Recently, in collaboration with Martin  and Sundaresan , I obtained a characterization of certain classes of non-linear functionals defined on spaces of measurable functions (see also ). The functionals in question had the form
with a continuous “kernel” φ: R → R,or
with a separately continuous kernel φ: R2 → R. There are direct applications of this work to the theory of generalized random processes in probability (see ) and to the theory of fading memory in continuum mechanics . However, the main motivation for these studies was an interest in possible application to the functional analytic study of non-linear differential equations. From the standpoint of this latter application it would also be desirable to characterize the broader class of functionals having the form
where the kernel φ: R × T → R satisfies “Carathéodory conditions”.