Vol. 92, No. 2, 1981

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Convergence theorems for some scalar valued integrals when the measure is Nemytskii

André (Piotrowsky) De Korvin and C. E. Roberts

Vol. 92 (1981), No. 2, 329–343

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 (t) = ϕ[x(t)] or (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.

Mathematical Subject Classification 2000
Primary: 47H15
Secondary: 28A20, 28A33, 60G99
Received: 24 October 1979
Revised: 12 April 1980
Published: 1 February 1981
André (Piotrowsky) De Korvin
C. E. Roberts