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The notion of (p, k)-epi mappings is introduced. The properties of such mappings are studied and the results obtained are applied to some differential equations.

The change of variables formula for the Riemann integral is discussed and a theorem is proved which perhaps compares favorably with its counterpart in Lebesgue theory.

1. The famous Hadamard three-circles theorem of the complex function theory has been generalized to solutions of elliptic and parabolic equations. For references as well as for some interesting applications we refer to [3]. The purpose of this note is to show that (a) three circles (spheres)-theorems lead naturally to a sharpened version of the boundary point maximum principle (see

A classical theorem of Marcinkiewicz states that a function is Perron integrable iff it has one continuous major and one continuous minor function. Using an elaboration of this remarkable theorem three applications are made; to obtain a new proof of a recent characterization of the Perron integral, to proofs of some theorems on interchange of limits and integration and to extend classical existence theorems for ordinary differential equations.