Existence, construction and extension of continuous solutions of an iterative equation with multiplication

The iterative equation is an equality with an unknown function and its iterates, most of which found from references are a linear combination of those iterates. In this paper, we work on an iterative equation with multiplication of iterates of the unknown function. First, we use an exponential conjugation to reduce the equation on R+ to the form of the linear combination on R, but those known results on the linear combination were obtained on a compact interval or a neighborhood near a fixed point. We use the Banach contraction principle to give the existence, uniqueness and continuous dependence of continuous solutions on R+ that are Lipschitzian on their ranges, and construct its continuous solutions on R+ sewing piece by piece. We technically extend our results on R+ to R- and show that none of the pairs of solutions obtained on R+ and R-can be combined at the origin to get a continuous solution of the equation on the whole R, but can extend those given on R+ to obtain continuous solutions on the whole R.