Int. Journal of Math. Analysis, Vol. 3, 2009, no. 38, 1881 - 1892 On the Convergence of a New Two-Step Iteration in the Class of Quasi-Contractive Operators ˙ Isa Yildirim 1 , Murat ¨ Ozdemir and H¨ ukmi Kiziltun¸ c Department of Mathematics, Faculty of Science Atat¨ urk University, Erzurum, 25240, Turkey Abstract In the class of quasi-contractive operators satisfying Zamfirescu’s conditions, the most used fixed point iterative methods, that is, the Krasnoselskij, Mann and Ishikawa iterations, are all known to be con- vergent to the unique fixed point. In this paper, we consider a new two- step iterative scheme for approximating fixed points of quasi-contractive operators and we show that Krasnoselskij, Mann, Ishikawa and new two- step iterations are equivalent for quasi-contractive operators satisfying Zamfirescu’s conditions. Mathematics Subject Classifications: 47H10 Keywords: Krasnoselskij iteration, Mann iteration, Ishikawa iteration, two-step iteration, quasi-contractive operators 1. Introduction and Preliminaries In the last three decades many papers have been published on the iterative approximation of fixed points for certain classes of operators, using the Kras- noselskij, Mann and Ishikawa iteration methods, see [2]. These papers were motivated by the fact that, under weaker contractive type conditions, the Picard iteration (or the method of successive approximations), need not con- verge to the fixed point of the operator in question. However, there exist large classes of operators, as for example that of quasi-contractive type operators introduced in [2, 3, 5, 6], for which not only the Picard iteration, but also the Ishikawa iterations can be used to approximate the fixed points. Let K be a nonempty convex subset of a normed space E and T : K → K be a mapping. F (T )= {x * ∈ K : Tx * = x * } is the set of fixed point of T . For 1 isayildirim@atauni.edu.tr