Continuous solutions of a second order iterative equation

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 $\mathbb{R}$.