Interval homeomorphic solutions of a functional equation of nonautonomous iterations

In this paper we study a functional equation of linear combination of nonautonomous iterations, which can be reduced from invariance of an interval homeomorphism under nonautonomous iteration. We discuss existence, uniqueness and continuous dependence for increasing Lipschitzian solutions on a compact interval, and also discuss for bounded increasing Lipschitzian solutions and unbounded ones on the whole \begin{document}$ \mathbb{R} $\end{document} .