- Research Article
- Open Access
Some Common Fixed Point Theorems in Menger PM Spaces
© M. Imdad et al. 2010
Received: 11 May 2010
Accepted: 11 August 2010
Published: 13 September 2010
The Erratum to this article has been published in Fixed Point Theory and Applications 2011 2011:28
Employing the common property (E.A), we prove some common fixed point theorems for weakly compatible mappings via an implicit relation in Menger PM spaces. Some results on similar lines satisfying quasicontraction condition as well as -type contraction condition are also proved in Menger PM spaces. Our results substantially improve the corresponding theorems contained in (Branciari, (2002); Rhoades, (2003); Vijayaraju et al., (2005)) and also some others in Menger as well as metric spaces. Some related results are also derived besides furnishing illustrative examples.
1. Introduction and Preliminaries
Sometimes, it is found appropriate to assign the average of several measurements as a measure to ascertain the distance between two points. Inspired from this line of thinking, Menger [1, 2] introduced the notion of Probabilistic Metric spaces (in short PM spaces) as a generalization of metric spaces. In fact, he replaced the distance function with a distribution function wherein for any number , the value describes the probability that the distance between and is less than . In fact the study of such spaces received an impetus with the pioneering work of Schweizer and Sklar . The theory of PM spaces is of paramount importance in Probabilistic Functional Analysis especially due to its extensive applications in random differential as well as random integral equations.
Fixed point theory is one of the most fruitful and effective tools in mathematics which has enormous applications within as well as outside mathematics. The theory of fixed points in PM spaces is a part of Probabilistic Analysis which continues to be an active area of mathematical research. By now, several authors have already established numerous fixed point and common fixed point theorems in PM spaces. For an idea of this kind of the literature, one can consult the results contained in [3–14].
In metric spaces, Jungck  introduced the notion of compatible mappings and utilized the same (as a tool) to improve commutativity conditions in common fixed point theorems. This concept has been frequently employed to prove existence theorems on common fixed points. However, the study of common fixed points of noncompatible mappings is also equally interesting which was initiated by Pant . Recently, Aamri and Moutawakil  and Liu et al.  respectively, defined the property (E.A) and the common property (E.A) and proved some common fixed point theorems in metric spaces. Imdad et al.  extended the results of Aamri and Moutawakil  to semimetric spaces. Most recently, Kubiaczyk and Sharma  defined the property (E.A) in PM spaces and used it to prove results on common fixed points wherein authors claim to prove their results for strict contractions which are merely valid up to contractions.
In 2002, Branciari  proved a fixed point result for a mapping satisfying an integral-type inequality which is indeed an analogue of contraction mapping condition. In recent past, several authors (e.g., [22–26]) proved various fixed point theorems employing relatively more general integral type contractive conditions. In one of his interesting articles, Suzuki  pointed out that Meir-Keeler contractions of integral type are still Meir-Keeler contractions. In this paper, we prove the fixed point theorems for weakly compatible mappings via an implicit relation in Menger PM spaces satisfying the common property (E.A). Our results substantially improve the corresponding theorems contained in [21, 24, 26, 28] along with some other relevant results in Menger as well as metric spaces. Some related results are also derived besides furnishing illustrative examples.
In the following lines, we collect the background material to make our presentation as self-contained as possible.
Definition 1.1 (see ).
Definition 1.2 (see ).
Every metric space can always be realized as a PM space by considering defined by for all . So PM spaces offer a wider framework (than that of the metric spaces) and are general enough to cover even wider statistical situations.
Definition 1.3 (see ).
Definition 1.5 (see ).
Definition 1.6 (see ).
Definition 1.7 (see ).
Definition 1.8 (see ).
Definition 1.9 (see ).
Clearly, a pair of compatible mappings as well as noncompatible mappings satisfies the property (E.A).
Inspired by Liu et al. , we introduce the following.
Definition 1.12 (see).
Definition 1.13 (see ).
2. Implicit Relation
3. Main Results
We begin with the following observation.
The converse of Lemma 3.1 is not true in general. For a counter example, one can see Example 3.17 (presented in the end).
Theorem 3.3 extends the main result of Ciric  to Menger PM spaces besides extending the main result of Kubiaczyk and Sharma  to two pairs of mappings without any condition on containment of ranges amongst involved mappings.
The conclusions of Theorem 3.5 remain true if condition (ii) of Theorem 3.5 is replaced by the following:
As an application of Theorem 3.3, we prove the following result for four finite families of self-mappings. While proving this result, we utilize Definition 1.13 which is a natural extension of commutativity condition to two finite families of mappings.
Let and be four finite families of self-mappings of a Menger PM space with , and satisfying condition (3.1). If the pairs and share the common property (E.A) and as well as are closed subsets of , then
By restricting four families as and in Theorem 3.9, we can derive improved versions of certain results according to Chugh and Rathi , Kutukcu and Sharma , Rashwan and Hedar , Singh and Jain , and some others. Theorem 3.9 also generalizes the main result of Razani and Shirdaryazdi  to any finite number of mappings.
Corollary 3.11 is a slight but partial generalization of Theorem 3.3 as the commutativity requirements (i.e., and ) in this corollary are stronger as compared to weak compatibility in Theorem 3.3. Corollary 3.11 also presents the generalized and improved form of a result according to Bryant  in Menger PM spaces.
Since the pairs and are weakly compatible and both the pairs have point of coincidence and , respectively. Following the lines of the proof of Theorem 3.3, one can easily prove the existence of unique common fixed point of mappings and . This concludes the proof.
Theorem 3.13 generalizes the main result of Kohli and Vashistha  to two pairs of self-mappings as Theorem 3.13 never requires any condition on the containment of ranges amongst involved mappings besides weakening the completeness requirement of the space to closedness of suitable subspaces along with suitable commutativity requirements of the involved mappings. Here one may also notice that the function is lower semicontinuous whereas all the involved mappings may be discontinuous at the same time.
Notice that results similar to Theorems 3.5 –3.9 and Corollaries 3.4–3.11 can also be outlined in respect of Theorem 3.13, but we omit the details with a view to avoid any repetition.
We conclude this paper with two illustrative examples which demonstrate the validity of the hypotheses of Theorem 3.3 and Theorem 3.13.
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