Characterization of Non-Linear Transformations Possessing Kernels

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Abstract
Recently, in collaboration with Martin [10] and Sundaresan [11], I obtained a characterization of certain classes of non-linear functionals defined on spaces of measurable functions (see also [12]). The functionals in question had the form (1.1) with a continuous “kernel” φ: R → R,or (1.2) 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 [8]) and to the theory of fading memory in continuum mechanics [3]. 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 (1.3) where the kernel φ: R × T → R satisfies “Carathéodory conditions”.

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