In this study we examine conditions for the convergence of two-step methods generated by point-to-point operators. Iterations of this type have a great importance in optimization theory, stability analysis of dynamic systems and many other fields of applied mathematics. The speed of convergence is also examined using the theory of majorants in a Banach space setting. The monotone convergence of these methods is also examined in a partially ordered topological space. Special cases of our results reduce to corresponding results already in the literature. In particular, relevant work can be found in I.K. Argyros, Appl. Math. Comput. 61 (1) (1994) 51–69, and F. Szidarovasky and O. Palusinski, Appl. Math. Comput. 64 (1994) 115–119 and references therein. Some applications are also given to the solution of systems of nonlinear integral equations of Uryson-type.