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Tables of Contents for Approximate Solutions of Operator Equations

Chapter/Section Title

Page #

Page Count

Preface

vii

Chapter 1 Introduction

1

26

1.1 Overview of Different Approximation Methods

2

4

1.2 Projection Operators and Their Properties

6

1

1.3 Projective Approximation Algorithms (I)

7

5

1.4 Projective Approximation Algorithms (II)

12

4

1.5 Examples of Projective Approximation Methods

16

5

Exercises

21

6

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

27

36

2.1 Compact Operators and Their Equations

28

4

2.2 Projection Algorithms: The Banach Space Setting

32

4

2.3 Approximate Solutions of Fredholm Integral Equation and Boundary Value Problems of Higher-Order Ordinary Differential Equations

36

10

2.3.1 Fredholm integral equation and its approximate solutions

36

6

2.3.2 Approximate solutions for boundary value problems of higher-order ordinary differential equations

42

4

2.4 Projection Algorithms: The Hilbert Space Setting

46

11

Exercises

57

6

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

63

50

3.1 Bounded Linear Operator Equations and Their Approximate Solvability

64

9

3.1.1 The approximate solvability problem

64

4

3.1.2 The perturbed operator equation

68

1

3.1.3 Operator equations on reflexive Banach space and Hilbert space

69

4

3.2 Densely Defined Linear Operators and Their Equations

73

21

3.2.1 Closable linear operator equation in a Banach space setting

74

2

3.2.2 Closed linear operator equation in a Hilbert space setting

76

4

3.2.3 Definite linear operator equation in a Hilbert space setting

80

7

3.2.4 K-positive definite operator equation in a Hilbert space setting

87

7

3.3 Stability of Approximation Schemes

94

2

3.4 Numerical Solutions of Boundary Value Problems

96

9

3.4.1 Ordinary differential equations

96

4

3.4.2 Partial differential equations

100

5

Exercises

105

8

Chapter 4 Topological Degrees and Fixed Point Equations

113

50

4.1 Topological Degrees of Continuous Operators in Euclidean Spaces

114

18

4.1.1 Topological degrees of regular operators and their integral representations

114

8

4.1.2 Basic properties of topological degrees

122

2

4.1.3 Topological degrees of continuous operators

124

8

4.2 Topological Degrees of Compact Fields

132

7

4.3 Generalized Topological Degrees of A-Proper Operators

139

5

4.4 Fixed Point Theorems

144

7

4.4.1 Brouwer fixed point and open-set invariant theorems

145

2

4.4.2 Schauder and Krasnosel'skii fixed point theorems

147

1

4.4.3 Leray-Schauder fixed point theorem

148

1

4.4.4 Boundary conditions and fixed point theorems

149

2

4.5 Approximate Solutions of Nonlinear Fixed Point Equations

151

7

4.5.1 Projective approximate solvability of fixed point equations

151

3

4.5.2 Projective solutions of nonlinear integral equations

154

4

Exercises

158

5

Chapter 5 Nonlinear Monotone Operator Equations and Their Approximate Solutions

163

64

5.1 Continuity, Derivative, and Differential of Operators

164

10

5.1.1 Continuity of operators

164

1

5.1.2 Derivative and differential of operators

165

9

5.2 Monotone Operators from a Banach Space to Its Dual Space

174

11

5.2.1 Monotone operators

174

6

5.2.2 Monotonicity and semicontinuities

180

3

5.2.3 Strongly monotone operators

183

2

5.3 Approximate Solvability of Monotone Operator Equations

185

10

5.3.1 Monotone operator equations

185

7

5.3.2 The perturbation problem

192

2

5.3.3 Some remarks on the complex Banach space setting

194

1

5.4 Solvability and Approximate Solutions of K-Monotone Operator Equations

195

7

5.4.1 K-monotone operator equations

195

5

5.4.2 The perturbation problem

200

2

5.5 Application Examples: Numerical Solutions of Boundary Value Problems

202

19

Exercises

221

6

Chapter 6 Operator Evolution Equations and Their Projective Approximate Solutions

227

58

6.1 Preliminaries

228

7

6.1.1 Strongly and weakly measurable functions

228

1

6.1.2 Bochner integrals

229

1

6.1.3 Abstract functions on the L(p) space

229

2

6.1.4 Smoothing operator and smooth approximation

231

2

6.1.5 Generalized derivatives of abstract functions and the H(m) space

233

2

6.2 Projective Solutions of First Order Evolution Equations

235

45

6.2.1 Initial-boundary value problem of a linear parabolic equation

235

2

6.2.2 Continuous-time projection methods

237

15

6.2.3 Discrete-time projection methods

252

7

6.2.4 Initial-boundary value problems of nonlinear parabolic equations

259

5

6.3 Projective Solutions of Second Order Evolution Equations

264

16

Exercises

280

5

References

285

4

Appendix A: Fundamental Functional Analysis

289

28

Appendix B: Introduction to Sobolev Spaces

317

22

Subject Index

339