一般四次方程的稳定性(英文)

On the Stability of the Generalized Quartic Functional Equation

关于泛函方程的稳定性研究源自Ulam提出的关于群同态一个问题.近几十年来关于Banach空间中的设值泛函问题取得众多成果.研究了一般的四次泛函方程f(ax+by)+f(ax-by)=a2b2f(x+y)+a2b2f(x-y)+2a2(a2-b2)f(x)-2b2(a2-b2)f(y)的通解及其在Banach空间上的Hyers-Ulam-Rassias稳定性(对任意固定的整数a≠-1,0,1,b≠0和a±b≠0).证明了上述方程等价于f(ax+y)+f(ax-y)=a2f(x+y)+a2f(x-y)+2a2(a2-1)f(x)-2(a2-1)f(y),并获得该泛函方程的Hyers-Ulam-Rassias稳定性定理.

The stability problem of functional equations originated from a question of Ulam concerning the stability of group homomorphisms. Set-valued functions in Banach spaces have been developed in the past decades. In this paper, we achieve the general solution and the generalized Hyers-Ulam-Rassias stability theorem for quartic functional equation for any fixed integers a, b with a ≠ 1,0,1, b # 0 and a ＋ b #0. We prove that the functional equation is equivalent to the equation