Thomas L. Saaty

Nonlinear equations have existed for hundreds of years; their systematic study, however, is a relatively recent phenomenon. This volume, together with its companion, Nonlinear Mathematics (Dover 64233-X), provides exceptionally comprehensive coverage of this recently formed area of study. It encompasses both older and more recent developments in the field of equations, with particular emphasis on nonlinear equations because, as Professor Saaty, maintains, "that is what is needed today."
Together the two volumes cover all the major types of classical equations (except partial differential equations, which require a separate volume). This volume includes material on seven types: operator equations, functional equations, difference equations, delay-differential equations, integral equations, integro-differential equations and stochastic differential equations. Special emphasis is placed on linear and nonlinear equations in function spaces and on general methods of solving different types of such equations.
Above all, this book is practical. It reviews the variety of existing types of equations and provides methods for their solution. It is meant to help the reader acquire new methods for formulating problems. Its clear organization and copious references make it suitable for graduate students as well as scientists, technologists and mathematicians.
"...a welcome contribution to the existing literature..." ? Math. Reviews.

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