二元混合五次函数方程的稳定性

The Hyers-Ulam-Rassias stability of a mixed quintic functional equation of two variables

设X和Y分别是实向量空间和实Banach空间,映射f:X2→Y称为二元混合五次函数是指任给x1,x2,y1,y2∈X都满足方程f(x1+x2,2y1+y2)+f(x1+x2,2y1-y2)+f(x1-x2,2y1+y2)+f(x1-x2,2y1-y2)=4f(x1,y1+y2)+4f(x2,y1+y2)+4f(x1,y1-y2)+4f(x2,y1-y2)+24f(x1,y1)+24f(x2,y1)。给出了二元混合五次方程的一般解,并证明了它的Hyers-Ulam-Rassias稳定性。

Let X be a vector space and Y be a Banach space over the real field, R. A mapping f： X2→Y from X2 into Y is called a mixed quintic functional equation of two variables if it satisfies that f（x1 ＋x E, 2yl ＋ Y2 ） ＋f（xl ＋ X2 ,2y1 -Y2 ） ＋ f（x1 -X2, 2Yl ＋Y2） ＋f（x1-x2, 2y1 -y2） =4f（x1 ,y1 ＋y2） ＋4f（x2, y1 ＋y2） ＋4f（x1 ,y1 -y2） ＋4f（x2, y1 -y2） ＋ 24f（x1, y1 ） ＋24f（x2, Yl ） for all x1 , x2, Yl, Y2 ∈X. The general solution of the mixed quintic functional equation of two variables is obtained and the Hyers-Ulam-Rassias stability for it is proved.