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sturm liouville and difference equation
Difference Equations and Inequalities: Theory, Methods, and Applications
Format Hardcover Edition 2ND Subject Mathematics / Differential Equations ISBN/SKU 0824790073
Author Ravi P. Agarwal
Publisher Marcel Dekker Inc., Publish Date January 2000 Price $215.00
Review
In a self-contained monograph synthesizing some 400 recent publications, Agarwal (mathematics, National U. of Singapore) documents the revival of interest in difference equations in nearly every field that uses mathematics. They have been thought of as only discrete analogs of differential equations since the latter developed from them, but interest is rising now that computers are solving differential equations by using their approximate difference equation formulations. After some preliminaries, he surveys their use in linear initial value problems, miscellaneous different equations, difference inequalities, qualitative properties of solutions of difference systems and higher order difference equations, boundary value problems for linear and nonlinear systems, and other areas. To the 1992 edition he here adds a chapter on the qualitative properties of solutions of neutral difference equations. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Table of Contents
Preface to the Second Edition v Preface to the First Edition v
Preliminaries
Notations 1 (1) Difference Equations 2 (2) Initial Value Problems 4 (2) Some Examples: Initial Value Problems 6 (5)
Boundary Value Problems 11 (2) Some Examples: Boundary Value Problems 13 (9) Some Examples: Real World Phenomena 22 (4) Finite Difference Calculus 26 (8) Problems 34 (10) Notes 44 (1) References 45 (4)
Linear Initial Value Problems
Introduction 49 (1) Preliminary Results from Algebra 49 (5) Linear Dependence and Independence 54 (1) Matrix Linear Systems 55 (3) Variation of Constants Formula 58 (1) Green's Matrix 59 (1) Adjoint Systems 60 (2) Systems with Constant Coefficients 62 (7) Periodic Linear Systems 69 (3) Almost Periodic Linear Systems 72 (2) Higher Order Linear Equations 74 (6) Method of Generating Functions 80 (5) Bernoulli's Method 85 (2) Poincare's and Perron's Theorems 87 (4) Regular and Singular Perturbations 91 (6) Problems 97 (16) Notes 113 (1) References 114 (3)
Miscellaneous Difference Equations
Clairaut's Equation 117 (1) Euler's Equation 118 (2) Riccati's Equation 120 (5) Bernoulli's Equation 125 (1) Verhulst's Equation 126 (2) Best Discrete Approximations: Harmonic Oscillator Equation 128 (2) Duffing's Equation 130 (5) van der Pol's Equation 135 (7) Hill's Equation 142 (5) Mathieu's Equation 147 (1) Weierstrass' Elliptic Equations 148 (2) Volterra's Equations 150 (2) Elementary Partial Difference Equations: Riccati's Extended Form 152 (6) Wave Equation 158 (1) FitzHugh-Nagumo's Equation 159 (1) Korteweg-de Vries' Equation 160 (1) Modified KdV Equation 161 (1) Lagrange's Equations 162 (5) Problems 167 (13) Notes 180 (1) References 181 (3)
Difference Inequalities
Gronwall Inequalities 184 (9) Nonlinear Inequalities 193 (7) Inequalities Involving Differences 200 (5) Finite Systems of Inequalities 205 (4) Opial Type Inequalities 209 (5) Wirtinger Type Inequalities 214 (5) Problems 219 (10) Notes 229 (1) References 230 (4)
Qualitative Properties of Solutions of Difference Systems
Dependence on Initial Conditions and Parameters 234 (4) Asymptotic Behavior of Linear Systems 238 (9)
Asymptotic Behavior of Nonlinear Systems 247 (3) Concepts of Stability 250 (5) Stability of Linear Systems 255 (7)
Stability of Nonlinear Systems 262 (7) Nonlinear Variation of Constants 269 (3) Dichotomies 272 (9) Lyapunov's Direct Method for Autonomous Systems 281 (8) Lyapunov's Direct Method for Non-Autonomous Systems 289 (3) Stability of Discrete Models in Population Dynamics 292 (6) Converse Theorems 298 (5) Total Stability 303 (2) Practical Stability 305 (1) Mutual Stability 306 (2) Problems 308 (18) Notes 326 (2) References 328 (7)
Qualitative Properties of Solutions of Higher Order Difference Equations
General Properties of Solutions of (6.1.1) 335 (2) Boundedness of Solutions of (6.1.1) 337 (5) Recessive and Dominant Solutions of (6.1.1) 342 (6) Oscillation and Nonoscillation for (6.1.1) 348 (1) Riccati Type Transformations for (6.1.1) 349 (6) Riccati Type Transformations for (6.6.1) 355 (7) Olver Type Comparison Results 362 (3) Sturm Type Comparison Results 365 (2) Variety of Properties of Solutions of (6.9.1) 367 (3) Variety of Properties of Solutions of (6.10.1) 370 (8) Oscillation and Nonoscillation for (6.11.1) 378 (6) Asymptotic Behavior of Solutions of (6.12.1) 384 (3) l2 and c0 Solutions of (6.13.1) 387 (4) Oscillation and Nonoscillation for (6.14.1) 391 (2) Oscillation and Nonoscillation for (6.15.1) 393 (5) Variety of Properties of Solutions of (6.16.1) 398 (5) Asymptotic Behavior of Solutions of (6.17.1) 403 (5) Asymptotic Behavior, Oscillation and Nonoscillation for (6.18.1) 408 (6) Oscillation and Nonoscillation for (6.19.1) 414 (6) Oscillation and Nonoscillation for (6.20.1) 420 (5) Oscillation and Nonoscillation for (6.21.1) 425 (2) Oscillation and Nonoscillation for (6.22.1) 427 (9) Oscillation and Nonoscillation for (6.23.1) 436 (7) Problems 443 (30) Notes 473 (1) References 474 (11)
Qualitative Properties of Solutions of Neutral Difference Equations
Oscillation and Nonoscillation for (7.1.1) 485 (8) Existence and Asymptotic Behavior of Nonoscillatory Solutions of (7.1.1) 493 (16) Oscillation and Comparison Theorems for (7.1.1) 509 (3) Global Asymptotic Stability Criterion for (7.1.1) 512 (5) Oscillation and Nonoscillation for (7.5.1) 517 (2) Oscillation and Nonocillation for (7.6.1) 519 (2) Oscillation and Nonoscillation for (7.7.1) 521 (6) Oscillation and Nonoscillation for (7.8.1) 527 (4) Oscillatory and Asymptotic Behavior for (7.9.1) 531 (13) Oscillation and Nonoscillation for (7.10.1) 544 (4) Oscillation and Asymptotic Behavior for (7.11.1) 548 (8) Classification of Solutions for (7.12.1) 556 (9) Existence of Solutions for (7.12.1) 565 (7) Oscillation of Mixed Difference Equations 572 (11)
Oscillation and Nonoscillation for (7.15.1) 583 (7) Oscillation and Nonoscillation for (7.16.1) 590 (3) Problems 593 (27) Notes
620 (1) References 621 (8)
Boundary Value Problems for Linear Systems
Existence and Uniqueness 629 (5) Method of Complementary Functions 634 (9) Method of Particular Solutions 643 (1) Method of Adjoints 643 (8) Method of Chasing 651 (5) Method of Imbedding: First Formulation 656 (6) Method of Imbedding: Second Formulation 662 (4) Method of Sweep 666 (5) Miller's and Olver's Algorithms 671 (3) Problems 674 (2) Notes 676 (1) References 677 (4)
Boundary Value Problems for Nonlinear Systems
Preliminary Results from Analysis 681 (3) Existence and Uniqueness 684 (7) Approximate Picard's Iterates 691 (4) Oscillatory State 695 (3) Stopping Criterion 698 (2) Application to the Perturbation Method 700 (3) Monotone Convergence 703 (4) Periodic Boundary Value Problems 707 (7) Newton's Method 714 (5) Approximate Newton's Method 719 (5) Initial-Value Methods 724 (7) Invariant Imbedding Method 731 (3) Problems 734 (5) Notes 739 (1) References 739 (6)
Miscellaneous Properties of Solutions of Higher Order Linear Difference Equations
Disconjugacy 745 (1) Right and Left Disconjugacy 746 (4) Adjoint Equations 750 (1) Right and Left Disconjugacy for the Adjoint Equation 751 (1) Right Disfocality 752 (2) Eventual Disconjugacy and Right Disfocality 754 (1) A Classification of Solutions 755 (2) Interpolating Polynomials 757 (5) Green's Functions 762 (10) Inequalities and Equalities for Green's Functions 772 (2) Maximum Principles 774 (1) Error Estimates in Polynomial Interpolation 775 (3) Problems 778 (11)
Notes 789 (1) References 790 (5)
Boundary Value Problems for Higher Order Difference Equations
Existence and Uniqueness 795 (8) Picard's and Approximate Picard's Methods 803 (6) Quasilinearization and Approximate Quasilinearization 809 (10) Monotone Convergence 819 (5) Initial--Value Methods 824 (9) Uniqueness Implies Existence 833 (6) Problems 839 (2) Notes 841 (1) References 842 (2)
Sturm--Liouville Problems and Related Inequalities
Sturm--Liouville Problems 844 (4) Eigenvalue Problems for Symmetric Matrices 848 (2) Matrix Formulation of Sturm--Liouville Problems 850 (2) Symmetric, Antisymmetric and Periodic Boundary Conditions 852 (4) Discrete Fourier Series 856 (3) Wirtinger Type Inequalities 859 (6) Generalized Wirtinger Type Inequalities 865 (2) Generalized Opial Type Inequalities 867 (2) Comparison Theorems for Eigenvalues 869 (4) Positive Solutions of (12.10.1) (12.10.2) 873 (9) Problems 882 (14) Notes 896 (1) References 897 (4)
Difference Inequalities in Several Independent Variables
Discrete Riemann's Function 901 (4) Linear Inequalities 905 (3) Wendroff Type Inequalities 908 (3) Nonlinear Inequalities 911 (3) Inequalities Involving Partial Differences 914 (7) Multidimensional Linear Inequalities 921 (4) Multidimensional Nonlinear nequalities 925 (5) Convolution Type Inequalities 930 (3) Opial and Wirtinger Type Inequalities in Two Variables 933 (8) Problems 941 (3) Notes 944 (1) References 945 (4)
Author Index 949 (14)
Subject Index 963
Files are correct now.
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makswell
Difference Equations and Inequalities: Theory, Methods, and Applications
Format Hardcover Edition 2ND Subject Mathematics / Differential Equations ISBN/SKU 0824790073
Author Ravi P. Agarwal
Publisher Marcel Dekker Inc., Publish Date January 2000 Price $215.00
Review
In a self-contained monograph synthesizing some 400 recent publications, Agarwal (mathematics, National U. of Singapore) documents the revival of interest in difference equations in nearly every field that uses mathematics. They have been thought of as only discrete analogs of differential equations since the latter developed from them, but interest is rising now that computers are solving differential equations by using their approximate difference equation formulations. After some preliminaries, he surveys their use in linear initial value problems, miscellaneous different equations, difference inequalities, qualitative properties of solutions of difference systems and higher order difference equations, boundary value problems for linear and nonlinear systems, and other areas. To the 1992 edition he here adds a chapter on the qualitative properties of solutions of neutral difference equations. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Table of Contents
Preface to the Second Edition v Preface to the First Edition v
Preliminaries
Notations 1 (1) Difference Equations 2 (2) Initial Value Problems 4 (2) Some Examples: Initial Value Problems 6 (5)
Boundary Value Problems 11 (2) Some Examples: Boundary Value Problems 13 (9) Some Examples: Real World Phenomena 22 (4) Finite Difference Calculus 26 (8) Problems 34 (10) Notes 44 (1) References 45 (4)
Linear Initial Value Problems
Introduction 49 (1) Preliminary Results from Algebra 49 (5) Linear Dependence and Independence 54 (1) Matrix Linear Systems 55 (3) Variation of Constants Formula 58 (1) Green's Matrix 59 (1) Adjoint Systems 60 (2) Systems with Constant Coefficients 62 (7) Periodic Linear Systems 69 (3) Almost Periodic Linear Systems 72 (2) Higher Order Linear Equations 74 (6) Method of Generating Functions 80 (5) Bernoulli's Method 85 (2) Poincare's and Perron's Theorems 87 (4) Regular and Singular Perturbations 91 (6) Problems 97 (16) Notes 113 (1) References 114 (3)
Miscellaneous Difference Equations
Clairaut's Equation 117 (1) Euler's Equation 118 (2) Riccati's Equation 120 (5) Bernoulli's Equation 125 (1) Verhulst's Equation 126 (2) Best Discrete Approximations: Harmonic Oscillator Equation 128 (2) Duffing's Equation 130 (5) van der Pol's Equation 135 (7) Hill's Equation 142 (5) Mathieu's Equation 147 (1) Weierstrass' Elliptic Equations 148 (2) Volterra's Equations 150 (2) Elementary Partial Difference Equations: Riccati's Extended Form 152 (6) Wave Equation 158 (1) FitzHugh-Nagumo's Equation 159 (1) Korteweg-de Vries' Equation 160 (1) Modified KdV Equation 161 (1) Lagrange's Equations 162 (5) Problems 167 (13) Notes 180 (1) References 181 (3)
Difference Inequalities
Gronwall Inequalities 184 (9) Nonlinear Inequalities 193 (7) Inequalities Involving Differences 200 (5) Finite Systems of Inequalities 205 (4) Opial Type Inequalities 209 (5) Wirtinger Type Inequalities 214 (5) Problems 219 (10) Notes 229 (1) References 230 (4)
Qualitative Properties of Solutions of Difference Systems
Dependence on Initial Conditions and Parameters 234 (4) Asymptotic Behavior of Linear Systems 238 (9)
Asymptotic Behavior of Nonlinear Systems 247 (3) Concepts of Stability 250 (5) Stability of Linear Systems 255 (7)
Stability of Nonlinear Systems 262 (7) Nonlinear Variation of Constants 269 (3) Dichotomies 272 (9) Lyapunov's Direct Method for Autonomous Systems 281 (8) Lyapunov's Direct Method for Non-Autonomous Systems 289 (3) Stability of Discrete Models in Population Dynamics 292 (6) Converse Theorems 298 (5) Total Stability 303 (2) Practical Stability 305 (1) Mutual Stability 306 (2) Problems 308 (18) Notes 326 (2) References 328 (7)
Qualitative Properties of Solutions of Higher Order Difference Equations
General Properties of Solutions of (6.1.1) 335 (2) Boundedness of Solutions of (6.1.1) 337 (5) Recessive and Dominant Solutions of (6.1.1) 342 (6) Oscillation and Nonoscillation for (6.1.1) 348 (1) Riccati Type Transformations for (6.1.1) 349 (6) Riccati Type Transformations for (6.6.1) 355 (7) Olver Type Comparison Results 362 (3) Sturm Type Comparison Results 365 (2) Variety of Properties of Solutions of (6.9.1) 367 (3) Variety of Properties of Solutions of (6.10.1) 370 (8) Oscillation and Nonoscillation for (6.11.1) 378 (6) Asymptotic Behavior of Solutions of (6.12.1) 384 (3) l2 and c0 Solutions of (6.13.1) 387 (4) Oscillation and Nonoscillation for (6.14.1) 391 (2) Oscillation and Nonoscillation for (6.15.1) 393 (5) Variety of Properties of Solutions of (6.16.1) 398 (5) Asymptotic Behavior of Solutions of (6.17.1) 403 (5) Asymptotic Behavior, Oscillation and Nonoscillation for (6.18.1) 408 (6) Oscillation and Nonoscillation for (6.19.1) 414 (6) Oscillation and Nonoscillation for (6.20.1) 420 (5) Oscillation and Nonoscillation for (6.21.1) 425 (2) Oscillation and Nonoscillation for (6.22.1) 427 (9) Oscillation and Nonoscillation for (6.23.1) 436 (7) Problems 443 (30) Notes 473 (1) References 474 (11)
Qualitative Properties of Solutions of Neutral Difference Equations
Oscillation and Nonoscillation for (7.1.1) 485 (8) Existence and Asymptotic Behavior of Nonoscillatory Solutions of (7.1.1) 493 (16) Oscillation and Comparison Theorems for (7.1.1) 509 (3) Global Asymptotic Stability Criterion for (7.1.1) 512 (5) Oscillation and Nonoscillation for (7.5.1) 517 (2) Oscillation and Nonocillation for (7.6.1) 519 (2) Oscillation and Nonoscillation for (7.7.1) 521 (6) Oscillation and Nonoscillation for (7.8.1) 527 (4) Oscillatory and Asymptotic Behavior for (7.9.1) 531 (13) Oscillation and Nonoscillation for (7.10.1) 544 (4) Oscillation and Asymptotic Behavior for (7.11.1) 548 (8) Classification of Solutions for (7.12.1) 556 (9) Existence of Solutions for (7.12.1) 565 (7) Oscillation of Mixed Difference Equations 572 (11)
Oscillation and Nonoscillation for (7.15.1) 583 (7) Oscillation and Nonoscillation for (7.16.1) 590 (3) Problems 593 (27) Notes
620 (1) References 621 (8)
Boundary Value Problems for Linear Systems
Existence and Uniqueness 629 (5) Method of Complementary Functions 634 (9) Method of Particular Solutions 643 (1) Method of Adjoints 643 (8) Method of Chasing 651 (5) Method of Imbedding: First Formulation 656 (6) Method of Imbedding: Second Formulation 662 (4) Method of Sweep 666 (5) Miller's and Olver's Algorithms 671 (3) Problems 674 (2) Notes 676 (1) References 677 (4)
Boundary Value Problems for Nonlinear Systems
Preliminary Results from Analysis 681 (3) Existence and Uniqueness 684 (7) Approximate Picard's Iterates 691 (4) Oscillatory State 695 (3) Stopping Criterion 698 (2) Application to the Perturbation Method 700 (3) Monotone Convergence 703 (4) Periodic Boundary Value Problems 707 (7) Newton's Method 714 (5) Approximate Newton's Method 719 (5) Initial-Value Methods 724 (7) Invariant Imbedding Method 731 (3) Problems 734 (5) Notes 739 (1) References 739 (6)
Miscellaneous Properties of Solutions of Higher Order Linear Difference Equations
Disconjugacy 745 (1) Right and Left Disconjugacy 746 (4) Adjoint Equations 750 (1) Right and Left Disconjugacy for the Adjoint Equation 751 (1) Right Disfocality 752 (2) Eventual Disconjugacy and Right Disfocality 754 (1) A Classification of Solutions 755 (2) Interpolating Polynomials 757 (5) Green's Functions 762 (10) Inequalities and Equalities for Green's Functions 772 (2) Maximum Principles 774 (1) Error Estimates in Polynomial Interpolation 775 (3) Problems 778 (11)
Notes 789 (1) References 790 (5)
Boundary Value Problems for Higher Order Difference Equations
Existence and Uniqueness 795 (8) Picard's and Approximate Picard's Methods 803 (6) Quasilinearization and Approximate Quasilinearization 809 (10) Monotone Convergence 819 (5) Initial--Value Methods 824 (9) Uniqueness Implies Existence 833 (6) Problems 839 (2) Notes 841 (1) References 842 (2)
Sturm--Liouville Problems and Related Inequalities
Sturm--Liouville Problems 844 (4) Eigenvalue Problems for Symmetric Matrices 848 (2) Matrix Formulation of Sturm--Liouville Problems 850 (2) Symmetric, Antisymmetric and Periodic Boundary Conditions 852 (4) Discrete Fourier Series 856 (3) Wirtinger Type Inequalities 859 (6) Generalized Wirtinger Type Inequalities 865 (2) Generalized Opial Type Inequalities 867 (2) Comparison Theorems for Eigenvalues 869 (4) Positive Solutions of (12.10.1) (12.10.2) 873 (9) Problems 882 (14) Notes 896 (1) References 897 (4)
Difference Inequalities in Several Independent Variables
Discrete Riemann's Function 901 (4) Linear Inequalities 905 (3) Wendroff Type Inequalities 908 (3) Nonlinear Inequalities 911 (3) Inequalities Involving Partial Differences 914 (7) Multidimensional Linear Inequalities 921 (4) Multidimensional Nonlinear nequalities 925 (5) Convolution Type Inequalities 930 (3) Opial and Wirtinger Type Inequalities in Two Variables 933 (8) Problems 941 (3) Notes 944 (1) References 945 (4)
Author Index 949 (14)
Subject Index 963
Files are correct now.
--
makswell