An extension of Chu-Moyer's theorem to two variable functions

In this paper we study the problem of global convergence of the successive approximation method for the pair ([0,1]2,F) under the hypothesis that F(x,y)=(f(x,y),x), where f is a continuous function increasing with respect to the second variable. We demonstrate that, if FixF is totally disconnected, then for the pair ([0,1]2,F) the successive approximation method converges globally if and only if F has no periodic point of prime period two.