Asymptotically periodic and continuous solutions of a polynomial-like iterative equation

Abstract

In the present paper, using Schauder’s fixed point theorem and the Banach contraction principle, we discuss the polynomial-like iterative equation

$$\begin{aligned} \lambda _1f(x)+\lambda _2f^2(x)+\cdots +\lambda _nf^n(x)=F(x),\,\,x\ge 0. \end{aligned}$$

The sufficient conditions for the existence, uniqueness, and stability of the asymptotically periodic and continuous solutions are presented. An example of the existence, uniqueness, and stability of asymptotically periodic and continuous solutions of an iterative equation is examined.