Chapter 25

Nonlinear OperatorsIn this chapter, is a discussion of various kinds of nonlinear operators. Some standardreferences on these operators are [39], [40], [22], [24], [13], [91], [116], [25] and referenceslisted there. The most important examples of these operators seem to be due to Brezis inthe 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 mainlyfrom Barbu [13]. I am trying to include all the necessary basic results such as fixed pointtheorems which are needed to prove the main theorems and also to re write in a mannerunderstandable to me.

It seems like the main issue is the following. When does ⟨ fn,xn⟩ converge to ⟨ f ,x⟩given that fn and xn both converge weakly to f and x respectively? There is no problemin finite dimensions because in finite dimensions, there is only one meaning for conver-gence. However, in infinite dimensions, there certainly is a problem as can be instantlyrealized by consideration of the Riemann Lebesgue lemma, for example. You know that∫

π

−πf (x)sin(nx)dx→ 0 so sin(nx) converges weakly to 0 but

∫π

−πsin2 (nx)dx certainly

does not converge to 0.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. Whenthe operator is linear, there usually is no problem because the graph is a subspace and soif 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. Considerationof when this happens is the purpose of this catalogue of nonlinear operators, and also togeneralize to set valued operators. First is a section on single valued nonlinear operatorsand then the case of set valued nonlinear operators is discussed.

25.1 Some Nonlinear Single Valued OperatorsHere is an assortment of nonlinear operators which are useful in applications to nonlinearpartial differential equations. Generalizations of the notion of a pseudomonotone map willbe presented later to include the case of set valued pseudomonotone maps. This is on thesingle valued version of some of these and these ideas originate with Brezis in the 1960’s.A good description is given in Lions [91].

Definition 25.1.1 For V a real Banach space, A : V → V ′ is a pseudomonotone map ifwhenever

un ⇀ u (25.1.1)

andlim sup

n→∞

⟨Aun,un−u⟩ ≤ 0 (25.1.2)

it follows that for all v ∈V,

lim infn→∞⟨Aun,un− v⟩ ≥ ⟨Au,u− v⟩. (25.1.3)

The half arrows denote weak convergence.

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