On a class of composite functional equations in a single variable

Summary We prove a general result on continuous functions of the typef: (0, ) (0, ) which satisfy the functional equationf(xf(x)]^p) = (f(x))^{p + 1}, wherep is an arbitrary fixed real number. Applying this result we determine all continuous solutionsf: 0, ) 0, ) forp > 0, as well as all the continuous solutionsf: for a positive integerp. Forp = 1 this equation is relevant to a division model of population.