Some further generalizations of the Hyers-Ulam-Rassias stability of functional equations

In this paper we study the Hyers-Ulam-Rassias stability theory by considering the cases where the approximate remainder phi is defined by f (x - y) - f ( x ) - f ( y ) = phi (x, y) (For Allx,y epsilon G), (1) f (x * y) - g ( x ) - h ( y ) = phi (x, y) (For Allx, y epsilon G),(2) 2f((x * Y) (1/2)) - f(x) - f( y ) = phi (x, y) (For Allx, y epsilon G), (3) where (G, *) is a certain kind of algebraic system, E is a real or complex Hausdorff topological vector space, and f, g, h are mappings from G into E. We prove theorems for the Hyers-Ulam-Rassias stability of the above three kinds of functional equations and obtain the corresponding error formulas. (C) 2001 Academic Press.