In this paper, we introduce the class of extended rectangular b-metric spaces as a generalization of both rectangular metric and rectangular b-metric spaces. In addition, some fixed point results connected with certain contractions are obtained and examples are given to illustrate these results.
Fixed points theory has become an important field in mathematics due to its variety of applications in science, economics and game theory. Brouwer’s fixed-point theorem states that any continuous mapping on a compact convex set to itself has a fixed point. In addition to their importance in differential and integral equations, Brouwer’s theorem and its extension Kakutani theorem for set valued function play a very important role in proving the existence of general equilibrium in market economics and the existence of Nash equilibria in game theory—for more details, see [1,2].
In 2000, Branciari  introduced the concept of generalized metric space (rectangular space) as a generalization of normal metric space. In 2015, George et al.  introduced the notion of rectangular b-metric space as a generalization of rectangular metric space and they presented some fixed point results for contractive mappings.
In this paper, we introduce the notion of extended rectangular b-metric spaces which is a combination of properties of rectangular metric spaces and extended b-metric spaces. In addition, we obtain some fixed point results dealing with -type contraction mappings. Furthermore, we present examples to support these results.
2. Preliminaries and Known Results
In 1993, Czerwik  introduced the concept of b-metric space as follows:
().Let X be a nonempty set,be a given real number and letbe a mapping such that for allthe following conditions hold:
Then,is called a b-metric space.
For some fixed points results in b-metric space and its properties, we refer the reader to [5,6,7,8,9,10,11,12,13,14,15,16,17,18]. Recently, Parvaneh  introduced the concept of extended b-metric spaces as follows.
().Let X be a nonempty set. A functionis a p-metric if there exists a strictly increasing continuous functionwithsuch that for all, the following conditions hold:
Then, the pairis called a p-metric space or an extended b-metric space.
A b-metric is a p-metric, when , while a metric is a p-metric, when .
().Letbe a metric space and letwhereis a strictly increasing continuous function withand. Then, ρ is a p-metric with.
The following example is constructed using the above proposition.
Letbe a metric space and let. Then, ρ is a p-metric with
In 2000, Branciari  introduced the concept of generalized metric space (rectangular space) as follows:
().Let X be a nonempty set and letbe a mapping such that for alland all distinct points, each distinct from a and b, the following satisfied:
Then,is called a generalized metric space (g.m.s.) or rectangular space.
This metric attracted many authors and they obtain many results in this setting (see the references cited in [20,21,22]). One can notice that each metric space is a generalized metric space, but the converse need not be true (see [23,24]). Several authors show that the topology of g.m.s. need not be Hausdorff—see examples in [25,26,27]. The topological structure of g.m.s. is not compatible with the topology of ordinary metric space (see  (Example 7)), so it is not easy to deal with g.m.s. Therefore, this concept is very interesting for researchers.
().Let X be a nonempty set,be a given real number and letbe a mapping such that for alland all distinct points, each distinct from a and b, the following satisfied:
Then,is called a rectangular b-metric space.
The following are some easy examples of rectangular b-metric spaces.
().Letbe aandbe a given real number. Let. Then,is a rectangular b-metric space with.
Example 1.1 in  clarifies that some properties of metric spaces need not be true in g.m.s. (and so in rectangular b-metric space).
Motivated by , Hussain et al.  used to stand for the set of all functions satisfying the following conditions:
for each sequence , if and only if ;
there exist and such that .
for all .
Then, they introduced new concepts of generalized contractive mappings and obtained sufficient conditions for the existence of fixed points for mappings from these classes on complete metric spaces and complete b-metric spaces. In particular, they claimed that their results extend theorems of Ćirić, Chatterjea, Kannan and Reich. On the other hand, Jiang et al., by removing the condition , proved that in a metric space , defines a metric on X (Lemma 1 of ). They, also proved that the results in  are not real generalizations of Ćirić contractive principle.
From now on, we denote by the set of all functions satisfying the following conditions:
is a continuous strictly increasing function;
for each sequence , if and only if .
Note that, in general, the conditions of do not guarantee that the metric d generates a new metric , due to the absence of condition .
().The following are some examples of functions in Θ:
Note thatdoes not belong to Ψ, since condition (4) does not satisfy.
We recall the following:
(, Corollary 2.1).Letbe a complete metric space and letbe a given map. Suppose that there existandsuch that
Then, T has a unique fixed point.
Observe that the Banach contraction principle follows immediately from the above theorem.
3. New Definition and Basic Properties
We start this section by introducing the definition of Extended rectangular b-metric space.
Let A be a nonempty set,be a strictly increasing continuous function withfor allandand letbe a mapping such that for alland all distinct points, each distinct from a and b satisfies the following conditions:
Then,is called an extended rectangular b-metric space (ERbMS).
Note that for all and . Obviously, each rectangular b-metric space is an ERbMS with , . The following gives some more examples of extended rectangular b-metric spaces.
Letbe a rectangular b-metric space with coefficientand letbe a strictly increasing continuous function withand. Let. Evidently, for alland for any two distinct points, each of which distinct from a and b, we obtain
Thus,is an ERbMS with.
The convergence of sequences in ERbMS is introduced in a standard way. The following lemma will be needed in forthcoming results.
Letbe an ERbMS with the function Ω. Then, we have the following:
Suppose thatand are two sequences in A such that,and the elements ofare totally distinct. Then, we have
Letbe a Cauchy sequence in A converging to a. Ifhas infinitely many distinct terms, then
(i) Using the -rectangular inequality, we get that
Taking the lower limit as in the first inequality and the upper limit as in the second inequality, we obtain the desired result:
(ii) If , then for infinitely many ,
Taking the lower limit as in the first inequality and the upper limit as in the second inequality, we obtain the desired result. □
As shown in Example 1.1 of , a sequence in an ERbMS may have more than one limits. However, under some constraints, the sequence has a unique limit if it exists. The following lemma is a variant of  (Lemma 1.10) and  (Lemma 1) and will be used in the main results.
Letbe an ERbMS and letbe a Cauchy sequence in A such thatwhenever. Then,can converge to at most one point.
Suppose that converges to two different points, say . Since and are distinct elements, as well as a and b, it is clear that there exists such that a and b are different from for all . For , the rectangular inequality implies that
Taking the limit as , it follows that , i.e., . A contradiction. □
In this paper, by an ordered ERbMS, understand a triple , where is a partially ordered set and is an ERbMS.
4. Main Results
The following Lemma is needed in a sequel.
For each, we have.
Suppose that and let be a sequence in such that . Since is a strictly increasing function, we have . Thus, which contradicts the property . Therefore, . □
Let be an ERbMS with nontrivial function (i.e., ). Throughout this paper, denotes the class of all functions .
Letbe a complete ordered ERbMS with nontrivial function Ω. Letbe an increasing mapping with respect to ⪯ such that there exists an elementwith. Suppose that
for some,and all comparable elements, where
Then, f has a fixed point.
Starting with the given , put . If for some , then . Thus, is a fixed point of f. Therefore, we will assume that for all . Since and f is an increasing function, we obtain by induction that
Step I: We will show that . Since for each , then, by Inequality (1), we have
Again, taking the upper limit as and combining Inequalities (10) and (11), the above inequality turns into
which is a contradiction. Consequently, is an -Cauchy sequence in A. Therefore, the sequence -converges to some , that is, .
Step IV: Now, we show that c is a fixed point of f. Suppose that . Then, it follows that differs from both and c for n sufficiently large and . Hence,
a contradiction. Thus, c is a fixed point of f. □
The following examples illustrate the above obtained result.
Letbe equipped with the order ⪯ given by
and letbe given as
andfor all. Then, one can easily check thatis a (complete) ordered ERbMS with. Defineby. Consider the mappingdefined as
One easily can check that all the constraints of Theorem 2 are achieved with. The contractive Condition (1) is trivial except the case when,(or vice versa) when it reduces to
which implies that f has a (unique) fixed point γ. Note thatis neither a metric space, nor a rectangular metric space. For instance,
Letbe equipped with the following partial order ⪯:
Define the rectangular metricby
and let. It is easy to see thatis a complete ERbMS. Define the self-map f by
We see that f is an ordered increasing mapping. Defineby,and.
One can easily check that f satisfies the Condition (1). We will present the nontrivial cases as follows:
Thus, all the conditions of Theorem 2 are satisfied and hence f has a fixed point. Indeed, 0 is the fixed point of f.
By introducing the following concept (which is adapted from Definition 2.2 of ), we extend the result of Jleli and Samet . Furthermore, we obtain some new generalizations of the Banach contraction principle.
Letbe an ERbMS. The mappingis said to be a-contraction, whenever there exists a functionand functionswith;such that
Our second main result is the following:
Letbe a complete ERbMS and letbe a-contraction. Then, f has a unique fixed point.
Let be arbitrary. Define the sequence by . Assume that (If for some , then is a fixed point of f), i.e., for all .
First, we will prove that . Since f is a -contraction, using the Condition (14), we obtain that
Let . Then,
Note that, for , . Thus, . Hence, . According to we conclude that
Now, we will prove that . Since f is a -contraction and using Condition (14), we obtain that
Taking the upper limit as , we get
Since , the above can only happen if
Thus, it must hold
In order to show that is an -Cauchy sequence, suppose the contrary. Then, there exists for which we can find two subsequences and such that is the smallest index where
This means that
Rectangle inequality implies
Taking the upper limit as , and using Equation (15) and Inequality (18), we get
Combining rectangle inequalities and Inequality (17), we get
Taking the upper limit as and using Equation (16), Inequalities (20) and (18), we get
On the other hand, we have
Now, taking the upper limit as in the above inequality and using and Inequality (22), we have
which further implies, from Equation (15), Inequalities (18) and (19), that
which is a contradiction. Thus, we have proved that is an -Cauchy sequence. The completeness of A ensures that there exists such that, as . Suppose that ; then, from Condition (14), we have
Taking the upper limit as in the above inequality and using and Equation (15), we have
which is a contradiction, that is, . Thus, f has a fixed point. The uniqueness is straightforward. □
Taking various functions (see Remark 1), several corollaries of the previous theorem are obtained. For instance, taking , , , or , we get
Letbe a complete ERbMS andbe a mapping such that one of the following conditions is satisfied for all:
whereare such thatfor alland. Then, f has a unique fixed point.
Taking and in (1) of the above result, we obtain Theorem 2.7 of  in the framework of non-ordered ERbMS.
Letbe endowed with the rectangular metric
for alland. Defineandbyand,for. Let. Then,. Therefore,, and
Let. Then,, and
Thus,. Hence, f is a-contraction and so all conditions of Corollary 1 are satisfied and f has a fixed point.
5. Some Consequences in Rectangular b-Metric Spaces
Letbe a rectangular b-metric space andbe such that
for all, whereandare positive real functions such thatfor alland. Then, f has a unique fixed point.
Letbe a rectangular b-metric space. Letbe such that
for allwhereandare positive real functions such thatfor alland. Then, f has a unique fixed point.
We have introduced the class of extended rectangular b-metric spaces (ERbMS) as a generalization of both rectangular metric and rectangular b-metric spaces, and we proved some fixed point results on complete ordered ERbMS with certain contractions. Moreover, one can easily see that the set of fixed points of a mapping is well ordered if and only if the mapping has a unique fixed point. Furthermore, by introducing the concept of -contraction, we extend the result of Jleli and Samet and we obtained some new generalizations of the Banach contraction principle. It would be very interesting to analyze the existing literature in light of the new defined metric (ERbMS).
Conceptualization, Z.M., V.P., M.M.M.J. and Z.K.; methodology, Z.M., V.P., M.M.M.J. and Z.K.; investigation, Z.M., V.P., M.M.M.J. and Z.K.; writing—original draft preparation, V.P. and Z.K.; writing—review and editing, Z.M. and M.M.M.J.
The publication of this article was funded by the Qatar National Library.
The publication of this article was funded by the Qatar National Library. The authors are highly appreciated the referees efforts of this paper who helped us to improve it in several places.
Conflicts of Interest
The authors declare no conflict of interest.
Border, K. Fixed Point Theorems with Applications to Economics and Game Theory; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
Kakutani, S. A generalization of Brouwer’s fixed point theorem. Duke Math. J.1941, 8, 457–459. [Google Scholar] [CrossRef]
Branciari, A. A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ. Math. Debr.2000, 57, 31–37. [Google Scholar]
George, R.; Redenovic, S.; Reshma, K.P.; Shukla, S. Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl.2015, 8, 1005–1013. [Google Scholar] [CrossRef]
Czerwik, S. Contraction mappings in b-metric spaces. Acta Math. Inf. Univ. Ostrav.1993, 1, 5–11. [Google Scholar]
Aydi, H.; Bota, M.; Karapinar, E.; Moradi, S. A Common Fixed Point For Weak -Phi-Contractions on b-Metric Spaces. Fixed Point Theory2012, 13, 337–346. [Google Scholar]
Czerwik, S. Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Fis. Univ. Modena1998, 46, 263–276. [Google Scholar]
Karapinar, E.; Piri, H.; AlSulami, H.H. Fixed Points of Generalized F-Suzuki Type Contraction in Complete b-Metric Spaces. Discret. Dyn. Nat. Soc.2015, 2015, 969726. [Google Scholar]
Mustafa, Z.; Jaradat, M.M.M.; Jaradat, H.M. Some common fixed point results of graphs on b-metric space. J. Nonlinear Sci. Appl.2016, 9, 4838–4851. [Google Scholar] [CrossRef]
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, 296. [Google Scholar] [CrossRef]
Jaradat, M.M.M.; Mustafa, Z.; Arshad, M.; Khan, S.U.; Ahmad, J. Some fixed point results on G-metric and Gb-metric spaces. Demonstr. Math.2017, 50, 190–207. [Google Scholar]
Mustafa, Z.; Roshan, J.R.; Parvaneh, V. Existence Of Tripled Coincidence Point In Ordered Gb-Metric Spaces and Applications To System Of Integral Equations. J. Inequal. Appl.2013, 2013, 453. [Google Scholar] [CrossRef]
Mustafa, Z.; Roshan, J.R.; Parvaneh, V.; Kadelburg, Z. Some common fixed point results in ordered partial b-metric space. J. Inequal. Appl.2013, 2013, 562. [Google Scholar] [CrossRef]
Mustafa, Z.; Roshan, J.R.; Parvaneh, V.; Kadelburg, Z. Fixed point theorems for weakly T-Chatterjea and weakly T-Kannan contractions in b-metric spaces. J. Inequal. Appl.2014, 2014, 46. [Google Scholar] [CrossRef]
Mustafa, Z.; Jaradat, M.M.M.; Ansari, A.; Popović, B.Z.; Jaradat, H. C-class functions with new approach to coincidence point results for generalized (ψ, ϕ)-weakly contractions in ordered b-metric spaces. SpringerPlus2016, 5, 802. [Google Scholar] [CrossRef] [PubMed]
Mustafa, Z.; Parvaneh, V.; Roshan, J.R.; Kadelburg, Z. b2-Metric Spaces and Some Fixed Point Theorems. Fixed Point Theory Appl.2014, 2014, 144. [Google Scholar] [CrossRef]
Ansari, A.H.; Chandok, S.; Ionescu, C. Fixed point theorems on b-metric spaces for weak contractions with auxiliary functions. J. Inequal. Appl.2014, 2014, 429. [Google Scholar] [CrossRef]
Zabihi, F.; Razani, A. Fixed Point Theorems for Hybrid Rational Geraghty Contractive Mappings in Ordered b-Metric Spaces. J. Appl. Math.2014, 2014, 929821. [Google Scholar] [CrossRef]
Parvaneh, V. Fixed points of (ψ, ϕ)Ω-contractive mappings in ordered p-metric spaces. submitted.
Kadelburg, Z.; Radenović, S. Fixed point results in generalized metric spaces without Hausdorff property. Math. Sci.2014, 8, 125. [Google Scholar] [CrossRef]
Kadelburg, Z.; Radenović, S. On generalized metric spaces: a survey. TWMS J. Pure Appl. Math.2014, 5, 3–13. [Google Scholar]
Karapinar, E.; Aydi, H.; Samet, B. Fixed points for generalized (alpha,psi)-contractions on generalized metric spaces. J. Inequal. Appl.2014, 2014, 229. [Google Scholar]
Kirk, W.; Shahzad, N. Fixed Point Theory in Distance Spaces; Springer: Cham, Switzerland, 2014; Volume XI, p. 173. [Google Scholar]
Suzuki, T. Generalized metric spaces do not have the compatible topology. Abstr. Appl. Anal.2014, 2014, 458098. [Google Scholar] [CrossRef]
Roshan, J.R.; Parvaneh, V.; Kadelburg, Z.; Hussain, N. New fixed point results in b-rectangular metric spaces. Nonlinear Anal. Model. Control2016, 21, 614–634. [Google Scholar] [CrossRef]
Samet, B. Discussion on ‘A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces’ by A. Branciari. Publ. Math. Debr.2010, 76, 493–494. [Google Scholar]
Sarma, I.R.; Rao, J.M.; Rao, S.S. Contractions over generalized metric spaces. J. Nonlinear Sci. Appl.2009, 2, 180–182. [Google Scholar] [CrossRef]
Jleli, M.; Samet, B. A new generalization of the Banach contraction principle. J. Inequal. Appl.2014, 2014, 38. [Google Scholar] [CrossRef]
Hussain, N.; Parvaneh, V.; Samet, B.; Vetro, C. Some fixed point theorems for generalized contractive mappings in complete metric spaces. Fixed Point Theory Appl.2015, 2015, 185. [Google Scholar] [CrossRef]
Jiang, S.; Li, Z.; Damjanović, B. A note on “Some fixed point theorems for generalized contractive mappings in complete metric spaces”. Fixed Point Theory Appl.2016, 2016, 62. [Google Scholar] [CrossRef]