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Modern nonlinear equations. (English) Zbl 0148.28202
New York etc.: McGraw-Hill Book Comp. xv, 473 p. (1967).
This is a survey of recent developments in a number of fields of analysis. It is a sequel to “Nonlinear Mathematics” by the author and J. Bram [New York: McGraw-Hill (1964; Zbl 0198.00102)], which dealt with algebraic and differential equations. The present volume treats the following types of equations: operator, functional, difference, delay-differential, integral, integro-differential, and stochastic differential. A great many specific topics are discussed. We shall mention a few of them.
The treatment of operator equations includes basic material on functional analysis, implicit function and fixed point theorems, monotone and potential operators, the Newton, Ritz and Galerkin methods.
A number of particular functional equations, e. g., \(g(x+1)-g(x) = \log x\), are studied. Some general theory and a connection with optimization are given. Functional inequalities are mentioned briefly.
The chapter on difference equations emphasizes analogies with differential equations. Principal topics are stability and optimization, which feature also in the treatment of delay-differential equations.
The chapter on integral equations, written by D. H. Hyers, gives a general introduction, with sections on Volterra equations, Hammerstein theory, parameter problems, and Zarantanello’s method of contractive averaging.
Several practical integro-differential equations, pertaining to queues, star brightness, aircraft wings, transport theory, and atomic scattering, are discussed. The methods of upper and lower functions, and of successive approximations, are presented.
The chapter on stochastic differential equations, written by R. Syski, covers a wide range of topics, including random initial conditions, random forcing functions, random coefficients, Brownian motion, and diffusion processes.
Throughout the book, the writing is clear and the analysis well organized. Although the discussions of individual topics are generally
Reviewer: P. M. Anselone

MSC:
00A05 Mathematics in general
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
39Axx Difference equations
39Bxx Functional equations and inequalities
45-02 Research exposition (monographs, survey articles) pertaining to integral equations
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
60Hxx Stochastic analysis
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis