Continuous solutions of a second order iterative equation

TL;DR

This paper investigates the existence and construction of continuous solutions for a second order iterative functional equation, employing contraction principles under Lipschitz conditions and recursive methods without them.

Contribution

It introduces new methods for constructing continuous solutions to second order iterative equations, both with and without Lipschitz conditions.

Findings

Existence of continuous solutions under Lipschitz conditions using contraction principle.

Recursive construction of solutions without Lipschitz conditions.

Solutions are valid on the entire real line.

Abstract

In this paper we study the existence of continuous solutions and their constructions for a second order iterative functional equation, which involves iterate of the unknown function and a nonlinear term. Imposing Lipschitz conditions to those given functions, we prove the existence of continuous solutions on the whole $\mathbb{R}$ by applying the contraction principle. In the case without Lipschitz conditions we hardly use the contraction principle, but we construct continuous solutions on $\mathbb{R}$ recursively with a partition of $\mathbb{R}$.