- Research
- Open Access
Contraction conditions using comparison functions on b-metric spaces
- Wasfi Shatanawi^{1},
- Ariana Pitea^{2}Email author and
- Rade Lazović^{3}
https://doi.org/10.1186/1687-1812-2014-135
© Shatanawi et al.; licensee Springer. 2014
- Received: 7 January 2014
- Accepted: 23 May 2014
- Published: 2 June 2014
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.
Keywords
- b-metric space
- common fixed point
- contraction condition
- comparison function
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
- (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 b-metric space.
It is obvious that a b-metric space with base $s=1$ 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].
- (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 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 $(X,d)$ and $({X}^{\prime},{d}^{\prime})$ be two b-metric 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 ${d}^{\prime}$.
- (a)
φ is non-decreasing,
- (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.
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.
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 $n\in \mathbb{N}$.
There are two cases which we have to consider.
Case I. $n=2k$, $k\in \mathbb{N}$.
Thus we proved that (3.3) holds for $n=2k$.
Case II. $n=2k+1$, $k\in \mathbb{N}$.
From (3.4) and (3.5) we conclude that the inequality (3.3) holds for all $n\in \mathbb{N}$.
for all $n\ge {n}_{0}$.
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.
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.
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 b-metric space, then $\{{x}_{n}\}$ converges to some $u\in X$ as $n\to +\mathrm{\infty}$.
a contradiction. Therefore, $d(u,Tu)=0$. Hence $Tu=u$. Thus we proved that u is a common fixed point of T and S.
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.
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.
then T and S have a unique common fixed point.
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.
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.
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.
Thus in this case the contraction condition (3.1) holds.
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 $u=0$.
Declarations
Acknowledgements
Rade Lazović was supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.
Authors’ Affiliations
References
- Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.Google Scholar
- Caristi J: Fixed point theorems for mapping satisfying inwardness conditions. Trans. Am. Math. Soc. 1976, 215: 241–251.View ArticleMathSciNetGoogle Scholar
- Hicks TL: Fixed point theorems for quasi-metric spaces. Math. Jpn. 1988, 33(2):231–236.MathSciNetGoogle Scholar
- 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-yView ArticleMathSciNetGoogle Scholar
- Choudhury BS, Metiya N: Coincidence point and fixed point theorems in ordered cone metric spaces. J. Adv. Math. Stud. 2012, 5(2):20–31.MathSciNetGoogle Scholar
- 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 359054Google Scholar
- Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28Google Scholar
- 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 271Google Scholar
- Shatanawi W, Pitea A: Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction. Fixed Point Theory Appl. 2013., 2013: Article ID 275Google Scholar
- Altun I, Simsek H: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 2008, 1(1–2):1–8.MathSciNetGoogle Scholar
- Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011, 4(2):1–12.MathSciNetGoogle Scholar
- 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 230708Google Scholar
- 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 54Google Scholar
- Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535View ArticleMathSciNetGoogle Scholar
- Cosentino M, Salimi P, Vetro P: Fixed point on metric-type spaces. Acta Math. Sci. 2014, 34(4):1–17.View ArticleMathSciNetGoogle Scholar
- Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/0165-0114(88)90064-4View ArticleMathSciNetGoogle Scholar
- Gregori V, Sapena A: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 2002, 125: 245–252. 10.1016/S0165-0114(00)00088-9View ArticleMathSciNetGoogle Scholar
- 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 168Google Scholar
- 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 658971Google Scholar
- 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.MathSciNetGoogle Scholar
- Bakhtin IA: The contraction mapping principle in almost metric spaces. 30. In Functional Analysis. Ul’yanovsk Gos. Ped. Inst., Ul’yanovsk; 1989:26–37.Google Scholar
- 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.MathSciNetGoogle Scholar
- 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 130Google Scholar
- Berinde V: Generalized contractions in quasimetric spaces. Preprint 3. In Seminar on Fixed Point Theory. “Babeş-Bolyai” University, Cluj-Napoca; 1993:3–9.Google Scholar
- 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.Google Scholar
- Hussain N, Shah MH: KKM mappings in cone b -metric spaces. Comput. Math. Appl. 2011, 61(4):1677–1684.View ArticleMathSciNetGoogle Scholar
- 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 112Google Scholar
- 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 126Google Scholar
- 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.2011.03.032View ArticleMathSciNetGoogle Scholar
- Shah MH, Hussain N: Nonlinear contractions in partially ordered quasi b -metric spaces. Commun. Korean Math. Soc. 2012, 27: 117–128. 10.4134/CKMS.2012.27.1.117View ArticleMathSciNetGoogle Scholar
- Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 1–8. 10.1080/00036810701714164View ArticleMathSciNetGoogle Scholar
- 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 131294Google Scholar
- Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136(5):1861–1869.View ArticleGoogle Scholar
- Singh SL, Prasad B: Some coincidence theorems and stability of iterative procedures. Comput. Math. Appl. 2008, 55: 2512–2520. 10.1016/j.camwa.2007.10.026View ArticleMathSciNetGoogle Scholar
- Boriceanu M: Strict fixed point theorems for multivalued operators in b -metric spaces. Int. J. Mod. Math. 2009, 4(3):285–301.MathSciNetGoogle Scholar
- 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 605781Google Scholar
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