Abstract
In this work, we obtain some fixed point results for generalized weakly T-Chatterjea-contractive and generalized weakly T-Kannan-contractive mappings in the framework of complete b-metric spaces. Examples are provided in order to distinguish these results from the known ones.
MSC:47H10, 54H25.
Similar content being viewed by others
1 Introduction and preliminaries
The theoretical framework of metric fixed point theory has been an active research field over the last nine decades. Of course, the Banach contraction principle [1] is the first important result on fixed points for contractive-type mappings. So far, there have been a lot of fixed point results dealing with mappings satisfying various types of contractive inequalities. In particular, the concepts of K-contraction and C-contraction were introduced by Kannan [2], respectively, Chatterjea [3] as follows.
Definition 1 Let be a metric space and .
-
1.
([2]) The mapping f is said to be a K-contraction if there exists such that for all the following inequality holds:
-
2.
([3]) The mapping f is said to be a C-contraction if there exists such that for all the following inequality holds:
In 1968, Kannan [2] proved that if is a complete metric space, then every K-contraction on X has a unique fixed point. In 1972, Chatterjea [3] established a fixed point theorem for C-contractions.
Definition 2 Let be a metric space, and be a continuous function such that if and only if .
-
1.
([4]) f is said to be weakly C-contractive (or a weak C-contraction) if for all ,
-
2.
([5]) f is said to be weakly K-contractive (or a weak K-contraction) if for all ,
In 2009, Choudhury [4] proved the following theorem.
Theorem 1 ([[4], Theorem 2.1])
Every weak C-contraction on a complete metric space has a unique fixed point.
For more details of weakly C-contractive mappings we refer to [6] and [7].
Definition 3 Let be a metric space and be two mappings.
-
1.
([8]) is said to be a T-Kannan-contraction if there exists such that for all the following inequality holds:
-
2.
([5]) is said to be a T-Chatterjea-contraction if there exists such that for all the following inequality holds:
T-Kannan-contractions (in short, T-K-contractions) and T-Chatterjea-contractions (in short, T-C-contractions) are special cases of T-Hardy-Rogers contractions [9]. Recently, existence and uniqueness of fixed points for these types of contractions in cone metric spaces have been investigated in [9] and [10].
Definition 4 ([11])
Let be a metric space. A mapping is said to be sequentially convergent (respectively, subsequentially convergent) if, for a sequence in X for which is convergent, is also convergent (respectively, has a convergent subsequence).
In [8], Moradi has extended Kannan’s theorem [2] as follows.
Theorem 2 (Extended Kannan’s theorem [8])
Let be a complete metric space and be mappings such that T is continuous, one-to-one and subsequentially convergent. If f is a T-K-contraction then f has a unique fixed point. Moreover, if T is sequentially convergent then, for every , the sequence of iterates converges to this fixed point.
The notion of an altering distance function was introduced by Khan et al. as follows.
Definition 5 ([12])
The function is called an altering distance function, if the following properties are satisfied:
-
1.
ψ is continuous and strictly increasing.
-
2.
.
In the following definitions and theorems, ψ is an altering distance function and is a continuous function such that if and only if .
Definition 6 ([5])
Let be a metric space and let be two mappings.
-
1.
f is said to be a generalized weak T-C-contraction if, for all ,
-
2.
f is said to be a generalized weak T-K-contraction if, for all ,
Putting in the above definition, we obtain the concepts of weak T-C-contraction and weak T-K-contraction.
The following are the main results of [5].
Theorem 3 [5]
Let be a complete metric space and let be two mappings such that T is one-to-one and continuous. Suppose that:
-
1.
f is a generalized weak T-C-contraction, or
-
2.
f is a generalized weak T-K-contraction.
Then we have the following.
-
(i)
For every the sequence is convergent.
-
(ii)
If T is subsequentially convergent then f has a unique fixed point.
-
(iii)
If T is sequentially convergent then for each the sequence converges to the fixed point of f.
The aim of this article is to extend the stated results to the framework of b-metric spaces, introduced in 1993 by Czerwik [13]. These form a nontrivial generalization of metric spaces and several fixed point results for single and multivalued mappings in such spaces have been obtained since then (see, e.g., [14–17] and the references cited therein). We recall the following.
Definition 7 ([13])
Let X be a (nonempty) set and be a given real number. A function is a b-metric if, for all , the following conditions are satisfied:
() iff ,
() ,
() .
The pair is called a b-metric space.
It should be noted that the class of b-metric spaces is effectively larger than that of metric spaces, since a b-metric is a metric if (and only if) . We present an easy example to show that in general a b-metric need not be a metric.
Example 1 Let be a metric space, and , where is a real number. Then d is a b-metric with .
However, is not necessarily a metric space. For example, if is the set of real numbers and is the usual Euclidean metric, then is a b-metric on ℝ with , but it is not a metric on ℝ.
Recently, Hussain et al. [15] have presented an example of a b-metric which is not continuous (see [[15], Example 2]). Thus, while working in b-metric spaces, the following lemma is useful.
Lemma 1 ([14])
Let be a b-metric space with , and suppose that the sequences and are b-convergent to x, y, respectively. Then we have
In particular, if , then we have . Moreover, for each , we have,
2 Fixed points of weakly T-Chatterjea contractions
From now on, we assume:
and
Our first result is the following.
Theorem 4 Let be a complete b-metric space with parameter , be such that, for some , and all ,
and let T be one-to-one and continuous. Then we have the following.
-
(1)
For every the sequence is convergent.
-
(2)
If T is subsequentially convergent, then f has a unique fixed point.
-
(3)
If T is sequentially convergent, then for each the sequence converges to the fixed point of f.
Proof Let be arbitrary. Consider the sequence given by , . We will complete the proof in three steps.
Step I. We will prove that .
Using condition (2.1), we obtain
Therefore, by the triangular inequality and since φ is nonnegative and ψ is an increasing function,
Again, since ψ is increasing, we have
wherefrom
Thus, is a decreasing sequence of nonnegative real numbers and hence it is convergent.
Assume that . From the above argument we have
Passing to the limit when , we obtain
We have proved in (2.2) that
Now, letting and using the continuity of ψ and the properties of φ we obtain
and consequently, . This yields
by our assumptions about φ.
Step II. is a b-Cauchy sequence.
Suppose that is not a b-Cauchy sequence. Then there exists for which we can find subsequences and of such that is the smallest index for which and
This means that
From (2.4), (2.5) and the triangular inequality,
Letting , and taking into account (2.3), we can conclude that
Further, from
and (2.5), and using (2.3), we get
Moreover, from
and
and using (2.3) and (2.6), we get
Similarly, we can show that
and
Using (2.1) and (2.7)-(2.10) we have
since . Hence, we have
By our assumption about φ, we have
which contradicts (2.9) and (2.10).
Since is b-complete and is a b-Cauchy sequence, there exists such that
Step III. f has a unique fixed point, assuming that T is subsequentially convergent.
As T is subsequentially convergent, has a b-convergent subsequence. Hence, there exist and a subsequence such that
Since T is continuous, by (2.12) we obtain
and by (2.11) and (2.13) we conclude that .
From Lemma 1 and (2.1) we have
since ψ is increasing. By the properties of , it follows that . By the triangular inequality we have
Letting we can conclude that . Hence, . As T is one-to-one, . Consequently, f has a fixed point.
If we assume that w is another fixed point of f, because of (2.1), we have
since and ψ is increasing. Hence, . Since T is one-to-one, it follows that . Consequently, f has a unique fixed point.
Finally, if T is sequentially convergent, replacing with we conclude that . □
Taking and , where in Theorem 4, the extended Chatterjea’s theorem in the setting of b-metric spaces is obtained.
Corollary 1 Let be a complete b-metric space and be mappings such that T is continuous, one-to-one and subsequentially convergent. If and
for all , then f has a unique fixed point. Moreover, if T is sequentially convergent, then for every the sequence of iterates converges to this fixed point.
Remark 1 In the case when , this corollary reduces to [[18], Corollary 3.8.3∘] (the case ), which is Chatterjea’s theorem [3] in the framework of b-metric spaces.
By taking and in Theorem 4, we derive an extension of Choudhury’s theorem (Theorem 1) to the setup of b-metric spaces.
If , Theorem 4 reduces to Theorem 3 (case (1)).
We demonstrate the use of the obtained results by the following.
Example 2 (Inspired by [8])
Let , and let for . Then d is a b-metric with the parameter and is a complete b-metric space. Consider the mappings given by
We will show that the mappings f, T satisfy the conditions of Corollary 1 with . Indeed, for , , we have
It is easy to prove that, for ,
It follows that
Now, implies that and . It follows that , and hence
If one of the points is equal to 0, the proof is even simpler.
By Corollary 1, it follows that f has a unique fixed point (which is ).
3 Fixed points of weakly T-Kannan contractions
Our second main result is the following.
Theorem 5 Let be a complete b-metric space with the parameter , be such that for some , and all ,
and let T be one-to-one and continuous. Then:
-
(1)
For every the sequence is convergent.
-
(2)
If T is subsequentially convergent, then f has a unique fixed point.
-
(3)
If T is sequentially convergent then, for each , the sequence converges to the fixed point of f.
Proof Let be arbitrary. Consider the sequence given by , . At first, we will prove that
Using condition (3.1), we obtain
Since φ is nonnegative and ψ is increasing, it follows that
that is,
Thus, is a decreasing sequence of nonnegative real numbers and hence it is convergent.
Assume that . If in (3.2) , using the properties of ψ and φ we obtain
which is possible only if
Now, we will show that is a b-Cauchy sequence.
Suppose that this is not true. Then there exists for which we can find subsequences and of such that is the smallest index for which and . This means that
Again, as in Step II of Theorem 4 one can prove that
Using (3.1) we have
Passing to the upper limit as in the above inequality and taking into account (3.3), we have
and so . By our assumptions about ψ, we have , which is a contradiction.
Since is b-complete and is a b-Cauchy sequence, there exists such that
Now, if T is subsequentially convergent, then has a convergent subsequence. Hence, there exist a point and a sequence such that
Since T is continuous, by (3.5) we obtain
and by (3.4) and (3.6) we conclude that .
From Lemma 1 and (3.1) we have
By the properties of , it follows that
Since T is one-to-one, we obtain . Consequently, f has a fixed point.
Uniqueness of the fixed point can be proved in the same manner as in Theorem 4.
Finally, if T is sequentially convergent, replacing with we conclude that . □
Taking and , where in Theorem 5, the extended Kannan’s theorem in the setting of b-metric spaces is obtained.
Corollary 2 Let be a complete b-metric space with the parameter , be such that for some and all ,
and let T be one-to-one and continuous. Then we have the following.
-
(1)
For every the sequence is convergent.
-
(2)
If T is subsequentially convergent then f has a unique fixed point.
-
(3)
If T is sequentially convergent then, for each , the sequence converges to the fixed point of f.
Remark 2 In the case when , this corollary reduces to [[18], Corollary 3.8.2∘] (the case ). If , Corollary 2 reduces to Theorem 2 (i.e., [[8], Theorem 2.1]). Of course, if both of these conditions are fulfilled, we get just the classical Kannan’s theorem [2].
The following example distinguishes our results from the previously known ones.
Example 3 Let and be defined by for , , , for . It is easy to check that is a b-metric space (with ) which is not a metric space. Consider the mapping given by
We first note that the b-metric version of classical weak Kannan’s theorem is not satisfied in this example. Indeed, for , , we have and , hence the inequality
cannot hold, whatever and are chosen.
Take now defined by
Obviously, all the properties of T given in Corollary 2 are fulfilled. We will check that the contractive condition (3.7) holds true if α is chosen such that
Only the following cases are nontrivial:
1∘ , . Then (3.7) reduces to
2∘ , . Then (3.7) reduces to
All the conditions of Corollary 2 are satisfied and f has a unique fixed point ().
References
Banach S: Sur les operateurs dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133-181.
Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 60: 71-76.
Chatterjea SK: Fixed point theorems. C. R. Acad. Bulgare Sci. 1972, 25: 727-730.
Choudhury BS: Unique fixed point theorem for weak C -contractive mappings. Kathmandu Univ. J. Sci. Eng. Technol. 2009,5(1):6-13.
Razani A, Parvaneh V: Some fixed point theorems for weakly T -Chatterjea and weakly T -Kannan-contractive mappings in complete metric spaces. Russ. Math. (Izv. VUZ) 2013,57(3):38-45.
Harjani J, Lopez 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
Shatanawi W: Fixed point theorems for nonlinear weakly C -contractive mappings in metric spaces. Math. Comput. Model. 2011, 54: 2816-2826. 10.1016/j.mcm.2011.06.069
Moradi, S: Kannan fixed-point theorem on complete metric spaces and on generalized metric spaces depended on another function. arXiv:0903.1577v1 [math.FA]
Filipović M, Paunović L, Radenović S, Rajović M: Remarks on ‘Cone metric spaces and fixed point theorems of T -Kannan and T -Chatterjea contractive mappings’. Math. Comput. Model. 2011, 54: 1467-1472. 10.1016/j.mcm.2011.04.018
Morales JR, Rojas E: Cone metric spaces and fixed point theorems of T -Kannan contractive mappings. Int. J. Math. Anal. 2010,4(4):175-184.
Beiranvand, A, Moradi, S, Omid, M, Pazandeh, H: Two fixed point theorems for special mapping. arXiv:0903.1504v1 [math.FA]
Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30: 1-9. 10.1017/S0004972700001659
Czerwik S: Contraction mappings in b -metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1: 5-11.
Aghajani, A, Abbas, M, Roshan, JR: Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces. Math. Slovaca (2014, in press)
Hussain N, Parvaneh V, Roshan JR, Kadelburg Z: Fixed points of cyclic weakly -contractive mappings in ordered b -metric spaces with applications. Fixed Point Theory Appl. 2013., 2013: Article ID 256
Roshan, JR, Shobkolaei, N, Sedghi, S, Abbas, M: Common fixed point of four maps in b-metric spaces. Hacet. J. Math. Stat. (2014, in press)
Sintunavarat W, Plubtieng S, Katchang P: Fixed point result and applications on b -metric space endowed with an arbitrary binary relation. Fixed Point Theory Appl. 2013., 2013: Article ID 296
Jovanović M, Kadelburg Z, Radenović S: Common fixed point results in metric-type spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 978121 10.1155/2010/978121
Acknowledgements
The authors are grateful to the referees for valuable remarks that helped them to improve the exposition of the paper. The fourth author is thankful to the Ministry of Education, Science and Technological Development of Serbia.
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
Mustafa, Z., Roshan, J.R., Parvaneh, V. et al. Fixed point theorems for weakly T-Chatterjea and weakly T-Kannan contractions in b-metric spaces. J Inequal Appl 2014, 46 (2014). https://doi.org/10.1186/1029-242X-2014-46
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-46