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

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).