International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 12, Pages 1889-1898
doi:10.1155/IJMMS.2005.1889

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

Włodzimierz Fechner

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

Abstract

We are going to consider the functional inequality f(x+y)−f(x)−f(y)≥ϕ(x,y), x,y∈X, where (X,+) is an abelian group, and ϕ:X×X→ℝ and f:X→ℝ 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 x∈X, where a is an additive function. In the present paper, we continue and develop our earlier studies published by the author (2004).