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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