I'm currently trying to read a paper on Fixed Points of Asymptotic Contractions" by W.A. Kirk. A small excerpt can be seen here. Those with accounts on the Elsevier page can see the whole content.
In essence, the field is nonlinear analysis. The paper (and other papers by the same author) contains terms like guage function, upper semicontinuous, ultrapower, nontrivial ultrafilter, etc.
Looking up wikipedia for all these terms is not helpful at all. I do not get any sort of understanding apart from the basic definitions, which I soon forget. Background: I have studied functional analysis (mostly linear bounded functions, etc). I want to be able to understand this paper, without doing tons of more studying (to be able to understand the arguments at least).
Could someone kindly point me to set of online notes or something of the sort that would quickly get me acquainted with these terms and their applications, so that I could proceed with understanding the paper?
EDIT: Book recommendations for non-linear functional analysis are also welcome.
Thanks in advance!