Global Solutions for Leading Coefficient Problem of Polynomial-like Iterative Equations

For some technical difficulties many known results on the existence of solutions of the polynomial-like iterative equation, even though they are given globally on a closed interval between two fixed points, require the coefficient of the first order iterate term to be large. Recently, great attentions have been paid to the so-called leading coefficient problem, i.e., the existence of solutions under the most natural assumption that the coefficient of the highest order iterate term, called the leading coefficient, does not vanish but plays the main role. Some results on solutions near a fixed point were obtained. In this paper we prove the existence of continuous solutions globally on a closed interval between two fixed points, which are approximated by sequences of solutions which are locally linear near an end-point.