Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces M. Imdad, Javid Ali * , M. Tanveer Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India article info Article history: Accepted 6 April 2009 abstract In this paper, we prove some existence results on coincidence and common fixed points of two pairs of self mappings without continuity under relatively weaker commutativity requirement in Menger PM spaces. Our results generalize many known results in Menger as well as metric spaces. Some related results are also derived besides furnishing illustra- tive examples. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction and preliminaries The study of probabilistic metric spaces (abbreviated as PM spaces in the sequel) was initiated by Schweizer and Sklar [17] and some of their coworkers. The history of PM spaces thus far is brief. In the original paper, Menger [11] gave postu- lates for the distribution function F p;q , which include a generalized triangle inequality. In this continuation, he constructed a theory of betweenness and indicated possible fields of application. In 1943, shortly after the appearance of Menger’s paper, Wald [23] published a paper in which he criticized Menger’s generalized triangle inequality and proposed an alternative one. On the basis of this new inequality, Wald constructed a theory of betweenness having certain advantages over Menger’s theory. In 1951, Menger [12] continued his study of PM spaces in a paper devoted to a resume of some of the earlier work which also includes construction of several specific examples and further possible applications of the theory. Subsequently, several authors studied such spaces (e.g. [2,8,9,17,18,20,23]) and above all the well known scientist, El Naschie [4–7] gave applications of this concept in quantum particle physics particularly in connections with both string and  1 theory. For instance, in order to analyse the probability involved in the two-slit experiment can be modelled in terms of a probabilistic metric. Like various other directions, the study of fixed point theorems in PM spaces is also a topic of recent interest and forms an active direction of research. The first ever effort in this direction appears to be made by Sehgal [18] when he initiated the study of fixed point theorems in PM spaces in his doctoral dissertation. Since then the subject has been further investigated by various authors. To mention a few, we cite Sehgal and Bharucha-Reid [19], Sherwood [20], Cain and Kasriel [2] and others whose contributions are relevant to the presentation of this paper. In what follows, we collect the relevant definitions, examples and results needed to make our presentation as self con- tained as possible. Definition 1.1 [17]. A mapping F : R ! R þ is called distribution function if it is nondecreasing, left continuous with inf fFðtÞ : t 2 Rg¼ 0 and supfFðtÞ : t 2 Rg¼ 1. Let L denote the set of all distribution functions and H denote the specific distribution function (also known as Heaviside function) is defined by 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.017 * Corresponding author. E-mail addresses: mhimdad@yahoo.co.in (M. Imdad), javid@math.com (J. Ali), tanveer_gouri@yahoo.co.in (M. Tanveer). Chaos, Solitons and Fractals 42 (2009) 3121–3129 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos