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Some Common Fixed Point Theorems in Menger PM Spaces

Fixed Point Theory and Applications20102010:819269

DOI: 10.1155/2010/819269

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

Abstract

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 [3]. 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 [314].

In metric spaces, Jungck [15] 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 [16]. Recently, Aamri and Moutawakil [17] and Liu et al. [18] 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. [19] extended the results of Aamri and Moutawakil [17] to semimetric spaces. Most recently, Kubiaczyk and Sharma [20] 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 [21] 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., [2226]) proved various fixed point theorems employing relatively more general integral type contractive conditions. In one of his interesting articles, Suzuki [27] 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 [3]).

A mapping is called distribution function if it is nondecreasing and left continuous with and .

Let be the set of all distribution functions whereas be the set of specific distribution function (also known as Heaviside function) defined by
(1.1)

Definition 1.2 (see [1]).

Let be a nonempty set. An ordered pair is called a PM space if is a mapping from into satisfying the following conditions:

(1) if and only if ,

(2)

(3) and , then , for all and

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 [3]).

A mapping is called a -norm if

(1)

(2)

(3) for

(4) for all .

Example 1.4.

The following are the four basic -norms.

(i)The minimum -norm: .

(ii)The product -norm: .

(iii)The Lukasiewicz -norm: .

(iv)The weakest -norm, the drastic product:

(1.2)
In respect of above mentioned -norms, we have the following ordering:
(1.3)

Throughout this paper, stands for an arbitrary continuous -norm.

Definition 1.5 (see [1]).

A Menger PM space is a triplet where is a PM space and is a -norm satisfying the following condition:
(1.4)

Definition 1.6 (see [6]).

A sequence in a Menger PM space is said to converge to a point in if for every and , there is an integer such that for all .

Definition 1.7 (see [10]).

A pair of self-mappings of a Menger PM space is said to be compatible if for all , whenever is a sequence in such that for some in as .

Definition 1.8 (see [23]).

A pair of self-mappings of a Menger PM space is said to be noncompatible if and only if there exists at least one sequence in such that
(1.5)

but for some is either less than 1 or nonexistent.

Definition 1.9 (see [6]).

A pair of self-mappings of a Menger PM space is said to satisfy the property (E.A) if there exist a sequence in such that
(1.6)

Clearly, a pair of compatible mappings as well as noncompatible mappings satisfies the property (E.A).

Inspired by Liu et al. [18], we introduce the following.

Definition 1.10.

Two pairs and of self-mappings of a Menger PM space are said to satisfy the common property (E.A) if there exist two sequences in and some in such that
(1.7)

Example 1.11.

Let be a Menger PM space with and,
(1.8)
for all . Define self-mappings and on as , , , and for all . Then with sequences and in , one can easily verify that
(1.9)

This shows that the pairs and share the common property

Definition 1.12 (see[29]).

A pair of self-mappings of a nonempty set is said to be weakly compatible if the pair commutes on the set of coincidence points, that is, for some implies that .

Definition 1.13 (see [8]).

Two finite families of self-mappings and of a set are said to be pairwise commuting if

(1) ,

(2) ,

(3) and .

2. Implicit Relation

Let be the set of all continuous functions satisfying the following conditions:

, for all ,

, for all ,

, for all .

Example 2.1.

Define as
(2.1)

where is increasing and continuous function such that for all Notice that

, for all ,

, for all ,

, for all .

Example 2.2.

Define as
(2.2)
where is an increasing and continuous function such that for all and is a Lebesgue integrable function which is summable and satisfies
(2.3)

Observe that

, for all ,

, for all ,

, for all .

Example 2.3.

Define as
(2.4)
where is an increasing and continuous function such that for all and is a Lebesgue integrable function which is summable and satisfies
(2.5)

Observe that

, for all ,

, for all ,

() , for all .

3. Main Results

We begin with the following observation.

Lemma 3.1.

Let and be self-mappings of a Menger space satisfying the following:

(i)the pair (or satisfies the property ;
  1. (ii)
    for any and for all ,
    (3.1)
     

(iii) (or

Then the pairs and share the common property

Proof.

Suppose that the pair owns the property (E.A), then there exists a sequence in such that
(3.2)
Since , hence for each there exists such that Therefore,
(3.3)
Thus in all, we have and Now we assert that . Suppose that then applying inequality (3.1), we obtain
(3.4)
which on making reduces to
(3.5)
or
(3.6)

which is a contradiction to , and therefore . Hence the pairs and share the common property (E.A).

Remark 3.2.

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.

Let and be self-mappings on a Menger PM space satisfying inequality (3.1). Suppose that

(i)the pair (or enjoys the property (E.A),

(ii) (or

(iii) (or is a closed subset of .

Then the pairs and have a point of coincidence each. Moreover, and have a unique common fixed point provided that both the pairs and are weakly compatible.

Proof.

In view of Lemma 3.1, the pairs and share the common property (E.A), that is, there exist two sequences and in such that
(3.7)
Suppose that is a closed subset of , then for some If , then applying inequality (3.1), we obtain
(3.8)
which on making reduces to
(3.9)

which is a contradiction to . Hence

Since , there exists such that

If , then using inequality (3.1), we have
(3.10)
or
(3.11)

which is a contradiction to , and therefore

Since the pairs and are weakly compatible and therefore
(3.12)
If , then using inequality (3.1), we have
(3.13)
or
(3.14)

which is a contradiction to , and therefore

Similarly, one can prove that Hence , and is a common fixed point of and . The uniqueness of common fixed point is an easy consequences of inequality (3.1).

By choosing and suitably, one can derive corollaries involving two or three mappings. As a sample, we deduce the following natural result for a pair of self-mappings by setting and (in Theorem 3.3).

Corollary 3.4.

Let and be self-mappings on a Menger space . Suppose that

(i)the pair enjoys the property (E.A),

(ii)for all and for all ,
(3.15)

(iii) is a closed subset of .

Then and have a coincidence point. Moreover, if the pair is weakly compatible, then and have a unique common fixed point.

Theorem 3.5.

Let and be self-mappings of a Menger PM space satisfying the inequality (3.1). Suppose that

(i)the pairs and share the common property (E.A),

(ii) and are closed subsets of .

If the pairs and are weakly compatible, then and have a unique common fixed point in .

Proof.

Suppose that the pairs and satisfy the common property (E.A), then there exist two sequences and in such that
(3.16)

Since and are closed subsets of , we obtain for some .

If then using inequality (3.1), we have
(3.17)
which on making reduces to
(3.18)

which is a contradiction to , and hence The rest of the proof can be completed on the lines of the proof of Theorem 3.3, hence it is omitted.

Remark 3.6.

Theorem 3.3 extends the main result of Ciric [30] to Menger PM spaces besides extending the main result of Kubiaczyk and Sharma [20] to two pairs of mappings without any condition on containment of ranges amongst involved mappings.

Theorem 3.7.

The conclusions of Theorem 3.5 remain true if condition (ii) of Theorem 3.5 is replaced by the following:

and

Corollary 3.8.

The conclusions of Theorems 3.3 and 3.5 remain true if conditions (ii) (of Theorem 3.3) and (iii) (of Theorem 3.7) are replaced by the following:

(iv) and are closed subsets of whereas and .

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.

Theorem 3.9.

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

(i)the pair as well as has a coincidence point,

(ii) and have a unique common fixed point provided that the pair of families and commute pairwise, where , , , and .

Proof.

The proof follows on the lines of Theorem   4.1 according to M. Imdad and J. Ali[31] and Theorem   3.1 according to Imdad et al. [19].

Remark 3.10.

By restricting four families as and in Theorem 3.9, we can derive improved versions of certain results according to Chugh and Rathi [4], Kutukcu and Sharma [32], Rashwan and Hedar [11], Singh and Jain [14], and some others. Theorem 3.9 also generalizes the main result of Razani and Shirdaryazdi [12] to any finite number of mappings.

By setting and in Theorem 3.9, we deduce the following.

Corollary 3.11.

Let and be self-mappings of a Menger space such that the pairs and share the common property (E.A) and also satisfy the condition
(3.19)

, and and are fixed positive integers.

If and are closed subsets of , then and have a unique common fixed point provided, and .

Remark 3.12.

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 [33] in Menger PM spaces.

Our next result involves a lower semicontinuous function such that for all along with and .

Theorem 3.13.

Let and be self-mappings of a Menger space satisfying conditions (i) and (ii) of Theorem 3.5 and for all
(3.20)

where , , , , .

Then the pairs and have point of coincidence each. Moreover, and have a unique common fixed point provided that both the pairs and are weakly compatible.

Proof.

As both the pairs share the common property , there exist two sequences such that
(3.21)
If is a closed subset of , then (3.21) Therefore, there exists a point such that Now we assert that If it is not so, then setting in (3.20), we get
(3.22)
which on making , reduces to
(3.23)
or
(3.24)

a contradiction. Therefore , and hence which shows that the pair has a point of coincidence.

If is a closed subset of , then (3.21) Hence, there exists a point such that Now we show that If it is not so, then using (3.20) with , we have
(3.25)
which on making reduces to
(3.26)
or
(3.27)

a contradiction. Therefore and hence which proves that the pair has a point of coincidence.

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.

Remark 3.14.

Theorem 3.13 generalizes the main result of Kohli and Vashistha [9] 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.

Remark 3.15.

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.

Example 3.16.

Let be a Menger space, where with a -norm defined by for all , for all and
(3.28)
Define and by:
(3.29)
Also define as
(3.30)
It is easy to see that for all and
(3.31)

Also . Thus all the conditions of Theorem 3.3 are satisfied, and 1 is the unique common fixed point of and

Example 3.17.

Let and be the same as in Example 3.16. Define and by
(3.32)
By a routine calculation, one can verify that for all and
(3.33)

Also , , . Thus all conditions of Theorem 3.13 are satisfied, and 1 is the unique common fixed point of and

Notes

Authors’ Affiliations

(1)
Department of Mathematics, Aligarh Muslim University
(2)
School of Computer & Systems Sciences, Jawaharlal Nehru University
(3)
Department of Applied Mathematics, Aligarh Muslim University

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