Research Article | Open Access

# Coupled Fixed Point Theorems for a Pair of Weakly Compatible Maps along with *CLRg* Property in Fuzzy Metric Spaces

**Academic Editor:**Yansheng Liu

#### Abstract

The aim of this paper is to extend the notions of E.A. property and *CLRg* property for coupled mappings and use these notions to generalize the recent results of Xin-Qi Hu (2011). The main result is supported by a suitable example.

#### 1. Introduction and Preliminaries

The concept of fuzzy set was introduced by Zadeh [1] and after his work there has been a great endeavor to obtain fuzzy analogues of classical theories. This problem has been searched by many authors from different points of view. In 1994, George and Veeramani [2] introduced and studied the notion of fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space.

Bhaskar and Lakshmikantham [3] introduced the notion of coupled fixed points and proved some coupled fixed point results in partially ordered metric spaces. The work [3] was illustrated by proving the existence and uniqueness of the solution for a periodic boundary value problem. These results were further extended and generalized by Lakshmikantham and CiriΔ [4] to coupled coincidence and coupled common fixed point results for nonlinear contractions in partially ordered metric spaces.

Sedghi et al. [5] proved some coupled fixed point theorems under contractive conditions in fuzzy metric spaces. The results proved by Fang [6] for compatible and weakly compatible mappings under -contractive conditions in Menger spaces that provide a tool to Hu [7] for proving fixed points results for coupled mappings and these results are the genuine generalization of the result of [5].

Aamri and Moutawakil [8] introduced the concept of E.A. property in a metric space. Recently, Sintunavarat and Kuman [9] introduced a new concept of (*CLRg*). The importance of *CLRg* property ensures that one does not require the closeness of range subspaces.

In this paper, we give the concept of E.A. property and (*CLRg*) property for coupled mappings and prove a result which provides a generalization of the result of [7].

#### 2. Preliminaries

Before we give our main result, we need the following preliminaries.

*Definition 2.1 (see [1]). *A fuzzy set in is a function with domain and values in .

*Definition 2.2 (see [10]). *A binary operation is continuous -norm, if is a topological abelian monoid with unit such that whenever and for all .

Some examples are below: (i),
(ii).

*Definition 2.3 (see [11]). * Let . A -norm is said to be of -type if the family of functions is equicontinuous at , where
A -norm is an -type -norm if and only if for any , there exists such that for all , when .

The -norm is an example of -norm, of -type.

*Definition 2.4 (see [2]). *The 3-tuple is said to be a fuzzy metric space if is an arbitrary set, is a continuous -norm and is a fuzzy set on satisfying the following conditions: (FM-1) for all ,(FM-2) if and only if , for all and ,(FM-3) for all and ,(FM-4) for all and ,(FM-5) is continuous for all .

In present paper, we consider to be fuzzy metric space with, the following condition:(FM-6),ββforββallββ and .

*Definition 2.5 (see [2]). *Let be a fuzzy metric space. A sequence is said to be: (i)convergent to a point , if for all ,
(ii)a Cauchy sequence, if for all and ,

A fuzzy metric space is said to be complete if and only if every Cauchy sequence in is convergent.

We note that is nondecreasing for all .

Lemma 2.6 (see [12]). *Let and , then for all : *(i)*,
*(ii)* if is continuous. *

*Definition 2.7 (see [7]). *Define , and each satisfies the following conditions:(-1) is nondecreasing;(-2) is upper semicontinuous from the right;(-3) for all , where .Clearly, if , then for all .

*Definition 2.8 (see [4]). *An element is called: (i)a coupled fixed point of the mapping if , ,(ii)a coupled coincidence point of the mappings and if ,ββ,(iii)a common coupled fixed point of the mappings and if ,ββ.

*Definition 2.9 (see [6]). *An element is called a common fixed point of the mappings and if .

*Definition 2.10 (see [6]). *The mappings and are called: (i)commutative if for all ,(ii)compatible if
for all whenever and are sequences in , such thatββ, and , for some .

*Definition 2.11 (see [13]). *The maps and are called -compatible if whenever .

We note that the maps and are called weakly compatible if
implies ,ββ, for all .

There exist pair of mappings that are neither compatible nor weakly compatible, as shown in the following example.

*Example 2.12. *Let be a fuzzy metric space, being a continuous norm with . Define for all ,ββ. Also define the maps and by and , respectively. Note that (0,β0) is the coupled coincidence point of and in . It is clear that the pair is weakly compatible on .

We next show that the pair is not compatible.

Consider the sequences and , , then
but
which is not convergent to 1 as .

Hence the pair () is not compatible.

We note that, if and are compatible then they are weakly compatible. But the converse need not be true, as shown in the following example.

*Example 2.13. *Let be a fuzzy metric space, being a continuous norm with . Define for all ,ββ. Define the maps and by
The only coupled coincidence point of the pair is . The mappings and are noncompatible, since for the sequences , we have , , , , as . But they are weakly compatible since they commute at their coupled coincidence point .

Now we introduce our notions.

Aamri and El Moutawakil [8] introduced the concept of E.A. property in a metric space as follows.

Let be a metric space. Self mappings and are said to satisfy E.A. property if there exists a sequence such that for some .

Now we extend this notion for a pair of coupled maps as follows.

*Definition 2.14. *Let be a metric space. Two mappings and are said to satisfy E.A. property if there exists sequences such that
for some .

In a similar mode, we state E.A. property for coupled mappings in fuzzy metric spaces as follows.

Let be a FM space. Two maps and satisfy E.A. property if there exists sequences and such that ,ββ converges to and , β converges to in the sense of Definition 2.5.

*Example 2.15. *Let be a usual metric space. Define mappings and by and for all . Consider the sequences and . Since
therefore, and satisfy E.A. property, since .

*Remark 2.16. *It is to be noted that property E.A. need not imply compatibility, since in Example 2.12, the maps and defined are not compatible, but satisfy property E.A., since for the sequences and we have
since .

Recently, Sintunavarat and Kuman [9] introduced a new concept of *the common limit in the range of g*, (*CLRg*) property, as follows.

*Definition 2.17. *Let be a metric space. Two mappings and are said to satisfy (*CLRg*) property if there exists a sequence ββsuch thatββ ββfor someββ.

Now we extend this notion for a pair of coupled mappings as follows.

*Definition 2.18. *Let be a metric space. Two mappings and are said to satisfy (*CLRg*) property if there exists sequences such that
for some .

Similarly, we state (*CLRg*) property for coupled mappings in fuzzy metric spaces.

Let be an FM space. Two maps and satisfy (*CLRg*) property if there exists sequences such that converge to and , converge to , in the sense of Definition 2.5.

*Example 2.19. *Let be a metric space under usual metric. Define mappings and by and for all . We consider the sequences and . Since
therefore, the maps and satisfy property.

In the next example, we show that the maps satisfying property need not be continuous, that is, continuity is not the necessary condition for self maps to satisfy property.

*Example 2.20. *Let be a metric space under usual metric. Define mappings and by
We consider the sequences and . Since
therefore, the maps and satisfy property but the maps are not continuous.

We next show that the pair of maps satisfying property may not be compatible.

*Example 2.21. *Let be a fuzzy metric space, being a continuous norm,, and for all and .

Define the maps and by and , respectively.

Consider the sequences and . Then
Further there exists the point 1/3 in such that , so that the pair satisfies property. But,
does not converge to 1 as .

Hence, the pair is not compatible.

#### 3. Main Results

For convenience, we denote(1) for all .

Hu [7] proved the following result.

Theorem 3.1. *Let be a complete fuzzy metric space where is a continuous t-norm of H-type. Let and be two mappings and there exists such that*(2)*
for all and . Suppose that , is continuous, and are compatible maps. Then there exists a unique point such that , that is, and have a unique common fixed point in .*

We now give our main result which provides a generalization of Theorem 3.1.

Theorem 3.2. *Let be a Fuzzy Metric Space, being continuous t-norm of H-type. Let and be two mappings and there exists satisfying (2) with the following conditions: *(3)* the pair is weakly compatible, *(4)* the pair satisfy property. ** Then and have a coupled coincidence point in . Moreover, there exists a unique point such that .*

*Proof . *Since and satisfy property, there exists sequences and in such that
for some .*Stepββ1.* To show that and have a coupled coincidence point. From (2),
Taking limit , we get , that is, .

Similarly, .

Since and are weakly compatible, so that (say) and (say) implies and , that is, and . Hence and have a coupled coincidence point.*Stepββ2*. To show that , andββ. Since is a -norm of -type, for any , there exists such that
for all .

Since for all , there exists such that
Also since using condition , we have .

Then for any , there exists such that . From (2), we have
Similarly, we can also get
Continuing in the same way, we can get for all ,
Then, we have
So, for any , we have for all .

This implies . Similarly, .*Stepββ3*. Next we shall show that . Since is a -norm of -type, for any there exists such that
for all .

Since for all , there exists such that .

Also since , using condition (-3), we have . Then for any , there exists such that
Using condition (2), we have
Continuing in the same way, we can get for all ,
Then we have
which implies that . Thus, we have proved that and have a common fixed point .*Stepββ4*. We now prove the uniqueness of . Let be any point in such that with . Since is a -norm of -type, for any , there exists such that
for all . Since for all , there exists such that . Also since and using condition (-3), we have . Then for any , there exists such that
Using condition (2), we have
Continuing in the same way, we can get for all ,
Then we have
which implies that .

Hence and have a unique common fixed point in .

*Remark 3.3. *We still get a unique common fixed point if weakly compatible notion is replaced by w-compatible notion.

Now we give another generalization of Theorem 3.1.

Corollary 3.4. *Let be a fuzzy metric space where is a continuous t-norm of H-type. Let and be two mappings and there exists satisfying (2) and (3) with the following condition:*(5)* the pair satisfy E.A. property.**If is a closed subspace of , then and have a unique common fixed point in .*

*Proof. *Since and satisfy E.A. property, there exists sequences and in such that
for some .

It follows from being a closed subspace of that , for some and then and satisfy the property. By Theorem 3.2, we get that and have a unique common fixed point in .

Corollary 3.5. *Let be a fuzzy metric space where is a continuous -norm of -type. Let and be two mappings and there exists satisfying (2), (3), and (5).**Suppose that , if range of one of the maps or is a closed subspace of , then and have a unique common fixed point in .*

*Proof. *It follows immediately from Corollary 3.5.

Taking in Theorem 3.2, the Corollary 3.6 follows immediately the following.

Corollary 3.6. *Let be a fuzzy metric space where is a continuous -norm of -type. Let and be two mappings and there exists satisfying the following conditions, for all and :*(6)*,
*(7)* there exists sequences and in such that
for some .*

Then, there exists a unique such that .

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#### Copyright

Copyright © 2012 Manish Jain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.