Non-Archimedean空间中三次方程的稳定性

Stability of the cubic functional equation in a non-Archimedean space

研究了三次方程Ef(x,y):=f(mx+y)+f(mx-y)-mf(x+y)-mf(x-y)-2m(m2-1)f(x)=0在non-Archimedean空间中的稳定性问题,其中m为任一实数且m≠1且m≠0.证明了如果G为一赋范空间,X为一完备的non-Archimedean空间.f:G→X是一个映射,δ>0,使得当x,y∈G时,有‖Ef(x,y)‖≤δ(ρ(‖x‖3)+ρ(‖y‖3)),其中ρ:[0,+∞)→[0,+∞)满足ρ(|m|t)≤ρ(|m|)ρ(t)及ρ(|m|)<|m|,那么存在惟一三次映射Q:G→X,使得‖f(x)-Q(x)‖≤δ2|m|3ρ(‖x‖3)(x∈G).

Abstract ： The stability of the cubic functional equation Ef（x,y）：=f（mx＋y）＋f（mx-y）-mf（x＋y）-mf（x-y）-2m（m2-1）f（x）=0 is investigated, m is a real constant with m≠1 and m≠0. It is proved that if G is a normed linear space, X is a complete non-Archimedean space.f：G→X is a mapping and δ 〉 0 such that ‖Ef（x,y）‖≤δ（ρ（‖x‖3）＋ρ（‖y‖3））for all x,y∈G, where ρ：[0,＋∞）→[0,＋∞）is a function with ρ （｜m｜）ρ（t）及ρ（｜m｜）〈｜m｜,then there exists a unique cubic mapping Q：G→X such that ‖f（x）-Q（x）‖≤δ2｜m｜3ρ（‖x‖3）（x∈G）.