…,xn-k),n=0,1,…
is the point that satisfies the condition
(1.8)x¯=F(x¯,x¯,…,x¯).
A solution {xn}n=-k∞ of (1.1) is called nonoscillatory if there exists N≥-k such that either
(1.9)xn>x¯∀n≥N
or
(1.10)xn<x¯∀n≥N.
A solution {xn}n=-k∞ of (1.1) is called oscillatory if it is not nonoscillatory. We say that a solution {xn}n=-k∞ of (1.1) is bounded and persists if there exist positive constant P and Q such that P≤xn≤Q for n=-k,-k+1,….The linearized equation for (1.1) about the positive equilibrium x¯ is
(1.11)yn+1+qx¯p-q-1yn-px¯p-q-1yn-k=0,n=0,1,….


## 2. Semicycle Analysis

A positive semicycle of a solution{xn}n=-k∞ of (1.1) consists of a “string” of terms {xl,xl+1,…,xm} all greater than or equal to x¯, with l≥-k and m≤∞, such that
(2.1)eitherl=-korl>-k,xl-1<x¯,eitherm=∞orm<∞,xm+1<x¯.
A negative semicycle of a solution {xn}n=-k∞ of (1.1) consists of a “string” of terms {xl,xl+1,…,xm} all less than x¯, with l≥-k and m≤∞, such that
(2.2)eitherl=-korl>-k,xl-1≥x¯,eitherm=∞orm<∞,xm+1≥x¯.Lemma 2.1.
Let{xn}n=-k∞ be a solution of (1.1). Then either {xn}n=-k∞ consists of a single semicycle or {xn}n=-k∞ oscillates about equilibrium x¯ with semicycles having at most k terms.Proof.
Suppose that{xn}n=-k∞ has at least two semicycles. Then there exists N≥-k such that either xN<x¯≤xN+1 or xN+1<x¯≤xN. We assume that the former case holds. The latter case is similar and will be omitted. Now suppose that the positive semicycle beginning with the term xN+1 has k terms. Then xN<x¯≤xN+k and so the case
(2.3)xN+k+1=α+xNpxN+kq<α+x¯px¯q=x¯
holds for every p, q∈(0,∞), from which the result follows.


## 3. Boundedness and Global Stability of (1.1)

In this section, we consider the casep∈(0,1) with no restriction on other parameters and we consider the case p>1 with some specified conditions. For these cases, we have the following results which give a complete picture as regards the boundedness character of positive solutions of (1.1).Theorem 3.1.
Suppose that(3.1)p∈(0,1),
then every positive solution of (1.1) is bounded.Proof.
We have(3.2)xN+1=α+xN-kpxNq≤α+xN-kpαq,N≥-k.
Hence we will prove that {xN} is bounded. If
(3.3)f(x)=α+xpαq,x>0,
then we have f′(x)>0, f′′(x)<0. Hence, the function f is increasing and concave. Thus, we get that there is a unique fixed point x¯ of the equation f(x)=x. Also the function f satisfies the condition
(3.4)(f(x)-x)·(x-x¯)<0,x>0.
Using [15, 2.6.2] we obtain that x¯ is a global attractor of all positive solutions of (1.1) and so {xN} is bounded, from which the result follows.Now we study the boundedness of (1.1) for the case p>1. We give better result than Theorem 3.1 for the boundedness of (1.1) and we prove that in this case, there exist unbounded solutions of (1.1).Theorem 3.2.
Consider (1.1) and assume that p>1, p↛∞, α>1 and q→∞. Then every positive solution of (1.1) is bounded and limn→∞xn=α.Proof.
Let(3.5)f(x)=α+xpxqforx∈(0,∞).
Suppose on the contrary that every positive solution of (1.1) is unbounded. Then, from (1.1), we obtain xn≥α>1 for n≥1. Therefore we get
(3.6)limx→∞f(x)=α.
Thus the proof is complete. We note that in here f(x) is a continuous function for x,p,q∈(0,∞).Theorem 3.3.
Consider (1.1) when the case k is odd and suppose that
(3.7)p>1,
then there exists unbounded solutions of (1.1).Proof.
Let{xn}n=-k∞ be a solution of (1.1) with initial values x-k,x-k+1,…,x-1,x0 such that
(3.8)x-k,x-k+2,…,x-1>max{(α+1)p/q,(α+1)q/p-1},(3.9)x-k+1,x-k+3,…,x0<α+1.
Then from (1.1), (3.7), and (3.8) we have
(3.10)x1=α+x-kpx0q>α+x-kp(α+1)q-x-k+x-k=α+x-k(x-kp-1(α+1)q-1)+x-k>α+x-k,(3.11)x2=α+x-k+1px1q<α+(α+1)px-kq<α+1,(3.12)x3=α+x-k+2px2q>α+x-k+2p(α+1)q-x-k+2+x-k+2>α+x-k+2,(3.13)x4=α+x-k+3px3q<α+(α+1)px-k+2q<α+1,(3.14)xk=α+x-1pxk-1q>α+x-1p(α+1)q-x-1+x-1>α+x-1.
Also, from (3.8) and (3.10)–(3.14), it is clear that
(3.15)x1,x3,…,xk>max{(α+1)p/q,(α+1)q/p-1},xk+1=α+x0pxkq<α+(α+1)px-1q<α+1.
Moreover from (1.1) and (3.8)–(3.14) and arguing as above we get
(3.16)xk+2=α+x1pxk+1q>α+x1p(α+1)q-x1+x1>α+x1.
Therefore working inductively we can prove that for n=0,1,2,…,
(3.17)x2n+1>α+x2n-k,x2n<α+1,
which implies that
(3.18)limn→∞x2n+1=∞.
So {xn}n=-k∞ is unbounded. From which the result follows.Now, in the next theorem, we will provide an alternative proof for the theorem above when1≤p<∞ and k is odd, whose proof can be used for some practical applications.Theorem 3.4.
Consider (1.1) when the case k is odd and suppose that 1≤p<∞. If 0≤α<1, then there exists solutions of (1.1) that are unbounded.Proof.
We assume that0<α<1 and choose the initial conditions such that
(3.19)x-k,x-k+2,…,x-1>1(1-α)1/q,α<x-k+1,x-k+3,…,x0<1.
So,
(3.20)x1=α+x-kpx0q>α+x-kp,x2=α+x-k+1px1q<α+1x1q<α+1x-kq<1.
Therefore, we obtain α<xk+1<1 and xk+2>2α+x-kp. By induction, for i=1,2,…, we have α<x(k+1)i<1 and x(k+1)i+1>(i+1)α+x-kp. Thus,
(3.21)limi→∞x(k+1)i+1=∞,limi→∞x(k+1)i=α.
Now, we assume that α=0 and choose the initial conditions such that
(3.22)x-k,x-k+2,…,x-1>1(1-ε)forsomeε∈(0,1),0<x-k+1,x-k+3,…,x0<1.
So, we have
(3.23)x1=x-kpx0q>x-kp,x2=x-k+1px1q<1x1q<1.
Further, we have
(3.24)xk+1=x0pxkq<1xkq<1,xk+2=x1pxk+1q>x1p>(x-kp)p.
By induction for i=1,2,…, we have 0<x(k+1)i<1 and x(k+1)i+1>(x-kp)(i+1). Thus,
(3.25)limi→∞x(k+1)i+1=∞,limi→∞x(k+1)i=0,
from which the result follows.The following result is essentially proved in [10, 11] for k=1. The result is satisfied for k∈{2,3,…} and its proof is omitted.Lemma 3.5.
If Either(3.26)0<q<p<1
or
(3.27)0<p<q,q↛∞
holds, then (1.1) has a unique equilibrium point x¯.The following result is essentially proved in [10, 11] for k=1. It is clear that the result is satisfied when k is odd and its proof is omitted.Lemma 3.6.
Consider (1.1) when the case k is odd. Suppose that
(3.28)0<p<1<q,q↛∞,α>(p+q-1)1/(q-p+1)
or
(3.29)0<q<p<1,(p+q)≤1
holds. Then the unique positive equilibrium of (1.1) is globally asymptotically stable.


## 4. Periodicity of the Solutions of (1.1)

In this section, we investigate the existence of a prime two periodic solution fork is odd. Moreover, when k is even, we prove that there are no positive prime two periodic solutions and lastly, we give a correction for Theorem 2.5 which was given in [2].The following result is given when the casek=1 in [10]. If k is odd, the result is still satisfied and its proof is omitted.Lemma 4.1.
Assume that k is odd. Then, (1.1) has prime two periodic solutions if and only if
(4.1)0<p<1<q
and there exists a sufficient small positive number ε1, such that
(4.2)1(α+ε1)q-p>ε1,(α+ε1)p/qε1-1/q<α+ε1-p/q(α+ε1)p2-q2/q.Now, let consider the case wherek is even.Theorem 4.2.
Consider (1.1) when the case k is even and the following conditions are satisfied separately:
(4.3)0<q<p<1,0<p<q<1,1<p,q<p+1,1<q,p<q+1.
Then, there are no positive prime two periodic solutions of (1.1).Proof.
Firstly, we consider the case0<q<p<1 and k is even of (1.1) and suppose that
(4.4)…,x,y,x,y,…,
where x, y∈(α,∞) is a prime two periodic solution of (1.1). Then it must be
(4.5)x=α+yp-q,(4.6)y=α+xp-q.
Substituting (4.6) into (4.5), it follows that
(4.7)x-α=(α+xp-q)p-q.
Taking logarithm on both sides of (4.7), we obtain that
(4.8)F(x)=ln(x-α)-(p-q)ln(xp-q+α).
So from (4.8)
(4.9)limx→α+F(x)=-∞,F(x¯)=ln(x¯-α)-(p-q)ln(x¯p-q+α)=ln(x¯p-q)-(p-q)ln(x¯)=0,F′(x)=1x-α-(p-q)2xp-q-1α+xp-q>1x-α-xp-q-1α+xp-q>1x-α-xp-q-1xp-q>αx(x-α).
From x>α, thus F′(x)>0, which implies that x¯ is a unique solution of (1.1).
We consider the case0<p<q<1 and k is even of (1.1). The proof of this case is similar to the first case’s proof and will be omitted.
Now, suppose that1<p, q<p+1 and k is even of (1.1). In this case we have that
(4.10)F′(x)=α+xp-q-(p-q)2(x-α)xp-q-1(x-α)(α+xp-q).
Considering the numerator on the right hand side in (4.10) let
(4.11)k(x)=α+xp-q-(p-q)2(x-α)xp-q-1=(1-(p-q)2)xp-q+α(p-q)2xp-q-1+α.
From x>α and 1<p, q<p+1(4.12)k(x)>(1-(p-q)2)αp-q+α(p-q)2αp-q-1+α>αp-q+α>0,
which implies that x¯ is a unique solution of (1.1).
Suppose that1<q, p<q+1 and k is even of (1.1). The proof of this case is similar to the third case’s proof and will be omitted.The following result was given in [2, Theorem 2.5] for (1.1) when the case p=1, k=1 and q∈(0,∞). But the authors make some mistakes in this theorem. Now, we give a correction and a conjecture for this result.Theorem A.
Consider (1.1). Let be p=1, k=1, q∈(0,∞), q↛0+ and q↛∞. Suppose that
(4.13)α>1,α>q(1+q)(1-q)/q
hold. Then the unique positive equilibrium x¯ of (1.1) is globally asymptotically stable.Correction B
Consider (1.1). Let be p=1, k=1, q∈(1,∞), and q↛∞. Suppose that
(4.14)α>q1/q
holds. Then the unique positive equilibrium x¯ of (1.1) is globally asymptotically stable.Proof.
It is easy to see the proof from Theorem  2.5 in [2].Conjecture 4.3.
Consider (1.1). Let be p=1, k=1, q∈(0,1), and q↛0+. Suppose that
(4.15)α>1
holds. Then the unique positive equilibrium x¯ of (1.1) is globally asymptotically stable.



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