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Nadler’s fixed point theorem in νgeneralized metric spaces
 Tomonari Suzuki^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366301706112
© The Author(s) 2017
 Received: 13 May 2017
 Accepted: 29 August 2017
 Published: 15 November 2017
Abstract
We extend Nadler’s fixed point theorem to νgeneralized metric spaces. Through the proof of the above extension, we understand more deeply the mathematical structure of a νgeneralized metric space. In particular, we study the completeness of the space. We also improve Caristi’s and Subrahmanyam’s fixed point theorems in the space.
Keywords
 Nadler’s fixed point theorem
 νgeneralized metric space
 completeness
 Caristi’s fixed point theorem
 Subrahmanyam’s fixed point theorem
MSC
 54H25
 54E25
 54E50
1 Introduction and preliminaries
In 1969, Nadler proved the following; the splendid fixed point theorem for setvalued contractions, which is one of generalizations of the Banach contraction principle [1, 2]. See also, e.g., [3–8].
Theorem 1
Theorem 5 in Nadler [9]
Remark
In 2000, Branciari introduced the following, very interesting concept.
Definition 2
Branciari [10]
 (N1)
\(d(x,y) = 0\) iff \(x = y\) for any \(x, y \in X\).
 (N2)
\(d(x,y) = d(y,x)\) for any \(x, y \in X\).
 (N3)\(d(x,y) \leq D(x,u_{1},u_{2},\ldots,u_{\nu},y)\) for any \(x, u_{1}, u_{2}, \ldots, u_{\nu}, y \in X\) such that \(x, u_{1}, u_{2}, \ldots, u_{\nu}, y\) are all different, where$$D(x,u_{1},u_{2},\ldots,u_{\nu},y) = d(x,u_{1}) + d(u_{1},u_{2}) + \cdots+ d(u_{\nu},y). $$
We have studied the topological structure of this space. Indeed, recent studies tell that 1 and 3generalized metric spaces have the compatible topology and that all νgeneralized metric spaces have the strongly compatible topology. Also we have proved several fixed point theorems in this space. See, e.g., [11–25]. However, we have not generalized Theorem 1. Motivated by this fact, in this paper, we generalize Theorem 1. Another purpose of this paper is to understand more deeply the mathematical structure of this space. In particular, we study the completeness of this space. We also improve Caristi’s and Subrahmanyam’s fixed point theorems in this space.
Throughout this paper we denote by \(\mathbb {N}\) the set of all positive integers and by \(\mathbb {R}\) the set of all real numbers. For an arbitrary set A, we also denote by #A the cardinal number of A.
2 Completeness
In this section, we begin with definitions. Some of them are new.
Definition 3
 (i)
\(\{ x_{n} \}\) is said to be Cauchy [10] if \(\lim_{n} \sup_{m > n} d(x_{n},x_{m}) = 0\) holds.
 (ii)\(\{ x_{n} \}\) is said to be κCauchy [11] ifholds.$$\lim_{n \to\infty} \sup\bigl\{ d(x_{n}, x_{n + 1 + j \kappa}) : j = 0, 1, 2, \ldots\bigr\} = 0 $$
 (iii)\(\{ x_{n} \}\) is said to be \((\sum,\neq)\)Cauchy if \(x_{n}\) \((n \in \mathbb {N})\) are all different andholds.$$\sum_{j=1}^{\infty}d(x_{j}, x_{j+1}) < \infty $$
 (iv)
\(\{ x_{n} \}\) is said to converge to x [10] if \(\lim_{n} d(x_{n}, x) = 0\) holds.
 (v)\(\{ x_{n} \}\) is said to converge only to x [11] ifhold for any \(y \in X \setminus\{ x \}\).$$\lim_{n \to\infty} d(x_{n}, x) = 0 \quad\text{and}\quad \limsup_{n \to\infty} d(x_{n}, y) > 0 $$
 (vi)\(\{ x_{n} \}\) is said to converge exclusively to x [25] ifhold for any \(y \in X \setminus\{ x \}\).$$\lim_{n \to\infty} d(x_{n}, x) = 0 \quad\text{and}\quad \liminf_{n \to\infty} d(x_{n}, y) > 0 $$
 (vii)
\(\{ x_{n} \}\) is said to converge to x in the strong sense [25] if \(\{ x_{n} \}\) is Cauchy and \(\{ x_{n} \}\) converges to x.
Remark
Definition 4
Remark

X is complete iff X is 1complete.

If X is 2complete, then X is complete; see Proposition 6(ii) in [11].
We next study \((\sum,\neq)\)completeness.
Lemma 5
Proposition 7 in [11]
Let \((X,d)\) be a νgeneralized metric space where ν is odd. Let \(\{ x_{n} \}\) be a νCauchy sequence such that \(x_{n}\) are all different. Then \(\{ x_{n} \}\) is Cauchy.
Lemma 6
Proposition 8 in [11]
Let \((X,d)\) be a νgeneralized metric space where ν is even. Let \(\{ x_{n} \}\) be a νCauchy sequence such that \(x_{n}\) are all different. Then \(\{ x_{n} \}\) is 2Cauchy.
Lemma 7
Lemma 9 in [11]
Let \((X,d)\) be a νgeneralized metric space. Then every \((\sum,\neq)\)Cauchy sequence is νCauchy.
Lemma 8
Let \((X,d)\) be a νgeneralized metric space and let \(\kappa\in \mathbb {N}\). Let \(\{ x_{n} \}\) be a κCauchy sequence converging to some \(z \in X\). Assume that \(x_{n}\) are all different. Then \(\{ x_{n} \}\) is Cauchy.
Remark
We need the difference of \(x_{n}\). See Example 28(v) below.
Proof
Lemma 9
 (i)
If ν is odd, then \(\{ x_{n} \}\) is Cauchy.
 (ii)
\(\{ x_{n} \}\) is 2Cauchy.
 (iii)
If \(\{ x_{n} \}\) converges, then \(\{ x_{n} \}\) is Cauchy, that is, \(\{ x_{n} \}\) converges in the strong sense.
Proof
(i) follows from Lemmas 5 and 7. Similarly, (ii) follows from (i), Lemmas 6 and 7. (iii) follows from (ii) and Lemma 8. □
Lemma 10

ν is odd and X is complete.

X is 2complete.
Proof
Let \(\{ x_{n} \}\) be a \((\sum,\neq)\)Cauchy sequence. Then from the assumption and Lemma 9(i) and (ii), \(\{ x_{n} \}\) converges. □
Lemma 11
Let \((X,d)\) be a νgeneralized metric space and let \(\{ x_{n} \}\) be a Cauchy sequence in X converging to some \(z \in X\). Let \(\{ y_{n} \}\) be a sequence in X satisfying \(\lim_{n} d(x_{n}, y_{n}) = 0\). Then \(\{ y_{n} \}\) also converges to z.
Proof
 (i)
\(\# \{ x_{n} : n \in \mathbb {N}\} < \infty\),
 (ii)
\(\# \{ x_{n} : n \in \mathbb {N}\} = \infty\).
 (ii1)
\(x_{n} = z\) or \(y_{n} = z\) or \(x_{n} = y_{n}\),
 (ii2)
\(x_{n} \neq z\), \(y_{n} \neq z\), \(x_{n} \neq y_{n}\).
Lemma 12
Let \((X,d)\) be a νgeneralized metric space and let \(\{ x_{n} \}\) be a Cauchy sequence in X satisfying \(\liminf_{n} d(x_{n}, z) = 0\) for some \(z \in X\). Then \(\{ x_{n} \}\) converges to z.
Proof
There exists a subsequence \(\{ f(n) \}\) of the sequence \(\{ n \}\) in \(\mathbb {N}\) such that \(\{ x_{f(n)} \}\) converges to z. We note that \(\{ x_{f(n)} \}\) is Cauchy and that \(\lim_{n} d(x_{f(n)}, x_{n}) = 0\) holds. So by Lemma 11, we obtain the desired result. □
Lemma 13
Let \((X,d)\) be a \((\sum,\neq)\)complete, νgeneralized metric space. Then X is complete.
Proof

\(\# \{ x_{n} : n \in \mathbb {N}\} < \infty\),

\(\# \{ x_{n} : n \in \mathbb {N}\} = \infty\).
Definition 14
see Example 1.1 in [15]
Let \((X,d)\) be a νgeneralized metric space. X is said to be Hausdorff if \(\lim_{n} d(x_{n},x) = \lim_{n} d(x_{n},y) = 0\) implies \(x = y\).
Lemma 15
Let \((X,d)\) be a 2complete, νgeneralized metric space. Then X is Hausdorff.
Proof
Lemma 16
Let \((X,d)\) be a \((\sum,\neq)\)complete, Hausdorff, νgeneralized metric space. Then X is 2complete.
Proof

\(\# A_{1} < \infty\) or \(\# A_{2} < \infty\),

\(\# A_{1} = \infty\) and \(\# A_{2} = \infty\).
Proposition 17

X is complete.

X is \((\sum,\neq)\)complete.
Proposition 18

X is 2complete.

X is \((\sum,\neq)\)complete and Hausdorff.
Proposition 19

X is complete.

X is \((\sum,\neq)\)complete.

X is 2complete.
3 Fixed point theorems
In this section, we first generalize Theorem 1.
Theorem 20

For any \(x \in X\), Tx is a nonempty subset of X.

If a sequence \(\{ y_{n} \}\) in Tx converges to y, then \(y \in Tx\) holds.

There exists \(r \in[0,1)\) satisfying \(\delta(Tx, Ty) \leq r d(x, y) \) for all \(x,y \in X\), where δ is defined by (2).
Proof

For any \(x, y \in X\) and \(u \in Tx\) with \(x \neq y\), there exists \(v \in Ty\) satisfying \(d(u,v) < r d(x,y) \).
As a direct consequence of Theorem 20, we obtain the following.
Corollary 21
Branciari [10]
We improve Caristi’s fixed point theorem; see [26, 27].
Definition 22

A function f from X into \((\infty,+\infty]\) is proper if \(\{ x \in X : f(x) \in \mathbb {R}\}\) is nonempty.

A function f from X into \((\infty,+\infty]\) is said to be sequentially lower semicontinuous if \(f(x) \leq\liminf_{n} f(x_{n})\) holds whenever \(\{ x_{n} \}\) converges to x.

A mapping T on X is said to be sequentially continuous if \(\{ T x_{n} \}\) converges to Tx whenever \(\{ x_{n} \}\) converges to x.
Theorem 23
Theorem 2 in [28], Theorem 14 in [11]
Proof
We use Lemma 9(iii) in this paper instead of Lemma 12 in [11]. Then we can prove the conclusion as in the proof of Theorem 14 in [11]. □
Remark

\(f(x) \leq\liminf_{n} f(x_{n})\) holds whenever \(\{ x_{n} \}\) converges to x in the strong sense.
We next improve Subrahmanyam’s fixed point theorem; see [29–32].
Theorem 24
Theorem 13 in [11]
Proof
We use Lemma 9(iii) in this paper instead of Lemma 12 in [11]. Then we can prove the conclusion as in the proof of Theorem 13 in [11]. □
Remark

\(\{ T x_{n} \}\) converges to Tx whenever \(\{ x_{n} \}\) converges to x in the strong sense.
4 Counterexamples
In this section, we give counterexamples on some results in Sections 2 and 3. The following example is a counterexample on Proposition 17 and Theorem 20.
Example 25
see Example 1 in [14]
 (i)
\((X,d)\) is a 2generalized metric space.
 (ii)
X is complete.
 (iii)
\(\sum_{j=1}^{\infty}d(j,j+1) = 1 < \infty\) holds, however, \(\{ j \}\) is not Cauchy.
 (iv)
X is not \((\sum,\neq)\)complete. Hence X is not 2complete.
 (v)
T satisfies the assumption of Theorem 20. However, T does not have a fixed point.
Proof
Lemma 26
Proposition 4.1 in [20]
Let \((X,d)\) be a νgeneralized metric space and let \(\lambda\in \mathbb {N}\) such that λ is divisible by ν. Then \((X,d)\) is a λgeneralized metric space.
The following is a slight generalization of Lemma 4.1 in [19].
Lemma 27
Remark
The proof below employs the methods in the proofs of Lemma 4 in [18] and Lemmas 4.2 and 4.3 in [20].
Proof
 (a)
\(\nu= 2\).
 (b)
ν is odd.
 (c)
ν is even.
The following example is a counterexample on Lemma 8 and Proposition 18. Also this example tells that Theorem 20 is a true generalization of Theorem 1.
Example 28
see Example 1.1 in [15]
 (i)
\((X,d)\) is a νgeneralized metric space for \(\nu\in \mathbb {N}\setminus\{ 1, 3, 5 \}\).
 (ii)
\(\{ 2^{n} \}\) converges to 0, 2 and 3. Therefore X is not Hausdorff.
 (iii)
X is \((\sum,\neq)\)complete.
 (iv)
\(\{ x_{n} \}\) is 2Cauchy, however, it does not converge. Therefore X is not 2complete.
 (v)
\(\{ y_{n} \}\) is 2Cauchy and it converges to 0. However, \(\{ y_{n} \}\) is not Cauchy.
 (vi)
All the assumptions of Theorem 20 are satisfied.
 (vii)
There does not exist a metric q on X satisfying (1) with \(d := q\).
Proof
5 Conclusions
Declarations
Acknowledgements
The author is supported in part by JSPS KAKENHI Grant Number 16K05207 from Japan Society for the Promotion of Science.
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
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