Functional inequalities associated with additive mappings

The functionally inequality parallel to f(x) + 2f(y) + 2f(z)parallel to <= parallel to 2f(x/2 + y +z)parallel to + phi(x,y,z) (x,y,z is an element of G) is investigated, where G is a group divisible by 2,f : G -> X and phi : G(3)->[0,infinity) are mapping, and X is a Banach space. The main result of the paper states that the assumptions above together with (1) phi(2x,-x,0) = 0 = phi(0,x,-x)(x is an element of G) and (2) lim(n ->infinity)(1/2(n))phi(2(n+1)x,2(n)y,2(n)z) = 0, or lim(n ->infinity)2(n)phi(x/2(n-1),y/2(n),z/2(n)) = 0(x,y,z is an element of G), imply that f is additive. In addition, some stability theorems are proved. (C) 2008 J. Roh and I.-S. Chang.