Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 213876, 19 pages doi:10.1155/2012/213876 Research Article Some Generalizations of Jungck’s Fixed Point Theorem J. R. Morales 1 and E. M. Rojas 2 1 Departamento de Matem´ aticas, Universidad de Los Andes, M´ erida 5101, Venezuela 2 Departamento de Matem´ aticas, Pontificia Universidad Javeriana, Bogot´ a, Colombia Correspondence should be addressed to E. M. Rojas, edixonr@gmail.com Received 28 March 2012; Revised 25 July 2012; Accepted 24 September 2012 Academic Editor: A. Zayed Copyright q 2012 J. R. Morales and E. M. Rojas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We are going to generalize the Jungck’s fixed point theorem for commuting mappings by mean of the concepts of altering distance functions and compatible pair of mappings, as well as, by using contractive inequalities of integral type and contractive inequalities depending on another function. 1. Introduction and Preliminary Facts In 1922, Banach introduced his famous result in the metric fixed point theory, the Banach contraction Principle BCP, as follows. Theorem 1.1 see 1. Let M, d be a complete metric space and let S : M → M be a self-map that satisfies the following condition: there is a ∈ 0, 1 such that d ( Sx, Sy ) ≤ ad ( x, y ) BC for all x, y ∈ M. Then, S has a unique fixed point z 0 ∈ M such that for each x ∈ M, lim n →∞ S n x z 0 . One says that a mapping S belongs to the class BC if it satisfies the condition BC. Since then, several generalizations of the BCP have been appeared, some of them can be found, for instance, in 2–7 and into the references therein. In this paper we will focus our attention on an extension of the BCP introduced in 1976 by Jungck. More precisely, we are going to improve and generalize the following extension of the BCP.