# The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation

Authors: Stevo Stević; Kenneth S. Berenhaut
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equationxn=f(xn−2)/g(xn−1), n∈ℕ0, where f,g∈C[(0,∞),(0,∞)]. It is shown that if f and g are nondecreasing, then for every solution of the equation the subsequences {x2n} and {x2n−1} are eventually monotone. For the case when f(x)=α+βx and g satisfies the conditions g(0)=1, g is nondecreasing, and x/g(x) is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then f(x)=c1/x and g(x)=c2x, for some positive c1 and c2.


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## Body

## 1. Introduction

Recently there has
been great interest in the study of nonlinear and rational difference
equations (cf. [1–35] and the references therein).In this paper, we study the boundedness, global
asymptotic stability, and periodicity for positive solutions of the
equation(1.1)xn=f(xn−2)g(xn−1),n∈ℕ0,
where  f,g∈C[(0,∞),(0,∞)].


## 2. Asymptotic Periodicity of (1.1)

In this section,we investigate asymptotic periodicity
of (1.1). The asymptotic periodicity of some difference equations has been
investigated, for example, in the papers
[3–6, 10, 12, 15, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 35]. Our first result is the
following theorem regarding eventual monotonicity, which is a natural extension
of [24, Theorem 2].Theorem 2.1.
Assume thatf and g
are
nondecreasing functions which map the interval (0,∞)
into itself,
and assume that {xn}
is a solution
of (1.1). Then, the sequences {x2n} and {x2n+1}
are eventually
monotone.Proof.
Suppose that{xn} is a solution
to (1.1), and set
(2.1)rn={1ifxnxn−2≥1,0otherwise,
for n≥0. Note that it suffices to show that {r2i}i≥0 and {r2i+1}i≥0 are eventually
constant.
From (1.1), we have that
(2.2)xnxn−2=f(xn−2)g(xn−1)g(xn−3)f(xn−4)=f(xn−2)f(xn−4)g(xn−3)g(xn−1).
If rn−2=1 and rn−1=0, then by (1.1) and the monotonicity of f and g we have that xn/xn−2≥1, and hence rn=1. Similarly, if rn−2=0 and rn−1=1, then rn=0. Hence, if rn0≠rn0+1 for some n0≥0, then the sequences {rn0+2i}i≥0 and {rn0+2i+1}i≥0 are both
constant, as required. This confirms the statement in the theorem, in this
case. Otherwise, {ri}i≥0 itself is a
constant sequence and the result again follows.Remark 2.2.
Note that Theorem2.1 guarantees only the eventually
monotonicity of the sequences {x2n} and {x2n+1}. Hence, for a
solution {xn} of (1.1), one of
these two subsequences can be infinite. See, for example, [27, Theorem 1],
where it was shown that for the case f(x)=g(x)=xp,p≥1, (1.1) has unbounded solutions. The problem
was previously treated in the papers [1, 11] but the proofs appearing there
have a gap (for more details see [27]).The first special case of the nonrational (1.1) was
considered in the paper [22], where the second author considered the
equation
(2.3)xn=α+βxn−21+g(xn−1),
where the
function g satisfies the
following conditions:
(i)
g(x)>0 for x≥0,(ii)
g(x) is increasing
on [0,∞),(iii)
x/g(x) is increasing
on [0,∞).The following two conjectures, which were posed by the
second author, have circulated among the experts in the field, since early
2001.Conjecture 2.3.
Assume thatβ=1. Show that every solution of (2.3) is bounded.Conjecture 2.4.
Assume thatβ=1. Show that every prime periodic solution of (2.3) has
period equal to one or two.Conjecture2.3 was confirmed in [28] where
the following lemma was proved.Lemma 2.5.
Supposeh
is a function
which satisfies the following conditions:
(a)
h(0)≥0,(b)
h(x) is increasing
on [0,∞),(c)
x/h(x) is
nondecreasing on [0,∞).
Then for given l,L,α>0 such that l<L, there exist l0 and L0 such that
(1)
0<l0≤l and L≤L0,(2)
l0h(L0)≤α≤h(l0)L0.By Lemma2.5, a result was proven concerning an
extension of (2.3). We present the proof of the theorem for the case of (2.3), for
the benefit of the reader, since the proof is instructive. For related results
regarding boundedness, see, for example, [1, 6, 10, 22, 27, 31, 33, 34].Theorem 2.6.
Assume thatg
is a function
which satisfies conditions (ii) and (iii) and that g(0)≥0. Consider (2.3) where α,x−2,x−1∈(0,∞).
Then every
solution of (2.3)
is bounded and
persists.Proof.
Choosel and L such that L>max⁡{x−2,x−1} and min⁡{x−2,x−1}>l>0. By Lemma 2.5, we
may also assume that lg(L)≤α≤g(l)L. Now we may use
mathematical induction to prove the result. Assume the statement is true for x−2,x−1,x0,…,xn, that is,
(2.4)l≤xi≤L∀i=−2,−1,0,1,…,n−1.
Then(2.5)xn=α+xn−21+g(xn−1)≤α+L1+g(l).
We claim that(α+L)/(1+g(l))≤L. But this is obvious since
(2.6)α+L≤(1+g(l))L⟺α≤g(l)L.
Similarly, we have that
(2.7)xn=α+xn−21+g(xn−1)≥α+l1+g(L),
from which it follows that xn≥l for n=−2,−1,0,…, completing the
proof of the theorem.By Theorems2.1 and 2.6 we confirm Conjecture 2.4. Indeed, by Theorem 2.6 we have that every solution {xn} of (2.3) is
bounded. On the other hand, by Theorem 2.1, the sequences {x2n} and {x2n−1} are eventually
monotone, thus convergent. Hence, if (2.3) has periodic solutions they have
period one or two, as conjectured.


## 3. Global Periodicity of (1.1)

Definition 3.1.
Letg be a function
defined on a subset of ℝn. Say that the
difference equation
(3.1)xn=g(xn−1,…,xn−k),n∈ℕ0,
where k∈ℕ, is periodic, if every solution of (3.1) is
periodic.Periodic equations have been investigated, for
example, in [2, 7, 8, 12, 14, 16–19, 26] (see also the references therein).In this section, we investigate periodic equations of
type (1.1). In order to facilitate notation we will write (1.1) in the equivalent form
(3.2)xn=h(xn−1)f(xn−2),n∈ℕ0.If every solution of (3.2) is periodic with period p=2, then it must
hold that
(3.3)x=h(y)f(x),x,y∈(0,∞),
that is x/f(x)=h(y), which implies that f(x)=cx and h(y)=c for some
positive constant c. Thus, (3.2) has the form xn=c2xn−2. Since every
solution of (3.2) must be two periodic, it follows that c=1. Hence, the
equation xn=xn−2 is a unique
equation of type (3.2) for which all solutions are periodic with period two.Further we consider those equations of type (3.2) for
which all solutions are periodic with period three. For a mapping f:X→X the sequence of
iterates {f[m]}m∈ℕ∪{0} of f, is defined by f[0]=I (I is the identity
function on X), f[1]=f and generally f[m+1]=f∘f[m] for any m∈ℕ.Before we prove the result concerning the case, we
need the following auxiliary result which is folklore.Lemma 3.2.
Assume thatf:(0,∞)→(0,∞) is a continuous function such that
(3.4)f[6](x)=x,x∈(0,∞).
Then, f(x)=x,x∈(0,∞) or f[2](x)=x.Proof.
Iff(x)=f(y), then from (3.4) it follows that
(3.5)x=f[6](x)=f[6](y)=y,
which implies that the function f must be 1−1. Since f is a continuous
function we have that f must be
strictly monotone.
First assume thatf is strictly
increasing. If there is a point x0∈I such that x0<f(x0), then by the monotonicity of f we have
(3.6)x0<f(x0)<f[2](x0)<⋯<f[6](x0)=x0,
which is a contradiction.
Ifx0>f(x0), then we have
(3.7)x0>f(x0)>f[2](x0)>⋯>f[6](x0)=x0,
arriving again at a contradiction.
From this it follows thatf(x)=x for every x∈(0,∞).
Assume now thatf is strictly decreasing. Then the function g(x)=f[2](x) is strictly increasing and
(3.8)g[3](x)=(f[2])[3](x)=x.
Similar to the first case, we obtain that f[2](x)≡x, finishing the proof of the lemma.Remark 3.3.
Note that iff[2](x)=x, then f is decreasing,
maps interval (0,∞), “1−1,” and onto itself, and its graph is symmetric with
respect to the line y=x, since f(x)=f−1(x).Theorem 3.4.
All solutions of (3.2)
are periodic with period three if and only if f(x)=c1/x and h(x)=c2/x,
for some
positive constants c1 and c2.Proof.
Assume thatf(x)=c1/x and h(x)=c2/x, for some
positive constants c1 and c2. Then (3.2)
becomes
(3.9)xn=c3xn−1xn−2,c3=c1c2.
It is easy to see that every solution of the equation is periodic with period three (see, e.g., [12]).
Assume now that every solution of (3.2) is periodic with period three. Then, we have that
(3.10)u=h(y)f(x),x=h(u)f(y),y=h(x)f(u),
for every x,y∈(0,∞).
Eliminatingu in (3.10) we
obtain that
(3.11)x=h(h(y)f(x))f(y),y=h(x)f(h(y)f(x)).
Now, in each of the two equations (3.11), we choose that a variable is arbitrary and the other is
equal to 1, and use the
changes
(3.12)h^(x)=h(x)f(1),f^(x)=f(x)h(1).
Then, we obtain
(3.13)h^(f^(z))=z,h^(h^(z))f^(z)=C,f^(f^(z))h^(z)=C,f^(h^(z))=z,
for every z∈(0,∞), where C=f(1)h(1).
From (3.13) we have
(3.14)f^(h^(z))=h^(f^(z))=z,
which implies that
(3.15)f^(z)=h^−1(z).
If we setz→h^(z) in the second identity in (3.13) and z→f^(z) in the third
identity, and then apply (3.14), we obtain
(3.16)f^[3](z)=h^[3](z)=Cz.
From (3.15) and (3.16) it follows that
(3.17)h^[6](z)=z.
Lemma2.5 implies that h^(z)=z or h^[2](z)=z. If h^(z)=z, then (3.13)
implies f^(z)=z, and by the
second identity in (3.13) we obtain z=C, which is a
contradiction.Hence,
(3.18)h^[2](z)=f^[2](z)=z.
Substituting (3.18) in (3.13) we obtain that
(3.19)h^(z)=f^(z)=Cz,
from which it follows that
(3.20)h(z)=h(1)z,f(z)=f(1)z,
as desired.Remark 3.5.
It is expected that Theorem3.4 can be generalized for the case when all solutions of (3.2) are periodic with period more than three.In the case when all solutions of (3.2) are periodic
with period four, the functions f and g must satisfy
the following system of functional equations:
(3.21)x=h[h(h(y)⋅f(x))⋅f(y)]⋅f(h(y)f(x)),y=h(x)⋅f[h(h(y)⋅f(x))⋅f(y)].
From the system as in the proof of Theorem 3.4, it can be obtained that the functions h^ and f^ satisfy the following identities:
h^(h^(h^(z))f^(z)C−1)f^(h^(z))=C,h^(h^(f^(z)))f^(f^(z))=Cz,h^(z)f^(h^(f^(z)))=C,f^(h^(h^(z))f^(z)C−1)=z,
where C=f(1)h(1).An obvious solution of the system ish(x)=c1 and f(x)=c2/x, where c1,c2∈(0,∞). We leave the
problem of finding all solutions of the system as a further direction for
investigation for
interested readers.
Note. An
early draft by the second author of the paper, containing only results in Section 3, has circulated among the experts since the end of 2005 and was the starting point for further important investigations in the research field, see, for example, [2, 30]. The paper in the present form is a slight modification of a version from March 2006. A minor publication mishap caused some wrong citations (see [30]), as well as a delay in publishing of it.



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