Periodic and continuous solutions of a polynomial-like iterative equation

Abstract

Schauder’s fixed point theorem and the Banach contraction principle are used to study the polynomial-like iterative functional equation $$\begin{aligned} \lambda _1f(x)+\lambda _2f^2(x)+\cdots +\lambda _n f^n(x)=F(x). \end{aligned}$$ We give sufficient conditions for the existence, uniqueness, and stability of the periodic and continuous solutions. We examine the monotonicity, convexity, and differentiability of the solutions of the family \(2f(x)+\lambda f^2(x)=\sin (x)\), (\(\lambda \in [0,1]\)).