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