Leading coefficient problem for polynomial-like iterative equations

Because of difficulties in applying fixed point theorems, most of known results on the polynomial-like iterative equation ∑ j = 1 n λ j f j ( x ) = F ( x ) were given by assuming λ 1 > 0 and the existence of solutions under the most natural assumption λ n > 0 is an interesting problem, called “Leading Coefficient Problem”. For this problem locally expansive invertible C 1 solutions are obtained in the expansive case and the non-hyperbolic case in [W. Zhang, On existence for polynomial-like iterative equations, Results Math. 45 (2004) 185–194] and C 0 increasing solutions are constructed in [B. Xu, W. Zhang, Construction of continuous solutions and stability for the polynomial-like iterative equation, J. Math. Anal. Appl. 325 (2007) 1160–1170]. In this paper we discuss C 1 solutions for more combinations between expansive mappings and contractive ones and combinations between increasing mappings and decreasing ones.