Front Cover; Techniques of Functional Analysis for Differential and Integral Equations; Copyright; Contents; Preface; Chapter 1: Some Basic Discussion of Differential and Integral Equations; 1.1 Ordinary Differential Equations; 1.1.1 Initial Value Problems; 1.1.2 Boundary Value Problems; 1.1.3 Some Exactly Solvable Cases; 1.2 Integral Equations; 1.3 Partial Differential Equations; 1.3.1 First Order PDEs and the Method of Characteristics; 1.3.2 Second Order Problems in R2; 1.3.3 Further Discussion of Model Problems; Wave Equation; Heat Equation; Laplace Equation.
1.3.4 Standard Problems and Side Conditions1.4 Well-Posed and Ill-Posed Problems; 1.5 Exercises; Chapter 2: Vector Spaces; 2.1 Axioms of a Vector Space; 2.2 Linear Independence and Bases; 2.3 Linear Transformations of a Vector Space; 2.4 Exercises; Chapter 3: Metric Spaces; 3.1 Axioms of a Metric Space; 3.2 Topological Concepts; 3.3 Functions on Metric Spaces and Continuity; 3.4 Compactness and Optimization; 3.5 Contraction Mapping Theorem; 3.6 Exercises; Chapter 4: Banach Spaces; 4.1 Axioms of a Normed Linear Space; 4.2 Infinite Series; 4.3 Linear Operators and Functionals.
4.4 Contraction Mappings in a Banach Space4.5 Exercises; Chapter 5: Hilbert Spaces; 5.1 Axioms of an Inner Product Space; 5.2 Norm in a Hilbert Space; 5.3 Orthogonality; 5.4 Projections; 5.5 Gram-Schmidt Method; 5.6 Bessel's Inequality and Infinite Orthogonal Sequences; 5.7 Characterization of a Basis of a Hilbert Space; 5.8 Isomorphisms of a Hilbert Space; 5.9 Exercises; Chapter 6: Distribution Spaces; 6.1 The Space of Test Functions; 6.2 The Space of Distributions; 6.3 Algebra and Calculus With Distributions; 6.3.1 Multiplication of Distributions; 6.3.2 Convergence of Distributions.
6.3.3 Derivative of a Distribution6.4 Convolution and Distributions; 6.5 Exercises; Chapter 7: Fourier Analysis; 7.1 Fourier Series in One Space Dimension; 7.2 Alternative Forms of Fourier Series; 7.3 More About Convergence of Fourier Series; 7.4 The Fourier Transform on RN; 7.5 Further Properties of the Fourier Transform; 7.6 Fourier Series of Distributions; 7.7 Fourier Transforms of Distributions; 7.8 Exercises; Chapter 8: Distributions and Differential Equations; 8.1 Weak Derivatives and Sobolev Spaces; 8.2 Differential Equations in D'; 8.3 Fundamental Solutions.
8.4 Fundamental Solutions and the Fourier Transform8.5 Fundamental Solutions for Some Important PDEs; Laplace Operator; Heat Operator; Wave Operator; Schrödinger Operator; Helmholtz Operator; Klein-Gordon Operator; Biharmonic Operator; 8.6 Exercises; Chapter 9: Linear Operators; 9.1 Linear Mappings Between Banach Spaces; 9.2 Examples of Linear Operators; 9.3 Linear Operator Equations; 9.4 The Adjoint Operator; 9.5 Examples of Adjoints; 9.6 Conditions for Solvability of Linear Operator Equations; 9.7 Fredholm Operators and the Fredholm Alternative; 9.8 Convergence of Operators; 9.9 Exercises.
