Cauchy product of iterative functional equation

This manuscript examines the existence and uniqueness of differentiable and continuous solutions of the iterative functional equation of the form

$$\begin{aligned} \sum \limits _{i=0}^{n}\lambda _{i}f^{i}(\varkappa )f^{n-i}(\varkappa )= F (\varkappa ), \quad \varkappa \in [a,b], \end{aligned}$$

where \(\lambda _{i}\)’s are real constants and \( F \) is a given function. The novelty of this work lies in the generalization of the iterative root problem when n is even and all \(\lambda _i\)’s are zero except for \(\lambda _{n/2}\). This generalization offers the advantage of covering a wider class of functional equations. Numerical examples are presented to validate the existence results, and the stability of each solution is thoroughly analyzed.