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Ordinary Differential Equations

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ISBN-10: 0898715105

ISBN-13: 9780898715101

Edition: 2nd 2002 (Revised)

Authors: Philip Hartman

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Book details

List price: $88.00
Edition: 2nd
Copyright year: 2002
Publisher: Society for Industrial and Applied Mathematics
Publication date: 3/31/2002
Binding: Paperback
Pages: 632
Size: 5.85" wide x 8.80" long x 1.10" tall
Weight: 1.936
Language: English

Foreword to the Classics Edition
Preface to the First Edition
Preface to the Second Edition
Basic theorems
Smooth approximations
Change of integration variables
The Picard-Lindelof theorem
Peano's existence theorem
Extension theorem
H. Kneser's theorem
Example of nonuniqueness
Differential inequalities and uniqueness
Gronwall's inequality
Maximal and minimal solutions
Right derivatives
Differential inequalities
A theorem of Wintner
Uniqueness theorems
van Kampen's uniqueness theorem
Egress points and Lyapunov functions
Successive approximations
Linear differential equations
Linear systems
Variation of constants
Reductions to smaller systems
Basic inequalities
Constant coefficients
Floquet theory
Adjoint systems
Higher order linear equations
Remarks on changes of variables
Analytic Linear Equations
Fundamental matrices
Simple singularities
Higher order equations
A nonsimple singularity
Dependence on initial conditions and parameters
Higher order differentiability
Exterior derivatives
Another differentiability theorem
S- and L-Lipschitz continuity
Uniqueness theorem
A lemma
Proof of Theorem 8.1
Proof of Theorem 6.1
First integrals
Total and partial differential equations
A theorem of Frobenius
Total differential equations
Algebra of exterior forms
A theorem of Frobenius
Proof of Theorem 3.1
Proof of Lemma 3.1
The system (1.1)
Cauchy's method of characteristics
A nonlinear partial differential equation
Existence and uniqueness theorem
Haar's lemma and uniqueness
The Poincare-Bendixson theory
Autonomous systems
Index of a stationary point
The Poincare-Bendixson theorem
Stability of periodic solutions
Rotation points
Foci, nodes, and saddle points
The general stationary point
A second order equation
Poincare-Bendixson theory on 2-manifolds
Analogue of the Poincare-Bendixson theorem
Flow on a closed curve
Flow on a torus
Plane stationary points
Existence theorems
Characteristic directions
Perturbed linear systems
More general stationary point
Invariant manifolds and linearlizations
Invariant manifolds
The maps T[superscript t]
Modification of F([xi])
Invariant manifolds of a map
Existence of invariant manifolds
Linearization of a map
Proof of Theorem 7.1
Periodic solution
Limit cycles
Smooth equivalence maps
Smooth linearizations
Proof of Lemma 12.1
Proof of Theorem 12.2
Smoothness of stable manifolds
Perturbed linear systems
The case E = 0
A topological principle
A theorem of Wazewski
Preliminary lemmas
Proof of Lemma 4.1
Proof of Lemma 4.2
Proof of Lemma 4.3
Asymptotic integrations. Logarithmic scale
Proof of Theorem 8.2
Proof of Theorem 8.3
Logarithmic scale (continued)
Proof of Theorem 11.2
Asymptotic integration
Proof of Theorem 13.1
Proof of Theorem 13.2
Corollaries and refinements
Linear higher order equations
Linear second order equations
Basic facts
Theorems of Sturm
Sturm-Liouville boundary value problems
Number of zeros
Nonoscillatory equations and principal solutions
Nonoscillation theorems
Asymptotic integrations. Elliptic cases
Asymptotic integrations. Nonelliptic cases
Disconjugate systems
Disconjugate systems
Use of implicit function and fixed point theorems
Periodic solutions
Linear equations
Nonlinear problems
Second order boundary value problems
Linear problems
Nonlinear problems
A priori bounds
General theory
Basic facts
Green's functions
Nonlinear equations
Asymptotic integration
Dichotomies for solutions of linear equations
General theory
Notations and definitions
Preliminary lemmas
The operator T
Slices of [double vertical line]Py(t)[double vertical line]
Estimates for [double vertical line]y(t)[double vertical line]
Applications to first order systems
Applications to higher order systems
P(B, D)-manifolds
Adjoint equations
Associate spaces
The operator T'
Individual dichotomies
P'-admissible spaces for T'
Applications to differential equations
Existence of P D-solutions
Miscellany on monotony
Monotone solutions
Small and large solutions
Monotone solutions
Second order linear equations
Second order linear equations (continuation)
A problem in boundary layer theory
The problem
The case [lambda] > 0
The case [lambda] < 0
The case [lambda] = 0
Asymptotic behavior
Global asymptotic stability
Global asymptotic stability
Lyapunov functions
Nonconstant G
On Corollary 11.2
On "J(y)x . x [less than or equal] 0 if x . f(y) = 0"
Proof of Theorem 14.2
Proof of Theorem 14.1
Hints for exercises