Difference Equations and Inequalities: Theory, Methods, and Applications

by Agarwal; Ravi P.

• ISBN13: 9780824790073
• ISBN10: 0824790073
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-01-27
• Publisher: CRC Press

Summary

A study of difference equations and inequalities. This second edition offers real-world examples and uses of difference equations in probability theory, queuing and statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical networks, quanta in radiation, genetics, economics, psychology, sociology, and other disciplines. It features 200 new problems, 400 additional references, and a new chapter on the qualitative properties of solutions of neutral difference equations.

Author Biography

Ravi P. Agarwal is an Associate Professor of Mathematics at the National University of Singapore. Formerly he was a Visiting Professor of Mathematics at the University of Florence, Italy, and an Alexander von Humboldt Foundation Fellow at Ludwig-Maximilians University, Munich, Germany. Dr. Agarwal received the Ph.D. degree (1973) in mathematics from the Indian Institute of Technology, Madras, India.

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 Inequalities	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