One of the main potential
applications of Hammerstein operators is a functional analytic study of nonlinear
differential equations. In fact, some connections have already been established with
equations of the form ẋ(t) = ϕ[x(t)] or ẋ(t) = ϕ[x(t),t]. Other applications have
been made to generalized random processes and the theory of fading memory in
continuum mechanics. The main purpose of the present paper is to establish and
study the representation of Hammerstein operators on continuous functions. A
“nonlinear” integral is introduced for this purpose. Convergence theorems for a.e. and
convergence in measure are established and contrasted. The last result of the paper
relates uniform integrability, a key concept in the study of martingales, to essential
ranges, an important concept used to establish the differentiability of some set
functions.
