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E-raamat: Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures

(University of Rhode Island, Kingston, USA), (University of Rhode Island, Kingston, USA)
  • Formaat: 232 pages
  • Ilmumisaeg: 30-Jul-2001
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781040200582
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Dynamics of Second Order Rational Difference Equations: With Open Problems and ConjecturesSuurem pilt
  • Formaat: 232 pages
  • Ilmumisaeg: 30-Jul-2001
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781040200582
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This self-contained monograph provides systematic, instructive analysis of second-order rational difference equations. After classifying the various types of these equations and introducing some preliminary results, the authors systematically investigate each equation for semicycles, invariant intervals, boundedness, periodicity, and global stability. Of paramount importance in their own right, the results presented also offer prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one. The techniques and results in this monograph are also extremely useful in analyzing the equations in the mathematical models of various biological systems and other applications.

Each chapter contains a section of open problems and conjectures that will stimulate further research interest in working towards a complete understanding of the dynamics of the equation and its functional generalizations-many of them ideal for research projects or Ph.D. theses. Clear, simple, and direct exposition combined with thoughtful uniformity in the presentation make Dynamics of Second Order Rational Difference Equations valuable as an advanced undergraduate or a graduate-level text, a reference for researchers, and as a supplement to every textbook on difference equations at all levels of instruction.
Preface ix
Acknowledgements xi
Introduction and Classification of Equation Types 1(4)
Preliminary Results
5(24)
Definitions of Stability and Linearized Stability Analysis
5(2)
The Stable Manifold Theorem in the Plane
7(2)
Global Asymptotic Stability of the Zero Equilibrium
9(1)
Global Attractivity of the Positive Equilibrium
9(5)
Limiting Solutions
14(3)
The Riccati Equation
17(7)
Semicycle Analysis
24(5)
Local Stability, Semicycles, Periodicity, and Invariant Intervals
29(18)
Equilibrium Points
29(1)
Stability of the Zero Equilibrium
30(2)
Local Stability of the Positive Equilibrium
32(1)
When is Every Solution Periodic with the Same Period?
33(2)
Existence of Prime Period-Two Solutions
35(1)
Local Asymptotic Stability of a Two Cycle
36(2)
Convergence to Period-Two Solutions When C = 0
38(3)
Invariant Intervals
41(2)
Open Problems and Conjectures
43(4)
(1, 1)-Type Equations
47(6)
Introduction
47(1)
The Case α = γ = A = B = 0 :
48(1)
The Case α = β = A = C = 0 :
49(1)
Open Problems and Conjectures
49(4)
(1, 2)-Type Equations
53(16)
Introduction
53(1)
The Case β = γ = C = 0 :
54(1)
The Case β = γ = A = 0 :
55(2)
The Case α = γ = B = 0 :
57(1)
The Case α = γ = A = 0 :
58(1)
The Case α = β = C = 0 :
59(1)
The Case α = β = A = 0 :
60(4)
The Case p < 1
61(1)
The Case p = 1
62(1)
The Case p > 1
62(2)
Open Problems and Conjectures
64(5)
(2, 1)-Type Equations
69(8)
Introduction
69(1)
The Case γ = A = B = 0 :
70(2)
The Case β = A = C = 0 :
72(1)
Open Problems and Conjectures
72(5)
(2, 2)-Type Equations
77(54)
Introduction
77(1)
The Case γ = C = 0 :
78(1)
The Case γ = B = 0 :
79(3)
The Case γ = A = 0 :
82(7)
Invariant Intervals
83(1)
Semicycle Analysis
84(3)
Global Stability and Boundedness
87(2)
The Case β = C = 0 :
89(3)
Semicycle Analysis
90(1)
The Case q < 1
90(1)
The Case q > 1
91(1)
The Case β = A = 0 :
92(9)
Existence and Local Stability of Period-Two Cycles
93(3)
Invariant Intervals
96(1)
Semicycle Analysis
97(3)
Global Behavior of Solutions
100(1)
The Case α = C = 0 :
101(8)
Invariant Intervals
102(1)
Semicycle Analysis of Solutions When q ≤ 1
103(2)
Semicycle Analysis and Global Attractivity When q = 1
105(2)
Global Attractivity When q ≤ 1
107(1)
Long-term Behavior of Solutions When q > 1
107(2)
The Case α = B = 0 :
109(4)
Invariant Intervals and Semicycle Analysis
110(2)
Global Stability of the Positive Equilibrium
112(1)
The Case α = A = 0 :
113(10)
Existence of a Two Cycle
114(3)
Semicycle Analysis
117(1)
Global Stability Analysis When p ≤ q
117(6)
Global Stability Analysis When p > q
123(1)
Open Problems and Conjectures
123(8)
(1, 3)-Type Equations
131(6)
Introduction
131(1)
The Case β = γ = 0 :
131(1)
The Case α = γ = 0 :
132(1)
The Case α = γ = 0 :
133(1)
Open Problems and Conjectures
134(3)
(3, 1)-Type Equations
137(4)
Introduction
137(1)
Open Problems and Conjectures
138(3)
(2, 3)-Type Equations
141(26)
Introduction
141(1)
The Case γ = 0 :
141(8)
Boundedness of Solutions
142(1)
Invariant Intervals
143(1)
Semicycle Analysis
144(3)
Global Asymptotic Stability
147(2)
The Case β = 0 :
149(9)
Existence and Local Stability of Period-Two Cycles
150(4)
Invariant Intervals
154(2)
Convergence of Solutions
156(1)
Global Stability
157(1)
Semicycle Analysis When pr = q + q2/r
157(1)
The Case α = 0 :
158(5)
Invariant Intervals
160(2)
Convergence of Solutions
162(1)
Open Problems and Conjectures
163(4)
(3, 2)-Type Equations
167(16)
Introduction
167(1)
The Case C = 0 :
167(5)
Proof of Part (b) of Theorem 10.2.1
169(3)
Proof of Part (c) of Theorem 10.2.1
172(1)
The Case B = 0 :
172(3)
Invariant Intervals
173(1)
Global Stability When p ≥ r
173(1)
Convergence of Solutions
174(1)
The Case A = 0 :
175(5)
Invariant Intervals
177(2)
Convergence of Solutions
179(1)
Open Problems and Conjectures
180(3)
The (3, 3)-Type Equation
183(18)
Linearized Stability Analysis
183(1)
Invariant Intervals
184(5)
Convergence Results
189(3)
Global Stability
191(1)
Open Problems and Conjectures
192(9)
Global Attractivity for Higher Order Equations 201(10)
Bibliography 211(6)
Index 217


Kulenovic\, Mustafa R.S.; Ladas\, G.
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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