Dynamics of second order rational difference equations: with open problems and conjectures

235 Pages·2001·1.687 MB·English

by  Mustafa R.S. Kulenovic, G. Ladas

DYNAMICS of
SECOND ORDER
RATIONAL DIFFERENCE
EQUATIONS
With Open Problems and Conjectures

DYNAMICS of
SECOND ORDER
RATIONAL DIFFERENCE
EQUATIONS
With Open Problems and Conjectures
M. R. S. Kulenovi´c
G. Ladas
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.
disclaimer Page 1 Monday, June 18, 2001 9:53 AM
Library of Congress Cataloging-in-Publication Data
Kulenovic´, M.R.S. (Mustafa R.S.)
Dynamics of second order rational difference equations : with
open problems and conjectures / M.R.S. Kulenovic´, G. Ladas.
p. cm.
Includes bibliographical references and index.
ISBN 1-58488-275-1
1. Difference equations—Numerical solutions. I. Ladas, G.E.
II. Title.
QA431 .K84 2001
515′.625—dc21 20010370

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Contents
Preface ix
Acknowledgements xi
Introduction and Classification of Equation Types 1
1 Preliminary Results 5
1.1 Definitions of Stability and Linearized Stability Analysis . . . . . . . . . 5
1.2 The Stable Manifold Theorem in the Plane . . . . . . . . . . . . . . . . . 7
1.3 Global Asymptotic Stability of the Zero Equilibrium . . . . . . . . . . . 9
1.4 Global Attractivity of the Positive Equilibrium . . . . . . . . . . . . . . . 9
1.5 Limiting Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 The Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Semicycle Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Local Stability, Semicycles, Periodicity, and Invariant Intervals 29
2.1 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Stability of the Zero Equilibrium . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Local Stability of the Positive Equilibrium . . . . . . . . . . . . . . . . . 32
2.4 When is Every Solution Periodic with the Same Period? . . . . . . . . . 33
2.5 Existence of Prime Period-Two Solutions . . . . . . . . . . . . . . . . . . 35
2.6 Local Asymptotic Stability of a Two Cycle . . . . . . . . . . . . . . . . . 36
2.7 Convergence to Period-Two Solutions When
C = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.8 Invariant Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.9 Open Problems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . 43
3 (1,1)-Type Equations 47
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 The Case α = γ = A = B = 0 : x = βxn . . . . . . . . . . . . . . . 48
n+1 Cxn 1
3.3 The Case α = β = A = C = 0 : x = γxn−1 . . . . . . . . . . . . . . . . 49
n+1 Bx−n
3.4 Open Problems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . 49
v
vi Contents
4 (1,2)-Type Equations 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 The Case β = γ = C = 0 : x = α . . . . . . . . . . . . . . . . . . 54
n+1 A+Bxn
4.3 The Case β = γ = A = 0 : x = α . . . . . . . . . . . . . . . . 55
n+1 Bxn+Cxn 1
4.4 The Case α = γ = B = 0 : x = βxn −- Pielou’s Equation . . . . . . 57
n+1 A+Cxn 1
4.5 The Case α = γ = A = 0 : x = βxn− . . . . . . . . . . . . . . . 58
n+1 Bxn+Cxn 1
4.6 The Case α = β = C = 0 : x = γxn 1 .−. . . . . . . . . . . . . . . . . 59
4.7 The Case α = β = A = 0 : xn+1 = A+γBx−xnn1 . . . . . . . . . . . . . . . . 60
n+1 Bxn+C−xn 1
4.7.1 The Case p < 1 . . . . . . . . . . .−. . . . . . . . . . . . . . . . . 61
4.7.2 The Case p = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.7.3 The Case p > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.8 Open Problems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . 64
5 (2,1)-Type Equations 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 The Case γ = A = B = 0 : x = α+βxn - Lyness’ Equation . . . . . . . 70
n+1 Cxn 1
5.3 The Case β = A = C = 0 : x = α+γ−xn 1 . . . . . . . . . . . . . . . . . 72
n+1 Bxn−
5.4 Open Problems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . 72
6 (2,2)-Type Equations 77
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 The Case γ = C = 0 : x = α+βxn - Riccati Equation . . . . . . . . . . 78
n+1 A+Bxn
6.3 The Case γ = B = 0 : x = α+βxn . . . . . . . . . . . . . . . . . . . 79
n+1 A+Cxn 1
6.4 The Case γ = A = 0 : x = α+βx−n . . . . . . . . . . . . . . . . . . 82
n+1 Bxn+Cxn 1
6.4.1 Invariant Intervals . . . . . . −. . . . . . . . . . . . . . . . . . . . 83
6.4.2 Semicycle Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.4.3 Global Stability and Boundedness . . . . . . . . . . . . . . . . . . 87
6.5 The Case β = C = 0 : x = α+γxn 1 . . . . . . . . . . . . . . . . . . . 89
n+1 A+Bx−n
6.5.1 Semicycle Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.5.2 The Case q < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.5.3 The Case q > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.6 The Case β = A = 0 : x = α+γxn 1 . . . . . . . . . . . . . . . . . . 92
n+1 Bxn+Cx−n 1
6.6.1 Existence and Local Stability of−Period-Two Cycles . . . . . . . . 93
6.6.2 Invariant Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.6.3 Semicycle Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.6.4 Global Behavior of Solutions . . . . . . . . . . . . . . . . . . . . . 100
6.7 The Case α = C = 0 : x = βxn+γxn 1 . . . . . . . . . . . . . . . . . . 101
n+1 A+Bxn−
6.7.1 Invariant Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.7.2 Semicycle Analysis of Solutions When q 1 . . . . . . . . . . . . 103
≤
6.7.3 Semicycle Analysis and Global Attractivity When q = 1. . . . . . 105
6.7.4 Global Attractivity When q < 1 . . . . . . . . . . . . . . . . . . . 107
Contents vii
6.7.5 Long-term Behavior of Solutions When q > 1 . . . . . . . . . . . 107
6.8 The Case α = B = 0 : x = βxn+γxn 1 . . . . . . . . . . . . . . . . . . 109
n+1 A+Cxn −1
6.8.1 Invariant Intervals and Semicy−cle Analysis . . . . . . . . . . . . . 110
6.8.2 Global Stability of the Positive Equilibrium . . . . . . . . . . . . 112
6.9 The Case α = A = 0 : x = βxn+γxn 1 . . . . . . . . . . . . . . . . . . 113
n+1 Bxn+Cxn−1
6.9.1 Existence of a Two Cycle . . . −. . . . . . . . . . . . . . . . . . . . 114
6.9.2 Semicycle Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.9.3 Global Stability Analysis When p < q . . . . . . . . . . . . . . . 117
6.9.4 Global Stability Analysis When p > q . . . . . . . . . . . . . . . 123
6.10 Open Problems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . 123
7 (1,3)-Type Equations 131
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2 The Case β = γ = 0 : x = α . . . . . . . . . . . . . . . . . 131
n+1 A+Bxn+Cxn 1
−
7.3 The Case α = γ = 0 : x = βxn . . . . . . . . . . . . . . . . . 132
n+1 A+Bxn+Cxn 1
7.4 The Case α = β = 0 : x = γxn 1 − . . . . . . . . . . . . . . . . . 133
n+1 A+Bxn+−Cxn 1
7.5 Open Problems and Conjectures . . . . .−. . . . . . . . . . . . . . . . . . 134
8 (3,1)-Type Equations 137
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.2 Open Problems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . 138
9 (2,3)-Type Equations 141
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.2 The Case γ = 0 : x = α+βxn . . . . . . . . . . . . . . . . . . . 141
n+1 A+Bxn+Cxn 1
9.2.1 Boundedness of Solutions .−. . . . . . . . . . . . . . . . . . . . . 142
9.2.2 Invariant Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.2.3 Semicycle Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9.2.4 Global Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . 147
9.3 The Case β = 0 : x = α+γxn 1 . . . . . . . . . . . . . . . . . . . 149
n+1 A+Bxn+C−xn 1
9.3.1 Existence and Local Stability−of Period-Two Cycles . . . . . . . . 150
9.3.2 Invariant Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.3.3 Convergence of Solutions . . . . . . . . . . . . . . . . . . . . . . . 156
9.3.4 Global Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9.3.5 Semicycle Analysis When pr = q + q2 . . . . . . . . . . . . . . . . 157
r
9.4 The Case α = 0 : x = βxn+γxn 1 . . . . . . . . . . . . . . . . . . . 158
n+1 A+Bxn+Cx−n 1
9.4.1 Invariant Intervals . . . . . −. . . . . . . . . . . . . . . . . . . . . 160
9.4.2 Convergence of Solutions . . . . . . . . . . . . . . . . . . . . . . 162
9.5 Open Problems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . 163
viii Contents
10 (3,2)-Type Equations 167
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
10.2 The Case C = 0 : x = α+βxn+γxn 1 . . . . . . . . . . . . . . . . . . . . 167
n+1 A+Bxn −
10.2.1 Proof of Part (b) of Theorem 10.2.1 . . . . . . . . . . . . . . . . . 169
10.2.2 Proof of Part (c) of Theorem 10.2.1 . . . . . . . . . . . . . . . . . 172
10.3 The Case B = 0 : x = α+βxn+γxn 1 . . . . . . . . . . . . . . . . . . . . 172
n+1 A+Cxn 1−
10.3.1 Invariant Intervals . . . .−. . . . . . . . . . . . . . . . . . . . . . 173
10.3.2 Global Stability When p r . . . . . . . . . . . . . . . . . . . . . 173
≥
10.3.3 Convergence of Solutions . . . . . . . . . . . . . . . . . . . . . . . 174
10.4 The Case A = 0 : x = α+βxn+γxn 1 . . . . . . . . . . . . . . . . . . . . 175
n+1 Bxn+Cxn −1
10.4.1 Invariant Intervals . . . . −. . . . . . . . . . . . . . . . . . . . . . 177
10.4.2 Convergence of Solutions . . . . . . . . . . . . . . . . . . . . . . . 179
10.5 Open Problems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . 180
11 The (3,3)-Type Equation x = α+βxn+γxn 1 183
n+1 A+Bxn+Cxn−1
11.1 Linearized Stability Analysis . . . . . . . −. . . . . . . . . . . . . . . . . . 183
11.2 Invariant Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
11.3 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
11.3.1 Global Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
11.4 Open Problems and Conjectures . . . . . . . . . . . . . . . . . . . . . . . 192
A Global Attractivity for Higher Order Equations 201
Bibliography 211
Index 217
Preface
In this monograph, we present the known results and derive several new ones on the
boundedness,theglobalstability,andtheperiodicityofsolutionsofallrationaldifference
equations of the form
α+βx +γx
n n 1
xn+1 = − , n = 0,1,... (1)
A+Bx +Cx
n n 1
−
where the parameters α,β,γ,A,B,C and the initial conditions x and x are nonneg-
1 0
−
ative real numbers.
We believe that the results about Eq(1) are of paramount importance in their own
right and furthermore we believe that these results offer prototypes towards the devel-
opment of the basic theory of the global behavior of solutions of nonlinear difference
equations of order greater than one. The techniques and results which we develop in
this monograph to understand the dynamics of Eq(1) are also extremely useful in ana-
lyzing the equations in the mathematical models of various biological systems and other
applications.
ItisanamazingfactthatEq(1)contains,asspecialcases, alargenumberofequations
whose dynamics have not been thoroughly understood yet and remain a great challenge
for further investigation. To this end we pose several Open Problems and Conjec-
tures which we believe will stimulate further interest towards a complete understanding
of the dynamics of Eq(1) and its functional generalizations.
Chapter 1 contains some basic definitions and some general results which are used
throughout the book. In this sense, this is a self-contained monograph and the main
prerequisite that the reader needs to understand the material presented here and to be
able to attack the open problems and conjectures is a good foundation in analysis. In an
appendix, at the end of this monograph, we present some global attractivity results for
higher order difference equations which may be useful for extensions and generalizations.
In Chapter 2 we present some general results about periodic solutions and invariant
intervals of Eq(1). In particular we present necessary and sufficient conditions for Eq(1)
to have period-two solutions and we also present a table of invariant intervals.
In Chapters 3 to 11 we present the known results and derive several new ones on the
various types of equations which comprise Eq(1).
Every chapter in this monograph contains several open problems and conjectures
related to the particular equation which is investigated and its generalizations.
ix