Smooth solutions of a class of iterative functional equations

TL;DR

This paper proves the existence of smooth solutions to a class of iterative functional equations involving derivatives, iterates, and nonlinear terms, using the Fiber Contraction Theorem under certain conditions.

Contribution

It introduces a method to establish $C^1$ solutions for complex iterative functional equations with nonlinear components, expanding the theoretical understanding.

Findings

Existence of $C^1$ solutions under specified conditions

Application of Fiber Contraction Theorem to functional equations

Framework for solving equations with iterates and nonlinear terms

Abstract

Imposing some conditions on derivatives of the known functions, using the Fiber Contraction Theorem we prove the existence of $C^1$ solutions of a class of iterative functional equations which involves iterates of the unknown functions and a nonlinear term.