A note on convex solutions to an equation on open intervals

Abstract

The note is concerned with the functional equation

$$\begin{aligned} \lambda _1H_1(f(x))+\lambda _2H_2(f^2(x))+\cdots +\lambda _nH_n(f^n(x))=F(x), \end{aligned}$$

which is a generalised form of the so-called polynomial-like iterative equation. We investigate the existence of nondecreasing convex (both usual and higher order) solutions to this equation on open intervals using the Schauder fixed point theorem. The results supplement those proved by Trif (Aquat Math, 79:315–327, 2010) for the polynomial-like iterative equation by generalising them to a greater extent. This assertion is supported by some examples illustrating their applicability.