A Topological Introduction to Nonlinear Analysis

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Springer Science & Business Media, Dec 12, 2003 - Mathematics - 184 pages
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"The book is highly recommended as a text for an introductory course in nonlinear analysis and bifurcation theory . . . reading is fluid and very pleasant . . . style is informal but far from being imprecise."

—MATHEMATICAL REVIEWS (Review of the First Edition)

Here is a book that will be a joy to the mathematician or graduate student of mathematics---or even the well-prepared undergraduate---who would like, with a minimum of background and preparation, to understand some of the beautiful results at the heart of nonlinear analysis. Based on carefully expounded ideas from several branches of topology, and illustrated by a wealth of figures that attest to the geometric nature of the exposition, the book will be of immense help in providing its readers with an understanding of the mathematics of the nonlinear phenomena that characterize our real world.

New to the second edition: New chapters will supply additional applications of the theory and techniques presented in the book. * Several new proofs, making the second edition more self-contained.

 

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Contents

The Topological Point of View
3
AscoliArzela Theory
9
Brouwer Fixed Point Theory
19
Schauder Fixed Point Theory
23
The Forced Pendulum
29
Equilibrium Heat Distribution
39
Generalized Bernstein Theory
45
Degree Theory
49
A Separation Theorem
95
Compact Linear Operators
99
The Degree Calculation
111
The KrasnoselskiiRabinowitz Bifurcation Theorem
119
Nonlinear SturmLiouville Theory
133
More SturmLiouville Theory
143
Euler Buckling
155
Appendices
161

Brouwer Degree
51
Properties of the Brouwer Degree
55
LeraySchauder Degree
63
Properties of the LeraySchauder Degree
69
The Mawhin Operator
79
The Pendulum Swings Back
85
Bifurcation Theory
93
Singular Homology
163
Additivity and Product Properties
169
Bounded Linear Transformations
175
References
181
Index
183
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