Camouzis e ladas g dynamics of third order rational difference equations with open problems and conjectures

Dynamics ofThird-Order Rational Difference Equations with Open Problems and Conjectures... nonneg-We are primarily concerned with the boundedness nature of solutions, thestability of the

Dynamics of

Third-Order

Rational Difference Equations with

Open Problems

and Conjectures

Analysis and Modelling of Discrete Dynamical Systems

Edited by Daniel Benest and Claude Froeschlé

Volume 2

Stability and Stable Oscillations in Discrete Time Systems

Aristide Halanay and Vladimir Rasvan

Volume 3

Partial Difference Equations

Sui Sun Cheng

Volume 4

Periodicities in Nonlinear Difference Equations

E A Grove and G Ladas

Dynamics of

Third-Order

Rational Difference Equations with

Open Problems

and Conjectures

Elias Camouzis Gerasimos Ladas

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Library of Congress Cataloging‑in‑Publication Data

Camouzis, Elias.

Dynamics of third‑order rational difference equations with open problems and conjectures / Elias Camouzis and G Ladas.

p cm ‑‑ (Discrete mathematics and applications ; 48)

Includes bibliographical references and index.

ISBN 978‑1‑58488‑765‑2 (alk paper)

1 Difference equations‑‑Numerical solutions I Ladas, G E II Title III

DYNAMICS OF THIRD-ORDER RATIONAL

Lina and Mary

Preface

Acknowledgments

1.0 Introduction 3

1.1 Definitions of Stability 3

1.2 Linearized Stability Analysis 4

1.3 Semicycle Analysis 6

1.4 A Comparison Result 7

1.5 Full Limiting Sequences 8

1.6 Convergence Theorems 9

2 Equations with Bounded Solutions 29 2.0 Introduction 29

2.1 Some Straightforward Cases 32

2.2 The Second-Order Rational Equation 37

2.3 Boundedness by Iteration 41

2.4 Boundedness of the Special Case #58 46

2.5 Boundedness of x n+1= α + βxn + x n−2 A + xn 48

2.6 Boundedness of the Special Case #63 54

2.7 Boundedness of x n+1= α + βx n + γx n−1 + x n−2 A + xn−1 57

2.8 Boundedness of x n+1= α + βxn + x n−1 xn−1 + Dx n−2 70

2.9 Boundedness of x n+1= α + βxn + x n−2 Cxn−1 + x n−2 72

3 Existence of Unbounded Solutions 75 3.0 Introduction 75 3.1 Unbounded Solutions of x n+1=α + βxn + γx n−1 + δx n−2

3.2 Unbounded Solutions of x n+1=α + βxn + γx n−1 + δx n−2

4 Periodic Trichotomies 1054.0 Introduction 1054.1 Existence of Prime Period-Two Solutions 1064.2 Period-Two Trichotomies of Eq.(4.0.1) 1104.3 Period-Two Trichotomy of x n+1=α + βxn + γx n−1 + δx n−2

4.4 Period-Three Trichotomy of x n+1= δxn−2

A + Bxn + Cx n−1 1274.5 Period-Four Trichotomy of x n+1= α + βxn + δx n−2

n+1 A

Bxn + Dx n−2 237

n+1 Cxn−1 + Dx n−2

nonneg-We are primarily concerned with the boundedness nature of solutions, thestability of the equilibrium points, the periodic character of the equation,and with convergence to periodic solutions including periodic trichotomies.However, our ultimate goal should be to extend and generalize the results ofrational equations to equations

xn+1 = f (x n, , xn−k ), n = 0, 1,

of the most general pattern

For Eq.(1) and for each of its 225 special cases, we present the known sults and/or derive some new ones We also pose a large number of thought-provoking open problems and conjectures on the boundedness character, theglobal stability, and the periodic behavior of solutions of various special cases

re-of Eq.(1) The open problems are quite challenging and the conjecturesare based on numerous computer observations and analytic investigations

We believe that research work on these open problems and conjectures is ofparamount importance for the development of the basic theory of the globalbehavior of solutions of nonlinear difference equations of order greater thanone

The large number of open problems and conjectures in rational differenceequations will be a great source of attraction for research investigators in thisdynamic area where, at the beginning of the third millennium, we know sosurprisingly little

The methods and techniques that we develop to understand the dynamics

of rational difference equations and the theory we obtain will be useful inanalyzing the equations in any mathematical model that involves differenceequations

throughout the monograph.

Chapter 2 deals with the special cases of Eq.(1) that have bounded solutionsonly and Chapter 3 deals with the remaining cases, where the equations haveunbounded solutions in some range of their parameters

Chapter 4 is about the seven nonlinear known periodic trichotomies of order rational difference equations

third-Chapter 5 presents the known results on each of the 225 special cases ofEq.(1) This chapter is the reason we wrote this book The four precedingchapters present general results needed in order to discuss the character ofeach equation and how it relates to the other special cases

Appendix A at the end of the book presents at a glance the boundednesscharacter of each of the 225 special cases of Eq.(1) and gives important resultsand references related to each special case

Appendix B contains information on the boundedness character for allfourth-order rational difference equations The large number of conjectureslisted in Appendix B on the boundedness character of fourth-order rationaldifference equations will help give new directions for future investigations inthis fascinating area

This book is the outgrowth of lectures and seminars given at the University

of Rhode Island during the last 10 years and is based on an extensive oration of the authors by e-mail during the last five academic years and inperson during the last five beautiful summers in Greece We are grateful toProfessors E A Grove, J Hoag, W Kosmala, M R S Kulenovi´c, O Merino,

collab-S Schultz, and W collab-S Sizer, who have participated in class discussions or nar presentations at the University of Rhode Island and to former and presentURI students A M Amleh, S Basu, M R Bellavia, A M Brett, W Briden,

semi-E Chatterjee, C A Clark, R C DeVault, H A El-Metwalli, J Feuer, C.Gibbons, E Janowski, C M Kent, Y Kostrov, Z A Kudlak, S Kuruklis,

L C McGrath, C Overdeep, F Palladino, M Predescu, N Prokup, E P.Quinn, M Radin, I W Rodrigues, C T Teixeira, S Valicenti, P Vlahos,and G Tzanetopoulos for their contributions to research in this area and fortheir enthusiastic participation in the development of this subject A partic-ular debt of gratitude is due to Dr Amal Amleh, who read with great carethe entire manuscript and its many revisions and whose numerous correctionsand criticisms have improved this book

We are primarily concerned with the boundedness nature of solutions, thestability of the equilibrium points, the periodic character of the equation, andwith convergence to periodic solutions including periodic trichotomies.

If we allow one or more of the parameters in Eq.(0.0.1) to be zero, then wecan see that Eq.(0.0.1) contains

in this dynamic area of research

Out of the 225 special cases of Eq.(0.0.1), 39 cases are about equations thatare linear or reducible to linear or Riccati difference equations, or equationsreducible to Riccati See Appendix A

1

Another 28 equations were investigated in the Kulenovic/Ladas book [175],which deals with the second-order rational difference equation

The methods and techniques we develop to understand the dynamics ofvarious special cases of rational difference equations and the theory that weobtain will also be useful in analyzing the equation in any mathematical modelthat involves difference equations

The results from Theorem 1.6.7 to the end of this chapter were recentlyobtained by the authors while working on various special cases of rationaldifference equations and provide useful generalizations and some unifications

in some special cases

1.1 Definitions of Stability

A difference equation of order (k + 1) is an equation of the form

xn+1 = F (x n, xn−1, , xn−k ), n = 0, 1, (1.1.1)

where F is a function that maps some set I k+1 into I The set I is usually an

interval of real numbers, or a union of intervals, or a discrete set such as the

set of integers Z = { , −1, 0, 1, }.

A solution of Eq.(1.1.1) is a sequence {x n} ∞

n=−kthat satisfies Eq.(1.1.1) for

all n ≥ 0.

A solution of Eq.(1.1.1) that is constant for all n ≥ −k is called an

equilib-rium solution of Eq.(1.1.1) If

xn= ¯x, for all n ≥ −k

is an equilibrium solution of Eq.(1.1.1), then ¯x is called an equilibrium point,

or simply an equilibrium of Eq.(1.1.1).

DEFINITION 1.1 (Stability)

3

(i) An equilibrium point ¯ x of Eq.(1.1.1) is called locally stable if, for every

ε > 0, there exists δ > 0 such that if {xn} ∞

n=−k is a solution of Eq.(1.1.1) with

| x−k − ¯ x | + | x 1−k − ¯ x | + · · · + | x0− ¯ x | < δ, then

| x n − ¯ x | < ε, for all n ≥ 0.

(ii) An equilibrium point ¯ x of Eq.(1.1.1) is called locally asymptotically

stable if, ¯ x is locally stable, and if in addition there exists γ > 0 such that if {xn } ∞

n=−k is a solution of Eq.(1.1.1) with

| x−k − ¯ x | + | x−k+1 − ¯ x | + · · · + | x0− ¯ x | < γ, then

(iv) An equilibrium point ¯ x of Eq.(1.1.1) is called globally asymptotically

stable if ¯ x is locally stable, and ¯ x is also a global attractor of Eq.(1.1.1) (v) An equilibrium point ¯ x of Eq.(1.1.1) is called unstable if ¯ x is not locally stable.

1.2 Linearized Stability Analysis

Suppose that the function F is continuously differentiable in some open

neigh-borhood of an equilibrium point ¯x Let

qi= ∂F

∂ui(¯x, ¯ x, , ¯ x), for i = 0, 1, , k

denote the partial derivative of F (u0, u1, , uk ) with respect to u ievaluated

at the equilibrium point ¯x of Eq.(1.1.1) Then the equation

yn+1 = q0yn + q1yn−1 + · · · + q kyn−k, n = 0, 1, (1.2.1)

is called the linearized equation of Eq.(1.1.1) about the equilibrium point ¯ x,

and the equation

λ k+1 − q0λ k − · · · − qk−1λ − qk= 0 (1.2.2)

is called the characteristic equation of Eq.(1.2.1) about ¯ x.

The following result, known as the Linearized Stability Theorem, is very

useful in determining the local stability character of the equilibrium point ¯x

of Eq.(1.1.1) See [13], [95], [131], and [202]

Theorem 1.2.1 (The Linearized Stability Theorem)

Assume that the function F is a continuously differentiable function defined

on some open neighborhood of an equilibrium point ¯ x Then the following statements are true:

1 When all the roots of Eq.(1.2.2) have absolute value less than one, then the equilibrium point ¯ x of Eq.(1.1.1) is locally asymptotically stable.

2 If at least one root of Eq.(1.2.2) has absolute value greater than one, then the equilibrium point ¯ x of Eq.(1.1.1) is unstable.

The equilibrium point ¯x of Eq.(1.1.1) is called hyperbolic if no root of

Eq.(1.2.2) has absolute value equal to one If there exists a root of Eq.(1.2.2)with absolute value equal to one, then the equilibrium ¯x is called nonhyper- bolic.

An equilibrium point ¯x of Eq.(1.1.1) is called a saddle point if it is hyperbolic

and if there exists a root of Eq.(1.2.2) with absolute value less than one andanother root of Eq.(1.2.2) with absolute value greater than one

An equilibrium point ¯x of Eq.(1.1.1) is called a repeller if all roots of

Eq.(1.2.2) have absolute value greater than one

A solution {x n} ∞

n=−k of Eq.(1.1.1) is called periodic with period p if there exists an integer p ≥ 1 such that

xn+p = x n, for all n ≥ −k. (1.2.3)

A solution is called periodic with prime period p if p is the smallest positive

integer for which Eq.(1.2.3) holds

The following three theorems state necessary and sufficient conditions forall the roots of a real polynomial of degree two, three, or four, respectively, tohave modulus less than one For every equation of order two, three, or fourthat we investigate in this book we have to use one of these three theorems

to determine the local asymptotic stability of the equilibrium points of theequation

Theorem 1.2.2 Assume that a1 and a0are real numbers Then a necessary and sufficient condition for all roots of the equation

n=−k of Eq.(1.1.1) is called nonoscillatory about ¯ x, or

sim-ply nonoscillatory, if there exists N ≥ −k such that either

xn ≥ ¯ x, for all n ≥ N

or

xn < ¯ x, for all n ≥ N.

Otherwise, the solution {x n } ∞

n=−k is called oscillatory about ¯ x, or simply oscillatory.

1.4 A Comparison Result

The following comparison result is a very useful tool in establishing boundsfor solutions of nonlinear equations in terms of the solutions of equations withknown behavior, for example, linear or Riccati

Theorem 1.4.1 Let I be an interval of real numbers, let k be a positive

in-teger, and let

F : I k+1 → I

be a function increasing in all of its arguments Assume that {xn} ∞

n=−k , {yn} ∞

xn ≤ yn ≤ zn, f or all − k ≤ n ≤ 0.

Then

xn ≤ yn ≤ zn, f or all n > 0 (1.4.1)PROOF Clearly,

and (1.4.1) follows by induction

1.5 Full Limiting Sequences

The following result about full limiting sequences sometimes is useful in tablishing that all solutions of a given difference equation converge to theequilibrium of the equation See [101], [144], [145], and [208]

es-Theorem 1.5.1 Consider the difference equation

n→∞ xn, and suppose that I, S ∈ J Let L0 be a limit point of the solution {x n } ∞

n=−k Then the following statements are true:

1 There exists a solution {Ln } ∞

n=−∞ of Eq.(1.5.1), called a full limiting sequence of {x n } ∞

n=−k , such that L0= L0, and such that for every N ∈ { , −1, 0, 1, }, LN is a limit point of {xn} ∞

n=−k In particular,

I ≤ LN ≤ S, for all N ∈ { , −1, 0, 1, }.

2 For every i0∈ { , −1, 0, 1, }, there exists a subsequence {xr i } ∞

i=0 of {xn } ∞

n=−k such that

LN = lim

i→∞ xr i +N , for every N ≥ i0.

1 F is increasing in each of its arguments.

2 F (z1, , zk+1 ) is strictly increasing in each of the arguments z i1, zi2, , zi l , where 1 ≤ i1< i2< < il ≤ k + 1, and the arguments i1, i2, , il are relatively prime.

3 Every point c in I is an equilibrium point of Eq.(1.1.1).

Then every solution of Eq.(1.1.1) has a finite limit.

The following convergence result is due to Hautus and Bolis See [132] andTheorem 2.6.2 in [157, p 53]

Theorem 1.6.2 Let I be an open interval of real numbers, let F ∈ C(I k+1 , I), and let ¯ x ∈ I be an equilibrium point of the Eq.(1.1.1) Assume that F satisfies the following two conditions:

1 F is increasing in each of its arguments.

2 F satisfies the negative feedback property:

(u − ¯ x)[F (u, u, , u) − u] < 0, for all u ∈ I − {¯ x}.

Then the equilibrium point ¯ x is a global attractor of all solutions of Eq.(1.1.1).

The next two global attractivity results were motivated by second-orderrational equations and have several applications

Theorem 1.6.3 [157, p 27] Assume that the following conditions hold:

(i) f ∈ C[(0, ∞) × (0, ∞), (0, ∞)].

(ii) f (x, y) is decreasing in x and strictly decreasing in y.

(iii) xf (x, x) is strictly increasing in x.

(iv) The equation

xn+1 = x nf (xn, xn−1 ), n = 0, 1, (1.6.1)

has a unique positive equilibrium ¯ x.

Then ¯ x is a global attractor of all positive solutions of Eq.(1.6.1).

Theorem 1.6.4 [106] Assume that the following conditions hold:

has a unique positive equilibrium ¯ x.

Then ¯ x is a global attractor of all positive solutions.

The following global attractivity result from [175] is very useful in lishing convergence results in many situations

estab-Theorem 1.6.5 Let [a, b] be a closed and bounded interval of real numbers and let F ∈ C([a, b] k+1 , [a, b]) satisfy the following conditions:

1 The function F (z1, , zk+1 ) is monotonic in each of its arguments.

2 For each m, M ∈ [a, b] and for each i ∈ {1, , k + 1}, we define

Mi (m, M ) =

(

M, if F is increasing in zi

m, if F is decreasing in zi and

The following period-two convergence result of Camouzis and Ladas wasmotivated by several period-two convergence results in rational equations.(See Chapters 4 and 5.) Thanks to this result, several open problems andconjectures posed in the Kulenovic and Ladas book have now been resolvedand the character of solutions of many rational equations has now been clar-ified See Theorems 4.2.2, 4.3.1, 5.74.2, 5.86.1, 5.109.1, 5.145.2.

Theorem 1.6.6 [61] Let I be a set of real numbers and let

(i) They are both monotonically increasing.

(ii) They are both monotonically decreasing.

(iii) Eventually, one of them is monotonically increasing and the other is monotonically decreasing.

x 2N +2 ≤ x 2N and x 2N +3 ≥ x 2N +1 (1.6.4)Assume that (1.6.3) holds The case where (1.6.4) holds is similar and will

be omitted Then

x 2N +4 = F (x 2N +3 , x 2N +2 ) ≥ F (x 2N +1 , x 2N ) = x 2N +2

and

x 2N +5 = F (x 2N +4 , x 2N +3 ) ≤ F (x 2N +2 , x 2N +1 ) = x 2N +3

and the result follows by induction

The results in the remainder of this chapter were recently obtained by theauthors while working on various special cases of rational difference equationsand provide useful generalizations and some unifications in some special cases

In order to simplify and unify several convergence results for the differenceequation

2) : For each m ∈ (0, ∞) and M > m, we assume that

implies

f (m1, , mk) > m. (1.6.9)(H3) : For each m ∈ [0, ∞) and M > m, we assume that

either

(f (M1, , Mk) − M)(f (m1, , mk) − m) > 0 (1.6.10)or

f (M1, , Mk) − M = f (m1, , mk) − m = 0. (1.6.11)

3) : For each m ∈ (0, ∞) and M > m, we assume that

either

(f (M1, , Mk) − M)(f (m1, , mk) − m) > 0 (1.6.12)or

f (M1, , Mk) − M = f (m1, , mk) − m = 0. (1.6.13)

We also define the following sets:

S = {is∈ {i1, , ik} : f strictly increases in x n−is} = {is 1, , is r}

(H4) : The set S consists of even indices only and the set J consists

of odd indices only

(H5) : Either the set S contains at least one odd index, or the set Jcontains at least one even index

(H6) : The greatest common divisor of the indices in the union ofthe sets S and J is equal to 1

The next few theorems can be used to establish global attractivity andperiod-two convergence results in many special cases of rational equationsincluding the following:

See Chapters 4 and 5

Theorem 1.6.7 The following statements are true:

(a) Assume that (H1) and (H2) hold for the function f (z1, , zk ) of Eq.(1.6.5).

Then every solution of Eq.(1.6.5) which is bounded from above converges to a finite limit.

(a 0 ) Assume that (H ∗ ) and (H 0

2) hold for the function f (z1, , zk ) of Eq.(1.6.5).

Then every solution of Eq.(1.6.5) which is bounded from above and from below

by positive constants converges to a finite limit.

PROOF (a) Let {x n} be a bounded solution of Eq.(1.6.5) Set

I = lim inf n→∞ xn and S = lim sup

n→∞ xn

and assume, for the sake of contradiction, that

S > I.

Clearly, there exists a sequence of indices {n m} and positive numbers L−r,

for r ∈ {1, , k}, such that

S = lim m→∞ x n m and L −r= lim

m→∞ x n m −i r

From Eq.(1.6.5) and the monotonic character of f we see that

S = f (L−1, , L−k ) ≤ f (M1(I, S), , M k (I, S)). (1.6.14)Similarly, we see that

I ≥ f (m1(I, S), , m k (I, S)). (1.6.15)

But from (1.6.14) and the Hypothesis (H2) we see that

f (m1(I, S), , m k (I, S)) − I > 0,

which contradicts (1.6.15) The proof is complete in this case

(a 0 ) The proof in this case is similar to the proof in part (a) and will be

3) are satisfied for the arguments shown in the

equation and, furthermore, assume that the function f is:

strictly increasing in xn or xn−2, or strictly decreasing in xn−1 Then every solution of this equation bounded from below and from above by positive constants converges to a finite limit.

PROOF Let {x n} be a solution bounded from above and from below by

positive constants Set

I = lim inf n→∞ x n and S = lim sup

n→∞ x n

Clearly, there exists a sequence of indices {n i} and positive numbers L−j, for

j ∈ {0, 1, }, such that

S = lim i→∞ x n i+1 and L −j = lim

i→∞ x n i −j

First we will consider Eq.(1.6.16) and give the proof when the function f (z1, z2, z3)

is strictly increasing in z3 The proof when the function f (z1, z2, z3) is strictly

decreasing in z2, or when the function f (z1, z2, z3) is strictly increasing in z1,

is similar and will be omitted

Case 1: The function f (z1, z2, z3) is strictly increasing in each argument.Actually in this case we can show that the Hypotheses of Theorem 1.6.1are satisfied from which the result follows However, we give the details of theproof for completeness and practice

From Eq.(1.6.16) and the monotonic character of f we see that

L0= L −1 = L −2 = S otherwise and, because of the strict monotonicity of f in all of its arguments,

S = f (L0, L−1, L−2 ) < f (S, S, S),

which is a contradiction

Clearly, for an arbitrarily small positive number ² there exists N sufficiently

large such that

S − ² < xn , xn −1, xn −2 < S + ².