- Research
- Open access
- Published:
Fixed point theorems for weak C-contractions in partially ordered 2-metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 161 (2013)
Abstract
The aim of this paper is to state some fixed point results for weak C-contractions in a partially ordered 2-metric space. Examples are given to illustrate the results.
1 Introduction and preliminaries
Chatterjea in [1] introduced the notion of a C-contraction.
Definition 1.1 [1]
Let be a metric space and be a map. Then T is called a C-contraction if there exists such that for all , .
This notion was generalized to a weak C-contraction by Choudhury in [2].
Definition 1.2 ([2], Definition 1.3)
Let be a metric space and be a map. Then T is called a weak C-contraction if there exists which is continuous, and if and only if such that
for all .
In [2], Choudhury proved that if X is a complete metric space, then every weak C-contraction has a unique fixed point; see [[2], Theorem 2.1]. This result was generalized to a complete, partially ordered metric space in [3]; see [[3], Theorems 2.1, 2.3 and 3.1].
There were some generalizations of a metric such as a 2-metric, a D-metric, a G-metric, a cone metric, and a complex-valued metric. The notion of a 2-metric has been introduced by Gähler in [4]. Note that a 2-metric is not a continuous function of its variables, whereas an ordinary metric is. This led Dhage to introduce the notion of a D-metric in [5]. But in [6] Mustafa and Sims showed that most of topological properties of D-metric were not correct. In [7] Mustafa and Sims introduced the notion of a G-metric to overcome flaws of a D-metric. After that, many fixed point theorems on G-metric spaces have been stated. However, it was shown in [8] and [9] that in several situations fixed point results in G-metric spaces can be in fact deduced from fixed point theorems in metric or quasi-metric spaces.
In [10] Huang and Zhang defined the notion of a cone metric. After that, many authors extended some fixed point theorems on metric spaces to cone metric spaces. However, it was shown later by various authors that in several cases the fixed point results in cone metric spaces can be obtained by reducing them to their standard metric counterparts; for example, see [11–14]. In [15] Azam, Fisher and Khan have introduced the notion of a complex-valued metric and some fixed point theorems have been stated. But in [16] Sastry, Naidu and Bekeshie showed that some fixed point theorems recently generalized to complex-valued metric spaces are consequences of their counterparts in the setting of metric spaces and hence are redundant.
Note that in the above generalizations, only a 2-metric space has not been known to be topologically equivalent to an ordinary metric. Then there was no easy relationship between results obtained in 2-metric spaces and metric spaces. In particular, the fixed point theorems on 2-metric spaces and metric spaces may be unrelated easily. For the fixed point theorems on 2-metric spaces, the readers may refer to [17–26].
The aim of this paper is to state some fixed point results for weak C-contractions in a partially ordered 2-metric space. Examples are given to illustrate the results.
First we recall some notions and lemmas which will be useful in what follows.
Definition 1.3 [4]
Let X be a non-empty set and let be a map satisfying the following conditions:
-
1.
For every pair of distinct points , there exists a point such that .
-
2.
If at least two of three points are the same, then .
-
3.
The symmetry: for all .
-
4.
The rectangle inequality: for all .
Then d is called a 2-metric on X and is called a 2-metric space which will be sometimes denoted by X if there is no confusion. Every member is called a point in X.
Definition 1.4 [4]
Let be a 2-metric space and , . The set
is called a 2-ball centered at a and b with radius r. The topology generated by the collection of all 2-balls as a subbasis is called a 2-metric topology on X.
Definition 1.5 [22]
Let be a sequence in a 2-metric space .
-
1.
is said to be convergent to x in , written , if for all , .
-
2.
is said to be Cauchy in X if for all , , that is, for each , there exists such that for all .
-
3.
is said to be complete if every Cauchy sequence is a convergent sequence.
Definition 1.6 ([24], Definition 8)
A 2-metric space is said to be compact if every sequence in X has a convergent subsequence.
Lemma 1.7 ([24], Lemma 3)
Every 2-metric space is a -space.
Lemma 1.8 ([24], Lemma 4)
in a 2-metric space if and only if in the 2-metric topological space X.
Lemma 1.9 ([24], Lemma 5)
If is a continuous map from a 2-metric space X to a 2-metric space Y, then in X implies in Y.
Remark 1.10
-
1.
It is straightforward from Definition 1.3 that every 2-metric is non-negative and every 2-metric space contains at least three distinct points.
-
2.
A 2-metric is sequentially continuous in one argument. Moreover, if a 2-metric is sequentially continuous in two arguments, then it is sequentially continuous in all three arguments, see [[27], p.975].
-
3.
A convergent sequence in a 2-metric space need not be a Cauchy sequence, see [[27], Remark 01 and Example 01].
-
4.
In a 2-metric space , every convergent sequence is a Cauchy sequence if d is continuous, see [[27], Remark 02].
-
5.
There exists a 2-metric space such that every convergent sequence is a Cauchy sequence but d is not continuous, see [[27], Remark 02 and Example 02].
2 Main results
First, we introduce the notion of a weak C-contraction on a partially ordered 2-metric space.
Definition 2.1 Let be a partially ordered 2-metric space and be a map. Then T is called a weak C-contraction if there exists which is continuous, and if and only if such that
for all and or .
The following example gives some examples of ψ in Definition 2.1. Note that in [[2], Example 2.1], Choudhury considered the function for all . Unfortunately, for this function, we have , which is a contradiction to the condition that if and only if in [[2], Definition 1.3], also in Definition 2.1.
Example 2.2
-
1.
for all .
-
2.
for all .
The first result is a sufficient condition for the existence of a fixed point of a weak C-contraction on a 2-metric space. For the preceding one in metric spaces, see [[3], Theorem 2.1].
Theorem 2.3 Let be a complete, partially ordered 2-metric space and be a weak C-contraction such that:
-
1.
T is continuous and non-decreasing.
-
2.
There exists with .
Then T has a fixed point.
Proof If , then the proof is finished. Suppose now that . Since T is a non-decreasing map, we have . Put . Then, for all , from (2.1) and noting that and are comparable, we get
for all . By choosing in (2.2), we obtain , that is,
It follows from (2.2) and (2.3) that
It implies that
Thus is a decreasing sequence of non-negative real numbers and hence it is convergent. Let
Taking the limit as in (2.4) and using (2.6), we get
That is,
Taking the limit as in (2.2) and using (2.6), (2.7), we get . It implies that , that is, . Then (2.6) becomes
From (2.5), we have if , then . Since , we have for all . Since , we have
for all . For , noting that , from (2.9) we have
It implies that
Since , from (2.10) we have
for all . From (2.9) and (2.11), we have for all .
Now, for all with , we have . Therefore,
This proves that for all
In what follows, we will prove that is a Cauchy sequence. Suppose to the contrary that is not a Cauchy sequence. Then there exists for which we can find subsequences and where is the smallest integer such that and
for all . Therefore,
By using (2.12), (2.13) and (2.14), we have
Taking the limit as in (2.15) and using (2.8), we have
Also, from (2.12), we have
and
Taking the limit as in (2.17), (2.18) and using (2.8), (2.16), we obtain
Since and , are comparable, by using (2.1), we have
Taking the limit as in (2.20) and using (2.16), (2.19) and the continuity of ψ, we have
This proves that , that is, . It is a contradiction. This proves that is a Cauchy sequence. Since X is complete, there exists such that . It follows from the continuity of T that . Then z is a fixed point of T. □
The next result is another one for the existence of the fixed point of a weak C-contraction on a 2-metric space. For the preceding one in metric spaces, see [[3], Theorem 2.2].
Theorem 2.4 Let be a complete, partially ordered 2-metric space and be a weak C-contraction such that:
-
1.
T is non-decreasing.
-
2.
If is non-decreasing such that , then for all .
-
3.
There exists with .
Then T has a fixed point.
Proof As in the proof of Theorem 2.3, we have a Cauchy sequence with in X. We only have to prove that . Since is non-decreasing and , we have for all . It follows from (2.1) that
Taking the limit as in (2.21), we have
It implies that for all , that is, . □
In what follows, we prove a sufficient condition for the uniqueness of the fixed point in Theorem 2.3 and Theorem 2.4.
Theorem 2.5 Suppose that:
-
1.
Either hypotheses of Theorem 2.3 or hypotheses of Theorem 2.4 hold.
-
2.
For each , there exists that is comparable to x and y.
Then T has a unique fixed point.
Proof As in the proofs of Theorem 2.3 and Theorem 2.4, we see that T has a fixed point.
Now we prove the uniqueness of the fixed point of T. Let x, y be two fixed points of T. We consider the following two cases.
Case 1. x is comparable to y. Then is comparable to for all . For all , we have
This proves that , that is, for all . Then .
Case 2. x is not comparable to y. Then there exists that is comparable to x and y. It implies that is comparable to and . For all and , we have
It implies that . Then there exists . Taking the limit as in (2.22) and noting that ψ is continuous, we have . This proves that . Then , that is, . Similarly, . By Lemma 1.7, we get . □
Remark 2.6 Note that if is totally ordered, then the condition (2) in Theorem 2.5 is always satisfied.
The following result is an analogue of [[3], Theorem 3.1].
Theorem 2.7 Let be a complete, partially ordered 2-metric space and be a weak C-contraction such that:
-
1.
For all , if then .
-
2.
For each , there exists that is comparable to x and y.
-
3.
There exists with or .
Then, for all , . In particular, .
Proof We consider the following two cases.
Case 1. . By the hypothesis (1), consecutive terms of the sequence are comparable. It follows from (2.1) that for all ,
As in the proof of (2.12) of Theorem 2.3, we have for all . Then (2.23) implies
That is, . Then there exists . As in the proof of Theorem 2.3, we get . Then . That is, .
Case 2. . The same as in Case 1. □
For each , if for all is a metric on X, then the formula (2.1) becomes (1.1). Also, the above proofs may be similar to the method used in [2] and [3]. The following example guarantees that this fact is not true in general.
Example 2.8 There exists a 2-metric space such that for each , the formula for all is not a metric on X.
Proof Let and
Then is a 2-metric space. For each , we have
If , then we have
If , then we have
This proves that is not a metric on X for all . □
The following example shows that hypotheses in Theorem 2.3 and Theorem 2.4 do not guarantee the uniqueness of the fixed point.
Example 2.9 Let with the order
Define a 2-metric d on X as follows:
Then is a partially ordered, complete 2-metric space whose different elements are not comparable. The identity map for all is continuous, non-decreasing, and contraction conditions in Theorem 2.3 and Theorem 2.4 are satisfied. Moreover, and T has more than one fixed point.
The following example is an illustration of Theorem 2.4 and Theorem 2.7.
Example 2.10 Let with the order if and only if for all . Let d be a 2-metric on X defined by the symmetry of all three variables and
Let be defined by , , . It is easy to see that Theorem 2.4 and Theorem 2.7 are applicable to T and c is a unique fixed point of T. Moreover, the condition (2) in Theorem 2.5 does not hold, then it is not a necessary condition of the uniqueness of the fixed point.
Note that, in [[3], Theorem 3.1], if X is a compact metric space and T is continuous, then T has a unique fixed point. The following example shows that, in Theorem 2.7, if X is a compact 2-metric space and T is continuous, then T may not have a unique fixed point.
Example 2.11 Let be a 2-metric space as in Example 2.10. Let be defined by
We see that all assumptions in Theorem 2.7 are satisfied but T has more than one fixed point.
Finally, Example 2.12 and Example 2.13 show that the above results can be used to prove the existence of a fixed point when standard arguments in metric spaces in [2] and [3] fail, even for trivial maps.
Example 2.12 Let X be the 2-metric space on [[4], p.145] with the usual order and for all . Then we have:
-
1.
X is a complete, totally ordered 2-metric space.
-
2.
X is not metrizable.
-
3.
T is a C-weak contraction on the 2-metric space X.
Proof (1) and (2) See [[4], p.145].
(3) By choosing for all , then the condition (2.1) holds. This proves that T is a C-weak contraction on the 2-metric space X. □
Example 2.13 Let with the usual order,
and for all . Then we have:
-
1.
is a complete, totally ordered 2-metric space.
-
2.
is not completely metrizable, that is, there does not exist any metric ρ on X such that the metric topology and the completeness on are coincident with the 2-metric topology and the completeness on , respectively.
-
3.
T is a C-weak contraction on the 2-metric space .
Proof (1) It is easy to see that is a partially ordered 2-metric space.
Let be a Cauchy sequence in . We have for all . Then, for each , there exists such that for all . We consider the following three cases.
Case 1. For all , . This proves that is convergent.
Case 2. For all , . It is a contradiction because a is an arbitrary point of X.
Case 3. For all and all , and . Then for all . This proves that is convergent.
By the above three cases, the Cauchy sequence is convergent in . This proves that is complete.
(2) Since , we have is a convergent sequence in . On the other hand, , then is not a Cauchy sequence in . This proves that is not completely metrizable.
(3) By choosing , the condition (2.1) holds. This proves that T is a C-weak contraction on the 2-metric space . □
References
Chatterjea SK: Fixed point theorems. C. R. Acad. Bulgare Sci. 1972, 25: 727–730.
Choudhury BS: Unique fixed point theorem for weakly C -contractive mappings. Kathmandu Univ. J. Sci. Eng. Technol. 2009, 5: 6–13.
Harjani J, López B, Sadarangani K: Fixed point theorems for weakly C -contractive mappings in ordered metric spaces. Comput. Math. Appl. 2011, 61: 790–796. 10.1016/j.camwa.2010.12.027
Gähler VS: 2-metrische Räume und ihre topologische struktur. Math. Nachr. 1963/64, 26: 115–118. 10.1002/mana.19630260109
Dhage, BC: A study of some fixed point theorems. PhD thesis, Marathwada, Aurangabad, India (1984)
Mustafa Z, Sims B: Some remarks concerning D -metric spaces. Proceedings of the International Conferences on Fixed Point Theory and Applications 2003, 189–198., Valencia, Spain
Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7(2):289–297.
Jleli M, Samet B: Remarks on G -metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012., 2012: Article ID 201
Samet B, Vetro C, Vetro F: Remarks on G -metric spaces. Int. J. Anal. 2013, 2013: 1–6.
Huang LG, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 2007, 332: 1468–1476. 10.1016/j.jmaa.2005.03.087
Du WS: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 2010, 72: 2259–2261. 10.1016/j.na.2009.10.026
Feng Y, Mao W: The equivalence of cone metric spaces and metric spaces. Fixed Point Theory 2010, 11(2):259–264.
Kadelburg Z, Radenović S, Rakočević V: A note on the equivalence of some metric and cone metric fixed point results. Appl. Math. Lett. 2011, 24(3):370–374. 10.1016/j.aml.2010.10.030
Khani M, Pourmahdian M: On the metrizability of cone metric spaces. Topol. Appl. 2011, 158(2):190–193. 10.1016/j.topol.2010.10.016
Azam A, Fisher B, Khan M: Common fixed point theorems in complex valued metric spaces. Numer. Funct. Anal. Optim. 2011, 32(3):243–253. 10.1080/01630563.2011.533046
Sastry KPR, Naidu GA, Bekeshie T: Metrizability of complex valued metric spaces and some remarks on fixed point theorems in complex valued metric spaces. Int. J. Pure Appl. Math. 2012, 3(7):2686–2690.
Aliouche A, Simpson C: Fixed points and lines in 2-metric spaces. Adv. Math. 2012, 229: 668–690. 10.1016/j.aim.2011.10.002
Deshpande B, Chouhan S: Common fixed point theorems for hybrid pairs of mappings with some weaker conditions in 2-metric spaces. Fasc. Math. 2011, 46: 37–55.
Freese RW, Cho YJ, Kim SS: Strictly 2-convex linear 2-normed spaces. J. Korean Math. Soc. 1992, 29(2):391–400.
Gähler VS: Lineare 2-normierte Räume. Math. Nachr. 1965, 28: 1–43.
Gähler VS: Über die uniformisierbarkeit 2-metrischer Räume. Math. Nachr. 1965, 28: 235–244.
Iseki K: Fixed point theorems in 2-metric spaces. Math. Semin. Notes 1975, 3: 133–136.
Iseki K: Mathematics on 2-normed spaces. Bull. Korean Math. Soc. 1976, 13(2):127–135.
Lahiri BK, Das P, Dey LK: Cantor’s theorem in 2-metric spaces and its applications to fixed point problems. Taiwan. J. Math. 2011, 15: 337–352.
Lai SN, Singh AK: An analogue of Banach’s contraction principle of 2-metric spaces. Bull. Aust. Math. Soc. 1978, 18: 137–143. 10.1017/S0004972700007887
Vats RK, Kumar S, Sihag V: Fixed point theorems in complete G -metric space. Fasc. Math. 2011, 47: 127–139.
Naidu SVR, Prasad JR: Fixed point theorems in 2-metric spaces. Indian J. Pure Appl. Math. 1986, 17(8):974–993.
Acknowledgements
The authors would like to thank the referees for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Dung, N.V., Le Hang, V.T. Fixed point theorems for weak C-contractions in partially ordered 2-metric spaces. Fixed Point Theory Appl 2013, 161 (2013). https://doi.org/10.1186/1687-1812-2013-161
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-161