Abstract and Applied Analysis
Volume 2008 (2008), Article ID 136592, 11 pages
doi:10.1155/2008/136592

Functional inequalities associated with additive mappings

Jaiok Roh1 and Ick-Soon Chang2

1Department of Mathematics, Hallym University, Chuncheon 200-702, South Korea
 2Department of Mathematics, Mokwon University, Daejeon 302-729, South Korea

Abstract

The functional inequality ‖f(x)+2f(y)+2f(z)‖≤‖2f(x/2+y+z)‖+ϕ  (x,y,z) (x,y,z∈G) is investigated, where G is a group divisible by 2,f:G→X and ϕ:G3→[0,∞) are mappings, and X is a Banach space. The main result of the paper states that the assumptions above together with (1) ϕ(2x,−x,0)=0=ϕ(0,x,−x) (x∈G) and (2) limn→∞(1/2n)ϕ(2n+1x,2ny,2nz)=0, or limn→∞2nϕ(x/2n−1,y/2n,z/2n)=0  (x,y,z∈G), imply that f is additive. In addition, some stability theorems are proved.