Approximate Solutions of Operator Equations

Mingjun Chen    (Zhongshan University)
Zhongying Chen (Zhomgshan University)
Guanrong Chen  (University of Houston)

World Scientific Pub. Co., Singapore, 1997.  ISBN 981-02-3064-8
 

Contents

Preface

Chapter 1   Introduction   1

1.1 Overview of Different Approximation Methods   2
1.2 Projection Operators and Their Properties   6
1.3 Projective Approximation Algorithms (I)   7
1.4 Projective Approximation Algorithms (II)  12
1.5 Examples of Projective Approximation Methods   16
Exercises   21

Chapter 2   Operator Equations and Their Approximate Solutions (I):
                 Compact Linear Operators   27

2.1 Compact Operators and Their Equations   28
2.2 Projection Algorithms: The Banach Space Setting   32
2.3 Approximate Solutions of Fredholm Integral Equation and Boundary
     Value Problems of Higher-Order Ordinary Differential Equations   36
2.3.1 Fredholm Integral Equation and Its Approximate Solutions   36
2.3.2 Approximate Solutions for Boundary Value Problems of Higher-Order
         Ordinary Differential Equations   42
2.4  Projection Algorithms: The Hilbert Space Setting   46
Exercises   57

Chapter 3  Operator Equations and Their Approximate Solutions (II):
                 Other Linear Operators  63

3.1  Bounded Linear Operator Equations and Their Approximate Solvability  64
3.1.1 The Approximate Solvability Problem   64
3.1.2 The Perturbed Operator Equation   68
3.1.3 Operator Equations on Reflexive Banach Space and Hilbert Space  69
3.2  Densely Defined Linear Operators and Their Equations   73
3.2.1 Closable Linear Operator Equation in a Banach Space Setting   74
3.2.2 Closed Linear Operator Equation in a Hilbert Space Setting   76
3.2.3 Definite Linear Operator Equation in a Hilbert Space Setting   80
3.2.4 K-Positive Definite Operator Equation in a Hilbert Space Setting   87
3.3 Stability of Approximation Schemes   94
3.4 Numerical Solutions of Boundary Value Problems   96
3.4.1 Ordinary Differential Equations   96
3.4.2 Partial Differential Equations   100
Exercises   105

Chapter 4  Topological Degrees and Fixed Point Equations   113

4.1 Topological Degrees of Continuous Operators in Euclidean Spaces   114
4.1.1 Topological Degrees of Regular Operators and Their Integral Representations  114
4.1.2 Basic Properties of Topological Degrees  122
4.1.3 Topological Degrees of Continuous Operators  124
4.2  Topological Degrees of Compact Fields   132
4.3 Generalized Topological Degrees of A-Proper Operators   139
4.4  Fixed Point Theorems  144
4.4.1 Brouwer Fixed Point and Open-Set Invariant Theorems   145
4.4.2 Schauder and Krasnosel'skii Fixed Point Theorems   147
4.4.3 Leray-Schauder Fixed Point Theorem   148
4.4.4 Boundary Conditions and Fixed Point Theorems   149
4.5  Approximate Solutions of Nonlinear Fixed Point Equations   151
4.5.1 Projective Approximate Solvability of Fixed Point Equations   151
4.5.2 Projective Solutions of Nonlinear Integral Equations   154
Exercises  158

Chapter 5    Nonlinear Monotone Operator Equations and Their Approximate Solutions  163

5.1 Continuity, Derivative, and Differential of Operators   164
5.1.1 Continuity of Operators  164
5.1.2 Derivative and Differential of Operators  165
5.2  Monotone Operators from a Banach Space to Its Dual Space   174
5.2.1 Monotone Operators   174
5.2.2 Monotonicity and Semicontinuities  180
5.2.3 Strongly Monotone Operators   183
5.3 Approximate Solvability of Monotone Operator Equations   185
5.3.1 Monotone Operator Equations  185
5.3.2 The Perturbation Problem   192
5.3.3 Some Remarks on the Complex Banach Space Setting   194
5.4  Solvability and Approximate Solutions of K-Monotone Operator Equations  195
5.4.1 K-Monotone Operator Equations   195
5.4.2 The Perturbation Problem   200
5.5  Application Examples: Numerical Solutions of Boundary Value Problems   202
Exercises  221

Chapter 6    Operator Evolution Equations and Their Projective Approximate Solutions   227

6.1 Preliminaries   228
6.1.1 Strongly and Weakly Measurable Functions   228
6.1.2 Bochner Integrals   229
6.1.3 Abstract Functions on the Lp Space   229
6.1.4 Smoothing Operator and Smooth Approximation   231
6.1.5 Generalized Derivatives of Abstract Functions and the Hm Space  233
6.2  Projective Solutions of First Order Evolution Equations   235
6.2.1 Initial-Boundary Value Problem of a Linear Parabolic Equation   235
6.2.2 Continuous-Time Projection Methods   237
6.2.3 Discrete-Time Projection Methods   252
6.2.4 Initial-Boundary Value Problems of Nonlinear Parabolic Equations   259
6.3  Projective Solutions of Second Order Evolution Equations   264
Exercises  280

References  285

Appendix A: Fundamental Functional Analysis   289
Appendix B: Introduction to Sobolev Spaces   317
Subject Index  339
 

World Scientific Pub. Co., Singapore, 1997. ISBN 981-02-3064-8

TO ORDER:
SG: Fax: 65-382-5919     Tel: 65-382-5663     Email: sales@wspc.com.sg
US: Fax: 1-888-977-2665  Tel: 1-800-227-7562    Email: sales@wspc.com
UK: Fax: 44-171-836-2020 Tel: 44-171-836-0888   Email: sales@wspc2.demon.co.uk
HK: Fax: 852-2-771-8155  Tel: 852-2-771-8791    Email: wsped@hk.super.net
IN: Fax: 91-80-334-4593  Tel: 91-80-220-5972
TW: Fax: 886-2-2366-0460 Tel: 886-2-2369      Email: wsptw@ms13.hinet.net