Abstract
In this paper, we consider the setting of b-metric spaces to establish results regarding the common fixed points of two mappings, using a contraction condition defined by means of a comparison function. An example is presented to support our results comparing with existing ones.
MSC:49H09, 47H10.
Similar content being viewed by others
1 Introduction
The contraction principle of Banach [1], proved in 1922, was followed by diverse works about fixed points theory regarding different classes of contractive conditions on some spaces such as: quasi-metric spaces [2, 3], cone metric spaces [4, 5], partially ordered metric spaces [6–8], G-metric spaces [9], partial metric spaces [10–13], Menger spaces [14], metric-type spaces [15], and fuzzy metric spaces [16–18]. Also, there have been developed studies on approximate fixed point or on qualitative aspects of numerical procedures for approximating fixed points see, for example [19, 20].
The concept of b-metric spaces was introduced by Bakhtin [21] in 1989, who used it to prove a generalization of the Banach principle in spaces endowed with such kind of metrics. Since then, this notion has been used by many authors to obtain various fixed point theorems. Aydi et al. in [22] proved common fixed point results for single-valued and multi-valued mappings satisfying a weak ϕ-contraction in b-metric spaces. Roshan et al. in [23] used the notion of almost generalized contractive mappings in ordered complete b-metric spaces and established some fixed and common fixed point results. Starting from the results of Berinde [24], Păcurar [25] proved the existence and uniqueness of fixed points of ϕ-contractions on b-metric spaces. Hussain and Shah in [26] introduced the notion of a cone b-metric space, generalizing both notions of b-metric spaces and cone metric spaces. In this paper they also considered topological properties of cone b-metric spaces and results on KKM mappings in the setting of cone b-metric spaces. Fixed point theorems of contractive mappings in cone b-metric spaces without the assumption of the normality of a corresponding cone are proved by Huang and Xu in [27]. The setting of partially ordered b-metric spaces was used by Hussain et al. in [28] to study tripled coincidence points of mappings which satisfy nonlinear contractive conditions, extending those results of Berinde and Borcut [29] for metric spaces to b-metric spaces. Using the concept of a g-monotone mapping, Shah and Hussain in [30] proved common fixed point theorems involving g-non-decreasing mappings in b-metric spaces, generalizing several results of Agarwal et al. [31] and Ćirić et al. [32]. Some results of Suzuki [33] are extended to the case of metric-type spaces and cone metric-type spaces.
The aim of this paper is to consider and establish results on the setting of b-metric spaces, regarding common fixed points of two mappings, using a contraction condition defined by means of a comparison function. An example is given to support our results.
2 Preliminaries
Definition 1 Let X be a nonempty set and . A function d is called a b-metric with constant (base) if:
-
(1)
iff .
-
(2)
for all .
-
(3)
for all .
The pair is called a b-metric space.
It is obvious that a b-metric space with base is a metric space. There are examples of b-metric spaces which are not metric spaces (see, e.g., Singh and Prasad [34]).
The notions of a Cauchy sequence and a convergent sequence in b-metric spaces are defined by Boriceanu [35].
Definition 2 Let be a sequence in a b-metric space .
-
(1)
A sequence is called convergent if and only if there is such that when .
-
(2)
is a Cauchy sequence if and only if , when .
As usual, a b-metric space is said to be complete if and only if each Cauchy sequence in this space is convergent.
Regarding the properties of a b-metric space, we recall that if the limit of a convergent sequence exists, then it is unique. Also, each convergent sequence is a Cauchy sequence. But note that a b-metric, in the general case, is not continuous (see Roshan et al. [23]).
The continuity of a mapping with respect to a b-metric is defined as follows.
Definition 3 Let and be two b-metric spaces with constant s and , respectively. A mapping is called continuous if for each sequence in X, which converges to with respect to d, then converges to Tx with respect to .
Definition 4 Let be a constant. A mapping is called comparison function with base , if the following two axioms are fulfilled:
-
(a)
φ is non-decreasing,
-
(b)
for all .
Clearly, if φ is a comparison function, then for each .
For different properties and applications of comparison functions on partial metric spaces, we refer the reader to [36].
3 Main results
Now we are ready to prove our main results.
Theorem 1 Let be a complete b-metric space with a constant s and two mappings on X. Suppose that there is a constant and a comparison function φ such that the inequality
holds for each . Suppose that one of the mappings T or S is continuous. Then T and S have a unique common fixed point.
Proof Let be arbitrary. We define a sequence as follows:
Suppose that there is some such that . If , then and from the contraction condition (3.1) with and we have
Hence, as we supposed that and as a comparison function φ is non-decreasing,
If we assume that , then we have, as for ,
a contradiction. Therefore, . Hence . Thus we have . By (3.2), it means , that is, is a common fixed point of T and S.
If , then using the same arguments as in the case , it can be shown that is a common fixed point of T and S.
From now on, we suppose that for all .
Now we shall prove that
There are two cases which we have to consider.
Case I. , .
From the contraction condition (3.1) with and we get
Since , we get
Now, if we suppose that , then by the property (a) of φ in Definition 4 we get
a contradiction. Therefore, from the above inequality we have
Thus we proved that (3.3) holds for .
Case II. , .
Using the same argument as in the Case I, it can be proved that (3.3) holds for , that is,
From (3.4) and (3.5) we conclude that the inequality (3.3) holds for all .
From (3.3), by the induction it is easy to prove that
Since for all , from (3.6) it follows that
Now we shall prove that is a Cauchy sequence. Let . Since implies and , from (3.7) we conclude that there exists such that
for all .
Let with . By induction on m, we shall prove that
Let and . Then from (3.3) and (3.8) we get
Thus (3.9) holds for .
Assume now that (3.9) holds for some . We have to prove that (3.9) holds for .
We have to consider four cases.
Case I. n is odd, is even.
From the contraction condition (3.1) we get
Hence we get, as and for all ,
If from (3.10) we have , then by (3.8),
If (3.10) implies , then by the (general) triangle inequality,
Hence we get, as implies ,
Now, by (3.8) and the induction hypothesis (3.9),
Thus we proved that in this case (3.9) holds for . Therefore, by induction, we conclude that in Case I the inequality (3.9) holds for all .
Case II. n is even, is odd. The proof of (3.9) in this case is similar to one given in Case I.
Case III. n is even, is even.
Using the (general) triangle inequality and the contraction condition (3.1), we obtain
Hence we get, as and for all ,
If the inequality (3.11) implies , then from (3.8) we get
If (3.11) implies
then by the (general) triangle inequality we have
Hence we get
Now, by (3.8) and the induction hypothesis (3.3), we have
Hence
Thus we proved that (3.9) holds for . Therefore, by induction, we conclude that in Case III the inequality (3.9) holds for all .
Case IV. n is odd, is odd. The proof of (3.9) in this case is similar to one given in Case III.
Therefore, we proved that in all of four cases the inequality (3.9) holds.
From (3.9) it follows that is a Cauchy sequence. Since is a complete b-metric space, then converges to some as .
Now we shall prove that if one of the mappings T or S is continuous, then . Without loss of generality, we can suppose that S is continuous. Clearly, as , then by (3.2) we have as . Since and S is continuous, then . Thus, by the uniqueness of the limit in a b-metric space, we have . Now, from the contraction condition (3.1),
If we suppose that , then we have
a contradiction. Therefore, . Hence . Thus we proved that u is a common fixed point of T and S.
Suppose now that u and v are different common fixed points of T and S, that is, . Then
Since , then we get , a contradiction. Thus we proved that S and T have a unique common fixed point in X. □
If in Theorem 1, then we have the following result.
Corollary 1 Let be a complete b-metric space with a constant s and two mappings on X. Suppose that there is a constant and a comparison function φ such that the inequality
holds for each . Suppose that a mapping T is continuous. Then T has a unique fixed point.
Omitting the continuity assumption of mapping T or S in Theorem 1, modifying the contraction condition (3.1) and imposing on a comparison function φ a corresponding condition, then we can prove the following theorem.
Theorem 2 Let be a complete b-metric space with a constant s and two mappings on X. Suppose that there is a constant and a comparison function φ such that the inequality
holds for all . If in addition a comparison function φ satisfies the following condition:
then T and S have a unique common fixed point.
Proof Since the contraction condition (3.13) implies the contraction condition (3.1) in Theorem 1, then from the proof of Theorem 1 it follows that a sequence , defined as in (3.3), converges to some , that is,
Now we prove that . From the contraction condition (3.13) and by the monotonicity of φ we obtain
Since φ is non-decreasing and , from (3.16) we get
Set
Then, in virtue of (3.15),
where . Let be a subsequence of such that as . For simplicity, denote again by . Then from (3.18),
Suppose that . Then from (3.19), (3.17), and the assumption (3.14) of φ, we have
a contradiction. Therefore,
Hence we have as . Since by (3.15), , and as the limit in a b-metric space is unique, it follows that . Now, by (3.13),
If we suppose that , then we have , a contradiction. Therefore, , that is, . Thus we proved that . □
If in Theorem 2, then we get the following result.
Corollary 2 Let be a complete b-metric space with a constant s and a mapping on X. Suppose that there is a constant and a comparison function φ such that the inequality
holds for all . If in addition a comparison function φ satisfies the inequality (3.14), then T has a unique fixed point.
Now we give an example to support our results.
Example 1 Let endowed with the b-metric
with constant . Consider mappings , , , and the comparison function , . Clearly, is a complete metric space, and S is continuous with respect to d, so we have to verify the contraction condition (3.1). There are three cases to be considered.
Case I. . Hence , , and, therefore, the inequality (3.1) holds.
Case II. . Then , and
Thus in this case the contraction condition (3.1) holds.
Case III. . Then
Therefore, we showed that the contraction condition (3.1) is satisfied in all cases. Thus we can apply our Theorem 1, and T and S have a unique common fixed point .
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
Caristi J: Fixed point theorems for mapping satisfying inwardness conditions. Trans. Am. Math. Soc. 1976, 215: 241–251.
Hicks TL: Fixed point theorems for quasi-metric spaces. Math. Jpn. 1988, 33(2):231–236.
Altun I, Durmaz G: Some fixed point results in cone metric spaces. Rend. Circ. Mat. Palermo 2009, 58: 319–325. 10.1007/s12215-009-0026-y
Choudhury BS, Metiya N: Coincidence point and fixed point theorems in ordered cone metric spaces. J. Adv. Math. Stud. 2012, 5(2):20–31.
Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N: Theorems for Boyd-Wong type contractions in ordered metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 359054
Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28
Shatanawi W, Postolache M: Common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 271
Shatanawi W, Pitea A: Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction. Fixed Point Theory Appl. 2013., 2013: Article ID 275
Altun I, Simsek H: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 2008, 1(1–2):1–8.
Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011, 4(2):1–12.
Khan AR, Abbas M, Nazir T, Ionescu C: Fixed points of multivalued contractive mappings in partial metric spaces. Abstr. Appl. Anal. 2014., 2014: Article ID 230708
Shatanawi W, Postolache M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde. Fixed Point Theory Appl. 2013., 2013: Article ID 54
Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535
Cosentino M, Salimi P, Vetro P: Fixed point on metric-type spaces. Acta Math. Sci. 2014, 34(4):1–17.
Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/0165-0114(88)90064-4
Gregori V, Sapena A: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 2002, 125: 245–252. 10.1016/S0165-0114(00)00088-9
Ionescu C, Rezapour S, Samei M: Fixed points of some new contractions on intuitionistic fuzzy metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 168
Haghi RH, Postolache M, Rezapour S: On T -stability of the Picard iteration for generalized ϕ -contraction mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 658971
Miandaragh MA, Postolache M, Rezapour S: Some approximate fixed point results for generalized alpha-contractive mappings. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys. 2013, 75(2):3–10.
Bakhtin IA: The contraction mapping principle in almost metric spaces. 30. In Functional Analysis. Ul’yanovsk Gos. Ped. Inst., Ul’yanovsk; 1989:26–37.
Aydi H, Bota MF, Karapinar E, Moradi S: A common fixed point for weak ϕ -contractions on b -metric spaces. Fixed Point Theory 2012, 13(2):337–346.
Roshan JR, Parvaneh V, Sedghi S, Shobkolaei N, Shatanawi W: Common fixed points of almost generalized -contractive mappings in ordered b -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 130
Berinde V: Generalized contractions in quasimetric spaces. Preprint 3. In Seminar on Fixed Point Theory. “Babeş-Bolyai” University, Cluj-Napoca; 1993:3–9.
Păcurar M: A fixed point result for ϕ -contractions and fixed points on b -metric spaces without the boundness assumption. Fasc. Math. 2010, 43(1):127–136.
Hussain N, Shah MH: KKM mappings in cone b -metric spaces. Comput. Math. Appl. 2011, 61(4):1677–1684.
Huang H, Xu S: Fixed point theorems of contractive mappings in cone b -metric spaces and applications. Fixed Point Theory Appl. 2013., 2013: Article ID 112
Hussain N, Dorić N, Kadelburg Z, Radenović S: Suzuki-type fixed point results in metric type spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 126
Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74(15):4889–4897. 10.1016/j.na.2011.03.032
Shah MH, Hussain N: Nonlinear contractions in partially ordered quasi b -metric spaces. Commun. Korean Math. Soc. 2012, 27: 117–128. 10.4134/CKMS.2012.27.1.117
Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 1–8. 10.1080/00036810701714164
Ćirić L, Cakić N, Rojović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294
Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136(5):1861–1869.
Singh SL, Prasad B: Some coincidence theorems and stability of iterative procedures. Comput. Math. Appl. 2008, 55: 2512–2520. 10.1016/j.camwa.2007.10.026
Boriceanu M: Strict fixed point theorems for multivalued operators in b -metric spaces. Int. J. Mod. Math. 2009, 4(3):285–301.
Hussain N, Kadelburg Z, Radenović S, Al-Solami F: Comparison functions and fixed point results in partial metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 605781
Acknowledgements
Rade Lazović was supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic 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 article. 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
Shatanawi, W., Pitea, A. & Lazović, R. Contraction conditions using comparison functions on b-metric spaces. Fixed Point Theory Appl 2014, 135 (2014). https://doi.org/10.1186/1687-1812-2014-135
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-135