A note on Ćirić type nonunique fixed point theorems
 Erdal Karapınar^{1, 2}Email authorView ORCID ID profile and
 Ravi P Agarwal^{3}
https://doi.org/10.1186/s136630170614z
© The Author(s) 2017
Received: 4 April 2017
Accepted: 11 September 2017
Published: 15 October 2017
Abstract
In this paper, we suggest some nonunique fixed results in the setting of various abstract spaces. The proposed results extend, generalize and unify many existing results in the corresponding literature.
Keywords
MSC
1 Introduction and preliminaries
In 1974, Ćirić [1] introduced the notion of nonunique fixed point and proposed criteria for certain operators which possess nonunique fixed points. Inspired by this pioneering work, a number of authors reported nonunique fixed point for the operators that provide different conditions, see e.g. [1–6].
In 2000, Branciari [7] introduced a new distance function that is obtained by replacing the quadrilateral inequality with the triangle inequality in the axioms of the standard metric notion. In what follows, we recall the notion of a Branciari metric space.
Definition 1
see e.g. [8]
In some sources, BMS was called ‘generalized metric space’. On the other hand, in the literature, the words ‘generalized metric’ space have been used for several different extensions of the notion of metric (see e.g. [7, 9–23]). For this reason, we prefer to use ‘Branciari metric’ to avoid the confusion.
From now onward, the set of positive integers and the set of nonnegative integers will be denoted by \(\mathbb{N}\) and \(\mathbb{N}_{0}\), respectively. Further, the symbols \(\mathbb{R}\), \(\mathbb{R}^{+}\) and \(\mathbb{R}^{+}_{0}\) indicate the real numbers, positive real numbers and nonnegative real numbers, respectively.
Notice that the concepts of open ball and closed ball are defined on BMS as the corresponding notions in the setting of the standard metric space. Hence, there is a proper topology on BMS \((X,\rho)\).
Definition 2
see e.g. [8]
 (1)
A sequence \(\{x_{n}\}\) in a BMS \((X,\rho)\) is BMS convergent to a limit x if and only if \(\rho(x_{n},x)\rightarrow0\) as \(n\rightarrow\infty\).
 (2)
A sequence \(\{x_{n}\}\) in a BMS \((X,\rho)\) is BMS Cauchy if and only if, for every \(\varepsilon>0\), there exists a positive integer \(N(\varepsilon)\) such that \(\rho (x_{n},x_{m})<\varepsilon\) for all \(n>m>N(\varepsilon)\).
 (3)
A BMS \((X,\rho)\) is called complete if every BMS Cauchy sequence in X is BMS convergent.
 (4)
A mapping \(T:(X,\rho)\rightarrow(X,\rho)\) is continuous if, for any sequence \(\{x_{n}\}\) in X such that \(\rho(x_{n},x)\rightarrow0\) as \(n\rightarrow\infty\), we have \(\rho(Tx_{n},Tx)\rightarrow0\) as \(n\rightarrow\infty\).
On the other hand, the topology of BMS \((X,\rho)\) brings some difficulties. We state the following example to illustrate the possible handicaps.
Example 3
 \((p1)\) :

Since \(\lim_{n\to\infty}\frac{1}{n^{2}+1}=0\), we have \(\lim_{n\to\infty}\rho (\frac{1}{ n^{2}+1},\frac{1}{5})\neq\rho(0,\frac{1}{5})\). Thus, the function ρ is not continuous;
 \((p2)\) :

There is no \(r>0\) such that \(B_{r}(0)\cap B_{r}(z_{i})=\emptyset \) for \(i=1,2,3\), and hence it is not Hausdorff;
 \((p3)\) :

It is clear that the ball \(B_{\frac{3}{5}}(\frac{1}{5})=\{ 0,\frac{1}{5},z_{1},z_{2},z_{3}\}\) since there is no \(r>0\) such that \(B_{r}(0)\subset B_{\frac{3}{5}}(\frac{1}{5})\), that is, open balls may not be an open set;
 \((p4)\) :

The sequence \(\{\frac{1}{n^{2}+1}: n \in\mathbb{N}\}\) converges to \(0,z_{1},z_{2},z_{3}\), and hence it is not Cauchy.
 \((p1)\) :

Branciari metric is not necessarily continuous (see e.g. Example 3);
 \((p2)\) :

BMS is not necessarily Hausdorff (limit is not necessarily unique) (see e.g. Example 3);
 \((p3)\) :

open ball need not be an open set (see e.g. Example 3);
 \((p4)\) :

a convergent sequence in BMS needs not be Cauchy (see e.g. Example 3);
 \((p5)\) :

the mentioned topologies are incompatible (see e.g. Example 7 in [23]).
Lemma 4
Let \((X, \rho)\) be a BMS, and let \(\{x_{n}\}\) be a Cauchy sequence in X such that \(x_{m}\ne x_{n}\) whenever \(m\ne n\). Then the sequence \(\{x_{n}\}\) can converge to at most one point.
Later, regarding the wellknown bmetric defined by Czerwik [25], the notion of Branciari metric is refined as bBranciari metric (see e.g. [26]).
Definition 5
Analogously, one can state the topological concepts for bBMS (see e.g. [26]).
Definition 6
 (1)
A sequence \(\{x_{n}\}\) in a bBMS \((X,\rho)\) is bBMS convergent to a limit x if and only if \(\rho(x_{n},x)\rightarrow0\) as \(n\rightarrow\infty\).
 (2)
A sequence \(\{x_{n}\}\) in a bBMS \((X,\rho)\) is bBMS Cauchy if and only if, for every \(\varepsilon>0\), there exists a positive integer \(N(\varepsilon)\) such that \(\rho (x_{n},x_{m})<\varepsilon\) for all \(n>m>N(\varepsilon)\).
 (3)
A bBMS \((X,\rho)\) is called complete if every bBMS Cauchy sequence in X is bBMS convergent.
 (4)
A mapping \(T:(X,\rho)\rightarrow(X,\rho)\) is continuous if, for any sequence \(\{x_{n}\}\) in X such that \(\rho(x_{n},x)\rightarrow0\) as \(n\rightarrow\infty\), we have \(\rho(Tx_{n},Tx)\rightarrow0\) as \(n\rightarrow\infty\).
As in the discussion on the topology of BMS, the topology of bBMS has the same difficulties (p1)(p5) above. Since these problems arise from the topology of BMS, Example 3 can be adopted for bBMS to illustrate that the same problems hold for the topology of bBMS (see e.g. [26]).
Inspired by the corresponding Lemma 4, we propose the following.
Lemma 7
Let \((X, d)\) be a bBMS, and let \(\{x_{n}\}\) be a Cauchy sequence in X such that \(x_{m}\ne x_{n}\) whenever \(m\ne n\). Then the sequence \(\{x_{n}\}\) can converge to at most one point.
Proof
Let Ψ be a family of increasing mappings \(\psi:[0,\infty )\rightarrow{}[ 0,\infty)\) satisfying \(\psi^{n}(t)\rightarrow0 \), \(n\rightarrow \infty\) for any \(t\in[0,\infty)\). In the literature such functions are called comparison functions (see e.g. [27]). The basic example of such mappings is \(\psi(t)=\frac{kt}{n}\), where \(k\in[0,1)\) and \(n \in\{2,3,\ldots\}\).
Lemma 8
see e.g. [27]
 (i)
ψ is continuous at 0;
 (ii)
\(\psi ( t ) < t\) for any \(t\in\mathbb{R}^{+}\).
In this manuscript, we investigate some nonunique fixed point results in the context of bBMS. Our results extend and generalize several results in the corresponding literature.
2 Nonunique fixed points on bBMS
First, we shall give the analog of the crucial topological notions, orbitally continuous and orbitally complete, in the context of bBMS.
Definition 9
see [1]
 (1)T is called orbitally continuous ifimplies$$ \lim_{i\rightarrow \infty} T^{n_{i}}x=z $$(2.1)for each \(x\in X\).$$ \lim_{i\rightarrow \infty}TT^{n_{i}}x =Tz $$(2.2)
 (2)
\((X,d)\) is called orbitally complete if every Cauchy sequence of type \(\{T^{n_{i}}x\}_{i\in\Bbb{N}}\) converges with respect to \(\tau_{d}\).
A point z is said to be a periodic point of a function T of period m if \(T^{m}(z)=z\), where \(T^{0}(x)=x\) and \(T^{m}(x)\) is defined recursively by \(T^{m}(x)=T(T^{m1}(x))\).
2.1 Ćirić type nonunique fixed point results
Theorem 10
Proof
On account of the Torbital completeness, we conclude that there is \(z\in X\) such that \(x_{n}\rightarrow z\). Due to the orbital continuity of T, we conclude that \(x_{n}\rightarrow Tz\). Hence, by taking Lemma 7 into account, we find \(z=Tz\), which terminates the proof. □
Corollary 11
Proof
It is sufficient to take \(\psi(t)=qt\), where \(q \in[0,1)\), in Theorem 10. □
Corollary 12
Proof
It is sufficient to take \(s=1\) in Theorem 10. □
Corollary 13
Proof
It is sufficient to take \(\psi(t)=qt\), where \(q \in[0,1)\), in Corollary 12. □
Example 14
2.2 ĆirićJotić type nonunique fixed point results [3]
Theorem 15
Proof
A verbatim repetition of the related lines in the proof of Theorem 10 completes the proof. □
Corollary 16
Corollary 17
Corollary 18
Corollary 19
Corollary 20
Corollary 21
Corollary 22
Theorem 23
Proof
A verbatim repetition of the related lines in the proof of Theorem 10 completes the proof. □
Corollary 24
Corollary 25
Corollary 26
Corollary 27
Corollary 28
2.3 Achari type nonunique fixed point results [3]
Theorem 29
Proof
Corollary 30
The following is an immediate consequence of Theorem 29 by letting \(\psi(t)=q t\), where \(q \in[0,1)\).
Corollary 31
The following is an immediate consequence of Theorem 29 by letting \(s=1\).
Corollary 32
The following is an immediate consequence of Corollary 32 by letting \(\psi(t)=q t\), where \(q \in[0,1)\).
Corollary 33
2.4 Pachpatte type nonunique fixed point results [2]
Theorem 34
Proof
Again by following line by line the proof of Theorem 10, we construct an iterative sequence \(\{x_{n}=Tx_{n1}\}_{n \in\mathbb {N}}\) whose terms are distinct from each other, by starting from an arbitrary initial value \(x_{0}:= x \in X\).
The rest of the proof is a verbatim repetition of the related lines in the proof of Theorem 10. □
If we take \(\psi(t)=q t\), then Theorem 34 implies the following result.
Corollary 35
If the statements of Theorem 34 are considered in the setting of BMS instead of bBMS, we get the following consequence.
Corollary 36
If we take \(\psi(t)=q t\) in Corollary 36, then the following consequence is obtained immediately.
Corollary 37
2.5 Karapınar type nonunique fixed point results [28]
Theorem 38
Proof
Consequently, we derive that \((x_{n})_{n \in\mathbb{N}}\) is a Cauchy sequence.
The rest of the proof is deduced by following the corresponding lines in the proof of Theorem 10. □
Corollary 39
Proof
Take \(s=1\) in the proof of Theorem 38. □
Notes
Declarations
Acknowledgements
The authors appreciate the support of their institutes.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Ćirić, LB: On some maps with a nonunique fixed point. Publ. Inst. Math. 17, 5258 (1974) MathSciNetMATHGoogle Scholar
 Pachpatte, BG: On Ćirić type maps with a nonunique fixed point. Indian J. Pure Appl. Math. 10(8), 10391043 (1979) MathSciNetMATHGoogle Scholar
 Achari, J: On Ćirić’s nonunique fixed points. Mat. Vesn. 13(28), 255257 (1976) MathSciNetMATHGoogle Scholar
 Gupta, S, Ram, B: Nonunique fixed point theorems of Ćirić type. Vijnana Parishad Anusandhan Patrika 41(4), 217231 (1998) MathSciNetMATHGoogle Scholar
 Liu, Z, Guo, Z, Kang, SM, Lee, SK: On Ćirić type mappings with nonunique fixed and periodic points. Int. J. Pure Appl. Math. 26(3), 399408 (2006) MathSciNetMATHGoogle Scholar
 Liu, ZQ: On Ćirić type mappings with a nonunique coincidence points. Mathematica 35(58), 221225 (1993) MathSciNetMATHGoogle Scholar
 Branciari, A: A fixed point theorem of BanachCaccioppoli type on a class of generalized metric spaces. Publ. Math. (Debr.) 57, 3137 (2000) MathSciNetMATHGoogle Scholar
 Branciari, A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 29(9), 531536 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Aydi, H, Karapınar, E, Samet, B: Fixed points for generalized \((\alpha,\psi)\)contractions on generalized metric spaces. J. Inequal. Appl. 2017, 229 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Aydi, H, Karapınar, E, Lakzian, H: Fixed point results on the class of generalized metric spaces. Math. Sci. 6, 46 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Azam, A, Arshad, M: Kannan fixed point theorems on generalized metric spaces. J. Nonlinear Sci. Appl. 1, 4548 (2008) MathSciNetMATHGoogle Scholar
 Bilgili, N, Karapınar, E: A note on ‘common fixed points for \((\psi, \alpha, \beta )\)weakly contractive mappings in generalized metric spaces’. Fixed Point Theory Appl. 2013, 287 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Das, P, Lahiri, BK: Fixed point of a Ljubomir Ćirić’s quasicontraction mapping in a generalized metric space. Publ. Math. (Debr.) 61, 589594 (2002) MATHGoogle Scholar
 Erhan, IM, Karapınar, E, Sekulić, T: Fixed points of (\(\psi,\phi\)) contractions on rectangular metric spaces. Fixed Point Theory Appl. 2012, 138 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Jleli, M, Samet, B: The Kannan’s fixed point theorem in a cone rectangular metric space. J. Nonlinear Sci. Appl. 2(3), 161167 (2009) MathSciNetMATHGoogle Scholar
 Kadeburg, Z, Radenovic̀, S: On generalized metric spaces: a survey. TWMS J. Pure Appl. Math. 5(1), 313 (2014) MathSciNetGoogle Scholar
 Karapınar, E: Discussion on \((\alpha,\psi)\) contractions on generalized metric spaces. Abstr. Appl. Anal. 2014 Article ID 962784 (2014) MathSciNetGoogle Scholar
 Karapınar, E: Fixed points results for alphaadmissible mapping of integral type on generalized metric spaces. Abstr. Appl. Anal. 2014 Article ID 141409 (2014) Google Scholar
 Karapınar, E: On \((\alpha,\psi)\) contractions of integral type on generalized metric spaces. In: Mityushevand, V, Ruzhansky, M (eds.) Proceedings of the 9th ISAAC Congress. Springer, Krakow, Poland (2013) Google Scholar
 Kikina, L, Kikina, K: A fixed point theorem in generalized metric space. Demonstr. Math. XLVI(1), 181190 (2013) MathSciNetMATHGoogle Scholar
 Mihet, D: On Kannan fixed point principle in generalized metric spaces. J. Nonlinear Sci. Appl. 2(2), 9296 (2009) MathSciNetMATHGoogle Scholar
 Samet, B: Discussion on: a fixed point theorem of BanachCaccioppoli type on a class of generalized metric spaces by A. Branciari. Publ. Math. (Debr.) 76(4), 493494 (2010) MathSciNetMATHGoogle Scholar
 Suzuki, T: Generalized metric spaces do not have the compatible topology. Abstr. Appl. Anal. 2014, Article ID 458098 (2014) MathSciNetGoogle Scholar
 Sarma, IR, Rao, JM, Rao, SS: Contractions over generalized metric spaces. J. Nonlinear Sci. Appl. 2(3), 180182 (2009) MathSciNetMATHGoogle Scholar
 Czerwik, S: Contraction mappings in bmetric spaces. Acta Math. Inform. Univ. Ostrav. 1, 511 (1993) MathSciNetMATHGoogle Scholar
 George, R, Radenovic, S, Reshma, KP, Shukla, S: Rectangular bmetric space and contraction principles. J. Nonlinear Sci. Appl. 8(6), 10051013 (2015) MathSciNetMATHGoogle Scholar
 Rus, IA: Generalized Contractions and Applications. Cluj University Press, ClujNapoca, Romania (2001) MATHGoogle Scholar
 Karapinar, E: A new nonunique fixed point theorem. J. Appl. Funct. Anal. 7(12, 9297 (2012) MathSciNetMATHGoogle Scholar