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1.

図書

図書
Saber Elaydi
出版情報: New York : Springer, c2005  xxii, 539 p. ; 25 cm
シリーズ名: Undergraduate texts in mathematics
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2.

図書

図書
Ravi P. Agarwal
出版情報: New York : Marcel Dekker, c2000  xiii, 971 p. ; 24 cm
シリーズ名: Monographs and textbooks in pure and applied mathematics ; 228
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目次情報: 続きを見る
Preface to the Second Edition
Preface to the First Edition
Preliminaries / Chapter 1:
Notations / 1.1.:
Difference Equations / 1.2.:
Initial Value Problems / 1.3.:
Some Examples: Initial Value Problems / 1.4.:
Boundary Value Problems / 1.5.:
Some Examples: Boundary Value Problems / 1.6.:
Some Examples: Real World Phenomena / 1.7.:
Finite Difference Calculus / 1.8.:
Problems / 1.9.:
Notes / 1.10.:
References / 1.11.:
Linear Initial Value Problems / Chapter 2:
Introduction / 2.1.:
Preliminary Results from Algebra / 2.2.:
Linear Dependence and Independence / 2.3.:
Matrix Linear Systems / 2.4.:
Variation of Constants Formula / 2.5.:
Green's Matrix / 2.6.:
Adjoint Systems / 2.7.:
Systems with Constant Coefficients / 2.8.:
Periodic Linear Systems / 2.9.:
Almost Periodic Linear Systems / 2.10.:
Higher Order Linear Equations / 2.11.:
Method of Generating Functions / 2.12.:
Bernoulli's Method / 2.13.:
Poincare's and Perron's Theorems / 2.14.:
Regular and Singular Perturbations / 2.15.:
Miscellaneous Difference Equations / 2.16.:
Clairaut's Equation / 3.1.:
Euler's Equation / 3.2.:
Riccati's Equation / 3.3.:
Bernoulli's Equation / 3.4.:
Verhulst's Equation / 3.5.:
Best Discrete Approximations: Harmonic Oscillator Equation / 3.6.:
Duffing's Equation / 3.7.:
van der Pol's Equation / 3.8.:
Hill's Equation / 3.9.:
Mathieu's Equation / 3.10.:
Weierstrass' Elliptic Equations / 3.11.:
Volterra's Equations / 3.12.:
Elementary Partial Difference Equations: Riccati's Extended Form / 3.13.:
Wave Equation / 3.14.:
FitzHugh-Nagumo's Equation / 3.15.:
Korteweg-de Vries' Equation / 3.16.:
Modified KdV Equation / 3.17.:
Lagrange's Equations / 3.18.:
Difference Inequalities / 3.19.:
Gronwall Inequalities / 4.1.:
Nonlinear Inequalities / 4.2.:
Inequalities Involving Differences / 4.3.:
Finite Systems of Inequalities / 4.4.:
Opial Type Inequalities / 4.5.:
Wirtinger Type Inequalities / 4.6.:
Qualitative Properties of Solutions of Difference Systems / 4.7.:
Dependence on Initial Conditions and Parameters / 5.1.:
Asymptotic Behavior of Linear Systems / 5.2.:
Asymptotic Behavior of Nonlinear Systems / 5.3.:
Concepts of Stability / 5.4.:
Stability of Linear Systems / 5.5.:
Stability of Nonlinear Systems / 5.6.:
Nonlinear Variation of Constants / 5.7.:
Dichotomies / 5.8.:
Lyapunov's Direct Method for Autonomous Systems / 5.9.:
Lyapunov's Direct Method for Non-Autonomous Systems / 5.10.:
Stability of Discrete Models in Population Dynamics / 5.11.:
Converse Theorems / 5.12.:
Total Stability / 5.13.:
Practical Stability / 5.14.:
Mutual Stability / 5.15.:
Qualitative Properties of Solutions of Higher Order Difference Equations / 5.16.:
Qualitative Properties of Solutions of Neutral Difference Equations / Chapter 7:
Boundary Value Problems for Linear Systems / Chapter 8:
Existence and Uniqueness / 8.1.:
Method of Complementary Functions / 8.2.:
Method of Particular Solutions / 8.3.:
Method of Adjoints / 8.4.:
Method of Chasing / 8.5.:
Method of Imbedding: First Formulation / 8.6.:
Method of Imbedding: Second Formulation / 8.7.:
Method of Sweep / 8.8.:
Miller's and Olver's Algorithms / 8.9.:
Boundary Value Problems for Nonlinear Systems / 8.10.:
Preliminary Results from Analysis / 9.1.:
Approximate Picard's Iterates / 9.2.:
Oscillatory State / 9.4.:
Stopping Criterion / 9.5.:
Application to the Perturbation Method / 9.6.:
Monotone Convergence / 9.7.:
Periodic Boundary Value Problems / 9.8.:
Newton's Method / 9.9.:
Approximate Newton's Method / 9.10.:
Initial-Value Methods / 9.11.:
Invariant Imbedding Method / 9.12.:
Miscellaneous Properties of Solutions of Higher Order Linear Difference Equations / 9.13.:
Disconjugacy / 10.1.:
Right and Left Disconjugacy / 10.2.:
Adjoint Equations / 10.3.:
Right and Left Disconjugacy for the Adjoint Equation / 10.4.:
Right Disfocality / 10.5.:
Eventual Disconjugacy and Right Disfocality / 10.6.:
A Classification of Solutions / 10.7.:
Interpolating Polynomials / 10.8.:
Green's Functions / 10.9.:
Inequalities and Equalities for Green's Functions / 10.10.:
Maximum Principles / 10.11.:
Error Estimates in Polynomial Interpolation / 10.12.:
Boundary Value Problems for Higher Order Difference Equations / 10.13.:
Picard's and Approximate Picard's Methods / 11.1.:
Quasilinearization and Approximate Quasilinearization / 11.3.:
Uniqueness Implies Existence / 11.4.:
Sturm-Liouville Problems and Related Inequalities / 11.7.:
Sturm-Liouville Problems / 12.1.:
Eigenvalue Problems for Symmetric Matrices / 12.2.:
Matrix Formulation of Sturm-Liouville Problems / 12.3.:
Symmetric, Antisymmetric and Periodic Boundary Conditions / 12.4.:
Discrete Fourier Series / 12.5.:
Generalized Wirtinger Type Inequalities / 12.6.:
Generalized Opial Type Inequalities / 12.8.:
Comparison Theorems for Eigenvalues / 12.9.:
Positive Solutions of (12.10.1) [Delta][superscript 3]u(k) + [lambda]a(k)f(u(k)) = 0, k [set membership] N(2, K + 2) (12.10.2) u(0) = u(1) = u(K + 3) = 0 / 12.10.:
Difference Inequalities in Several Independent Variables / 12.11.:
Discrete Riemann's Function / 13.1.:
Linear Inequalities / 13.2.:
Wendroff Type Inequalities / 13.3.:
Inequalities Involving Partial Differences / 13.4.:
Multidimensional Linear Inequalities / 13.6.:
Multidimensional Nonlinear Inequalities / 13.7.:
Convolution Type Inequalities / 13.8.:
Opial and Wirtinger Type Inequalities in Two Variables / 13.9.:
Author Index / 13.10.:
Subject Index
Preface to the Second Edition
Preface to the First Edition
Preliminaries / Chapter 1:
3.

図書

図書
Saber N. Elaydi
出版情報: Boca Raton, Fla. : Chapman & Hall/CRC, c2000  xiii, 355 p. ; 25 cm
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目次情報: 続きを見る
List of Symbols
Preface
The Stability of One-Dimensional Maps / 1:
Maps vs. Difference Equations / 1.1:
Maps vs. Differential Equations / 1.2:
Linear Maps/Difference Equations / 1.3:
Fixed (Equilibrium) Points / 1.4:
Graphical Iteration and Stability / 1.5:
Criteria for Stability / 1.6:
Hyperbolic Fixed Points / 1.6.1:
Nonhyperbolic Fixed Points / 1.6.2:
Periodic Points and their Stability / 1.7:
The Period-Doubling Route to Chaos / 1.8:
Fixed Points / 1.8.1:
2-Cycles / 1.8.2:
2[superscript 2]-Cycles / 1.8.3:
Beyond [mu][characters not reproducible] / 1.8.4:
Applications / 1.9:
A Genotype Selection Model / 1.9.1:
Sharkovsky's Theorem and Bifurcation / 2:
The Mystery of Period 3 / 2.1:
Converse of Sharkovsky's Theorem / 2.2:
Basin of Attraction / 2.3:
The Schwarzian Derivative / 2.4:
Bifurcation / 2.5:
The Lorenz Map / 2.6:
Appendix
Chaos in One Dimension / 3:
Introduction / 3.1:
Metric Spaces / 3.2:
Transitivity / 3.3:
Sensitive Dependence and Liapunov Exponents / 3.4:
Definition of Chaos / 3.5:
Symbolic Dynamics / 3.6:
Conjugacy / 3.7:
Stability of Two-Dimensional Maps / 4:
Linear Maps vs. Linear Systems / 4.1:
Computing A[superscript n] / 4.2:
Fundamental Set of Solutions / 4.3:
Second-Order Difference Equations / 4.4:
Phase Space Diagrams / 4.5:
Stability Notions / 4.6:
Stability of Linear Systems / 4.7:
Liapunov Functions for Nonlinear Maps / 4.8:
Linear Systems Revisited / 4.9:
Stability via Linearization / 4.10:
The Hartman-Grobman Theorem / 4.10.1:
The Stable Manifold Theorem / 4.10.2:
The Kicked Rotator and the Henon Map / 4.11:
The Henon Map / 4.11.2:
Discrete Epidemic Model for Gonorrhea / 4.11.3:
Perennial Grass / 4.11.4:
Chaos in Two Dimensions / 5:
Hyperbolic Anosov Toral Automorphism / 5.1:
Subshifts of Finite Type / 5.2:
The Horseshoe and Henon Maps / 5.3:
The Henon Map Revisited / 5.3.1:
Center Manifolds / 5.4:
Neimark-Sacker (Hopf) Bifurcation / 5.5:
Fractals / 6:
Examples of Fractals / 6.1:
The Dimension of a Fractal / 6.2:
Iterated Function System / 6.3:
Mathematical Foundation of Fractals / 6.4:
The Collage Theorem and Image Compression / 6.5:
The Julia and Mandelbrot Sets / 7:
Mapping by Functions on the Complex Domain / 7.1:
The Riemann Sphere / 7.2:
The Julia Set / 7.3:
Topological Properties of the Julia Set / 7.4:
Newton's Method in the Complex Plane / 7.5:
The Mandelbrot Set / 7.6:
Bibliography
Answers to Selected Problems
Index
List of Symbols
Preface
The Stability of One-Dimensional Maps / 1:
4.

図書

図書
Marius van der Put, Michael F. Singer
出版情報: Berlin ; New York : Springer, c1997  vii, 178 p. ; 24 cm
シリーズ名: Lecture notes in mathematics ; 1666
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Algebraic Theory
Picard-Vessiot rings / 1:
Existence and uniqueness of Picard-Vessiot rings / 1.1:
The Galois group / 1.2:
Galois correspondence for difference equations / 1.3:
Difference modules and fibre functors / 1.4:
Algorithms for difference equations / 2:
Difference equations of order one / 2.1:
Difference equations in diagonal form / 2.2:
Difference equations of order two / 2.3:
The inverse problem for difference equations / 3:
The ring S of sequences / 4:
An excursion in positive characteristic / 5:
Generalities / 5.1:
Modules over K[T. T superscript -1] / 5.2:
Difference Galois groups / 5.3:
Comparing characteristic 0 and p / 5.4:
Difference modules over P / 6:
Classification of difference modules over P / 6.1:
The universal Picard-Vessiot ring of P / 6.2:
Fields of constants which are not algebraically closed / 6.3:
Automorphisms of the universal Picard-Vessiot ring of P / 6.4:
Difference equations over C((z[superscript -1])) and the formal Galois group / 6.5:
Analytic Theory
Classification and canonical forms / 7:
A classification of singularities / 7.1:
Canonical forms / 7.2:
Semi-regular difference equations / 8:
Introduction / 8.1:
Some easy asymptotics / 8.2:
The connection matrix of a semi-regular equation / 8.3:
The theorem of Malgrange and Sibuya / 8.4:
Regular difference equations / 8.5:
Inverse problems for semi-regular equations / 8.6:
Mild difference equations / 9:
Asymptotics for mild equations / 9.1:
Connection matrices of mild equations / 9.2:
Tame differential modules / 9.3:
Inverse problems for mild equations / 9.4:
Examples of equations and Galois groups / 10:
Calculating connection matrices / 10.1:
Classification of order one equations / 10.2:
More on difference Galois groups / 10.3:
Mild difference and differential equations / 10.4:
Very mild difference modules and multisummability / 10.5:
Very mild differential modules / 10.6:
Wild difference equations / 11:
Multisummability of formal solutions / 11.1:
The Quadrant Theorem / 11.3:
On the Gamma function / 11.4:
An example / 11.5:
Solutions on a right half plane / 11.6:
Solutions on an upper half plane / 11.7:
Analytic equivalence classes of difference equations / 11.8:
q-difference equations / 11.9:
Formal aspects / 12.1:
Analytic properties / 12.2:
Regular singular equations over k[subscript 0] / 12.2.1:
Equations over C(z) / 12.2.2:
Construction of the connection map / 12.3:
Meromorphic vector bundles / 12.3.1:
The connection map of a regular equation / 12.3.2:
The connection map of a regular singular equation / 12.3.3:
Inverse problems / 12.3.4:
Bibliography
Index
Notations
Algebraic Theory
Picard-Vessiot rings / 1:
Existence and uniqueness of Picard-Vessiot rings / 1.1:
5.

図書

図書
Louis Brand
出版情報: New York : Wiley, c1966  xvi, 698 p. ; 24 cm
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6.

図書

図書
by Ravi P. Agarwal and Patricia J.Y. Wong
出版情報: Dordrecht ; Boston : Kluwer, c1997  507 p. ; 25 cm
シリーズ名: Mathematics and its applications ; v. 404
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7.

図書

図書
by V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan
出版情報: Dordrecht : Kluwer, c1996  x, 285 p. ; 25 cm
シリーズ名: Mathematics and its applications ; v. 370
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8.

図書

図書
by George Boole ; edited by J.F. Moulton
出版情報: New York : Chelsea Pub. Co., [1958]  xii, 336 p. ; 21 cm
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9.

図書

図書
Frank Chorlton
出版情報: London ; New York : Van Nostrand, c1965  xii, 284 p ; 24 cm
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10.

図書

図書
by Ravi P. Agarwal and Peter Y.H. Pang
出版情報: Dordrecht ; Boston : Kluwer Academic Publishers, c1995  x, 393 p. ; 25 cm
シリーズ名: Mathematics and its applications ; v. 320
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