A SOLUTION OF ONE OPEN PROBLEM OF RASSIAS FOR MAPPINGS

Abstract

In this paper we give a solution of the following open problem given by Rassias: Find all the mappings f : R3 -R3, such that V_OABC = V_OA'B'C' for arbitrary points A,B,C in R^3, where A' = f(A), B' = f(B), and C' = f(C). The set of all such mappings can be described on the following way: Choose an arbitrary unimodular 3*3 matrix M, and then for arbitrary point P in R^3 we put f(P) = P' if x' = Mx or x' =-Mx, where x = OP and x' =OP'.