A primer of nonlinear analysis

✍ Scribed by A Ambrosetti; G Prodi

• Publisher: Cambridge University Press
• Year: 1995
• Tongue: English
• Leaves: 179
• Series: Cambridge studies in advanced mathematics, 34
• Category: Library

✦ Synopsis

This is an introduction to nonlinear functional analysis, in particular to those methods based on differential calculus in Banach spaces. It is in two parts; the first deals with the geometry of Banach spaces and includes a discussion of local and global inversion theorems for differential mappings. In the second part, the authors are more concerned with bifurcation theory, including the Hopf bifurcation. They include plenty of motivational and illustrative applications, which indeed provide much of the justification of nonlinear analysis. In particular, they discuss bifurcation problems arising from such areas as mechanics and fluid dynamics

✦ Table of Contents

Content: Preliminaries and notation --
Differential calculus --
Frechet and Gateaux derivatives --
Continuity and differentiability of Nemitski operators --
Higher derivatives --
Partial derivatives, Taylor's formula --
Local inversion theorems --
The Local Inversion Theorem --
The Implicit Function Theorem --
A stability property of orbits --
Global inversion theorems --
The Global Inversion Theorem --
Global inversion with singularities --
Semilinear Dirichlet problems --
Problems at resonance --
Problems with asymmetric nonlinearities --
Bifurcation results --
Some elementary examples --
The Lyapunov-Schmidt reduction --
Bifurcation from the simple eigenvalue --
A bifurcation theorem from a multiple eigenvalue --
Bifurcation problems --
The rotating heavy string --
The Benard problem --
Small oscillations for second-order dynamical systems --
Water waves --
Periodic solutions of a semilinear hyperbolic equation --
Bifurcation of periodic solutions --
The Hopf bifurcation --
Nonlinear oscillations of autonomous systems --
The Lyapunov Centre Theorem --
The restricted three-body problem.