Common Fixed Point Theorems for Probabilistic Nearly Densifying Mappings

and Applied Analysis 3 p ∈ R 2 (B), so there is a w ∈ B such that p = R(w). Suppose that neither P and R nor Q and R have a coincidence point. Then


Introduction and Preliminaries
Banach contraction mapping principle is one of the most interesting and useful tools in applied mathematics.In recent years many generalizations of Banach contraction mapping principle have appeared.The notion of probabilistic metric spaces (in short PM-spaces) is a probabilistic generalization of metric spaces which are appropriate to carry out the study of those situations wherein distances are measured in the sense of distribution functions rather than nonnegative real numbers.The study of PM-spaces was initiated by Menger [1].Since then, Schweizer and Sklar [2] enriched this concept and provided a new impetus by proving some fundamental results on this theme.The first result on fixed point theory in PMspaces was given by Sehgal and Bharucha-Reid [3] wherein the notion of probabilistic contraction was introduced as a generalization of the classical Banach fixed point principle in terms of probabilistic settings.Some recent fixed point results can be studied in [4][5][6][7].
The aim of this paper is to prove some coincidence and common fixed point theorems for certain classes of nearly densifying mappings in complete Menger spaces.First, we give some topological definitions and terminology defined in [8,[15][16][17].Definition 1.A semigroup  is said to be left reversible if for any ,  ∈  there exist ,  ∈  such that  = .
It is easy to see that the notion of left reversibility is equivalent to the statement that any two right ideals of  have nonempty intersection.We restate the notion of probabilistic diameter for the sake of quick reference.Definition 4. Let  be a nonempty subset of .A function   (⋅) defined by is called probabilistic diameter of . is said to be bounded if The following definition is due to Bocsan and Constantin [15].
The following properties of Kuratowski's functions are proved in [8]: (e) let  be the closure of  in the (, )-topology on ; then where  denotes the specific distribution function defined by

Main Results
First, we prove some fixed point theorems for probabilistic nearly densifying mappings in Menger spaces.
Note that which implies () =  or  2 () = .Now, assume that  1 is upper semicontinuous.Then the function  :  → I, defined by () =  1 (, ), is u.s.c.So  assumes its maximal value at some point  in .Clearly,  ∈  2 (), so there is a  ∈  such that  =  2 ().Suppose that neither  and  nor  and  have a coincidence point.Then
The same result holds good if  2 is upper semicontinuous.This completes the proof of the theorem.Theorem 11.Let , , , and  be as in Theorem 9. Further, let , , and  satisfying ( 5) and ( 6) have a coincidence point ; then  is a unique common fixed point of , , and .
The uniqueness of  as a common fixed point of , , and  follows from ( 5) and ( 6).

Remark 10 .
The above theorem extends the results of Khan and Liu [25, Theorem 3.1 and Corollary 3.3] to PM-spaces.

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Definition 6.Let (, F) be a PM-space.A continuous mapping  of  into  is called a probabilistic densifying mapping if and only if, for every subset  of ,   <  implies  () >   .