The functional equations
are considered for functions mapping a real normed space into a real Banach space . The stability of the equation (1) was proved by J. M. Rassias [Geometry, Analysis and Mechanics, World Scientific, 365-376 (1994; Zbl 0842.39014)]. For a fixed the stability of equations (1), (2) on a restricted domain is proved. As a consequence the following results are obtained.
Corollary 1. A mapping fulfils (1) iff the asymptotic condition
Corollary 2. A mapping fulfils (2) iff the asymptotic conditions