Korean J. Math.  Vol 19, No 1 (2011)  pp.
DOI: https://doi.org/10.11568/kjm.2011.19.1.

GENERALIZED QUADRATIC MAPPINGS IN $2d$ VARIABLES

Yeol Je Cho, Sang Han Lee, Choonkil Park

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. 


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ISSN: 1976-8605 (Print), 2288-1433 (Online)

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