Common Fixed Point Theorems in Menger Probabilistic Quasimetric Spaces
© Shaban Sedghi et al. 2009
Received: 21 November 2008
Accepted: 19 April 2009
Published: 1 June 2009
We consider complete Menger probabilistic quasimetric space and prove common fixed point theorems for weakly compatible maps in this space.
1. Introduction and Preliminaries
K. Menger introduced the notion of a probabilistic metric space in 1942 and since then the theory of probabilistic metric spaces has developed in many directions . The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to the situations when we do not know exactly the distance between two points, we know only probabilities of possible values of this distance. Such a probabilistic generalization of metric spaces appears to be well adapted for the investigation of physiological thresholds and physical quantities particularly in connections with both string and theory; see [2–5]. It is also of fundamental importance in probabilistic functional analysis, nonlinear analysis and applications [6–10].
Definition 1.1 (see ).
There is a nice characterization of continuous of the class .
(ii)If is continuous and , then there exists a sequence as in (i).The is an trivial example of a of but there are of Hadžić-type with (see, e.g., ).
Definition 1.3 (see ).
If is a and , then is defined recurrently by 1, if and for all . If is a sequence of numbers from then is defined as (this limit always exists) and as . In fixed point theory in probablistic metric spaces there are of particular interest the -norms and sequences such that and . Some examples of with the above property are given in the following proposition.
Note [14, Remark 13] that if is a for which there exists such that and , then Important class of is given in the following example.
In  the following results are obtained.
A Menger Probabilistic Quasimetric space (briefly, Menger PQM space) is a triple , where is a nonempty set, is a continuous , and is a mapping from into such that, if denotes the value of at the pair , then the following conditions hold, for all in ,
(2)A sequence in is called Cauchy sequence  if, for every and , there exists positive integer such that whenever ( ).
In 1998, Jungck and Rhoades  introduced the following concept of weak compatibility.
2. The Main Result
Similarly, we have
Hence, it follows that
By Proposition 1.6 all the conditions of the Theorem 2.2 are satisfied.
The second author is supported by MNTRRS 144012.
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