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 Sigma(n)(j=1) lambda(j)f(j)(x) = F(x) were given by assuming lambda(1) > 0 and the existence of solutions under the most natural assumption lambda(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. (c) 2008 Elsevier Inc. All rights reserved.