  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 $\mathbb{R}$.
