 Research
 Open Access
 Published:
Contraction conditions using comparison functions on bmetric spaces
Fixed Point Theory and Applications volume 2014, Article number: 135 (2014)
Abstract
In this paper, we consider the setting of bmetric 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.
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: quasimetric spaces [2, 3], cone metric spaces [4, 5], partially ordered metric spaces [6–8], Gmetric spaces [9], partial metric spaces [10–13], Menger spaces [14], metrictype 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 bmetric 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 singlevalued and multivalued mappings satisfying a weak ϕcontraction in bmetric spaces. Roshan et al. in [23] used the notion of almost generalized contractive mappings in ordered complete bmetric 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 ϕcontractions on bmetric spaces. Hussain and Shah in [26] introduced the notion of a cone bmetric space, generalizing both notions of bmetric spaces and cone metric spaces. In this paper they also considered topological properties of cone bmetric spaces and results on KKM mappings in the setting of cone bmetric spaces. Fixed point theorems of contractive mappings in cone bmetric spaces without the assumption of the normality of a corresponding cone are proved by Huang and Xu in [27]. The setting of partially ordered bmetric 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 bmetric spaces. Using the concept of a gmonotone mapping, Shah and Hussain in [30] proved common fixed point theorems involving gnondecreasing mappings in bmetric 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 metrictype spaces and cone metrictype spaces.
The aim of this paper is to consider and establish results on the setting of bmetric 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 d:X\times X\to [0,+\mathrm{\infty}). A function d is called a bmetric with constant (base) s\ge 1 if:

(1)
d(x,y)=0 iff x=y.

(2)
d(x,y)=d(y,x) for all x,y\in X.

(3)
d(x,y)\le s(d(x,z)+d(z,y)) for all x,y,z\in X.
The pair (X,d) is called a bmetric space.
It is obvious that a bmetric space with base s=1 is a metric space. There are examples of bmetric spaces which are not metric spaces (see, e.g., Singh and Prasad [34]).
The notions of a Cauchy sequence and a convergent sequence in bmetric spaces are defined by Boriceanu [35].
Definition 2 Let \{{x}_{n}\} be a sequence in a bmetric space (X,d).

(1)
A sequence \{{x}_{n}\} is called convergent if and only if there is x\in X such that d({x}_{n},x)\to 0 when n\to +\mathrm{\infty}.

(2)
\{{x}_{n}\} is a Cauchy sequence if and only if d({x}_{n},{x}_{m})\to 0, when n,m\to +\mathrm{\infty}.
As usual, a bmetric space is said to be complete if and only if each Cauchy sequence in this space is convergent.
Regarding the properties of a bmetric 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 bmetric, in the general case, is not continuous (see Roshan et al. [23]).
The continuity of a mapping with respect to a bmetric is defined as follows.
Definition 3 Let (X,d) and ({X}^{\prime},{d}^{\prime}) be two bmetric spaces with constant s and {s}^{\prime}, respectively. A mapping T:X\to {X}^{\prime} is called continuous if for each sequence \{{x}_{n}\} in X, which converges to x\in X with respect to d, then T{x}_{n} converges to Tx with respect to {d}^{\prime}.
Definition 4 Let s\ge 1 be a constant. A mapping \phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) is called comparison function with base s\ge 1, if the following two axioms are fulfilled:

(a)
φ is nondecreasing,

(b)
{lim}_{n\to +\mathrm{\infty}}{\phi}^{n}(t)=0 for all t>0.
Clearly, if φ is a comparison function, then \phi (t)<t for each t>0.
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 (X,d) be a complete bmetric space with a constant s and T,S:X\to X two mappings on X. Suppose that there is a constant L<\frac{1}{1+s} and a comparison function φ such that the inequality
holds for each x,y\in X. Suppose that one of the mappings T or S is continuous. Then T and S have a unique common fixed point.
Proof Let {x}_{0}\in X be arbitrary. We define a sequence \{{x}_{n}\} as follows:
Suppose that there is some n\in \mathbb{N} such that {x}_{n}={x}_{n+1}. If n=2k, then {x}_{2k}={x}_{2k+1} and from the contraction condition (3.1) with x={x}_{2k} and y={x}_{2k+1} we have
Hence, as we supposed that {x}_{2k}={x}_{2k+1} and as a comparison function φ is nondecreasing,
If we assume that d({x}_{2k+1},{x}_{2k+2})>0, then we have, as \phi (t)<t for t>0,
a contradiction. Therefore, d({x}_{2k+1},{x}_{2k+2})=0. Hence {x}_{2k+1}={x}_{2k+2}. Thus we have {x}_{2k}={x}_{2k+1}={x}_{2k+2}. By (3.2), it means {x}_{2k}=T{x}_{2k}=S{x}_{2k}, that is, {x}_{2k} is a common fixed point of T and S.
If n=2k+1, then using the same arguments as in the case {x}_{2k}={x}_{2k+1}, it can be shown that {x}_{2k+1} is a common fixed point of T and S.
From now on, we suppose that {x}_{n}\ne {x}_{n+1} for all n\in \mathbb{N}.
Now we shall prove that
There are two cases which we have to consider.
Case I. n=2k, k\in \mathbb{N}.
From the contraction condition (3.1) with x={x}_{2k} and y={x}_{2k1} we get
Since L<1/2, we get
Now, if we suppose that max\{sd({x}_{2k},{x}_{2k1}),sd({x}_{2k},{x}_{2k+1})\}=sd({x}_{2k},{x}_{2k+1}), 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 n=2k.
Case II. n=2k+1, k\in \mathbb{N}.
Using the same argument as in the Case I, it can be proved that (3.3) holds for n=2k+1, that is,
From (3.4) and (3.5) we conclude that the inequality (3.3) holds for all n\in \mathbb{N}.
From (3.3), by the induction it is easy to prove that
Since {lim}_{n\to +\mathrm{\infty}}{\phi}^{n}(t)=0 for all t>0, from (3.6) it follows that
Now we shall prove that \{{x}_{n}\} is a Cauchy sequence. Let \u03f5>0. Since L<\frac{1}{1+s} implies s2L>0 and 1L(1+s)>0, from (3.7) we conclude that there exists {n}_{0}\in \mathbb{N} such that
for all n\ge {n}_{0}.
Let m,n\in \mathbb{N} with m>n. By induction on m, we shall prove that
Let n\ge {n}_{0} and m=n+1. Then from (3.3) and (3.8) we get
Thus (3.9) holds for m=n+1.
Assume now that (3.9) holds for some m\ge n+1. We have to prove that (3.9) holds for m+1.
We have to consider four cases.
Case I. n is odd, m+1 is even.
From the contraction condition (3.1) we get
Hence we get, as d({x}_{m},{x}_{m+1})<d({x}_{n1},{x}_{n}) and \phi (t)<t for all t>0,
If from (3.10) we have sd({x}_{n},{x}_{m+1})<sd({x}_{n1},{x}_{n}), then by (3.8),
If (3.10) implies sd({x}_{n},{x}_{m+1})<L[d({x}_{n1},{x}_{m+1})+d({x}_{n},{x}_{m})], then by the (general) triangle inequality,
Hence we get, as L<1/(1+s) implies L/(1L)<2L<1\le s,
Now, by (3.8) and the induction hypothesis (3.9),
Thus we proved that in this case (3.9) holds for m+1. Therefore, by induction, we conclude that in Case I the inequality (3.9) holds for all m>n.
Case II. n is even, m+1 is odd. The proof of (3.9) in this case is similar to one given in Case I.
Case III. n is even, m+1 is even.
Using the (general) triangle inequality and the contraction condition (3.1), we obtain
Hence we get, as d({x}_{m},{x}_{m+1})<d({x}_{n1},{x}_{n}) and \phi (t)<t for all t>0,
If the inequality (3.11) implies d({x}_{n},{x}_{m+1})<sd({x}_{n},{x}_{n+1})+sd({x}_{n},{x}_{n+1}), 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 m+1. Therefore, by induction, we conclude that in Case III the inequality (3.9) holds for all m>n.
Case IV. n is odd, m+1 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 \{{x}_{n}\} is a Cauchy sequence. Since (X,d) is a complete bmetric space, then \{{x}_{n}\} converges to some u\in X as n\to +\mathrm{\infty}.
Now we shall prove that if one of the mappings T or S is continuous, then Tu=Su=u. Without loss of generality, we can suppose that S is continuous. Clearly, as {x}_{n}\to u, then by (3.2) we have S{x}_{2n+1}={x}_{2n+2}\to u as n\to +\mathrm{\infty}. Since {x}_{2n+1}\to u and S is continuous, then S{x}_{2n+1}\to Su. Thus, by the uniqueness of the limit in a bmetric space, we have Su=u. Now, from the contraction condition (3.1),
If we suppose that d(u,Tu)>0, then we have
a contradiction. Therefore, d(u,Tu)=0. Hence Tu=u. 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, d(u,v)>0. Then
Since 2L<1\le s, then we get sd(u,v)\le \phi (sd(u,v))<sd(u,v), a contradiction. Thus we proved that S and T have a unique common fixed point in X. □
If S=T in Theorem 1, then we have the following result.
Corollary 1 Let (X,d) be a complete bmetric space with a constant s and T:X\to X two mappings on X. Suppose that there is a constant L<\frac{1}{2} and a comparison function φ such that the inequality
holds for each x,y\in X. 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 (X,d) be a complete bmetric space with a constant s and T,S:X\to X two mappings on X. Suppose that there is a constant L<\frac{1}{1+s} and a comparison function φ such that the inequality
holds for all x,y\in X. 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 \{{x}_{n}\}, defined as in (3.3), converges to some u\in X, that is,
Now we prove that Su=u. From the contraction condition (3.13) and by the monotonicity of φ we obtain
Since φ is nondecreasing and L<1, from (3.16) we get
Set
Then, in virtue of (3.15),
where r\ge 0. Let \{{t}_{{n}_{k}}\} be a subsequence of \{{t}_{n}\} such that {t}_{{n}_{k}}\to r as k\to \mathrm{\infty}. For simplicity, denote \{{t}_{{n}_{k}}\} again by \{{t}_{n}\}. Then from (3.18),
Suppose that r>0. Then from (3.19), (3.17), and the assumption (3.14) of φ, we have
a contradiction. Therefore,
Hence we have {x}_{2n+1}\to Su as n\to \mathrm{\infty}. Since by (3.15), {x}_{2n+1}\to u, and as the limit in a bmetric space is unique, it follows that Su=u. Now, by (3.13),
If we suppose that d(u,Tu)>0, then we have sd(Tu,u)\le \phi (sd(u,Tu))<sd(u,Tu), a contradiction. Therefore, d(Tu,u)=0, that is, Tu=u. Thus we proved that Tu=Su=u. □
If S=T in Theorem 2, then we get the following result.
Corollary 2 Let (X,d) be a complete bmetric space with a constant s and T:X\to X a mapping on X. Suppose that there is a constant L<\frac{1}{1+s} and a comparison function φ such that the inequality
holds for all x,y\in X. 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 X=[0,1] endowed with the bmetric
with constant s=2. Consider mappings T,S:X\to X, Tx=\frac{1}{4}x, Sx=\frac{1}{8}x, and the comparison function \phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}), \phi (t)=\frac{t}{t+1}. Clearly, (X,d) 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. y=2x. Hence Tx=Sy, d(Tx,Sy)=0, and, therefore, the inequality (3.1) holds.
Case II. y>2x. Then \frac{1}{8}y>\frac{1}{4}x, and
Thus in this case the contraction condition (3.1) holds.
Case III. y<2x. 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 u=0.
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 quasimetric 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/s122150090026y
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 BoydWong type contractions in ordered metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 359054
Chandok S, Postolache M: Fixed point theorem for weakly Chatterjeatype 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 omegadistance 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 metrictype spaces. Acta Math. Sci. 2014, 34(4):1–17.
Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/01650114(88)900644
Gregori V, Sapena A: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 2002, 125: 245–252. 10.1016/S01650114(00)000889
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 alphacontractive 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 {(\psi ,\phi )}_{s} 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. “BabeşBolyai” University, ClujNapoca; 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: Suzukitype 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, ElGebeily 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, AlSolami 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 bmetric spaces. Fixed Point Theory Appl 2014, 135 (2014). https://doi.org/10.1186/168718122014135
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122014135
Keywords
 bmetric space
 common fixed point
 contraction condition
 comparison function