Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations

384 Pages·1996·7.401 MB·English

by  Calvin D. Ahlbrandt, Allan C. Peterson (auth.)

Discrete Hamiltonian Systems
Kluwer Texts inthe Mathematical Sciences
VOLUME 16
A Graduate-Level Book Series
Thetitlespublishedinthisseries arelistedattheendofthisvolume.
Discrete
Hamiltonian Systems
Difference Equations,
Continued Fractions,
and Riccati Equations
by
Calvin D. Ahlbrandt
UniversityofMissouri
and
Allan C.Peterson
UniversityofNebraska
Springer-Science+Business Media, B.Y.
AC.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress.
ISBN978-1-4419-4763-5 ISBN978-1-4757-2467-7(eBook)
DOI 10.1007/978-1-4757-2467-7
Printedonacid-freepaper
AllRightsReserved
©1996 SpringerScience+BusinessMediaDordrecht
OriginallypublishedbyKluwerAcademicPublishersin1996.
SoftcoverreprintofthehardcoverIstedition1996
Nopartofthematerialprotectedbythiscopyrightnoticemaybereproducedor
utilizedinanyformorbyanymeans,electronicormechanical,
includingphotocopying,recordingorbyanyinformationstorageand
retrievalsystem,withoutwrittenpermissionfromthecopyrightowner.
Dedicated to our families
Evelyn, Robert, William, and Michael
and
Tina, Carla, David, and Carrie
CONTENTS
PREFACE xi
1 SECOND ORDER SCALAR DIFFERENCE
EQUATIONS 1
1.1 Difference Equations and Recurrence Relations 1
1.2 Second Order Scalar Equations as Symplectic Systems 4
1.3 Wronskians ofSolutions 9
1.4 Prepared Solutions 10
1.5 Generalized Zeros ofSolutions 11
1.6 Disconjugacy and the Reid Roundabout Theorem 12
1.7 Disfocality 36
1.8 Comparison Theorems for Disfocality 39
1.9 Notes 43
2 CONTINUED FRACTIONS 45
2.1 The Scalar Case 45
2.2 Scalar Symplectic Continued Fractions 53
2.3 Companion Matrices 55
2.4 Symplectic C.F.s versus Companion Matrix C.F.s 58
2.5 Ratios of Bessel Functions 60
2.6 Matrix Continued Fractions 61
2.7 Continued Fractions in a Normed Ring 67
2.8 Notes 69
3 SYMPLECTIC SYSTEMS 71
3.1 Linear Systems and the Lagrange Identity 71
3.2 Symplectic Matrices 73
vii
viii Discrete Hamiltonian Systems
3.3 Solutions ofSymplectic Systems 76
3.4 Discrete Linear Hamiltonian Systems 82
3.5 Even Order DifferenceEquations as Systems 85
3.6 Discrete Jacobi Equations 91
3.7 Discrete Wronskians - Liouville's Theorem 93
3.8 Prepared Families - Lagrangian Subspaces 96
3.9 Linear Independence and the Wronskian Test 101
3.10 Reduction of Order 105
3.11 Dominant and RecessiveSolutions 113
3.12 The Normal Basis Theorem 119
3.13 The Connection Theorem 125
3.14 Essential Uniqueness of RecessiveSolutions 130
3.15 Asymptotic Behavior of Solutions 132
3.16 The Olver-Reid Construction 133
3.17 Associated Riccati Equations 135
3.18 Transformations of Symplectic Systems 144
3.19 Discrete Floquet Theory 145
3.20 Exponential Dichotomies 146
3.21 Periodic Symplectic Systems 149
3.22 Notes 150
4 DISCRETE VARIATIONAL THEORY 153
4.1 The Discrete Variational Problem 153
4.2 The Second Variation 160
4.3 Legendre's Necessary Condition 163
4.4 Discrete Hamiltonian Systems 165
4.5 Higher Order DifferenceEquations 168
4.6 Disconjugacy and Generalized Zeros 172
4.7 Variable Step Variational Problems 183
4.8 Discrete Hamiltonian Systems Yield Symplectic Integrators 189
4.9 Existence and Uniqueness of Local Solutions of Discrete
Hamiltonian Systems 191
4.10 Notes 197
Contents ix
5 SYMMETRIC THREE TERM
RECURRENCE RELATIONS 199
5.1 A Discrete Reid Roundabout Theorem 199
5.2 Discrete Legendre Conditions 213
5.3 A Sturmian Comparison Theorem 215
5.4 Prepared Bases 216
5.5 An Associated Bilinear Form 218
5.6 A Discrete Sturm Separation Theorem 221
5.7 Discrete Jacobi Conditions 226
5.8 Reduction of Order 228
5.9 Backwards Reduction ofOrder 231
5.10 Dominant and RecessiveSolutions 233
6 DISCRETE RICCATI EQUATIONS FOR
THREE TERM RECURRENCES 263
6.1 A Riccati Equation for £U = O. 263
6.2 Distinguished Solutions of Riccati Equations 269
6.3 Periodic Coefficient Riccati Equations 270
6.4 Constant Coefficient Riccati Equations 271
6.5 The Characteristic Equation 273
6.6 Minimality ofthe Distinguished Solution at 00 275
6.7 The Reverse Riccati Equation 278
6.8 Upper Solutions ofthe Reverse Riccati Equation 285
7 GREEN'S FUNCTIONS FOR
NONHOMOGENEOUS SECOND ORDER
DIFFERENCE EQUATIONS 295
7.1 Introduction 295
7.2 Variation of Constants Formula 295
7.3 The Green's Matrix Function forthe Conjugate Problem 298
7.4 Green's matrix function forthe right focalBVP 303
7.5 A Green's Matrix Function for a General Two-Point BVP 308
7.6 Notes 317
x Discrete Hamiltonian Systems
8 DISCONJUGACY CRITERIA 319
8.1 Introduction 319
8.2 A SufficientCondition for Disconjugacy 319
8.3 A SufficientCondition for Right Disfocality 326
8.4 Notes 329
9 DISCRETE LINEAR HAMILTONIAN
SYSTEMS 331
9.1 Preliminaries 331
9.2 The Moore-Penrose Inverse 337
9.3 The Quadratic Form 343
9.4 Notes 356
REFERENCES 357
INDEX
371