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Common Fixed Point Theorems for Probabilistic Nearly Densifying Mappings
The aim of this paper is to prove some coincidence and common fixed point theorems for probabilistic nearly densifying mappings in complete Menger spaces. Our results improve the results of Chamola et al. (1991), Dimri and Pant (2002), and Pant et al. (2004) and extend the results of Khan and Liu (1997) in the framework of probabilistic settings.
1. 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 . Since then, Schweizer and Sklar  enriched this concept and provided a new impetus by proving some fundamental results on this theme. The first result on fixed point theory in PM-spaces was given by Sehgal and Bharucha-Reid  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 [4–7].
Kuratowski  introduced the notion of measure of noncompactness of a bounded subset of a metric space. Further, this study was carried on by Furi and Vignoli . They introduced the notion of densifying (also called condensing) mapping in terms of Kuratowski’s measure of noncompactness and obtained some fixed point theorems. Following Furi and Vignoli , a number of mathematicians worked on densifying mappings and proved some metrical fixed point theorem (cf. [10–14]). As a generalization of Kuratowski’s measure of noncompactness, Bocsan and Constantin  introduced the notion of Kuratowski’s measure of noncompactness in PM-spaces. Subsequently, Bocşan  studied the notion of probabilistic densifying mappings. Later, Hadžić , Tan , Chamola et al. , Dimri and Pant , Pant et al. , Pant et al. , and Singh and Pant  proved some results for such mappings. In , Ganguly et al. introduced the notion of probabilistic nearly densifying mappings and proved some interesting results in this setting.
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, 15–17].
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.
Definition 2. Let be a family of self-mappings in . A subset of is called -invariant if for all .
Definition 3. Let be the semigroup generated by under composition . Clearly, for any and for .
We restate the notion of probabilistic diameter for the sake of quick reference.
Definition 4. Let be a nonempty subset of . A function defined byis called probabilistic diameter of . is said to be bounded if
The following definition is due to Bocsan and Constantin .
Definition 5. For a probabilistic bounded subset of , defined by is called Kuratowski’s function.
The following properties of Kuratowski’s functions are proved in :(a), the set of distribution functions;(b);(c)if , then ;(d);(e)let be the closure of in the -topology on ; then (f) is probabilistic precompact (totally bounded) if , where denotes the specific distribution function defined by
Definition 6. Let be a PM-space. A continuous mapping of into is called a probabilistic densifying mapping if and only if, for every subset of , implies .
Definition 7. A self-mapping is probabilistic nearly densifying if , whenever , , and is -invariant.
Definition 8. Suppose is an upper semicontinuous function with and for all .
2. Main Results
First, we prove some fixed point theorems for probabilistic nearly densifying mappings in Menger spaces.
Theorem 9. Let , , and be three continuous and nearly densifying self-mappings on a complete Menger space such that and commutes with and . If, for all , , the following conditions are satisfied:where and are real valued mappings from to , the collection of all distribution functions, with either or being upper semicontinuous (u.s.c.) and for all .
Further, if, for some , is bounded, then and or and have a coincidence point.
Proof. For , let and .
If , thena contradiction. It implies that is precompact.
Then it is easy to see that and is nonempty compact subset of . By the continuity of , , and , it follows that , , and . Further, it is clear that , , and .
Note thatwhich implies or .
Now, assume that is upper semicontinuous. Then the function , defined by , is u.s.c. So assumes its maximal value at some point in . Clearly, , so there is a such that . Suppose that neither and nor and have a coincidence point. Thena contradiction to the selection of . Hence, and or and must have a coincidence point.
The same result holds good if is upper semicontinuous. This completes the proof of the theorem.
Remark 10. The above theorem extends the results of Khan and Liu [25, Theorem 3.1 and Corollary 3.3] to PM-spaces.
Proof. We have . By commutativity of with and , and , or .
Now let ; then by (5) and (6), we havewhich is a contradiction. Hence, . Thus, is a fixed point of . Thus, . Therefore, is a common fixed point of , , and .
The uniqueness of as a common fixed point of , , and follows from (5) and (6).
Theorem 12. Let and be commuting, continuous, and nearly densifying self-mappings on a complete Menger space satisfyingfor , , and , where is u.s.c. and , . If, for some in , is bounded, then and have a unique common fixed point.
Proof. Let . Since and are commuting and continuous, we have and and then .
If , thenwhich is a contradiction. It implies that is precompact.
Now define . Since is a decreasing sequence of nonempty compact subset of , it follows that is nonempty set such that , .
Suppose that ; then for all . Hence, there exists . Since is compact and closed for all , and are continuous and nearly densifying; therefore, there exists a point for all so that . Hence, and . Thus, we haveLet us define a real valued function on by . It is u.s.c. and hence attains its maximum at some point . Then there exists a such that .
Suppose that there is no point in such that ; then we have by (11)which is a contradiction to the selection of . Hence, there exists a such that or .
Suppose ; then we havewhich is a contradiction. Hence, . Therefore, is common fixed point of and . Now we will prove the uniqueness of . Let be the other fixed point of and ; then, by (11), we haveHence, is unique. This completes the proof of the theorem.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
All authors contributed equally to this paper. The guidance of Aeshah Hassan Zakri is very important and she helped in revising the paper according to reviewers reports.
The authors thank the anonymous referees for their careful reading and useful suggestions on the paper. The authors are highly thankful to Deanship of Scientific Research, Jazan University, for financial support of this paper.
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