摘要
设λ的二次三项式λ２－ａλ－ｂ的两个零点为λ１＝ｒ，λ２＝ｓ（ａ及ｂ为实数）．对０＜ｒ＜ｓ，ｒ＜０＜ｓ≠－ｒ及ｒ＝ｓ≠０这三种情形，Ｊ．Ｍａｔｋｏｗｓｋｉ与ＷｅｉｎｉａｎＺｈａｎｇ在“Ｍｅｔｈｏｄｏｆｃｈａｒａｃｔｅｒｉｓｔｉｃｓｆｏｒｆｕｎｃｔｉｏｎａｌｅｑｕａｔｉｏｎｓｉｎｐｏｌｙｎｏｍｉａｌｆｏｒｍ”一文中给出了迭代函数方程ｆ２（ｘ）＝ａｆ（ｘ）＋ｂｘ，对任ｘ∈Ｒ；ｆ∈Ｃ０（Ｒ，Ｒ）（１）的通解，并证明了当ｒ及ｓ非实数时方程（１）无解．对ｒ＝－ｓ≠０的情形，Ｍ．Ｋｕｃｚｍａ已给出了方程（１）的通解．本文则对ｒ＜ｓ＜０及ｒｓ＝０这两种情形给出了方程（１）的通解．此外。
Let a and b be real numbers, and let the two zero points of the quadratic polynomial λ 2-aλ-b of λ be λ 1=r and λ 2=s. For the three cases 0<r<s,r<0<s≠-r, and r=s≠0, J.Matkowski and Weinian Zhang obtained general solutions of the iterated functional equationf 2(x)=af(x)+bx, for all x∈R;f∈C 0(R,R)(1)in their paper “Method of characteristics for functional equation in polynomial form”, and proved that there are no solutions of equation (1) when r and s are not real numbers. For the case r=-s≠0, M.Kuczma has given general solutions of (1). And in this paper, for the remaining two cases r<s<0 and rs=0, we give general solutions of (1). Moreover, we give a simple proof about general solutions of (1) in the case r<0<s≠-r.
基金
国家自然科学基金
关键词
迭代函数方程
动力系统
通解
函数方程
continuous function, iterated functional equation, dynamical system, general solution.
