Research Article | Open Access

# Common Fixed Point Theorems Using the E.A. and CLR Properties in 2-Menger Spaces

**Academic Editor:**Chuanxi Qian

#### Abstract

First, we prove a common fixed point theorem using weakly compatible maps in 2-Menger space with *t*-norm of Hadzic type. Second, we prove a common fixed point theorem using the E.A. property along with weakly compatible maps. Further, we obtained a common fixed point theorem using the CLR property along with weakly compatible maps. At the end, we provide an application of our main theorem for four finite families of mappings.

#### 1. Introduction

The theory of probabilistic metric spaces is an important part of stochastic analysis, and so it is of interest to develop the fixed point theory in such spaces. The first result from the fixed point theory in probabilistic metric spaces was obtained by Sehgal and Bharucha-Reid [1]. Since then many fixed points theorems for single valued and multivalued mappings in probabilistic metric spaces have been proved in [2–8]. The study of 2-metric spaces was initiated by Gähler [9] and some fixed point theorems in 2-metric spaces were proved in [2, 8, 10–13]. In 1987, Zeng [14] gave the generalization of 2-metric to probabilistic 2-metric as follows.

A probabilistic 2-metric space is an ordered pair , where is an arbitrary set and is a mapping from into the set of distribution functions. The distribution function will denote the value of at the positive real number . The function is assumed to satisfy the following conditions:(i) for all ;(ii) for all if and only if at least two of the three points are equal;(iii)for distinct points , there exists a point such that for ;(iv) for all and ;(v)if , , and , then for all and .

In 2003, Shi et al. [15] gave the notion of th order -norm as follows.

*Definition 1. *A mapping is called an th order -norm if the following conditions are satisfied:(i), for all ;(ii);
(iii), implies ;(iv) .

For , we have a binary -norm, which is commonly known as -norm.

Basic examples of -norm are the Lukasiewicz -norm -norm , , and -norm , .

*Definition 2 (see [6]). *A special class of *t*-norms (called a Hadzic-type -norm) is introduced as follows.

Let be a -norm and let be defined in the following way:

We say that the -norm is of type if is continuous and the family is equicontinuous at .

The family is equicontinuous at if for every there exists such that the following implication holds:

A trivial example of *t*-norm of type is .

*Remark 3. *Every -norm is of Hadzic type but the converse need not be true; see [7].

There is a nice characterization of continuous -norm.(i)If there exists a strictly increasing sequence such that and for all , then is of Hadzic type.(ii)If is continuous and is of Hadzic type, then there exists a sequence as in (i).

*Definition 4 (see [7]). *If is a -norm and , then is defined recurrely by 1, if and for all . If is a sequence of numbers from , then is defined as (this limit always exists) and as .

*Definition 5. *Let be any nonempty set and the set of all left-continuous distribution functions. A triplet is said to be a 2-Menger space if the probabilistic 2-metric space satisfies the following condition: , , ), where , ; ; and is the 3rd order -norm.

*Definition 6. *A sequence in a 2-Menger space is said to be(i)converge with limit if for all and for every ,(ii)Cauchy sequence in , if given , , there exists a positive integer such that
(iii)complete if every Cauchy sequence in is convergent in .In 1996, Jungck’s [16] introduced the notion of weakly compatible as follows.

*Definition 7. *Two maps and are said to be weakly compatible if they commute at their coincidence points.

In 2002, Aamri and Moutawakil [17] generalized the notion of noncompatible mapping to the E.A. property. It was pointed out in [17] that the property E.A. buys containment of ranges without any continuity requirements besides minimizing the commutativity conditions of the maps to the commutativity at their points of coincidence. Moreover, the E.A. property allows replacing the completeness requirement of the space with a more natural condition of closeness of the range. Recently, some common fixed point theorems in probabilistic metric spaces/fuzzy metric spaces by the E.A. property under weak compatibility have been obtained in [18–20].

*Definition 8 (see [17]). *Let and be two self-maps of a metric (, ). The maps and are said to satisfy the E.A. property if there exists a sequence in such that

Now in a similar mode, we can state the E.A. property in 2-Menger space as follows.

*Definition 9. *A pair of self-mappings of 2-Menger spaces is said to hold the E.A. property if there exists a sequence in such that

*Example 10. *Let be the usual metric space. Define by and for all . Consider the sequence . Since , then and satisfy the E.A. property.

Although E.A property is generalization of the concept of noncompatible maps, yet it requires either completeness of the whole space or any of the range spaces or continuity of maps. But on the contrary, the new notion of the CLR property (common limit range property) recently given by Sintunavarat and Kumam [21] does not impose such conditions. The importance of the CLR property ensures that one does not require the closeness of range subspaces.

*Definition 11 (see [21]). *Two maps and on 2-Menger spaces satisfy the common limit in the range of (CLRg) property if for some .

*Example 12. *Let be the usual metric space. Define by and for all . Consider the sequence . Since , and satisfy the CLRg property.

Now we state a lemma which is useful in our study.

Lemma 13 (see [22]). *Let be a 2-Menger space. If there exists such that for all with , and , then .*

#### 2. Weakly Compatible Maps

Theorem 14. *Let be a complete 2-Menger space with continuous t-norm of type. Let and be self-mappings of . Then and have a unique common fixed point in if and only if there exist two self-mappings of satisfying the following:*(2.1)

*and ;*(2.2)

*the pairs and are weakly compatible;*(2.3)

*there exists such that for every and , , ;*(2.4)

*one of the subsets , , , or is a closed subset of .*

*Indeed, , and have a unique common fixed point in .*

*Proof. *Suppose that and have a unique common fixed point, say .

Define by for all and by for all .

Then one can see that (2.1)–(2.4) are satisfied.

Conversely, assume that there exist two self-mappings of satisfying conditions (2.1)–(2.4). From condition (2.1) we can construct two sequences and of such that
Putting and in (2.3), we have that for all and
implies , because is nondecreasing.

Also, letting and in (2.3), we have that
In general, we have
Thus for all

We now show that is a Cauchy sequence in .

Let be given. Since the -norm is of type, there exists such that for all , with
Since , there exists such that for all and , for all .

From (10) we have for all and , for all .

Let . Then for all and , we have

Since
From (13), we get
Inductively, we obtain

From (11) and (13) we get, for all and for all . Thus is a Cauchy sequence in . Since is complete, there exists a point in such that and this gives
Without loss of generality, we assume that is a complete subspace of ; therefore, for some .

Subsequently, we have
Next, we claim that .

For this purpose, we put and in (2.3); we have

Taking limit as
By Lemma 13, we have .

Hence .

Since , there exists a point such that .

Next we claim that .

Putting and in (2.3), we have

That is, .

By Lemma 13, we have .

Thus .

Since the pairs and are weakly compatible and and are their points of coincidence, respectively, then

Now we prove that is a common fixed point of , and .

For this purpose, puting and in (2.3), we get

By Lemma 13, we have .

Hence , and is a common fixed point of and . One can prove that is also a common fixed point of and .*Uniqueness.* Suppose is another fixed point of , and .

Then, for all with and and ,
which implies that .

Hence is a unique common fixed point of , and .

This completes the proof of the theorem.

Corollary 15. *Let be a 2-Menger space with continuous t-norm of type. Let and be self-mappings of . Then and have a unique common fixed point in if and only if there exist two self-mappings of satisfying the following:*(2.5)

*;*(2.6)

*pair is weakly compatible;*(2.7)

*there exists such that for every and ,*(2.8)

*one of the subspaces or is a closed subspace of .*

*Indeed, and have a unique common fixed point in .*

*Proof . *We can easily prove the theorem by setting and , in the proof of the Theorem 14.

Corollary 16. *Let be a 2-Menger space with continuous t-norm of type. Let and be self-mappings of . Then and have a unique common fixed point in if and only if there exist two self-mappings of satisfying (2.1), (2.2), (2.4), and the following:*

*there exists such that for every and , .*

*Indeed, , and have a unique common fixed point in .*

*Proof. *Proof easily follows by setting , , , in the proof of Theorem 14.

#### 3. E.A. Property and Weakly Compatible Maps

Now we prove our main result for weakly compatible maps along with the E.A. property as follows.

Theorem 17. *Let be a 2-Menger space with continuous t-norm of type. Let , , , and be self-mappings of satisfying (2.1), (2.2), (2.3), and (2.4) and the following:*(3.1)

*pair or pair satisfies E.A. property.*

*Then , and have a unique common fixed point in .*

*Proof. *If the pair satisfies the property E.A., then there exists a sequence in such that for some .

Since , there exists a sequence in such that . Hence . Also ; there exists a sequence in such that . Hence .

Suppose that is a complete subspace of . Then for some ; subsequently; we have

Next, we claim that .

For this purpose, we put and in (2.3); this gives

Taking limit as

By Lemma 13, we have .

Hence .

Since , there exists a point such that .

Next we claim that .

Putting and in (2.3), we have
That is, .

By Lemma 13, we have .

Thus .

Since the pairs and are weakly compatible and and are their points of coincidence, respectively, then and .

Now we prove that is a common fixed point of , and .

For this purpose, we put and in (2.3); we get

By Lemma 13, we have .

Hence and is a common fixed point of and . One can prove that is also a common fixed point of and .*Uniqueness.* Suppose is another fixed point of , and .

Then, for all with and and ,
which implies that .

Hence is a unique common fixed point of , and .

Corollary 18. *Let be a 2-Menger space with continuous t-norm of type. Let , and be self-mappings of satisfying (2.1), (2.2), (2.4), and (2.9).*

*Then , and have a unique common fixed point in .*

#### 4. CLR Property and Weakly Compatible Maps

Now we prove our main result for weakly compatible maps along with the CLR_{S} property as follows.

Theorem 19. *Let be a 2-Menger space with continuous t-norm of type. Let , and be self mapping of satisfying (2.1), (2.2), (2.3), and the following:*(4.1)* pair or pair satisfies CLR _{S} property;*(4.2)

*one of the subspaces , or is a closed subspace of .*

*Then , and have a unique common fixed point in .*

*Proof. *If the pair satisfies the CLR_{S} property then there exists a sequence in such that , where . Therefore, there exists a point such that . Since is a closed subspace of and , for each there corresponds a sequence such that .

Therefore, , where .

Thus in all, we have

Now we are required to show that .

Putting and in (2.3), we get

Let for .Then taking limit as , we have

By Lemma 13, we have , then .

Hence = , for some .

Uniqueness follows from Theorem 14.

Corollary 20. *Let be a 2-Menger space with continuous t-norm of H type. Let , and be self-mappings of X satisfying (2.1), (2.2), (2.9), (4.1), and (4.2).*

*Then , and have a unique common fixed point in .*

#### 5. Application

As an application of Theorem 14, we prove a common fixed point theorem for four finite families of mappings which runs as follows.

Theorem 21. *Let , , , and be four finite families of self-mappings of a 2-Menger space with continuous t-norm of Hadzic type such that , , , and satisfy condition (2.1), (2.2), (2.3), and (2.4); then*(i)*A and have a point of coincidence;*(ii)*B and have a point of coincidence.**Moreover, if , , , , , and for all , , , , , and , , then (for all , and ), , , and have a common fixed point.*

*Proof. *The conclusions (i) and (ii) are immediate as , and satisfy all the conditions of Theorem 14. Now appealing to component wise commutativity of various pairs, one can immediately prove that and , and hence, obviously both pairs and are coincidently commuting. Note that all the conditions of Theorem 14 (for mappings , and ) are satisfied ensuring the existence of a unique common fixed point, say . Now one needs to show that remains the fixed point of all the component maps. For this consider
Similarly, one can show that
which show that (for all , and ) are other fixed points of the pair whereas are other fixed points of the pair . Now appealing to the uniqueness of common fixed points of both pairs separately, we get
which shows that is a common fixed point of ,,, and for all , and .

By setting , and , one deduces the following for certain iterates of maps, which run as follows.

Corollary 22. *Let , and be four self-mappings of a 2-Menger space such that ,,, and satisfy the conditions (2.1) and (2.3). If one of ,,, or is a closed subset of , then , and have a unique common fixed point provided and commute.*

#### Acknowledgments

One of the authors (S. Kumar) would like to acknowledge the UGC for providing financial grant through Major Research Project under Ref. 39-41/2010 (SR). I am also thankful to the referee for careful reading of the paper and giving useful comments to improve it.

#### References

- V. M. Sehgal and A. T. Bharucha-Reid, “Fixed points of contraction mappings on probabilistic metric spaces,”
*Mathematical Systems Theory*, vol. 6, pp. 97–102, 1972. View at: Google Scholar | Zentralblatt MATH | MathSciNet - O. Hadžić, “On common fixed point theorems in 2-metric spaces,”
*Univerzitet u Novom Sadu, Zbornik Radova Prirodno-Matematičkog Fakulteta*, vol. 12, pp. 7–18, 1982. View at: Google Scholar | MathSciNet - O. Hadžić, “Some theorems on the fixed points in probabilistic metric and random normed spaces,”
*Bollettino dell'Unione Matematica Italiana B*, vol. 1, no. 6, pp. 381–391, 1982. View at: Google Scholar | Zentralblatt MATH | MathSciNet - O. Hadžić, “Fixed point theorems for multivalued mappings in some classes of fuzzy metric spaces,”
*Fuzzy Sets and Systems*, vol. 29, no. 1, pp. 115–125, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - O. Hadzic, “A fixed point theorem for multivalued mappings in 2-Menger spaces,”
*Univerzitet u Novom Sadu, Zbornik Radova Prirodno-Matematičkog Fakulteta*, vol. 24, pp. 1–7, 1994. View at: Google Scholar - O. Hadžić, “A fixed point theorem in Menger spaces,”
*Institut Mathématique*, vol. 40, pp. 107–112, 1979. View at: Google Scholar | Zentralblatt MATH | MathSciNet - O. Hadžić and E. Pap,
*Fixed Point Theory in Probabilistic Metric Spaces*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. View at: MathSciNet - T. L. Hicks, “Fixed point theory in probabilistic metric spaces,”
*Univerzitet u Novom Sadu, Zbornik Radova Prirodno-Matematičkog Fakulteta*, vol. 13, pp. 63–72, 1983. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Gähler, “2-metrische Räume und ihre topologische Struktur,”
*Mathematische Nachrichten*, vol. 26, pp. 115–148, 1963. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Iseki, “Fixed point theorems in 2-metric spaces,”
*Mathematics Seminar Notes*, vol. 3, pp. 133–136, 1975. View at: Google Scholar - B. G. Pachpatte, “Fixed point theorems for contraction type mappings on a 2-metric space,”
*Proceedings of the National Academy of Sciences, India A*, vol. 48, no. 2, pp. 94–102, 1978. View at: Google Scholar | MathSciNet - B. E. Rhoades, “Contraction type mappings on a 2-metric space,”
*Mathematische Nachrichten*, vol. 91, pp. 151–155, 1979. View at: Publisher Site | Google Scholar | MathSciNet - S.-S. Chang and N. J. Huang, “On the generalized 2-metric spaces and probabilistic 2-metric spaces with applications to fixed point theory,”
*Mathematica Japonica*, vol. 34, no. 6, pp. 885–900, 1989. View at: Google Scholar | MathSciNet - W. Z. Zeng, “Probabilistic 2-metric spaces,”
*Journal of Mathematical Research and Exposition*, vol. 7, no. 2, pp. 241–245, 1987. View at: Google Scholar | MathSciNet - Y. Shi, L. Ren, and X. Wang, “The extension of fixed point theorems for set valued mapping,”
*Journal of Applied Mathematics & Computing*, vol. 13, no. 1-2, pp. 277–286, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Jungck, “Common fixed points for noncontinuous nonself maps on nonmetric spaces,”
*Far East Journal of Mathematical Sciences*, vol. 4, no. 2, pp. 199–215, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 181–188, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Imdad and J. Ali, “Jungck's common fixed point theorem and E.A property,”
*Acta Mathematica Sinica*, vol. 24, no. 1, pp. 87–94, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Jungck and B. E. Rhoades, “Fixed points for set valued functions without continuity,”
*Indian Journal of Pure and Applied Mathematics*, vol. 29, no. 3, pp. 227–238, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. K. Pathak, “Common fixed point theorems using property (E.A) in complex-valued metric spaces,”
*Thai Journal of Mathematics*. In press. View at: Google Scholar - W. Sintunavarat and P. Kumam, “Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces,”
*Journal of Applied Mathematics*, vol. 2011, Article ID 637958, 14 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Han, “A common fixed point theorem on Fuzzy 2-metric spaces,”
*Journal of the Chung Cheong Mathematical Society*, vol. 23, no. 4, 2010. View at: Google Scholar

#### Copyright

Copyright © 2013 Balbir Singh 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.