On the general theory of the Cauchy type functional equations with applications in analysis

Abstract

The work deals with the questions of the solvability of the Cauchy type functional equations introduced recently and proved to be not only very much interesting objects of functional analysis but also an important technical tool in solving quite different problems in such diverse fields as integral geometry, partial differential equations, the measure theory, ergodic theory, functional equations. The major part of the results is obtained on the base of the new dynamical method which has been worked out especially for studying functional equations in question. Two of the possible applications in Analysis of the results obtained (in Integral geometry and in the theory of hyperbolic differential equations) are discussed in the final part of the paper.

Remarks on the Cauchy functional equation and variations of it

“…In both cases one may conclude the existence of nonlinear solutions to (1) (even under "strong" regularity conditions) or that (in the case where f is assumed to be defined on the whole space) f must satisfy (1) for all possible pairs (x, y). For a very partial list of related works see Pisot-Schoenberg [87], Aczel-Erdös [4], Jurkat [60], de Bruijn [30], Zdun [105], Kuczma [66], Dhombres-Ger [32], Dhombres [31], Forti [37], Matkowski [73], Sablik [93], Szabo [99], Choczewski et al [20], Alsina and Garc Roig [7], Ger-Sikorska [44], Skof [98], Paneah [82,83], Shalit [94,95], Laohakosol-Pimsert [70] and the references therein.…”