On General Solutions of the Iterated Functional Equation f2(x)=af(x)+bx

DOI：10.3770/j.issn:1000-341X.1997.01.017

 作者 单位 麦结华 汕头大学数学研究所

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Ｗｅｉｎｉａｎ Ｚｈａｎｇ在“Ｍｅｔｈｏｄ ｏｆ ｃｈａｒａｃｔｅｒｉｓｔｉｃｓ ｆｏｒ ｆｕｎｃｔｉｏｎａｌ ｅｑｕａｔｉｏｎｓ ｉｎ ｐｏｌｙｎｏｍｉａｌ ｆｏｒｍ”一文中给出了迭代函数方程ｆ2（ｘ）＝ａｆ（ｘ）＋ｂｘ，对任ｘ∈Ｒ；ｆ∈Ｃ0

Let a and b be real numbers, and let the two zero points of the quadratic polynomial λ2-aλ-b of λ be λ1＝ｒandλ2=s. For the three cases 0 < r < s, r < 0 < s ≠- r , and r = s≠ 0 , J. Matkow skiand Weinian Zhang obtained general solutions of the iterated functional equation f2（ｘ）＝ａｆ（ｘ）＋ｂｘ, fo r a ll x ∈ R ; f ∈ C0(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