Approximate Solutions of Operator Equations

The following sections are included:

• Approximations and Decompositions

• Preface

• Contents

The following sections are included:

• Overview of Different Approximation Methods

• Projection Operators and Their Properties

• Projective Approximation Algorithms (I)

• Projective Approximation Algorithms (II)

• Examples of Projective Approximation Methods

• Exercises

The following sections are included:

• Compact Operators and Their Equations

• Projection Algorithms: The Banach Space Setting

• Approximate Solutions of Fredholm Integral Equation and Boundary Value Problems of Higher-Order Ordinary Differential Equations
  • Fredholm integral equation and its approximate solutions
  • Approximate solutions for boundary value problems of higher-order ordinary differential equations

• Projection Algorithms: The Hilbert Space Setting

• Exercises

The following sections are included:

• Bounded Linear Operator Equations and Their Approximate Solvability
  • The approximate solvability problem
  • The perturbed operator equation
  • Operator equations on reflexive Banach space and Hilbert space

• Densely Defined Linear Operators and Their Equations
  • Closable linear operator equation in a Banach space setting
  • Closed linear operator equation in a Hilbert space setting
  • Definite linear operator equation in a Hilbert space setting
  • K-positive definite operator equation in a Hilbert space setting

• Stability of Approximation Schemes

• Numerical Solutions of Boundary Value Problems
  • Ordinary differential equations
  • Partial differential equations

• Exercises

The following sections are included:

• Topological Degrees of Continuous Operators in Euclidean Spaces
  • Topological degrees of regular operators and their integral representations
  • Basic properties of topological degrees
  • Topological degrees of continuous operators

• Topological Degrees of Compact Fields

• Generalized Topological Degrees of A-Proper Operators

• Fixed Point Theorems
  • Brouwer fixed point and open-set invariant theorems
  • Schauder and Krasnosel'skii fixed point theorems
  • Leray-Schauder fixed point theorem
  • Boundary conditions and fixed point theorems

• Approximate Solutions of Nonlinear Fixed Point Equations
  • Projective approximate solvability of fixed point equations
  • Projective solutions of nonlinear integral equations

• Exercises

The following sections are included:

• Continuity, Derivative and Differential of Operators
  • Continuity of operators
  • Derivative and differential of operators

• Monotone Operators from a Banach Space to Its Dual Space
  • Monotone operators
  • Monotonicity and semicontinuities
  • Strongly monotone operators

• Approximate Solvability of Monotone Operator Equations
  • Monotone operator equations
  • The perturbation problem
  • Some remarks on the complex Banach space setting

• Solvability and Approximate Solutions of K-Monotone Operator Equations
  • K-monotone operator equations
  • The perturbation problem

• Application Examples: Numerical Solutions of Boundary Value Problems

• Exercises

The following sections are included:

• Preliminaries
  • Strongly and weakly measurable functions
  • Bochner integrals
  • Abstract functions on the L_p space
  • Smoothing operator and smooth approximation
  • Generalized derivatives of abstract functions and the H^m space

• Projective Solutions of First Order Evolution Equations
  • Initial-boundary value problem of a linear parabolic equation
  • Continuous-time projection methods
  • Discrete-time projection methods
  • Initial-boundary value problems of nonlinear parabolic equations

• Projective Solutions of Second Order Evolution Equations

• Exercises

The following sections are included:

• References

• APPENDIX A Fundamental Functional Analysis
  • Linear Topological Spaces
    • Topological spaces
    • Metric spaces
    • Completeness
    • Separability
    • Sequential compactness
    • Linear topological spaces
  • Linear Normed Spaces and Inner-Product Spaces
    • Linear normed spaces and their completion
    • Inner product spaces and their completion
    • Function spaces and L_p(Ω) (1 ≤ p ≤ ∞)
    • Function spaces and H^m(Ω) (m ≥ 1)
    • Equivalent inner products and the Poincaré inequality
    • Semi-norms and topologization of spaces C^m(Ω) (0 ≤ m ≤ ∞) and
  • The Open Mapping Theorem
    • Linear operators
    • Open mapping theorem and its corollaries
    • Strong extensions of linear operators
    • Example: Strong extensions of differential operators
  • Uniform Boundedness Theorem
    • Uniform boundedness theorem
    • Corollaries of the uniform boundedness theorem
    • Banach-Sake theorem
  • Riesz Representation Theorem
    • The shortest distance problem
    • Riesz representation theorem
    • Inflexibility of Hilbert spaces
    • The Lax-Milgram theorem
  • Hahn-Banach Extension Theorem
    • Hahn-Banach theorem and its corollaries
    • Reflexive Banach spaces
  • Weak and Weak*-Convergence
    • Weak convergence and weak completeness
    • Weak*-convergence and weak*-completeness
    • Weak* and weak sequential compactness
  • Dual and Adjoint Operators and Their Weak Extensions
    • Dual and adjoint operators of bounded linear operators
    • Adjoint operators of linear operators in Hilbert spaces
    • Closed extension of linear operators in Hilbert spaces
    • Weak extension of differential operators
  • Linear Equations in Hilbert Spaces
    • Linear compact operator equations
    • Eigenvalues of compact linear operators

• APPENDIX B Introduction to Sobolev Spaces
  • L_p(Ω) Spaces
    • L_p(Ω) (1 ≤ p ≤ ∞) spaces and their completeness
    • Separability of L_p(Ω) (1 ≤ p ≤ ∞)
    • Reflexiveness of L_p(Ω) (1 ≤ p ≤ ∞)
    • Sequential compactness of bounded sets in L_p(Ω) (1 ≤ p < ∞)
  • Smoothing Operator and Mean-Value Approximation
    • Definition of smooth operators
    • Mean-value approximation in L_p(Ω)
    • Fundamental lemma of the variational principle
    • Smooth approximation in
    • Partition of unity
  • Generalized Derivatives
    • Weak generalized derivatives
    • Strong generalized derivatives
    • Local dependence of generalized derivatives
    • Operational rules of generalized derivatives
  • Sobolev Spaces
    • Sobolev spaces: definition and properties
    • Interpolation inequality of intermediate derivatives and the density of in W^m,p(Ω)
    • Boundary values of functions in W^m,p(Ω)
    • Embedding theorem

• Subject Index