Piecewise linear difference equations and convexity

G.F. Liddell*

Department of Mathematics, University of Otago, Dunedin, New Zealand

(Received 4 May 2010; final version received 26 August 2010)

This note presents a new method for analysing piecewise linear difference equations.The equations are considered in their natural phase space and interpreted via theirassociated semigroups and number theoretic graphs.

Keywords: piecewise-linear difference equation; convexity; asymptotic behaviour;periodic solution

AMS Subject Classification: 39A11; 52A07

1. Introduction

Piecewise linear generalizations of the well-known Collatz problem have been investigated

by a number of authors [1–6]. Proofs of convergence typically involve induction

arguments that are specific to each case considered. In this note, we present a geometric

method for establishing convergence, and provide a description of the bounds that arise in

geometric terms. As we will see, two aspects of the equations determine much of their

behaviour, namely, what we have called the mod n projection of the system considered as a

multiedge graph, and the semigroup generated by the linear transformations that arise from

the equation, in particular, the action of the semigroup on convex sets.

Our main result is actually a method. The method depends, however, on a general

theorem concerning semigroups kHl generated by sets H of real d £ d matrices

kHl ¼ {I} <[1j¼1

H j; ð1Þ

where H 0 ¼ {I} and H i ¼ {h1 + h2 + · · · + hi : hj [ H; 1 # j # i}, i $ 1. Before stating our

theorem, we briefly review some notation and definitions.

The topological interior of a set X in a vector space V is denoted by X +, the closure by �X,

the boundary by ›X, the convex hull by G(X) and the extreme points by ex(X). We call the

setF ðXÞ :¼ ›XnexðXÞ the face of X.X is a body if it contains a neighbourhood of the origin,

and we say that X is H-invariant if hðXÞ # X for all h [ H. Recall that linear combinations

of H-invariant sets are H-invariant, convex hulls of H-invariant sets are H-invariant and in

general HðGðXÞÞ # GðHðXÞÞ. Finally, we say that X is strongly connected under H if

X # >x[XkHlðxÞ:

We will prove the following.

ISSN 1023-6198 print/ISSN 1563-5120 online

q 2011 Taylor & Francis

DOI: 10.1080/10236198.2010.524214

http://www.informaworld.com

*Email: gliddell@maths.otago.ac.nz

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Theorem 1. Let H be a finite set of real non-singular d £ d matrices and suppose that H

has a non-trivial compact convex H-invariant body C. If X is a strongly connected set

under H, then there is a number k . 0 such that kX is contained in either exðCÞ or F ðCÞ.

The plan of the paper is as follows: first, we apply Theorem 1 in a setting in which the

analysis is particularly straightforward – the case of an equation for which the solution

happens to be known [1] – on the way introducing ideas on which the application of

Theorem 1 depends. We then prove Theorem 1. In Section 2, we establish a conjecture of

Feuer and Ladas [6] (Conjecture 5.1) when the parameter ( p in the present paper) is 4.

(The conjecture was made for p $ 3 and is known to be true for p ¼ 3 [5].) The conjecture

can also be proved using our methods when p ¼ 6; the proof is similar to the p ¼ 4 case,

however, and we omit the details. (In fact, though it is beyond the scope of the present

paper, there is a concept of ‘modularly bounded’ that characterizes those piecewise linear

difference equations to which our method applies, and covers all published proofs of

convergence for this class of equations. When p ¼ 5 or 7, our method is inconclusive: it

can be shown that the system is not modularly bounded. But then neither is any system

that is not modularly bounded amenable to methods that have been developed elsewhere.)

We conclude with some general remarks on Theorem 1.

2. Phase space, graph and semigroup: an example

Given a;b; g; d, all of which are ^1, consider the equation ([1], p. 196)

xnþ1 ¼ðaxn þ bxn21Þ=2 if axn þ bxn21 is even;

gxn þ dxn21 otherwise:

(ð2Þ

We denote by kxn the column vector

� xn21

xn

�and by xn the corresponding row vector, and

introduce piecewise linear maps A1 and A2 defined on their natural phase space Z2

A1 ¼0 1

b=2 a=2

!; A2 ¼

0 1

d g

!: ð3Þ

We may rewrite (2) as follows:

kxnþ1 :¼ Tkxn ¼A1kxn if axn þ bxn21 ;2 0;

A2kxn otherwise;

(ð4Þ

where ;2 denotes congruence mod 2. Each piece of T has a corresponding domain

D1 ¼ {ðx; yÞ [ Z2 : ayþ bx ;2 0}; D2 ¼ {ðx; yÞ [ Z2 : ayþ bx �2 0};

and function graph

Gi ¼ {ðkx;AikxÞ : kx [ Di}; i ¼ 1; 2:

Taking values mod 2 (which we indicate by a subscript: ;2), these induce relations

Ai ¼ {ðkx ;Aikx Þ;2[ Z2

2 £ Z22 : kx [ Di};

where Z2 is the group of integers mod 2.

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To determine A1 and A2, note first that since a and b are ^1, the condition ayþ

bx ;2 0 is equivalent to yþ x ;2 0, as can be easily checked. Thus, if yþ x ;2 0, we have

Tx

y

!¼ A1

x

y

!¼

y

ðayþ bxÞ=2

!:

Since ðayþ bxÞ=2 may be even or odd,

A1 ¼ ð0; 0Þ; ð0; 0Þ� �

; ð0; 0Þ; ð0; 1Þ� �

; ð1; 1Þ; ð1; 0Þ� �

; ð1; 1Þ; ð1; 1Þ� �� �

:

Similarly,

A2 ¼ ð1; 0Þ; ð0; 1Þ� �

; ð0; 1Þ; ð1; 1Þ� �� �

:

It is convenient to write this as follows:

ð0;0Þ!A1ð0;0Þ; ð0;0Þ!

A1ð0;1Þ; ð1;1Þ!

A1ð1;0Þ; ð1;1Þ!

A1ð1;1Þ; ð1;0Þ!

A2ð0;1Þ; ð0;1Þ!

A2ð1;1Þ:

These relations can be viewed as a multiedge graph, which we call the mod 2 graph of

the difference equation. Using e and o to denote ‘even’ and ‘odd’, respectively, it can be

visualized in the following diagram:

ðe; oÞ !A2

ðo; oÞ $A1

"A1c

A2 #A1

ðe; eÞ ðo; eÞ

lA1

Any solution sequence of the difference equation kxnþ1 ¼ Tkxn can be projected mod 2 to

give a path in the mod 2 graph. It is clear from the diagram that these trajectories are

eventually composed of the relations A1 and A† :¼ A22A1. Thus, the sequence of matrices

corresponding to a solution sequence is eventually composed of A1 and A† :¼ A22A1.

Let H ¼ {A1;A†}. From our analysis so far, any solution is eventually of the form

kxn [ kA1lkxm for all n . m, or else kxn [ kHlkxm for all n . m, for some m.

We look first at the case kxn [ kA1lkxm for all n . m, and consider two representative

subcases; the remaining cases may be approached similarly.

If a ¼ 1;b ¼ 21, then jsðA1Þj , 1, where s denotes the spectrum. Thus, kxn ! 0,

which is impossible under the injection A1 for integer values, starting from a non-zero

initial value.

If a ¼ b ¼ 1, then

A1 ¼0 1

1=2 1=2

!

has spectrum sðA1Þ ¼ {1;21=2} and the eigenvector for the eigenvalue l ¼ 1 is ku ¼� 1

1

�: Since we do not have kxn ! 0, we must have

kxn ¼ kxm ¼ x1

1

!; x [ Z:

In the remaining case, when kxn [ kHlkxm for all n . m, for some m, we certainly have

kxnþ1 [ HðkxnÞ for all n . m. To proceed, we look in detail at one illustrative subcase, when

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a ¼ 1, b ¼ 1, g ¼ 1 and d ¼ 21. This is the equation of [2] and equation (2) of [1]. It can

be verified that for these a;b; g and d,

H ¼ {A1;A†} ¼

0 1

1=2 1=2

!1=2 21=2

0 21

!8<:

9=;;

and it is trivial to show that the closed square D :¼ {ðx; yÞ : jxj; jyj # 1} is H-invariant. It

follows that kHlðkxnÞ # D> Z2 for all n $ m, and this is a finite set. Thus, eventually the kxnare a cycle under H, which we relabel X ¼ {kx0; kx1; . . . ; kxp21}.

Evidently, X is strongly connected under H. It follows from Theorem 1.1 that for some

k . 0, kX is a cycle on either ex(D) or F ðDÞ. Now, the only points of exðDÞ that map to

points of exðDÞ are

ð1; 1Þ!A1

ð1; 1Þ; 2ð1; 1Þ!A1

2 ð1; 1Þ; ð21; 1Þ!A†

ð21;21Þ; ð1;21Þ!A†

ð1; 1Þ;

and thus the only cycles in the extreme points are those of A1 on its eigenvectors ^ð1; 1Þ.

These correspond to the fixed points kxn21 ¼ kxn and are odd multiples of the fixed points

^1 of [2].

The only faces of F ðDÞ that map into F ðDÞ are

Gð{ð21; 1Þ; ð1; 1Þ}Þ !A1

Gð{ð1;21Þ; ð1; 1Þ}Þ

lA†

Gð{ð21; 1Þ; ð1; 1Þ}Þ !A1

Gð{ð21; 1Þ; ð21;21Þ}Þ:

So, the only facial cycles come from the map

A† : Gð{ð21; 1Þ; ð1; 1Þ}Þ! Gð{ð21;21Þ; ð1;21Þ}Þ:

The fixed points are kx ¼

� x

1

�such that A2

†kx ¼ kx. This gives x ¼ 1=3, and the cycle of

vectors is

1

3

!!A1

3

2

!!A2

2

21

!!A2

21

23

!!A1

23

22

!!A2

22

1

!!A2

1

3

!:

This is the 6-cycle {22; 1; 3; 2;21;23} of Theorem 3.3 of [1].

2.1 Induction hypothesis

We conclude with a brief comment on another subcase, when a ¼ 1, b ¼ 1, g ¼ 21,

d ¼ 21, which is the subject of Lemma 2.4 of [1]. Here, we have

H ¼ {A1;A†} ¼

0 1

1=2 1=2

!21=4 23=4

21=8 21=8

!8<:

9=;:

We could proceed as in the case we have just analysed, but instead make the following

remark. In their proof of Lemma 2.4, the authors use an induction hypothesis in terms of

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k ¼ kx0k1

xn ;2 ð1; 0Þ ) jxnj # k;

xn ;2 ð0; 1Þ ) jxnj # 2k and jxn21 þ xnj # k;

xn ;2 ð1; 1Þ ) jxnj # k and jxn21 þ xnj # 2k:

If we apply the linear maps to the region

C ¼ {ðxn21; xnÞ : jxnj # 1 and jxn21 þ xnj # 2};

for the ðo; oÞ quadrant, we obtain the regions A1ðCÞ, A2A1ðCÞ and A†ðCÞ shown in their

appropriate mod 2 quadrants (with thick black outlines) in Figure 1.

Also, shown with dashed grey lines are the constraint regions of the induction. In the

ðo; eÞ quadrant, the condition jxnj # 1 does not constrain the xn21 values, and this is

indicated by the jagged line. In the ðe; oÞ quadrant, the induction condition is too large.

As a result of these choices, the proof in [1] needs to use course-of-values induction and

backtrack to bound values of xn22.

This shows that the natural induction hypotheses for a simple inductive proof are

xn ;2 ð1; 0Þ ) kxnk1 # k;

xn ;2 ð0; 1Þ ) jxn21j # k and jxn21 þ xnj # k;

xn ;2 ð1; 1Þ ) jxnj # k and jxn21 þ xnj # 2k:

The method introduced in this paper is equivalent to using the simpler hypothesis

xn ;2 ð1; 1Þ ) jxnj # k and jxn21 þ xnj # k

and just a one-step induction argument to show that this region is invariant under both A1

and A†.

To summarize, the method we have developed is as follows:

. form the mod n graph for the difference equation;

. construct the corresponding generators H;

. find an H-invariant compact convex body C;

1

1

1

1

1

1

1

1

A2

A2

A1

(even, even) (odd, even)

(even, odd) (odd, odd)

C

A2 A1(C )

A1

A1(C )

A•(C )

Figure 1. Mod 2 regions for a ¼ 1, b ¼ 1, g ¼ 2 1, d ¼ 2 1.

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. find all the cycles in the extreme points or the faces of C;

. check that the cycles occur as solutions of the difference equation.

3. Proof of Theorem 1

First, we prove the following.

Lemma 1. Let H be a set of real, non-singular d £ d matrices and suppose that H has a

non-trivial compact convex H-invariant body C. Then, CnexðCÞ and C + are H-invariant.

Proof. If x [ CnexðCÞ then, for some l [ ð0; 1Þ and a; b [ C, with a – b, we have

x ¼ ð1 2 lÞaþ lb. Thus for any A [ H, AðxÞ ¼ ð1 2 lÞAðaÞ þ lAðbÞ [ CnexðCÞ since,

because A is an injection, AðaÞ – AðbÞ.

Since A is injective, AðC +Þ is open, and since C is H-invariant, AðC +Þ # C. Thus,

AðC +Þ # C +. A

Let us note in passing that from Lemma 2,

HðF ðCÞÞ # F ðCÞ< C +: ð5Þ

Theorem 1 follows from Lemma 2 and the following lemma.

Lemma 2. Let H be a set of real, non-singular d £ d matrices and suppose that H has a

non-trivial compact convex H-invariant body C. For any finite set X that is strongly

connected under H, there exists a positive number k such that either kX # exðCÞ or

kX # F ðCÞ.

Proof. Choose k . 0 such that kX # C and kX > ›C – Y. This is possible since C is a

body and, if k is the supremum of all numbers such that kX # C, then for some x [ X we

have ðk þ eÞx � C for certain arbitrarily small e . 0, and therefore kx [ �C. Thus,

kx [ ›C.

Write Y ¼ kX and let y [ Y > ›C. We show first that Y # ›C. Suppose on the

contrary that there exists some z [ Y > C +. From the strong connectivity of X, we have

y [ <1i¼1H

iðzÞ, and since H iðzÞ # C + for each i [ N, by Lemma 2, it follows that y [ C +,

a contradiction.

Suppose that Y > F ðCÞ – Y and let z [ Y > F ðCÞ # CnexðCÞ. By strong

connectivity,

Y # <1i¼1H

iðzÞ # <1i¼1H

iðCnexðCÞÞ # CnexðCÞ;

using the H-invariance of CnexðCÞ. Hence, Y # F ðCÞ.

On the other hand, if Y > exðCÞ – Y, then from what we have done, Y > F ðCÞ ¼ Yand thus Y # exðCÞ. A

4. An application

Consider equation (2) of [5]

xnþ1 ¼ðxn 2 xn21Þ=p if p divides xn 2 xn21;

2xn 2 xn21 otherwise;

(ð6Þ

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where p [ N, p $ 2. This can be rewritten as

kxnþ1 :¼ Tkxn ¼A1kxn if p divides xn þ xn21;

A2kxn otherwise;

(ð7Þ

where

A1 ¼0 1

21=p 1=p

!; A2 ¼

0 1

21 21

!:

When p ¼ 4, the mod 4 graph is as shown in Figure 2.

Here, the vertices ði; jÞ;4 are labelled 1; 2; . . . ; 16; the edges for A1 are shown with

solid arrows, those for A2 with dotted arrows and reflexive edges ði; iÞ!A1

ði; iÞ are shown as

circles.

It is clear from Figure 2 that there are five strongly connected components

ð3; 0Þ!A2

ð0; 1Þ!A2

ð1; 3Þ!A2

ð3; 0Þ; ð2; 2Þ!A1

ð2; 0Þ!A2

ð0; 2Þ!A2

ð2; 2Þ;

ð1; 0Þ!A2

ð0; 3Þ!A2

ð3; 1Þ!A2

ð1; 0Þ; ð1; 1Þ!A1

ð1; 2Þ!A2

ð2; 1Þ!A1

ð1; 1Þ;

ð3; 2Þ!A1

ð3; 1Þ!A2

ð2; 2Þ!A2

ð3; 2Þ:

Thus, any solutions are eventually cycles of A1 or of A2 or are generated by the

semigroup {A1;A†}, where

A† ¼ A22A1 ¼

1=4 25=4

0 1

!:

The set C ¼ G {ð1; 2=3Þ; ð25=3; 1Þ ð21;22=3; ð5=3;21Þ}� �

is an invariant, shown in

Figure 3 below, together with its images under A1 (the solid parallelogram) and A† (the

dashed parallelogram); the arrows indicate which vertex goes to which place.

Clearly, the only cycle is that around the vertex (25/3,1), which is an eigenvector

of A†. Restricted to Z2, the possibilities are {mð5;23Þ : m [ Z}. Since these are cycles

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Figure 2. Mod 4 graph for equation (6).

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from the mod 4 vertices ð1; 1Þ, ð2; 2Þ or ð3; 3Þ, we have mð5;23Þ ;4 ðk; kÞ, k ¼ 1; 2; 3.

Hence, mð1; 1Þ ;4 ðk; kÞ. But we must also check the intermediate points under the factors

of the product A†.

The solution sequence from (1,1) is ð1; 1Þ!A1

ð1; 0Þ!A1

ð0;21Þ!A1

ð21; 1Þ!A1

ð1; 0Þ, so it

falls into an A2 cycle. The intermediate conditions require mð5;23Þ ;4 ðk; kÞ, k ¼ 1; 2; 3

and mð21; 1ÞA1

� 1

1

��4 0 which reduce to m �4 0 for m [ Z. Considering (5,–3) itself,

we have ð5;23Þ ;4 ð1; 1Þ and

A1

5

23

!¼

23

22

!

and ð21; 1Þð23;22Þ �4 0, so ð23;22Þ!A2

ð22; 5Þ. Now, ð21; 1Þð22; 5Þ �4 0 so

ð22; 5Þ!A2ð5;23Þ and we have the three cycles {5;23;22} and all cycles that are

integer multiples m of this cycle, where m �4 0.

The other possible solutions are A2 3-cycles on the mod 4 points ð3; 0Þ, ð0; 1Þ, ð1; 3Þ and

ð1; 0Þ, ð3; 1Þ, ð0; 3Þ, giving cycles {0; 1; 3} and {0; 3; 1} and all cycles that are mod 4

equivalent to these. For example, {13; 3;216} is one such.

5. Concluding remarks

5.1 The existence of compact convex H-invariant bodies

If H has bounded action and C is a compact body, then kHlðCÞ is bounded and GðkHlðCÞÞ is

an H-invariant compact convex body. Under certain circumstances, H-invariant bodies

can be constructed from smaller sets.

For example, suppose that H has bounded action and has no non-trivial invariant

subspaces. If C is a non-trivial compact set such that 2C # C, then GðkHlðCÞÞ is an

H-invariant compact convex body.

To see this, note that since C is compact, its orbit kHlðCÞ is bounded, and therefore

GðkHlðCÞÞ ¼ GðkHlðCÞÞ is compact. Furthermore, the set RGðkHlðCÞÞ is an H-invariant

subspace containing C, and so, since C – {0}, RGkHlðCÞ ¼ Rd. There are thus d linearly

independent vectors X ¼ {x1; . . . ; xd} in GðkHlðCÞÞ, and the ball GðX <2XÞ is a

neighbourhood of zero contained in GðkHlðCÞÞ. A

–1.5 –1.0 –0.5 0.5 1.0 1.5

–1.0

–0.5

0.5

1.0

Figure 3. Action of A1 and A† on an invariant set for equation (6).

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5.2 Strictly contractive H

We will show that if H is not too ‘contractive’, then the boundary has non-trivial

H-invariant sets. If the linear parts of a piecewise linear difference equation are

‘contractive’, then all solutions collapse to zero and are of little interest. If any of the maps

has a unit eigenvector in its domain, then H ¼ {A1;A2} is not ‘contractive’, and in this

situation any cycle will be captured in exðCÞ or F ðCÞ for any compact convex body C.

Here are the definitions we will use in our discussion.

We call H strictly contractive if kHk , 1, where

kHk ¼ sup{kAk : A [ H};

and eventually strictly contractive if there exists i [ N such that H i is strictly contractive.

For an eventually strictly contractive H, all orbits are asymptotic to the origin. Note that if

C is a compact body and HðCÞ # uC for some u , 1, then kH iðCÞk , u ikCk, so that H is

eventually strictly contractive.

More generally, if H is finite and there exists a compact neighbourhood of the origin C

such that HðCÞ , C +, then H is eventually strictly contractive.

Suppose that x [ C +. Since there is an open set U such that x [ U # C +, we can find

r [ Q, r . 1, such that rx [ C +, and thus x [ r21C + # r21C. It follows that

C + # <u[Q;0,u,1uC. Now, HðCÞ is a compact subset of C + and so has a finite subcover

<uj[Q;0,uj,1ujC. There is thus a u [ Q> ð0; 1Þ such that HðCÞ # uC. Now, HnðCÞ #unC and eventually kunCk , 1, and therefore H is eventually strictly contractive. A

It follows from what we have shown, on replacing H by H i, that if H is not eventually

strictly contractive, and C is an H-invariant compact convex body, then for all i [ N,

H iðCÞ> ›C – Y: ð8Þ

5.3 Non-eventually strictly contractive H

We establish the existence of sets in ›C on which H has a more ‘cyclic’ action. This has

not been used in the examples we have dealt with, because the action of H can be seen

explicitly in those cases. It does, however, confirm that in general the restriction of the

action of H to ›C is non-trivial.

If H is a set of linear injections that is not eventually strictly contractive, and C is an

H-invariant compact convex body, then there exists an H-invariant subset K – Y of the

boundary such that K ¼ HðKÞ> ›C.

Let Xi ¼ H iðCÞ, i $ 0, so that Xi is a decreasing sequence of compact H-invariant sets

with non-empty compact intersection X ¼ >i$0Xi.

We show first that HðXÞ ¼ X. We have

HðXÞ ¼ Hð>i$0XiÞ # >i$0HðXiÞ # >i$1Xi ¼ X:

Conversely, if x [ X then x [ Xiþ1 ¼ HðXiÞ for all i, and thus we can find Ai [ H and

xi [ Xi such that AiðxiÞ ¼ x, for all i $ 1. Since H, being finite, is compact, and C is also

compact, there is a subsequence ij and there are A [ H and y [ C such that Aij ! A and

xij ! y. For any fixed n, we have xij [ Xn for all large j, and thus y [ Xn since Xn is

compact. Thus, y [ X and x ¼ AðyÞ [ HðXÞ, as required.

Let K ¼ X > ›C. Note that K – Y, from (8). Now,

HðKÞ # HðXÞ> Hð›CÞ # HðXÞ ¼ X;

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so

HðKÞ> ›C # X > ›C ¼ K: ð9Þ

On the other hand, since X # C,

HðXÞ ¼ HðX > ð›C < C +ÞÞ ¼ HðX > ›CÞ< HðX > C +Þ # HðKÞ< C +;

using Lemma 2, and thus

K # HðKÞ< C +� �

> ›C ¼ HðKÞ> ›C: ð10Þ

The result follows from (9) and (10).

Acknowledgement

The author acknowledges helpful conversations with Peter Fenton during the preparation of the paper.

References

[1] A. Amleh, E. Grove, C. Kent, and G. Ladas, On some difference equations with eventuallyperiodic solutions, J. Math. Anal. Appl. 223 (1998), pp. 196–215.

[2] D. Clark and J. Lewis, A Collatz-type difference equation, Congr. Numer. 111 (1995),pp. 129–135.

[3] J. Feuer, Two classes of piecewise-linear difference equations with eventual periodicity six,J. Math. Anal. Appl. 295 (2004), pp. 570–575.

[4] J. Feuer, Some equations with a periodic parameter and eventually periodic solutions, J. Differ.Equ. Appl. 13 (2007a), pp. 1005–1010.

[5] J. Feuer, Two classes of piecewise-linear difference equations with eventual periodicity three,J. Math. Anal. Appl. 332 (2007b), pp. 564–569.

[6] J. Feuer and G. Ladas, Open problems and conjectures, J. Differ. Equ. Appl. 10 (2004),pp. 447–451.

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