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International Journal of Mathematics and Mathematical Sciences
Volume 2005, Issue 12, Pages 1889-1898

On functions with the Cauchy difference bounded by a functional. Part II

Institute of Mathematics, Faculty of Mathematics, Physics and Chemistry, University of Silesia, 14 Bankowa Street, Katowice 40-007, Poland

Received 10 December 2004; Revised 17 May 2005

Copyright © 2005 Włodzimierz Fechner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We are going to consider the functional inequality f(x+y)f(x)f(y)ϕ(x,y), x,yX, where (X,+) is an abelian group, and ϕ:X×X and f:X are unknown mappings. In particular, we will give conditions which force biadditivity and symmetry of ϕ and the representation f(x)=(1/2)ϕ(x,x)+a(x) for xX, where a is an additive function. In the present paper, we continue and develop our earlier studies published by the author (2004).