A Learning Roadmap to the "foundations" of Nonlinear Analysis (and certain specific topics)
Nonlinear analysis is a very large field, and you'd be hard-pressed to find a resource that deals with its many methods in a comprehensive manner. Most resources I know of deal with a subset of methods or methods applied to particular situations (nonlinear elliptic equations, for example).
That said, if you're only looking for an introduction to the subject then there may be some nice books to get you started. Ambrosetti and Prodi's A Primer of Nonlinear Analysis is an introductory text in nonlinear functional analysis and bifurcation theory. I haven't read Primer myself, but I have read a sort-of-sequel by Ambrosetti and Malchiodi, Nonlinear Analysis and Semilinear Elliptic Equations, which builds on the material in Primer and discusses degree theory, fixed-point theory, critical point and Morse methods. My experience with the exposition in that book was very positive, so I think Primer should also live up to that standard.
Another book I might recommend is Chang's Methods in Nonlinear Analysis. It focuses on more topological methods and variational principles.
These books don't even come close to a comprehensive view of nonlinear analysis. If you're really interested in the subject then you'll probably come across problems and tools from all areas of mathematics.
There are the four enormous tomes collectively titled Nonlinear Functional Analysis and Its Applications by Zeidler:
- Part I - Fixed Point Theorems
- Part II(a) - Linear Monotone Operators
- Part II(b) - Non-linear Monotone Operators
- Part III - Variational Methods and Optimization
- Part IV - Applications to Mathematical Physics
The usual (linear) functional analysis prerequisites are contained in the appendix to the first part.
I have only skimmed (even that is an overstatement) over these books and they seem to be thoroughly peppered with real-world examples. They look amazing and I think they author has a nice style.