The authors consider several problems concerning iterative functional equations. More exactly the existence and uniqueness of iterative roots is investigated in different classes of functions, such as strictly increasing homeomorphisms of a real interval, piecewise strictly monotonic, differentiable or Morse-Smale diffeomorphisms. There is also a generalization of the iterative roots problem to the question of solving iterative equations of polynomial type, i.e. equations of the form , where are constant coefficients. The paper is concluded by some open problems.
Apparently, the authors are not aware of the existence of the monographic book devoted to similar topics by M. Kuczma, B. Choczewski and R. Ger [Iterative functional equations (1990; Zbl 0703.39005)].