In this chapter, is a discussion of various kinds of nonlinear operators. Some standard references on these operators are [?], [?], [?], [?], [?], [?], [?], [?] and references listed there. The most important examples of these operators seem to be due to Brezis in the 1960’s and these things have been generalized and used by many others since this time. I am following many of these, but the stuff about maximal monotone operators is mainly from Barbu [?]. I am trying to include all the necessary basic results such as fixed point theorems which are needed to prove the main theorems and also to re write in a manner understandable to me.
It seems like the main issue is the following. When does
The idea behind all of these considerations is that fn is to come from some nonlinear operator which has properties which will allow one to successfully pass to a limit. When the operator is linear, there usually is no problem because the graph is a subspace and so if it is closed, it will also be weakly closed. Thus, if xn → x weakly and Lxn → f weakly, then f = Lx. However, nothing like this happens with nonlinear operators. Consideration of when this happens is the purpose of this catalogue of nonlinear operators, and also to generalize to set valued operators. First is a section on single valued nonlinear operators and then the case of set valued nonlinear operators is discussed.