A note on Ćirić type nonunique fixed point theorems
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
abstract metric space nonunique fixed point selfmappingsMSC
46T99 47H10 54H251 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, 2, 3, 4, 5, 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, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 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
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.
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