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Journal of Mathematics
Volume 2013 (2013), Article ID 719324, 7 pages
http://dx.doi.org/10.1155/2013/719324
Research Article

Some Common Fixed Point Theorems in Generalized Metric Spaces

1Department of Mathematics, Delhi Institute of Technology and Management, Gannaur 131101, Sonipat, India
2Guru Jambheshwar University of Science and Technology, Hisar 125001, India
3Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar 125001, India
4Department of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology, Murthal 131039, India

Received 30 January 2013; Accepted 18 March 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Manoj Kumar 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.

Abstract

First we prove common fixed point theorems for weakly compatible maps which generalize the results of Chen (2012). Secondly, we prove common fixed point theorems using property E.A. along with weakly compatible maps. At the end, we prove common fixed point theorems using common limit range property (CLR property) along with weakly compatible maps.

1. Introduction

In 2000, Branciari [1] introduced the notion of generalized metric space as follows.

Definition 1. Let be a nonempty set and let be a mapping such that for all and for all distinct points    each of them different from and , one has the following:(i) if and only if ,(ii), (iii) (rectangular inequality).
Then is called a generalized metric space (or shortly g.m.s.).

Definition 2. Let be a g.m.s. Let be a sequence in and  . We say that is (i) g.m.s. convergent to if and only if   as  ,(ii) g.m.s. Cauchy sequence if and only if, for each  , there exists such that, for all ,(iii) complete g.m.s. if every g.m.s. Cauchy sequence is g.m.s. convergent in .
In 1986, Jungck [2] introduced the notion of compatible maps as follows.
Let be a metric space. Let be two single-valued functions. We say that and are compatible if  , whenever is a sequence in such that .
In 1998, Jungck and Rhoades [3] introduced the notion of weakly compatible maps as follows.
Two maps are said to be weakly compatible if they commute at the coincidence point. Note that compatible maps are weakly compatible but the converse is not true in general.
In 2002, Aamri and El Moutawakil [4] introduced the notion of property E.A. as follows.
Let and be two self-mappings of a metric space . We say that and satisfy the property E.A. if there exists a sequence such that for some in .
In 2011, Sintunavarat and Kumam [5] introduced the notion of (CLRg) property as follows.
Let be a metric space. Two mappings and are said to satisfy (CLRg) property if there exists sequences in such that Similarly, we can state () and () properties if the mapping    has been replaced by the mapping and  , respectively.
In 2012, Chen [6] introduced the notion of function and function as follows.

Definition 3. A function   is said to be a   function if the function satisfies the following conditions: () for all and , () for all .

Definition 4. A function   is said to be a function if the function satisfies the following conditions:() is a strictly increasing, continuous function in each coordinate, () for all , ,  ,  , and .
Before proving our results, we need the following lemma.

Lemma 5 (see [6]). Let be a function. Then for all , where denotes the th iteration of .

2. Weakly Compatible Maps

In 2012, Chen [6] proved the following fixed point theorems for compatible maps.

Theorem 6. Let be Hausdorff and complete g.m.s. and let be a function. Let ,  ,  ,  and    be self-maps on satisfying the following: If or is continuous, then , , , and have a unique common fixed point in .

Theorem 7. Let be Hausdorff and complete g.m.s. and let be a function. Let  ,  , , and be self-maps on satisfying (2), (4), and the following: If one of , , , and , is a complete subset of then , , , and have a unique common fixed point in .

Now, we prove our results for weakly compatible maps.

Theorem 8. Let be Hausdorff and complete g.m.s. and let be a function. Let , , , and be self-maps on satisfying (2), (3) and the following: If one of , , , and is a complete subset of , then ,  ,  , and have a unique common fixed point in .

Proof. Let . Define the sequence recursively as follows: By Theorem 6 [6], is g.m.s. Cauchy sequence in . Now suppose that is complete subset of , then the subsequence must get a limit in . Call it to be and . Then . As is a Cauchy sequence containing a convergent subsequence , therefore the sequence also converges implying, thereby, the convergence of being a subsequence of the convergent sequence .
Taking and in (3), we get Letting , we have Therefore, implies that the pair has a point of coincidence.
As , implies that . Letting , then .
Now, from (3), we have Letting , we have Therefore, .
Thus, we have shown that , which amounts to say that both pairs have point of coincidence. If one assumes to be complete, then an analogous argument establishes this claim.
The remaining two cases pertain essentially to the previous cases. Indeed, if is complete, then , and if is complete, then . Thus and ,  , and have a point of coincidence.
Since the pairs and are weakly compatible at and , respectively, then If , then Therefore, .
Similarly, one can show that .
Thus is the common fixed point of , , , and .
Finally, we prove that , , , and have a unique common fixed point. Let be another common fixed point of , , , and .
From (3), we have Hence is the unique common fixed point of , , , and in .
We give the following example to illustrate Theorem 8.

Example 9. Let , where , , , , and are positive constants. We define by, for all ,, for all ,, , , ,
where is a constant.
If , , then is a function. We next define by the identity mapping, Then all the conditions of Theorem 8 are satisfied and is a unique common fixed point of , , , and .

Theorem 10. Let be Hausdorff and complete g.m.s. and let be a function. Let , , , and be self-maps on satisfying (2), (5) and (6). If one of , , , and is a complete subset of , then , , , and have a unique common fixed point in .

Proof. Given that , define the sequence recursively as follows: Then due to Theorem 7 [6], is g.m.s. Cauchy sequence in . Now, suppose that is complete subset of ; then the subsequence must get a limit in . Call it to be and . Then . As is a Cauchy sequence containing a convergent subsequence , therefore the sequence also converges implying thereby the convergence of being a subsequence of the convergent sequence . On taking and in (5) one gets Letting , we have Therefore, , which shows that the pair has a point of coincidence.
As , implies that . Letting , then .
From (5), we have Letting , we have Therefore, . Thus, we have shown that , which amounts to say that both pairs have point of coincidence. If one assumes to be complete, then an analogous argument establishes this claim.
The remaining two cases pertain essentially to the previous cases. Indeed if is complete, then , and if is complete, then .
Thus and , , and have a point of coincidence.
Since the pairs and are weakly compatible at and , respectively, then If , then Therefore, .
Similarly, one can show that .
Thus is the common fixed point of , , , and .
Finally, we prove that , , , and    have a unique common fixed point. Let be another common fixed point of , , , and . Then using (5), we have Hence is the unique common fixed point of , , , and in .

We give the following example to illustrate Theorem 10.

Example 11. Let , where , , , , and are positive constants. We define by, for all ,, for all ,,,, ,
where is a constant.
If ,   , then is a function. We next define , , ,   by Then all the conditions of Theorem 10 are satisfied and is a unique common fixed point of , , , and .

3. Property E.A.

Theorem 12. Let be Hausdorff and complete g.m.s. and let be a function. Let , , , and be self-maps on satisfying (2), (3), (6), and the following: If the range of one of the maps , , , or is a complete subspace of , 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 and for some as . Since , there exists a sequence in such that . Hence as . Also, since , there exists a sequence in such that . Hence as .
Suppose that is a complete subspace of . Then, for some .
Subsequently, we have , , as .
From (3), we have Letting , we have which implies that .
The weak compatibility of and implies that and then .
On the other hand, since , there exists a such that .
Now, we show that .
From (3), we have The weak compatibility of and implies that and .
Let us show that is the common fixed point of ,  ,  ,  and  .
From (3), we have Therefore, and is the common fixed point of and .
Similarly, we can prove that is the common fixed point of and .
Since , we conclude that is the common fixed point of ,  ,  ,  and .
The proof is similar when is assumed to be complete subspace of . The cases in which or is a complete subspace of are similar to the cases in which or , respectively, is complete since If and , then (3) gives Therefore, and the common fixed point is unique.

Theorem 13. Let be Hausdorff and complete g.m.s. and let    be a function. Let ,  , , and be self-maps on satisfying (2), (5), (6), and (26). If the range of one of the maps ,  , , or is a complete subspace of , 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 and for some as .
Since , there exists a sequence in such that. Hence as . Also, since , there exists a sequence in such that . Hence as .
Suppose that is a complete subspace of . Then, for some .
Subsequently, we have , , as .
From (5), we have Letting , we have
The weak compatibility of and implies that and then .
On the other hand, since , there exists a such that .
Now, we show that .
From (5), we have The weak compatibility of and implies that and .
Let us show that is the common fixed point of , , and .
From (5), we have Therefore, is the common fixed point of and .
Similarly, one can prove that is the common fixed point of and .
Since , we conclude that is the common fixed point of , , , and .
The proof is similar when is assumed to be complete subspace of . The cases in which or is a complete subspace of are similar to the cases in which or , respectively, is complete since and .
If and , then (5) gives
Therefore, and the common fixed point is unique.

4. (CLR) Property

Theorem 14. Let be Hausdorff and complete g.m.s. and let be a function. Let ,  ,  , and   be self-maps on satisfying (3), (6), and the following: or Then , , , and have a unique common fixed point.

Proof. Without loss of generality, we assume that and the pair satisfies property; then there exists some sequence in such that and converge to , for some in as .
Since there exists a sequence in such that ; hence as .
Now, we claim that .
Let .
Taking and in (3), we have Letting , we have Subsequently, we have , , , and that converge to .
Now, we claim that .
Taking and in (3), we have Letting , we have Since the pair is weakly compatible, it follows that .
Also, since , there exists some in such that .
We next show that
Taking , in (3), we have Letting , we have But the pair is weakly compatible; it follows that .
Next, we claim that .
Taking ,   in (3), we have
Now, we show that .
Taking , in (3), we have Hence is the common fixed point of , , , and .
If and , then (3) gives
Therefore, uniqueness follows.

Theorem 15. Let be Hausdorff and complete g.m.s. and let be a function. Let , , , and be self-maps on satisfying (5), (6), and (38). Then , , , and have a unique common fixed point.

Proof. Without loss of generality, we assume that and the pair satisfies property; then there exists some sequence in such that and converge to , for some in as .
Since there exists a sequence in such that ; hence as .
Now, we claim that .
Let .
Taking and in (5), we have Letting  , we have Subsequently, we have , , , and that converge to .
Now, we claim that .
Taking and in (5), we have Letting , we have Since the pair is weakly compatible, it follows that .
Also, since , there exists some in such that .
We next show that .
Taking , in (5), we have Letting , we have But the pair is weakly compatible; it follows that .
Next, we claim that .
Taking , in (5), we have Hence, .
Now, we show that .
Taking , in (5), we have Hence is the common fixed point of ,  , and .
If and , then (5) gives Therefore, and the common fixed point is unique.

Acknowledgment

One of the authors (S. Kumar) is very grateful to UGC for providing MRP under Ref. F. no. 39-41/2010(SR).

References

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