How should one begin one's study of fixed point theory, especially of multi-valued maps?
What background --- in topology, analysis, functional analysis, algebra, and set theory --- should one have?
Should one begin with metric fixed point theory or topological fixed point theory?
What background is needed for discrete fixed point theory?
Which book(s) should one start with?
Is fixed point theory taught at the undergrad level anywhere in North America, Europe, or Australia?
On
Andrew's McLennan paper Advanced Fixed Point Theory for Economics was removed but still available on Web Archive at https://web.archive.org/web/20130430063940/http://cupid.economics.uq.edu.au/mclennan/Advanced/advanced_fp.pdf
When it comes to topological fixed point theories, such as Brouwer fixed point theorem, Borsuk-Ulam fixed point theorem and Lefschetz fixed point theory, a great start is Guillemin & Pollack's Differential Topology. It's a wonderful book, and you can read it if you have some background in multivariable calculus or a bit of differential geometry.