Previous results by the author on the connection between three measures of noncompactness obtained for ${{\mathcal{L}}_{p}}$ are extended to regular spaces of measurable functions. An example is given of the advantages of some cases in comparison with others. Geometric characteristics of regular spaces are determined. New theorems for $\left( k,\,\beta \right)$-boundedness of partially additive operators are proved.

A new semilocal convergence result for the Picard method is presented, where the main required condition in the contraction mapping principle is relaxed.

This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a class of functions on [0,1 ]. The attracting orbit is identified and various properties of convergence to this orbit are derived. The results imply that the space-time scaling limit of a certain infinite system of interacting diffusions has universal behavior independent of model parameters.

By now the literature concerning the representation of disjointly additive functionals and operators is quite extensive. A few entries on the subject are [6, 7, 8, 11, 20, 21]. In [7, 8, 17] further references can be found, in [7] the “prehistory” of the subject is also discussed.

To quote a typical result, we may take a 1967 theorem of Drewnowski and Orlicz ([6] Th. 3.2, [17] 12.4) which asserts that, under proper assumptions, an abstract modular (= disjointly countably additive functional) p on a “substantial“ subspace D of L° can be realized by the formula .

If (X, β, μ) is a σ-finite, non-atomic measure space, and ϕ is an increasing non-negative concave function defined on the positive real numbers, we give a set of necessary and sufficient conditions for an additive functional T on the Lorentz space Nϕ to have an integral representation with a Caratheodory kernel. In the special case when T is statistical we classify the functional properties (enjoyed by the kernels) in terms of the Lorentz norm on the space.