GENERALIZED QUADRATIC MAPPINGS IN $2d$ VARIABLES

Abstract

Let X, Y be vector spaces. It is shown that if an even

mapping f:X→Y satisfies f(0)=0,and

2( {2d−2}_C_{d−1} − {2d−2}_C_{d})

f= ( \sum_{j=1}^{2d} x_j ) +

\sum_{ i(j) = 0,1, \sum_{j=1}^{2d} i)j)=d }

f ( \sum_{j=1}^{2d} (-1)^{ i(j)} x_j )

= 2( {2d-1_C_d + {2d-2}_C_{d-1} - {2d-2}_C_d )

\sum_{j=1}^{2d} f(x_j)

for all x_1, ··· , x_{2d} ∈ X, then the even mapping f : X → Y is quadratic.

Furthermore, we prove the Hyers-Ulam stability of the above functional equation in Banach spaces.