Open Access

Common property (E.A) and existence of fixed points in Menger spaces

  • Sunny Chauhan1Email author,
  • Sumitra Dalal2,
  • Wutiphol Sintunavarat3 and
  • Jelena Vujaković4
Journal of Inequalities and Applications20142014:56

https://doi.org/10.1186/1029-242X-2014-56

Received: 30 October 2013

Accepted: 6 January 2014

Published: 4 February 2014

Abstract

The aim of this paper is to prove some common fixed-point theorems for weakly compatible mappings in Menger spaces satisfying common property (E.A). Some examples are also given which demonstrate the validity of our results. As an application of our main result, we present a common fixed-point theorem for four finite families of self-mappings in Menger spaces. Our result is an improved probabilistic version of the result of Sedghi et al. [Gen. Math. 18:3-12, 2010].

MSC:54H25, 47H10, 54E70.

Keywords

t-norm Menger space weakly compatible mappings property (E.A) common property (E.A)

1 Introduction

In 1922, Banach proved the principal contraction result [1]. As we know, there have been published many works about fixed-point theory for different kinds of contractions on some spaces such as quasi-metric spaces [2], cone metric spaces [3], convex metric spaces [4], partially ordered metric spaces [59], G-metric spaces [1014], partial metric spaces [15, 16], quasi-partial metric spaces [17], fuzzy metric spaces [18], and Menger spaces [19]. Also, studies either on approximate fixed point or on qualitative aspects of numerical procedures for approximating fixed points are available in the literature; see [4, 20, 21].

Jungck and Rhoades [22] weakened the notion of compatibility by introducing the notion of weakly compatible mappings (extended by Singh and Jain [23] to probabilistic metric space) and proved common fixed-point theorems without assuming continuity of the involved mappings in metric spaces. In 2002, Aamri and Moutawakil [24] introduced the notion of property (E.A) (extended by Kubiaczyk and Sharma [25] to probabilistic metric space) for self-mappings which contained the class of noncompatible mappings due to Pant [26]. Further, Liu et al. [27] defined the notion of common property (E.A) (extended by Ali et al. [28] to probabilistic metric space) which contains the property (E.A) and proved several fixed-point theorems under hybrid contractive conditions. Since then, there has been continuous and intense research activity in fixed-point theory and by now there exists an extensive literature (e.g. [2933] and the references therein).

Many mathematicians proved several common fixed-point theorems for contraction mappings in Menger spaces by using different notions viz. compatible mappings, weakly compatible mappings, property (E.A), common property (E.A) (see [28, 3451]).

In the present paper, we prove some common fixed-point theorems for weakly compatible mappings in Menger space using the common property (E.A). Some examples are also derived which demonstrate the validity of our results. As an application of our main result, we extend the related results to four finite families of self-mappings in Menger spaces.

2 Preliminaries

In the sequel, , R + , and denote the set of real numbers, the set of nonnegative real numbers, and the set of positive integers, respectively.

Definition 2.1 [52]

A triangular norm (shortly t-norm) is a binary operation on the unit interval [ 0 , 1 ] such that for all a , b , c , d [ 0 , 1 ] the following conditions are satisfied:
  1. (1)

    a 1 = a ,

     
  2. (2)

    a b = b a ,

     
  3. (3)

    a b c d whenever a c and b d ,

     
  4. (4)

    a ( b c ) = ( a b ) c .

     

Examples of t-norms are a b = min { a , b } , a b = a b , and a b = max { a + b 1 , 0 } .

Definition 2.2 [52]

A mapping F : R R + is called a distribution function if it is nondecreasing and left continuous with inf { F ( t ) : t R } = 0 and sup { F ( t ) : t R } = 1 . We shall denote the set of all distribution functions on ( , ) by , while H will always denotes the specific distribution function defined by
H ( t ) = { 0 , if  t 0 ; 1 , if  t > 0 .

If X is a nonempty set, F : X × X is called a probabilistic distance on X and F ( x , y ) is usually denoted by F x , y .

Definition 2.3 [52]

The ordered pair ( X , F ) is called a probabilistic metric space (shortly, PM-space) if X is a nonempty set and is a probabilistic distance satisfying the following conditions:
  1. (1)

    F x , y ( t ) = 1 for all t > 0 if and only if x = y ,

     
  2. (2)

    F x , y ( 0 ) = 0 for all x , y X ,

     
  3. (3)

    F x , y ( t ) = F y , x ( t ) for all x , y X and for all t > 0 ,

     
  4. (4)

    F x , z ( t ) = 1 , F z , y ( s ) = 1 F x , y ( t + s ) = 1 for x , y , z X and t , s > 0 .

     

Every metric space ( X , d ) can always be realized as a probabilistic metric space defined by F x , y ( t ) = H ( t d ( x , y ) ) for all x , y X and t > 0 . So probabilistic metric spaces offer a wider framework (than that of the metric spaces) and are general enough to cover even wider statistical situations.

Definition 2.4 [19]

A Menger space ( X , F , ) is a triplet where ( X , F ) is a probabilistic metric space and is a t-norm satisfying the following condition:
F x , y ( t + s ) F x , z ( t ) F z , y ( s ) ,

for all x , y , z X and t , s > 0 .

Throughout this paper, ( X , F , ) is considered to be a Menger space with condition lim t F x , y ( t ) = 1 for all x , y X . Every fuzzy metric space ( X , M , ) may be a Menger space by considering F : X × X defined by F x , y ( t ) = M ( x , y , t ) for all x , y X .

Definition 2.5 [52]

Let ( X , F , ) be a Menger space and be a t-norm. Then
  1. (1)

    a sequence { x n } in X is said to converge to a point x in X if and only if for every ϵ > 0 and λ ( 0 , 1 ) , there exists an integer N N such that F x n , x ( ϵ ) > 1 λ for all n N ;

     
  2. (2)

    a sequence { x n } in X is said to be Cauchy if for every ϵ > 0 and λ ( 0 , 1 ) , there exists an integer N N such that F x n , x m ( ϵ ) > 1 λ for all n , m N .

     

A Menger space in which every Cauchy sequence is convergent is said to be complete.

Definition 2.6 [53]

A pair ( A , S ) of self-mappings of a Menger space ( X , F , ) is said to be compatible if lim n F A S x n , S A x n ( t ) = 1 for all t > 0 , whenever { x n } is a sequence in X such that lim n A x n = lim n S x n = z for some z X .

Definition 2.7 [28]

A pair ( A , S ) of self-mappings of a Menger space ( X , F , ) is said to be noncompatible if there exists at least one sequence { x n } in X such that lim n A x n = z = lim n S x n for some z X , but, for some t > 0 , either lim n F A S x n , S A x n ( t ) 1 or the limit does not exist.

Definition 2.8 [25]

A pair ( A , S ) of self-mappings of a Menger space ( X , F , ) is said to satisfy property (E.A) if there exists a sequence { x n } in X such that
lim n A x n = lim n S x n = z ,

for some z X .

Remark 2.1 From Definition 2.8, it is easy to see that any two noncompatible self-mappings of ( X , F , ) satisfy property (E.A) but the reverse need not be true (see [[40], Example 1]).

Definition 2.9 [34]

Two pairs ( A , S ) and ( B , T ) of self-mappings of a Menger space ( X , F , ) are said to satisfy the common property (E.A) if there exist two sequences { x n } , { y n } in X such that
lim n A x n = lim n S x n = lim n B y n = lim n T y n = z ,

for some z X .

Definition 2.10 [22]

A pair ( A , S ) of self-mappings of a nonempty set X is said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if A z = S z for some z X , then A S z = S A z .

Remark 2.2 If self-mappings A and S of a Menger space ( X , F , ) are compatible then they are weakly compatible but the reverse need not be true (see [[23], Example 1]).

Remark 2.3 It is noticed that the notion of weak compatibility and the (E.A) property are independent to each other (see [[54], Example 2.2]).

Definition 2.11 [41]

Two families of self-mappings { A i } and { S j } are said to be pairwise commuting if:
  1. (1)

    A i A j = A j A i , i , j { 1 , 2 , , m } ,

     
  2. (2)

    S k S l = S l S k , k , l { 1 , 2 , , n } ,

     
  3. (3)

    A i S k = S k A i , i { 1 , 2 , , m } , k { 1 , 2 , , n } .

     

3 Main results

Let Φ is a set of all increasing and continuous functions ϕ : ( 0 , 1 ] ( 0 , 1 ] , such that ϕ ( t ) > t for every t ( 0 , 1 ) .

Example 3.1 Let ϕ : ( 0 , 1 ] ( 0 , 1 ] defined by ϕ ( t ) = t 1 2 . It is easy to see that ϕ Φ .

Before proving our main theorems, we begin with the following lemma.

Lemma 3.1 Let A, B, S and T be self-mappings of a Menger space ( X , F , ) , where is a continuous t-norm. Suppose that
  1. (1)

    A ( X ) T ( X ) or B ( X ) S ( X ) ,

     
  2. (2)

    the pair ( A , S ) or ( B , T ) satisfies property (E.A),

     
  3. (3)

    B ( y n ) converges for every sequence { y n } in X whenever T ( y n ) converges or A ( x n ) converges for every sequence { x n } in X whenever S ( x n ) converges,

     
  4. (4)
    there exist ϕ Φ and 1 k < 2 such that
    F A x , B y ( t ) ϕ ( min { F S x , T y ( t ) , sup t 1 + t 2 = 2 k t min { F S x , A x ( t 1 ) , F T y , B y ( t 2 ) } , sup t 3 + t 4 = 2 k t max { F S x , B y ( t 3 ) , F T y , A x ( t 4 ) } } )
    (3.1)
     

holds for all x , y X , t > 0 . Then the pairs ( A , S ) and ( B , T ) share the common property (E.A).

Proof Suppose the pair ( A , S ) satisfies property (E.A), then there exists a sequence { x n } in X such that
lim n A x n = lim n S x n = z ,
(3.2)
for some z X . Since A ( X ) T ( X ) , hence for each { x n } X there corresponds a sequence { y n } X such that A x n = T y n . Therefore,
lim n T y n = lim n A x n = z .
(3.3)
Thus in all, we have A x n z , S x n z and T y n z . By (3), the sequence { B y n } converges and in all we need to show that B y n z as n . Let B y n l for t > 0 as n . Then, it is enough to show that z = l . Suppose that z l , then there exists t 0 > 0 such that
F z , l ( 2 k t 0 ) > F z , l ( t 0 ) .
(3.4)
In order to establish the claim embodied in (3.4), let us assume that (3.4) does not hold. Then we have F z , l ( 2 k t ) F z , l ( t ) for all t > 0 . Repeatedly using this equality, we obtain
F z , l ( t ) F z , l ( 2 k t ) F z , l ( ( 2 k ) n t ) 1 ,
as n . This shows that F z , l ( t ) = 1 for all t > 0 , which contradicts z l , and hence (3.4) is proved. Using inequality (3.1), with x = x n , y = y n , we get
F A x n , B y n ( t 0 ) ϕ ( min { F S x n , T y n ( t 0 ) , sup t 1 + t 2 = 2 k t 0 min { F S x n , A x n ( t 1 ) , F T y n , B y n ( t 2 ) } , sup t 3 + t 4 = 2 k t 0 max { F S x n , B y n ( t 3 ) , F T y n , A x n ( t 4 ) } } ) ϕ ( min { F S x n , T y n ( t 0 ) , min { F S x n , A x n ( ϵ ) , F T y n , B y n ( 2 k t 0 ϵ ) } , max { F S x n , B y n ( 2 k t 0 ϵ ) , F T y n , A x n ( ϵ ) } } ) ,
for all ϵ ( 0 , 2 k t 0 ) . As n , it follows that
F z , l ( t 0 ) ϕ ( min { F z , z ( t 0 ) , min { F z , z ( ϵ ) , F z , l ( 2 k t 0 ϵ ) } , max { F z , l ( 2 k t 0 ϵ ) , F z , z ( ϵ ) } } ) = ϕ ( F z , l ( 2 k t 0 ϵ ) ) > F z , l ( 2 k t 0 ϵ ) ,
as ϵ 0 , we have
F z , l ( t 0 ) F z , l ( 2 k t 0 ) ,

which contradicts (3.4). Therefore, z = l . Hence the pairs ( A , S ) and ( B , T ) share the common property (E.A). □

Remark 3.1 In general, the converse of Lemma 3.1 is not true (see [[28], Example 3.1]).

Now we prove a common fixed-point theorem for two pairs of mappings in Menger space which is an extension of the main result of Sedghi et al. [55] in a version of Menger space.

Theorem 3.1 Let A, B, S and T be self-mappings of a Menger space ( X , F , ) , where is a continuous t-norm satisfying inequality (3.1) of Lemma  3.1. Suppose that
  1. (1)

    the pairs ( A , S ) and ( B , T ) share the common property (E.A),

     
  2. (2)

    S ( X ) and T ( X ) are closed subsets of X.

     

Then the pairs ( A , S ) and ( B , T ) have a coincidence point each. Moreover, A, B, S, and T have a unique common fixed point provided both pairs ( A , S ) and ( B , T ) are weakly compatible.

Proof Since the pairs ( A , S ) and ( B , T ) share the common property (E.A), there exist two sequences { x n } and { y n } in X such that
lim n A x n = lim n S x n = lim n B y n = lim n T y n = z ,
(3.5)
for some z X . Since S ( X ) is a closed subset of X, hence lim n S x n = z S ( X ) . Therefore, there exists a point u X such that S u = z . Now we assert that A u = S u . Suppose that A u S u , then there exists t 0 > 0 such that
F A u , S u ( 2 k t 0 ) > F A u , S u ( t 0 ) .
(3.6)
In order to establish the claim embodied in (3.6), let us assume that (3.6) does not hold. Then we have F A u , S u ( 2 k t ) F A u , S u ( t ) for all t > 0 . Repeatedly using this equality, we obtain
F A u , S u ( t ) F A u , S u ( 2 k t ) F A u , S u ( ( 2 k ) n t ) 1 ,
as n . This shows that F A u , S u ( t ) = 1 for all t > 0 , which contradicts A u S u and hence (3.6) is proved. Using inequality (3.1), with x = u , y = y n , we get
F A u , B y n ( t 0 ) ϕ ( min { F S u , T y n ( t 0 ) , sup t 1 + t 2 = 2 k t 0 min { F S u , A u ( t 1 ) , F T y n , B y n ( t 2 ) } , sup t 3 + t 4 = 2 k t 0 max { F S u , B y n ( t 3 ) , F T y n , A u ( t 4 ) } } ) ϕ ( min { F z , T y n ( t 0 ) , min { F z , A u ( 2 k t 0 ϵ ) , F B y n , T y n ( ϵ ) } , max { F z , B y n ( ϵ ) , F T y n , z ( 2 k t 0 ϵ ) } } ) ,
for all ϵ ( 0 , 2 k t 0 ) . As n , it follows that
F A u , z ( t 0 ) ϕ ( min { F z , z ( t 0 ) , min { F z , A u ( 2 k t 0 ϵ ) , F z , z ( ϵ ) } , max { F z , z ( ϵ ) , F z , z ( 2 k t 0 ϵ ) } } ) = ϕ ( F z , A u ( 2 k t 0 ϵ ) ) > F A u , z ( 2 k t 0 ϵ ) ,
as ϵ 0 , we have
F A u , z ( t 0 ) F A u , z ( 2 k t 0 ) ,

which contradicts (3.6). Therefore A u = S u = z and hence u is a coincidence point of ( A , S ) .

If T ( X ) is a closed subset of X. Therefore there exists a point v X such that T v = z . Now we assert that B v = T v = z . Let, on the contrary, B v T v . As earlier, there exists t 0 > 0 such that
F B v , T v ( 2 k t 0 ) > F B v , T v ( t 0 ) .
(3.7)
To support the claim, let it be untrue. Then we have F B v , T v ( 2 k t ) F B v , T v ( t ) for all t > 0 . Repeatedly using this equality, we obtain
F B v , T v ( t ) F B v , T v ( 2 k t ) F B v , T v ( ( 2 k ) n t ) 1 ,
as n . This shows that F B v , T v ( t ) = 1 for all t > 0 , which contradicts B v T v and hence (3.7) is proved. Using inequality (3.1), with x = x n , y = v , we get
F A x n , B v ( t 0 ) ϕ ( min { F S x n , T v ( t 0 ) , sup t 1 + t 2 = 2 k t 0 min { F S x n , A x n ( t 1 ) , F T v , B v ( t 2 ) } , sup t 3 + t 4 = 2 k t 0 max { F S x n , B v ( t 3 ) , F T v , A x n ( t 4 ) } } ) ϕ ( min { F S x n , z ( t 0 ) , min { F S x n , A x n ( ϵ ) , F z , B v ( 2 k t 0 ϵ ) } , max { F S x n , B v ( 2 k t 0 ϵ ) , F z , A x n ( ϵ ) } } ) ,
for all ϵ ( 0 , 2 k t 0 ) . As n , it follows that
F z , B v ( t 0 ) ϕ ( min { F z , z ( t 0 ) , min { F z , z ( ϵ ) , F z , B v ( 2 k t 0 ϵ ) } , max { F z , B v ( 2 k t 0 ϵ ) , F z , z ( ϵ ) } } ) = ϕ ( F z , B v ( 2 k t 0 ϵ ) ) > F z , B v ( 2 k t 0 ϵ ) ,
as ϵ 0 , we have
F z , B v ( t 0 ) F z , B v ( 2 k t 0 ) ,

which contradicts (3.7). Therefore B v = T v = z , which shows that v is a coincidence point of the pair ( B , T ) .

Since the pair ( A , S ) is weakly compatible, therefore A z = A S u = S A u = S z . Now we assert that z is a common fixed point of ( A , S ) . If z A z , then on using (3.1) with x = z , y = v , we get, for some t 0 > 0 ,
F A z , B v ( t 0 ) ϕ ( min { F S z , T v ( t 0 ) , sup t 1 + t 2 = 2 k t 0 min { F S z , A z ( t 1 ) , F T v , B v ( t 2 ) } , sup t 3 + t 4 = 2 k t 0 max { F S z , B v ( t 3 ) , F T v , A z ( t 4 ) } } ) , F A z , z ( t 0 ) ϕ ( min { F A z , z ( t 0 ) , min { F A z , A z ( ϵ ) , F z , z ( 2 k t 0 ϵ ) } , max { F A z , z ( ϵ ) , F z , A z ( 2 k t 0 ϵ ) } } ) ,
for all ϵ ( 0 , 2 k t 0 ) . As ϵ 0 , we have
F A z , z ( t 0 ) ϕ ( min { F A z , z ( t 0 ) , F z , A z ( 2 k t 0 ) } ) = ϕ ( F A z , z ( t 0 ) ) > F A z , z ( t 0 ) ,
which is a contradiction. Hence A z = S z = z , i.e. z is a common fixed point of ( A , S ) . Also the pair ( B , T ) is weakly compatible, therefore B z = B T v = T B v = T z . Now we show that z is also a common fixed point of ( B , T ) . If z B z , then on using (3.1) with x = u , y = z , we get, for some t 0 > 0 ,
F A u , B z ( t 0 ) ϕ ( min { F S u , T z ( t 0 ) , sup t 1 + t 2 = 2 k t 0 min { F S u , A u ( t 1 ) , F T z , B z ( t 2 ) } , sup t 3 + t 4 = 2 k t 0 max { F S u , B z ( t 3 ) , F T z , A u ( t 4 ) } } ) , F z , B z ( t 0 ) ϕ ( min { F z , B z ( t 0 ) , min { F z , z ( ϵ ) , F B z , B z ( 2 k t 0 ϵ ) } , max { F z , B z ( ϵ ) , F B z , z ( 2 k t 0 ϵ ) } } ) ,
for all ϵ ( 0 , 2 k t 0 ) . As ϵ 0 , we have
F z , B z ( t 0 ) ϕ ( min { F z , B z ( t 0 ) , F B z , z ( 2 k t 0 ) } ) = ϕ ( F z , B z ( t 0 ) ) > F z , B z ( t 0 ) ,

which is a contradiction. Therefore B z = z = T z , which shows that z is a common fixed point of the pair ( B , T ) . Therefore z is a common fixed point of both pairs ( A , S ) and ( B , T ) . The uniqueness of common fixed point is an easy consequence of inequality (3.1). □

Remark 3.2 Theorem 3.1 is an improved probabilistic version of the result of Sedghi et al. [[55], Theorem 1] for two pairs of self-mappings without any requirement on containment of ranges amongst the involved mappings.

The following example illustrates Theorem 3.1.

Example 3.2 Let ( X , F , ) be a Menger space, where X = [ 2 , 19 ] , with continuous t-norm is defined by a b = a b for all a , b [ 0 , 1 ] and
F x , y ( t ) = ( t t + 1 ) | x y |
for all x , y X . The function ϕ is defined as in Example 3.1. Define the self-mappings A, B, S, and T by
A ( x ) = { 2 , if  x { 2 } ( 3 , 19 ] ; 3 , if  x ( 2 , 3 ] , B ( x ) = { 2 , if  x { 2 } ( 3 , 19 ] ; 2.5 , if  x ( 2 , 3 ] , S ( x ) = { 2 , if  x = 2 ; 10 , if  x ( 2 , 3 ] ; x + 77 40 , if  x ( 3 , 19 ] , T ( x ) = { 2 , if  x = 2 ; 13 , if  x ( 2 , 3 ) ; 14 , if  x = 3 ; x + 77 40 , if  x ( 3 , 19 ] .
We take { x n } = { 3 + 1 n } , { y n } = { 2 } or { x n } = { 2 } , { y n } = { 3 + 1 n } . We have
lim n A x n = lim n S x n = lim n B y n = lim n T y n = 2 X .

Therefore, both pairs ( A , S ) and ( B , T ) satisfy the common property (E.A).

It is noted that A ( X ) = { 2 , 3 } [ 2 , 2.4 ] { 13 , 14 } = T ( X ) and B ( X ) = { 2 , 2.5 } [ 2 , 2.4 ] { 10 } = S ( X ) . On the other hand, S ( X ) and T ( X ) are closed subsets of X. Thus, all the conditions of Theorem 3.1 are satisfied and 2 is a unique common fixed point of the pairs ( A , S ) and ( B , T ) , which also remains a point of coincidence as well. Also, all the involved mappings are even discontinuous at their unique common fixed point 2.

Remark 3.3 In fact, the mapping in Example 3.2 is also a fuzzy metric. However, the result of Sedghi et al. [[55], Theorem 1] cannot be used for this case since A ( X ) T ( X ) and B ( X ) S ( X ) .

Theorem 3.2 The conclusion of Theorem  3.1 remains true if the condition (2) of Theorem  3.1 is replaced by the following:

(2)′ A ( X ) ¯ T ( X ) and B ( X ) ¯ S ( X ) , where A ( X ) ¯ is the closure range of A and B ( X ) ¯ is the closure range of B.

Proof Since the pairs ( A , S ) and ( B , T ) satisfy the common property (E.A), there exist two sequences { x n } and { y n } in X such that
lim n A x n = lim n S x n = lim n B y n = lim n T y n = z ,

for some z X . Then since z A ( X ) ¯ and A ( X ) ¯ T ( X ) there exists a point v X such that z = T v . By the proof of Theorem 3.1, we can show that the pair ( B , T ) has a coincidence point, call it v, i.e. B v = T v . Since z B ( X ) ¯ and B ( X ) ¯ S ( X ) there exists a point u X such that z = S u . Similarly we can also prove that the pair ( A , S ) has a coincidence point, call it u, i.e. A u = S u . The rest of the proof is on the lines of the proof of Theorem 3.1, hence it is omitted. □

Corollary 3.1 The conclusions of Theorems 3.1-3.2 remain true if condition (2) of Theorem  3.1 and condition (2)′ of Theorem  3.2 are replaced by the following:

(2)″ A ( X ) and B ( X ) are closed subsets of X provided A ( X ) T ( X ) and B ( X ) S ( X ) .

Theorem 3.3 Let ( X , F , ) be a Menger space, where is a continuous t-norm. Let A, B, S and T be mappings from X into itself and satisfying the conditions (1)-(4) of Lemma  3.1. Suppose that
  1. (5)

    S ( X ) (or T ( X ) ) is a closed subset of X.

     

Then the pairs ( A , S ) and ( B , T ) have a coincidence point each. Moreover, A, B, S and T have a unique common fixed point provided both pairs ( A , S ) and ( B , T ) are weakly compatible.

Proof In view of Lemma 3.1, the pairs ( A , S ) and ( B , T ) share the common property (E.A), i.e. there exist two sequences { x n } and { y n } in X such that
lim n A x n = lim n S x n = lim n B y n = lim n T y n = z ,

for some z X .

If S ( X ) is a closed subset of X, then on the lines of Theorem 3.1, we can show that the pair ( A , S ) has coincidence point, say u, i.e. A u = S u = z . Since A ( X ) T ( X ) and A u A ( X ) , there exists v X such that A u = T v . The rest of the proof runs along the lines of the proof of Theorem 3.1, therefore details are omitted. □

Remark 3.4 Theorem 3.3 is also a partial improvement of Theorem 3.1 besides relaxing the closedness of one of the subspaces.

Example 3.3 In setting of Example 3.2, replace the self-mappings A, B, S and T by
A ( x ) = { 2 , if  x { 2 } ( 3 , 19 ] ; 3 , if  x ( 2 , 3 ] , B ( x ) = { 2 , if  x { 2 } ( 3 , 19 ] ; 4 , if  x ( 2 , 3 ] , S ( x ) = { 2 , if  x = 2 ; 14 , if  x ( 2 , 3 ] ; x + 1 2 , if  x ( 3 , 19 ] , T ( x ) = { 2 , if  x = 2 ; 11 + x , if  x ( 2 , 3 ] ; x + 1 2 , if  x ( 3 , 19 ] .

It is noted that A ( X ) = { 2 , 3 } [ 2 , 10 ] ( 13 , 14 ] = T ( X ) and B ( X ) = { 2 , 4 } [ 2 , 10 ] { 14 } = S ( X ) . Also the pairs ( A , S ) and ( B , T ) commute at 2 which is their common coincidence point. Thus all the conditions of Theorems 3.2-3.3 and Corollary 3.1 are satisfied and 2 is a unique common fixed point of A, B, S and T. Here, it may be pointed out that Theorem 3.1 is not applicable to this example as S ( X ) is not a closed subset of X. Also, notice that some mappings in this example are even discontinuous at their unique common fixed point 2.

By choosing A, B, S, and T suitably, we can drive a multitude of common fixed-point theorems for a pair or triod of self-mappings. If we take A = B and S = T in Theorem 3.1 then we get the following natural result which is an improved probabilistic version of the result of Sedghi et al. [[55], Theorem 1].

Corollary 3.2 Let ( X , F , ) be a Menger space, where is a continuous t-norm. Let A and S be mappings from X into itself and satisfying the following conditions:
  1. (1)

    The pair ( A , S ) shares property (E.A),

     
  2. (2)

    S ( X ) is a closed subset of X,

     
  3. (3)
    there exist ϕ Φ and 1 k < 2 such that
    F A x , A y ( t ) ϕ ( min { F S x , S y ( t ) , sup t 1 + t 2 = 2 k t min { F S x , A x ( t 1 ) , F S y , A y ( t 2 ) } , sup t 3 + t 4 = 2 k t max { F S x , A y ( t 3 ) , F S y , A x ( t 4 ) } } )
    (3.8)
     

holds for all x , y X and t > 0 . Then the pair ( A , S ) has a coincidence point. Moreover, A and S have a unique common fixed point provided the pair ( A , S ) is weakly compatible.

Our next theorem is proved for six self-mappings in Menger space, which extends earlier proved Theorem 3.1.

Theorem 3.4 Let ( X , F , ) be a Menger space, where is a continuous t-norm. Let A, B, R, S, H and T be mappings from X into itself and satisfying the following conditions:
  1. (1)

    The pairs ( A , S R ) and ( B , T H ) share the common property (E.A),

     
  2. (2)

    S R ( X ) and T H ( X ) are closed subsets of X,

     
  3. (3)
    there exist ϕ Φ and 1 k < 2 such that
    F A x , B y ( t ) ϕ ( min { F S R x , T H y ( t ) , sup t 1 + t 2 = 2 k t min { F S R x , A x ( t 1 ) , F T H y , B y ( t 2 ) } , sup t 3 + t 4 = 2 k t max { F S R x , B y ( t 3 ) , F T H y , A x ( t 4 ) } } )
    (3.9)
     

holds for all x , y X and t > 0 . Then the pairs ( A , S R ) and ( B , T H ) have a coincidence point each. Moreover, A, B, R, S, H, and T have a unique common fixed point provided the pairs ( A , S R ) and ( B , T H ) commute pairwise (i.e. A S = S A , A R = R A , S R = R S , B T = T B , B H = H B , and T H = H T ).

Proof By Theorem 3.1, A, B, TH and SR have a unique common fixed point z in X. We show that z is a unique common fixed point of the self-mappings A, R and S. If z R z , then on using (3.9) with x = R z , y = z , we get, for some t 0 > 0 ,
F A ( R z ) , B z ( t 0 ) ϕ ( min { F S R ( R z ) , T H z ( t 0 ) , sup t 1 + t 2 = 2 k t 0 min { F S R ( R z ) , A ( R z ) ( t 1 ) , F T H z , B z ( t 2 ) } , sup t 3 + t 4 = 2 k t 0 max { F S R ( R z ) , B z ( t 3 ) , F T H z , A ( R z ) ( t 4 ) } } ) , F R z , z ( t 0 ) ϕ ( min { F R z , z ( t 0 ) , min { F R z , R z ( ϵ ) , F z , z ( 2 k t 0 ϵ ) } , max { F R z , z ( ϵ ) , F z , R z ( 2 k t 0 ϵ ) } } ) ,
for all ϵ ( 0 , 2 k t 0 ) . As ϵ 0 , we have
F R z , z ( t 0 ) ϕ ( min { F R z , z ( t 0 ) , F z , R z ( 2 k t 0 ϵ ) } ) = ϕ ( F R z , z ( t 0 ) ) > F R z , z ( t 0 ) ,

which is a contradiction. Therefore, R z = z and so S ( R z ) = S ( z ) = z . Similarly, we get T z = H z = z . Hence z is a unique common fixed point of self-mappings A, B, R, S, H and T in X. □

Corollary 3.3 Let ( X , F , ) be a Menger space, where is a continuous t-norm. Let { A i } i = 1 m , { B r } r = 1 n , { S k } k = 1 p and { T g } g = 1 q be four finite families from X into itself such that A = A 1 A 2 A m , B = B 1 B 2 B n , S = S 1 S 2 S p and T = T 1 T 2 T q , which satisfy the inequality (3.1). If the pairs ( A , S ) and ( B , T ) share the common property (E.A) along with closedness of S ( X ) and T ( X ) , then ( A , S ) and ( B , T ) have a point of coincidence each.

Moreover, { A i } i = 1 m , { B r } r = 1 n , { S k } k = 1 p and { T g } g = 1 q have a unique common fixed point provided the pairs of families ( { A i } , { S k } ) and ( { B r } , { T g } ) are commute pairwise, where i { 1 , 2 , , m } , k { 1 , 2 , , p } , r { 1 , 2 , , n } and g { 1 , 2 , , q } .

Proof The proof of this theorem is similar to that of Theorem 3.1 contained in Imdad et al. [41], hence details are omitted. □

Remark 3.5 Corollary 3.3 extends the result of Sedghi et al. [[55], Theorem 2] to four finite families of self-mappings.

By setting A 1 = A 2 = = A m = A , B 1 = B 2 = = B n = B , S 1 = S 2 = = S p = S , and T 1 = T 2 = = T q = T in Corollary 3.3, we deduce the following.

Corollary 3.4 Let ( X , F , ) be a Menger space, where is a continuous t-norm. Let A, B, S and T be mappings from X into itself such that the pairs ( A m , S p ) and ( B n , T q ) share the common property (E.A). Then there exist ϕ Φ , 1 k < 2 and m , n , p , q N such that
F A m x , B n y ( t ) ϕ ( min { F S p x , T q y ( t ) , sup t 1 + t 2 = 2 k t min { F S p x , A m x ( t 1 ) , F T q y , B n y ( t 2 ) } , sup t 3 + t 4 = 2 k t max { F S p x , B n y ( t 3 ) , F T q y , A m x ( t 4 ) } } )
(3.10)

holds for all x , y X and t > 0 . If S p ( X ) and T q ( X ) are closed subsets of X, then the pairs ( A , S ) and ( B , T ) have a point of coincidence each. Further, A, B, S, and T have a unique common fixed point provided both pairs ( A m , S p ) and ( B n , T q ) commute pairwise.

Conclusion

Theorem 3.1 is proved for two pairs of weakly compatible mappings in Menger spaces using common property (E.A). Theorem 3.1 is an improved probabilistic version of the result of Sedghi et al. [[55], Theorem 1] for two pairs of mappings without any requirement on containment of ranges amongst the involved mappings. Several results (Theorem 3.2, Theorem 3.3 and Corollary 3.1) are also defined for the existence of fixed points in Menger spaces. Example 3.2 and Example 3.3 are furnished in support of our results. As an extension of our main result, Theorem 3.4 is proved for six self-mappings using the notion of pairwise commuting whereas Corollary 3.3 extends Theorem 3.1 to four finite families of self-mappings.

Declarations

Acknowledgements

The authors would like to thank the referees for their useful comments on the manuscript.

Authors’ Affiliations

(1)
Near Nehru Training Centre
(2)
Department of Mathematics, Jazan University
(3)
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center
(4)
Faculty of Sciences and Mathematics

References

  1. Banach S: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundam. Math. 1922, 3: 133–181.MATHGoogle Scholar
  2. Hicks TL: Fixed point theorems for quasi-metric spaces. Math. Jpn. 1988,33(2):231–236.MATHMathSciNetGoogle Scholar
  3. Choudhury BS, Metiya N: Coincidence point and fixed point theorems in ordered cone metric spaces. J. Adv. Math. Stud. 2012,5(2):20–31.MATHMathSciNetGoogle Scholar
  4. Olatinwo MO, Postolache M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 2012,218(12):6727–6732. 10.1016/j.amc.2011.12.038MATHMathSciNetView ArticleGoogle Scholar
  5. Aydi H, Karapınar E, Postolache M: Tripled coincidence point theorems for weak φ -contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 44Google Scholar
  6. Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N: Theorems for Boyd-Wong type contractions in ordered metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 359054Google Scholar
  7. Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28Google Scholar
  8. Choudhury BS, Metiya N, Postolache M: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. 2013., 2013: Article ID 152Google Scholar
  9. Shatanawi W, Postolache M: Common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 60Google Scholar
  10. Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for ( ψ , ϕ ) -weakly contractive mappings in ordered G -metric spaces. Comput. Math. Appl. 2012,63(1):298–309. 10.1016/j.camwa.2011.11.022MATHMathSciNetView ArticleGoogle Scholar
  11. Chandok S, Mustafa Z, Postolache M: Coupled common fixed point theorems for mixed g -monotone mappings in partially ordered G -metric spaces. U. Politeh. Buch. Ser. A 2013,75(4):11–24.MathSciNetGoogle Scholar
  12. Shatanawi W, Pitea A: Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction. Fixed Point Theory Appl. 2013., 2013: Article ID 275Google Scholar
  13. Shatanawi W, Postolache M: Some fixed point results for a G -weak contraction in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 815870Google Scholar
  14. Shatanawi W, Pitea A: Omega-distance and coupled fixed point in G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 208Google Scholar
  15. Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011,4(2):1–12.MATHMathSciNetGoogle Scholar
  16. Shatanawi W, Postolache M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 54Google Scholar
  17. Shatanawi W, Pitea A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 153Google Scholar
  18. Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/0165-0114(88)90064-4MATHMathSciNetView ArticleGoogle Scholar
  19. Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535MATHMathSciNetView ArticleGoogle Scholar
  20. Haghi RH, Postolache M, Rezapour S: On T -stability of the Picard iteration for generalized φ -contraction mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 658971Google Scholar
  21. Miandaragh MA, Postolache M, Rezapour S: Some approximate fixed point results for generalized α -contractive mappings. U. Politeh. Buch. Ser. A 2013,75(2):3–10.MATHMathSciNetGoogle Scholar
  22. Jungck G, Rhoades BE: Fixed points for set valued functions without continuity. Indian J. Pure Appl. Math. 1998,29(3):227–238. MR1617919MATHMathSciNetGoogle Scholar
  23. Singh B, Jain S: A fixed point theorem in Menger space through weak compatibility. J. Math. Anal. Appl. 2005,301(2):439–448. 10.1016/j.jmaa.2004.07.036MATHMathSciNetView ArticleGoogle Scholar
  24. Aamri M, Moutawakil DEl: Some new common fixed point theorems under strict contractive conditions. J. Math. Anal. Appl. 2002,270(1):181–188. 10.1016/S0022-247X(02)00059-8MATHMathSciNetView ArticleGoogle Scholar
  25. Kubiaczyk I, Sharma S: Some common fixed point theorems in Menger space under strict contractive conditions. Southeast Asian Bull. Math. 2008,32(1):117–124. MR2385106 Zbl 1199.54223MATHMathSciNetGoogle Scholar
  26. Pant RP: Common fixed point theorems for contractive maps. J. Math. Anal. Appl. 1998, 226: 251–258. MR1646430 10.1006/jmaa.1998.6029MATHMathSciNetView ArticleGoogle Scholar
  27. Liu Y, Wu J, Li Z: Common fixed points of single-valued and multi-valued maps. Int. J. Math. Math. Sci. 2005, 19: 3045–3055.MathSciNetView ArticleGoogle Scholar
  28. Ali J, Imdad M, Bahuguna D: Common fixed point theorems in Menger spaces with common property (E.A). Comput. Math. Appl. 2010,60(12):3152–3159. MR2739482 (2011g:47124) Zbl 1207.54050 10.1016/j.camwa.2010.10.020MATHMathSciNetView ArticleGoogle Scholar
  29. Abbas M, Nazir T, Radenović S: Common fixed point of power contraction mappings satisfying (E.A) property in generalized metric spaces. Appl. Math. Comput. 2013, 219: 7663–7670. 10.1016/j.amc.2012.12.090MATHMathSciNetView ArticleGoogle Scholar
  30. Cakić N, Kadelburg Z, Radenović S, Razani A: Common fixed point results in cone metric spaces for a family of weakly compatible maps. Adv. Appl. Math. Sci. 2009,1(1):183–201.MATHMathSciNetGoogle Scholar
  31. Janković S, Golubović Z, Radenović S: Compatible and weakly compatible mappings in cone metric spaces. Math. Comput. Model. 2010, 52: 1728–1738. 10.1016/j.mcm.2010.06.043MATHView ArticleGoogle Scholar
  32. Kadelburg Z, Radenović S, Rosić B: Strict contractive conditions and common fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 173838Google Scholar
  33. Long W, Abbas M, Nazir T, Radenović S: Common fixed point for two pairs of mappings satisfying (E.A) property in generalized metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 394830Google Scholar
  34. Ali J, Imdad M, Miheţ D, Tanveer M: Common fixed points of strict contractions in Menger spaces. Acta Math. Hung. 2011,132(4):367–386. 10.1007/s10474-011-0105-3MATHView ArticleGoogle Scholar
  35. Altun I, Tanveer M, Miheţ D, Imdad M: Common fixed point theorems of integral type in Menger PM spaces. J. Nonlinear Anal. Optim., Theory Appl. 2012,3(1):55–66.Google Scholar
  36. Beg I, Abbas M: Common fixed points of weakly compatible and noncommuting mappings in Menger spaces. Int. J. Mod. Math. 2008,3(3):261–269.MATHMathSciNetGoogle Scholar
  37. Chauhan S, Pant BD: Common fixed point theorem for weakly compatible mappings in Menger space. J. Adv. Res. Pure Math. 2011,3(2):107–119. 10.5373/jarpm.585.100210MathSciNetView ArticleGoogle Scholar
  38. Cho YJ, Park KS, Park WT, Kim JK: Coincidence point theorems in probabilistic metric spaces. Kobe J. Math. 1991,8(2):119–131.MATHMathSciNetGoogle Scholar
  39. Fang JX: Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. Nonlinear Anal. 2009,71(5–6):1833–1843. 10.1016/j.na.2009.01.018MATHMathSciNetView ArticleGoogle Scholar
  40. Fang JX, Gao Y: Common fixed point theorems under strict contractive conditions in Menger spaces. Nonlinear Anal. 2009,70(1):184–193. 10.1016/j.na.2007.11.045MATHMathSciNetView ArticleGoogle Scholar
  41. Imdad M, Ali J, Tanveer M: Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces. Chaos Solitons Fractals 2009,42(5):3121–3129. MR2562820 (2010j:54064) Zbl 1198.54076 10.1016/j.chaos.2009.04.017MATHMathSciNetView ArticleGoogle Scholar
  42. Imdad M, Tanveer M, Hassan M: Some common fixed point theorems in Menger PM spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 819269Google Scholar
  43. Kumar, S, Chauhan, S, Pant, BD: Common fixed point theorem for noncompatible maps in probabilistic metric space. Surv. Math. Appl. (in press)Google Scholar
  44. Kumar S, Pant BD: Common fixed point theorems in probabilistic metric spaces using implicit relation and property (E.A). Bull. Allahabad Math. Soc. 2010,25(2):223–235.MATHMathSciNetGoogle Scholar
  45. Kutukcu S: A fixed point theorem in Menger spaces. Int. Math. Forum 2006,1(32):1543–1554.MATHMathSciNetGoogle Scholar
  46. Miheţ D: A note on a common fixed point theorem in probabilistic metric spaces. Acta Math. Hung. 2009,125(1–2):127–130. 10.1007/s10474-009-8238-3MATHView ArticleGoogle Scholar
  47. O’Regan D, Saadati R: Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 2008,195(1):86–93. MR2379198 Zbl 1135.54315 10.1016/j.amc.2007.04.070MATHMathSciNetView ArticleGoogle Scholar
  48. Pant BD, Abbas M, Chauhan S: Coincidences and common fixed points of weakly compatible mappings in Menger spaces. J. Indian Math. Soc. 2013,80(1–2):127–139.MATHMathSciNetGoogle Scholar
  49. Pant BD, Chauhan S: A contraction theorem in Menger space. Tamkang J. Math. 2011,42(1):59–68. MR2815806 Zbl 1217.54053MATHMathSciNetView ArticleGoogle Scholar
  50. Pant BD, Chauhan S: Common fixed point theorems for two pairs of weakly compatible mappings in Menger spaces and fuzzy metric spaces. Sci. Stud. Res. Ser. Math. Inform. 2011,21(2):81–96.MATHMathSciNetGoogle Scholar
  51. Saadati R, O’Regan D, Vaezpour SM, Kim JK: Generalized distance and common fixed point theorems in Menger probabilistic metric spaces. Bull. Iran. Math. Soc. 2009,35(2):97–117.MATHMathSciNetGoogle Scholar
  52. Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 313–334. 10.2140/pjm.1960.10.313MATHMathSciNetView ArticleGoogle Scholar
  53. Mishra SN: Common fixed points of compatible mappings in PM-spaces. Math. Jpn. 1991,36(2):283–289.MATHGoogle Scholar
  54. Pathak HK, López RR, Verma RK: A common fixed point theorem using implicit relation and property (E.A) in metric spaces. Filomat 2007,21(2):211–234. 10.2298/FIL0702211PMATHMathSciNetView ArticleGoogle Scholar
  55. Sedghi S, Shobe N, Aliouche A: A common fixed point theorem for weakly compatible mappings in fuzzy metric spaces. Gen. Math. 2010,18(3):3–12.MATHMathSciNetGoogle Scholar

Copyright

© Chauhan et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.