Ebook: Advanced Topics in Difference Equations

. The theory of difference equations, the methods used in their solutions and their wide applications have advanced beyond their adolescent stage to occupy a central position in Applicable Analysis. In fact, in the last five years, the proliferation of the subject is witnessed by hundreds of research articles and several monographs, two International Conferences and numerous Special Sessions, and a new Journal as well as several special issues of existing journals, all devoted to the theme of Difference Equations. Now even those experts who believe in the universality of differential equations are discovering the sometimes striking divergence between the continuous and the discrete. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. In 1992, the first author published a monograph on the subject entitled Difference Equations and Inequalities. This book was an in-depth survey of the field up to the year of publication. Since then, the subject has grown to such an extent that it is now quite impossible for a similar survey, even to cover just the results obtained in the last four years, to be written. In the present monograph, we have collected some of the results which we have obtained in the last few years, as well as some yet unpublished ones.

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This monograph is a collection of the results the authors have obtained on difference equations and inequalities. In the last few years this discipline has gone through such a dramatic development that it is no longer feasible to present an exhaustive survey of all research. However, this state-of-the-art volume offers a representative overview of the authors' recent work, reflecting some of the major advances in the field as well as the diversity of the subject.
Audience: This book will be of interest to graduate students and researchers in mathematical analysis and its applications, concentrating on finite differences, ordinary and partial differential equations, real functions and numerical analysis.

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This monograph is a collection of the results the authors have obtained on difference equations and inequalities. In the last few years this discipline has gone through such a dramatic development that it is no longer feasible to present an exhaustive survey of all research. However, this state-of-the-art volume offers a representative overview of the authors' recent work, reflecting some of the major advances in the field as well as the diversity of the subject.
Audience: This book will be of interest to graduate students and researchers in mathematical analysis and its applications, concentrating on finite differences, ordinary and partial differential equations, real functions and numerical analysis.
Content:
Front Matter....Pages i-ix
Periodic Solutions of First Order Linear Equations....Pages 1-11
Periodic Solutions of Generalized Difference Systems....Pages 11-26
Periodic Boundary Value Problems: First Order Systems....Pages 26-33
Periodic Boundary Value Problems: Second Order Systems....Pages 33-40
Periodic Solutions of Higher Order Partial Difference Equations....Pages 41-61
Nonlinear Variation of Parameter Methods: One Independent Variable....Pages 61-74
Nonlinear Variation of Parameter Methods: Several Independent Variables....Pages 74-84
Convergence to Equilibria....Pages 84-96
Asymptotic Behavior of Generalized Difference Systems....Pages 96-105
Stability of Generalized Difference Systems....Pages 105-112
Oscillation via Lyapunov’s Second Method....Pages 112-121
Oscillation via Difference Inequalities....Pages 121-134
Oscillation via Summation Averages....Pages 134-149
Oscillation for Nonlinear Difference Equations....Pages 149-166
Oscillation for Quasilinear Difference Equations....Pages 167-172
Oscillations Generated by Deviating Arguments....Pages 172-189
Positive Monotone Solutions for Nonlinear Difference Equations....Pages 189-197
Oscillation for nth Order Nonlinear Difference Equations....Pages 197-209
Comparison Theorems....Pages 209-219
Oscillation for Second Order Neutral Delay Difference Equations....Pages 219-227
Oscillation for Second Order Neutral Delay Difference Equations (Contd.)....Pages 227-233
Oscillation for Higher Order Neutral Delay Difference Equations....Pages 233-241
Oscillation for Partial Difference Equations I....Pages 241-271
Oscillation for Partial Difference Equations II....Pages 271-276
Oscillation for Partial Difference Equations III....Pages 277-296
Oscillation for Partial Difference Equations IV....Pages 296-310
A Three-Point Boundary Value Problem....Pages 311-318
A Singular Boundary Value Problem....Pages 318-329
Eigenvalues of Boundary Value Problems....Pages 329-350
Two Solutions of a Boundary Value Problem....Pages 350-361
(n, p) Boundary Value Problems....Pages 361-384
Boundary Value Problems Arising in Transport Phenomena....Pages 384-398
Solutions of m-Point Boundary Value Problems....Pages 398-413
Two-Point Boundary Value Problems for Generalized Difference Systems....Pages 414-425
Multi-Point Boundary Value Problems for Generalized Difference Systems....Pages 425-448
Resonant Boundary Value Problems for Generalized Difference Systems....Pages 449-464
Boundary Value Problems in Abstract Spaces....Pages 464-470
Linear Square Optimal Control Problem....Pages 471-478
Symmetries of Difference Systems on Manifolds....Pages 478-483
Discrete Polar Coordinates....Pages 483-491
Back Matter....Pages 492-510

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This monograph is a collection of the results the authors have obtained on difference equations and inequalities. In the last few years this discipline has gone through such a dramatic development that it is no longer feasible to present an exhaustive survey of all research. However, this state-of-the-art volume offers a representative overview of the authors' recent work, reflecting some of the major advances in the field as well as the diversity of the subject.
Audience: This book will be of interest to graduate students and researchers in mathematical analysis and its applications, concentrating on finite differences, ordinary and partial differential equations, real functions and numerical analysis.
Content:
Front Matter....Pages i-ix
Periodic Solutions of First Order Linear Equations....Pages 1-11
Periodic Solutions of Generalized Difference Systems....Pages 11-26
Periodic Boundary Value Problems: First Order Systems....Pages 26-33
Periodic Boundary Value Problems: Second Order Systems....Pages 33-40
Periodic Solutions of Higher Order Partial Difference Equations....Pages 41-61
Nonlinear Variation of Parameter Methods: One Independent Variable....Pages 61-74
Nonlinear Variation of Parameter Methods: Several Independent Variables....Pages 74-84
Convergence to Equilibria....Pages 84-96
Asymptotic Behavior of Generalized Difference Systems....Pages 96-105
Stability of Generalized Difference Systems....Pages 105-112
Oscillation via Lyapunov’s Second Method....Pages 112-121
Oscillation via Difference Inequalities....Pages 121-134
Oscillation via Summation Averages....Pages 134-149
Oscillation for Nonlinear Difference Equations....Pages 149-166
Oscillation for Quasilinear Difference Equations....Pages 167-172
Oscillations Generated by Deviating Arguments....Pages 172-189
Positive Monotone Solutions for Nonlinear Difference Equations....Pages 189-197
Oscillation for nth Order Nonlinear Difference Equations....Pages 197-209
Comparison Theorems....Pages 209-219
Oscillation for Second Order Neutral Delay Difference Equations....Pages 219-227
Oscillation for Second Order Neutral Delay Difference Equations (Contd.)....Pages 227-233
Oscillation for Higher Order Neutral Delay Difference Equations....Pages 233-241
Oscillation for Partial Difference Equations I....Pages 241-271
Oscillation for Partial Difference Equations II....Pages 271-276
Oscillation for Partial Difference Equations III....Pages 277-296
Oscillation for Partial Difference Equations IV....Pages 296-310
A Three-Point Boundary Value Problem....Pages 311-318
A Singular Boundary Value Problem....Pages 318-329
Eigenvalues of Boundary Value Problems....Pages 329-350
Two Solutions of a Boundary Value Problem....Pages 350-361
(n, p) Boundary Value Problems....Pages 361-384
Boundary Value Problems Arising in Transport Phenomena....Pages 384-398
Solutions of m-Point Boundary Value Problems....Pages 398-413
Two-Point Boundary Value Problems for Generalized Difference Systems....Pages 414-425
Multi-Point Boundary Value Problems for Generalized Difference Systems....Pages 425-448
Resonant Boundary Value Problems for Generalized Difference Systems....Pages 449-464
Boundary Value Problems in Abstract Spaces....Pages 464-470
Linear Square Optimal Control Problem....Pages 471-478
Symmetries of Difference Systems on Manifolds....Pages 478-483
Discrete Polar Coordinates....Pages 483-491
Back Matter....Pages 492-510
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