Higher regularity of solutions of an iterative functional equation

TL;DR

This paper proves the existence of highly regular solutions for a class of nonlinear second-order iterative functional equations using advanced mathematical tools.

Contribution

It establishes the existence of bounded $C^n$ solutions with bounded derivatives for nonlinear iterative functional equations, extending regularity results.

Findings

Existence of bounded $C^n$ solutions proven

Solutions have bounded derivatives up to order $n$

Application of Fiber Contraction Theorem and Faà di Bruno's Formula

Abstract

In this paper, we investigate the existence of $C^n$, $n\in \mathbb{N}^+$, solutions for a class of second-order iterative functional equations involving iterates of the unknown function and a nonlinear term. Applying the Fiber Contraction Theorem and Fa\`a di Bruno's Formula, we establish the existence of bounded $C^n$ solutions with bounded derivatives of order from $1$ to $n$.