# On the Recursive Sequencexn=A+xn−kp/xn−1r

Authors: Fangkuan Sun; Xiaofan Yang; Chunming Zhang
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

This paper studies the dynamic behavior of the positive solutions to the difference equationxn=A+xn−kp/xn−1r, n=1,2,…,  where A,p, and r are positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation for p∈(0,1).


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## Body

## 1. Introduction

In recent years, there has been intense interest in the dynamic behavior of the positive solutions to a class of difference equations of the form(1.1)xn=A+xn-kpxn-1p,n∈ℕ,
where A and p are positive real numbers. Now, let us make a brief review on the advances in this class of difference equations.In 1999, Amleh et al. [1] studied the second-order rational difference equation
(1.2)xn=A+xn-2xn-1,n∈ℕ.Later, Berenhaut and Stević [2], Stević [3], and El-Owaidy et al. [4] extended this work to the following more general second-order difference equation:
(1.3)xn=A+xn-2pxn-1p,n∈ℕ.On the other hand, DeVault et al. [5] investigated the following higher-order version of (1.2):
(1.4)xn=A+xn-kxn-1,n∈ℕ.By combining (1.3) and (1.4), Berenhaut and Stević [6] examined a larger class of difference equations, which are of the form
(1.5)xn=A+xn-kpxn-1p,n∈ℕ.Very recently, Berenhaut et al. [7] studied the following generalization of (1.5):
(1.6)xn=A+xn-kpxn-mp,n∈ℕ.For some related work, the interested reader is referred to [1, 3, 8–19].Inspired by the previous work and by the work owing to Stević [15], this paper studies the behavior of the recursive equation
(1.7)xn=A+xn-kpxn-1r,n∈ℕ.
We establish some interesting results regarding the stability and oscillation character of this equation for p∈(0,1).


## 2. Stability Character

In this section we investigate the stability character of the positive solutions to (1.7).A pointx̅∈ℝ is an equilibrium point of (1.7) if and only if it is a root for the function
(2.1)g(x)=x-xp-r-A,
that is,
(2.2)x̅=x̅p-r+A.Lemma 2.1.
Let0<p<r+1, then (1.7) has a unique equilibrium point x̅>1.Proof
Case 1.
p=r. Then x̅=A+1>1.Case 2.
r<p<r+1. Then g defined by (2.1) is decreasing on [0,(p-r)1/(r-p+1)] and increasing on [(p-r)1/(r-p+1),∞). Since g(1)=-A and limx→∞g(x)=∞, then g has a unique zero x̅>1.Case 3.
0<p<r. Since g is increasing on [0,∞),g(1)=-A and limx→∞g(x)=∞, then g has a unique zero x̅>1.Lemma 2.2.
Let0<p<r+1. Assume that x̅ is the equilibrium point of (1.7). If (p+r)(p-r)/(r+1-p)(p+r-1)<A, then x̅ is locally asymptotically stable.Proof.
By the Linearized Stability Theorem [11], x̅ is locally asymptotically stable if and only if x̅r+1-p>p+r. A simple calculations shows that
(2.3)g((p+r)1/(r+1-p))=(p+r)(p-r)/(r+1-p)(p+r-1)-A<0,
where g is defined by (2.1). Then since limx→∞g(x)=∞, we have x̅>(p+r)1/(r+1-p) and x̅r+1-p>p+r. The proof is complete.Lemma 2.3.
Ifp∈(0,1), then every positive solution to (1.7) is bounded.Proof.
Note that eachn∈ℕ can be written in the form lk+i for some l∈ℕ0 and i∈{0,1,…,k-1}. From (1.7) and since xn>A for every n≥0, we have that
(2.4)xlk+i=A+x(l-1)k+ipxlk+i-1r<A+x(l-1)k+ipAr,
for every l∈ℕ0 and i∈{0,1,…,k-1}. Let (ul(i))l∈ℕ0 be the solution to the difference equation
(2.5)ul(i)=A+(ul-1(i))pAr,u0(i)=x-k+i.
From (2.4) and by induction we see that x(l-1)k+i≤ul(i),l∈ℕ0. Hence it is enough to prove that the sequences (ul(i))l≥0,i∈{0,1,…,k-1} are bounded.
Since the functionf(x)=A+xp/Ar,x∈(0,∞) is increasing and concave for p∈(0,1), it follows that there is a unique fixed point x̅ of the equation f(x)=x and that the function f satisfies
(2.6)(f(x)-x)(x-x̅)<0,x∈(0,∞).
Using this fact it is easy to see that iful(i)∈(0,x̅], the sequence is nondecreasing and bounded from above by x̅, and if ul(i)≥x̅, it is nonincreasing and bounded from below by x̅. Hence for every u0(i)∈(0,∞), each of the sequences ul(i),i∈{0,1,…,k-1} is bounded. The claimed result follows.Lemma 2.4 (see [18]).
Lets,t be distinct nonnegative integers. Consider the difference equation
(2.7)xn=f(xn-s,xn-t),n=1,2,3,…,x1-max(s,t),x2-max(s,t),…,x0∈[a,b].
Suppose f satisfies the following conditions.(H1)
f:[a,b]2→[a,b] is a continuous function that is nondecreasing in the first argument and is nonincreasing in the second argument.(H2)
The system
(2.8)x=f(x,y),y=f(y,x)
has a unique solution (x̅,x̅)∈[a,b]×[a,b].Then x̅ is the global attractor of all solutions to (2.7).Theorem 2.5.
Letp+r≤1,  then the unique equilibrium x̅ to (1.7) is globally asymptotically stable.Proof.
By Lemma2.3, there must exist positive constants P and Q such that P≤xn≤Q. Let f(u,v)=A+up/vr,u,v∈[P,Q], it is easy to verify that (H1) holds. In addition, if
(2.9)x=A+xpyr,y=A+ypxr,
then
(2.10)x-Ay-A=xp+ryp+r.
Assume that x≠y, then x>y or x<y.
In casex>y, we have (x-A)/(y-A)>x/y≥xp+r/yp+r, which contradicts with (2.10).
In casex<y, we have (x-A)/(y-A)<x/y≤xp+r/yp+r, again a contradiction.
Thusx=y=x̅. By Lemma 2.4, the required result follows.Theorem 2.6.
Let0<p≤r<1 and Ar-p+1≥p/r. Then every positive solution to (1.7) converges to the unique equilibrium x̅.Proof.
By Lemma2.3, every positive solution {xn} to (1.7) is bounded, which implies that there are finite lim infxn=I and lim supxn=S. Assume that I≠S(I<S). Taking the lim inf and lim sup in (1.7), it follows that
(2.11)A+IpSr≤I<S≤A+SpIr.
From this and r∈(0,1), it follows that
(2.12)ASr+Ip≤ISr<SIr≤AIr+Sp,
yielding
(2.13)ASr-Sp<AIr-Ip.
Define functionf(x)=Axr-xp,x∈(A,∞). Since
(2.14)f′(x)=Arxr-1-pxp-1=xp-1(Arxr-p-p)>xp-1(rAr-p+1-p)≥0,
we deduce that f is increasing, and thus (2.13) cannot hold. Therefore we have I=S, which implies the result.Theorem 2.7.
Let0<p<1,r≥1, and Ar-p+1≥r+p-1. Then every positive solution to (1.7) converges to the unique equilibrium x̅.Proof.
From (2.11) we have
(2.15)AIr-1Sr+Ip+r-1≤IrSr≤AIrSr-1+Sp+r-1.
Consequently, we obtain (AIr-1Sr-1)(S-I)≤(Sr+p-1-Ir+p-1). Suppose that I≠S, we get
(2.16)AIr-1Sr-1≤Sr+p-1-Ir+p-1S-I=(r+p-1)γp+r-2,
where γ∈(I,S), leading to
(2.17)ArSr-1≤AIr-1Sr-1≤(r+p-1)γp+r-2<(r+p-1)Ap-1Sr-1.
This implies thatAr-p+1<r+p-1, which is a contradiction. Hence, I=S=x̅.


## 3. Oscillation Character

In this section we investigate the oscillation character of the positive solutions to (1.7).Theorem 3.1.
Let{xn}n=-k∞ be a positive solution to (1.7). Then either {xn}n=-k∞ consists of a single semicycle or {xn}n=-k∞ oscillates about the equilibrium x̅ with semicycles having at most k-1 terms.Proof.
Suppose that{xn}n=-k∞ has at least two semicycles. Then there exists N≥-k such that either xN<x̅≤xN+1 or xN+1<x̅≤xN. Assume that xN<x̅≤xN+1. (The argument for the case xN+1<x̅≤xN is similar and is omitted). Now suppose that the positive semicycle beginning with the term xN+1 has k-1 terms. Then xN<x̅≤xN+k-1 and so
(3.1)xN+k=A+xNpxN+k-1r<A+x̅px̅r=A+x̅p-r=x̅.
This completes the proof.Theorem 3.2.
Suppose thatk is even and let {xn}n=-k∞ be a solution to (1.7), which has k-1 consecutive semicycles of length one, then every semicycle after this point is of length one.Proof.
There existsN≥-k such that either
(3.2)xN,xN+2,…,xN+k-2<x̅≤xN+1,xN+3,…,xN+k-1
or
(3.3)xN+1,xN+3,…,xN+k-1<x̅≤xN,xN+2,…,xN+k-2.
We prove the former case. The proof for the latter is similar and is omitted. Now, we have
(3.4)xN+k=A+xNpxN+k-1r<A+x̅px̅r=A+x̅p-r=x̅,xN+k+1=A+xN+1pxN+kr>A+x̅px̅r=A+x̅p-r=x̅.
The result then follows by induction.Lemma 3.3.
Let0<p<r+1. Then (1.7) has no nontrivial periodic solutions of (not necessarily prime) period k-1.Proof.
Suppose that{xn}n=-k∞ is a positive solution to (1.7) satisfying xn-1=xn-k for all n≥1, then xn=A+xn-kp/xn-1r=A+xn-1p-r implies that xn-1=xn=x̅ for all n>-k. The proof is complete.Theorem 3.4.
Assume thatp≤r.  Let {xn}n=-k∞ be a positive solution to (1.7), which consists of a single semicycle, then{xn}n=-k∞ converges to the equilibrium x̅.Proof.
Supposexn≥x̅ (the case for xn≤x̅ is similar and is omitted) for all n≥-k, then
(3.5)xn+1=A+xn-(k-1)pxnr≥x̅=A+x̅p-r,
implying that
(3.6)xn-(k-1)≥x̅(p-r)/pxnr/p≥xn(p-r)/pxnr/p=xn,
and so
(3.7)xn-(k-1)≥xn≥x̅forn=1,2,….
From here it is clear that for i=0,…,k-2 there exists αi such that
(3.8)limn→∞xn(k-1)+i=αi.
But then α0,α1,…,αk-2 is a periodic solution of (not necessarily prime) period k-1. By Lemma 3.3 the result holds.



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*Source: 608976-2009-07-01.xml*