Resumen de Continuous solutions of an iterative-difference equation and Brillou¨et-Belluot�s

Yingying Zeng, Weinian Zhang

• It is an open problem proposed by N. Brillou¨et-Belluot to solve the equation f2(x) = f(x+a)..x. Although some related results have been obtained, the problem has remained open. In this paper we prove that it has no continuous real solutions, finally answering Brillou¨et-Belluot�s problem. Furthermore, we give existence of continuous real solutions for the general equation f2(x) = f(x + a) + x on the whole R in some cases which neither include the equation f2(x) = f(x + a) .. x nor are considered in [J. Difference Equ. Appl. 16(11) (2010), 1237�1255].