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1. Basic Concepts in the Solution of Equations

1.1 Operator Equations

1.2 Review of Basic Ideas

1.3 Inverse Operators and the Solvability of Equations

1.4 Existence Theorems

2. Some Iterative and Direct Techniques for Nonlinear Operator Equations

2.1 Introduction--Remarks on the Theory of Convergence

2.2 Iterative Methods (useful for Bounded Operators)

2.3 Direct Methods

3. Functional Equations

3.1 Introduction

3.2 Examples of Functional Equations

3.3 Continuous, Discontinuous, and Measurable Solutions--Cauchy's Additive Equation

3.4 Some Generalizations

3.5 Measurable and Bounded Solutions--A Generalization of an Equation due to Banach

3.6 Analytic Solutions of a Generalization of a Trigonometric Identity

3.7 A Continuous Strictly Increasing Solution

3.8 Various Types of Solutions--Even, Positive, Entire Exponential, Bounded, and Periodic

3.9 Convex Solution of g(x+1) - g(x)= log x

3.10 Three Examples--Series Expansion, Reduction to Simpler Forms, and Successive Approximations

3.11 A General Method of Solution

3.12 Functional Inequalities

3.13 Optimization and Functional Equations

4. Nonlinear Difference Equations

4.1 Introduction

4.2 Linear Difference Equations

4.3 A General Difference Equation of the First Order

4.4 Solutions of Some Nonlinear Equations

4.5 Stability of Some Difference Approximations

4.6 Stability

4.7 Differential-difference Equations--An Example

4.8 Optimization and Difference Equations

5. Delay-Differential Equations

5.1 Introduction

5.2 A Linear Equation of Neutral Type

5.3 What is a Solution?

5.4 Linear Delay-Differential Equations

5.5 Existence and Some Methods of Solution

5.6 Perturbation Methods

5.7 Solution and Stability of a Nonlinear Delay Equation

5.8 A Brief Discussion of Stability, with Examples

5.9 Optimization Problems with Delay

6. Integral Equalities

6.1 Introduction

6.2 Some Examples of Physical Problems Leading to Integral Equations

6.3 Linear Integral Equations of Fredholm Type

6.4 Linear Equations with Symmetric Kernels

6.5 Nonlinear Volterra Equations

6.6 Hammerstein's Theory

6.7 Nonlinear Integral Equations Containing a Parameter--Branching of Solutions

6.8 Some Results on Nonlinear Operator Equations

7. Integrodifferential Equations

7.1 Introduction

7.2 Examples

7.3 An Example of an Integrodifference Equation

7.4 Brief Illustration of Existence

7.5 A Nonlinear Equation--Boltzmann's Equation

7.6 Some Methods of Solution

7.7 Stability

8. Stochastic Differential Equations

8.1 Introduction

8.2 Random Initial Conditions

8.3 Random Forcing Function

8.4 Random Coefficients

8.5 General Properties

8.6 Appendix: Probability Theory

Index

1.1 Operator Equations

1.2 Review of Basic Ideas

1.3 Inverse Operators and the Solvability of Equations

1.4 Existence Theorems

2. Some Iterative and Direct Techniques for Nonlinear Operator Equations

2.1 Introduction--Remarks on the Theory of Convergence

2.2 Iterative Methods (useful for Bounded Operators)

2.3 Direct Methods

3. Functional Equations

3.1 Introduction

3.2 Examples of Functional Equations

3.3 Continuous, Discontinuous, and Measurable Solutions--Cauchy's Additive Equation

3.4 Some Generalizations

3.5 Measurable and Bounded Solutions--A Generalization of an Equation due to Banach

3.6 Analytic Solutions of a Generalization of a Trigonometric Identity

3.7 A Continuous Strictly Increasing Solution

3.8 Various Types of Solutions--Even, Positive, Entire Exponential, Bounded, and Periodic

3.9 Convex Solution of g(x+1) - g(x)= log x

3.10 Three Examples--Series Expansion, Reduction to Simpler Forms, and Successive Approximations

3.11 A General Method of Solution

3.12 Functional Inequalities

3.13 Optimization and Functional Equations

4. Nonlinear Difference Equations

4.1 Introduction

4.2 Linear Difference Equations

4.3 A General Difference Equation of the First Order

4.4 Solutions of Some Nonlinear Equations

4.5 Stability of Some Difference Approximations

4.6 Stability

4.7 Differential-difference Equations--An Example

4.8 Optimization and Difference Equations

5. Delay-Differential Equations

5.1 Introduction

5.2 A Linear Equation of Neutral Type

5.3 What is a Solution?

5.4 Linear Delay-Differential Equations

5.5 Existence and Some Methods of Solution

5.6 Perturbation Methods

5.7 Solution and Stability of a Nonlinear Delay Equation

5.8 A Brief Discussion of Stability, with Examples

5.9 Optimization Problems with Delay

6. Integral Equalities

6.1 Introduction

6.2 Some Examples of Physical Problems Leading to Integral Equations

6.3 Linear Integral Equations of Fredholm Type

6.4 Linear Equations with Symmetric Kernels

6.5 Nonlinear Volterra Equations

6.6 Hammerstein's Theory

6.7 Nonlinear Integral Equations Containing a Parameter--Branching of Solutions

6.8 Some Results on Nonlinear Operator Equations

7. Integrodifferential Equations

7.1 Introduction

7.2 Examples

7.3 An Example of an Integrodifference Equation

7.4 Brief Illustration of Existence

7.5 A Nonlinear Equation--Boltzmann's Equation

7.6 Some Methods of Solution

7.7 Stability

8. Stochastic Differential Equations

8.1 Introduction

8.2 Random Initial Conditions

8.3 Random Forcing Function

8.4 Random Coefficients

8.5 General Properties

8.6 Appendix: Probability Theory

Index

Library of Congress subject headings for this publication: Difference equations, Integral equations, Functional equations, Nonlinear theories