A generalization of HegedüsSzilágyi’s fixed point theorem in complete metric spaces
 Tomonari Suzuki^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366301706259
© The Author(s) 2018
Received: 7 July 2017
Accepted: 12 December 2017
Published: 8 January 2018
Abstract
In 1980, Hegedüs and Szilágyi proved some fixed point theorem in complete metric spaces. Introducing a new contractive condition, we generalize HegedüsSzilágyi’s fixed point theorem. We discuss the relationship between the new contractive condition and other contractive conditions. We also show that we cannot extend HegedüsSzilágyi’s fixed point theorem to MeirKeeler type.
Keywords
MSC
1 Introduction and preliminaries
Throughout this paper we denote by \(\mathbb {N}\) the set of all positive integers and by \(\mathbb {R}\) the set of all real numbers.
Hegedüs and Szilágyi in [1] proved the following fixed point theorem. The author thinks that the proof in [1] is splendid.
Theorem 1
(Theorem 5 in [1])
 (i)
\(\varphi(t) < t\) holds for all \(t \in(0, \infty)\);
 (ii)
φ is upper semicontinuous from the right;
 (iii)
\(d(Tx,Ty) \leq\varphi\circ D_{T}(x,y) \) holds for all \(x, y \in X\).
Remark 1
See also [2–4]. Note that in the proof of Theorem 1 in [3], we need an additional assumption such as the nondecreasingness of φ.
We state BoydWong’s [5], MeirKeeler’s [6] and Matkowski’s [7] fixed point theorems.
Theorem 2
(Theorem 1 in [5])
 (iii)
\(d(Tx,Ty) \leq\varphi\circ d(x,y) \) holds for all \(x, y \in X\).
Theorem 3
([6])
Theorem 4
(Theorem 1.2 in [7])
 (i)
φ is nondecreasing;
 (ii)
\(\lim_{n} \varphi^{n}(t) = 0\) holds for all \(t \in(0, \infty)\);
 (iii)
\(d(Tx,Ty) \leq\varphi\circ d(x,y) \) holds for all \(x, y \in X\).
From the above, we can tell that Theorem 1 is of BoydWong [5] type (see Definition 8). So it is a very natural question of whether we can extend Theorem 1 to MeirKeeler [6] type. It is also a natural question of whether we can prove a Matkowski [7] type fixed point theorem.
In this paper, we answer the above two questions; one is negative and the other is affirmative. Indeed, we generalize Theorem 1. The assumption of the new theorem (Theorem 5) is weaker than a Matkowski type condition (see Corollary 7). We also give a counterexample for a MeirKeeler type condition (Example 16). We further discuss the relationship between the assumption of Theorem 5 and other contractive conditions.
2 Main results
In this section, we generalize Theorem 1.
Theorem 5
 (i)
\(\varphi(t) < t\) holds for all \(t \in(0, \infty)\);
 (ii)For any \(\varepsilon> 0\), there exists \(\delta> 0\) such that, for any \(t \in(0,\infty)\),$$\varepsilon< t < \varepsilon+\delta \quad\textit{implies}\quad \varphi(t) \leq \varepsilon. $$
 (iii)For any \(x, y \in X\),holds.$$d(Tx,Ty) \leq\varphi\circ D_{T}(x,y) $$
Remark 2

\(D_{T}(x,y) < \infty\) obviously holds for any \(x, y \in X\).

Since \(D_{T}(x,y) = 0\) implies \(d(Tx,Ty) = 0\), without loss of generality, we may assume \(\varphi(0) = 0\).

We do not assume that φ is nondecreasing. So, in general, \(D_{T}(Tx,Ty) \leq\varphi\circ D_{T}(x,y)\) does not hold.
Before proving Theorem 5, we need one lemma.
Lemma 6
 (a)
\(x = y\);
 (b)
\(\lim_{n} D_{T}(T^{n} x) = \lim_{n} D_{T}(T^{n} y) = 0\).
Proof

\(\varepsilon< D_{T}(T^{n} x,T^{n} y) \) holds for any \(n \in \mathbb {N}\);

\(\varepsilon= D_{T}(T^{n} x,T^{n} y) \) holds for some \(n \in \mathbb {N}\).
Proof of Theorem 1
By Theorem 5, we obtain a Matkowski type fixed point theorem.
Corollary 7
 (i)
φ is nondecreasing;
 (ii)
\(\lim_{n} \varphi^{n}(t) = 0\) holds for all \(t \in(0, \infty)\);
 (iii)
\(d(Tx,Ty) \leq\varphi\circ D_{T}(x,y) \) holds for all \(x, y \in X\).
3 Comparison
In this section, using subsets of \((0,\infty)^{2}\), we discuss the relationship between the new contractive condition in Theorem 5 and other contractive conditions. See [1, 8–11] and the references therein.
Definition 8
 (1)
Q is said to be contractive (Cont for short) [12, 13] if there exists \(r \in(0,1)\) such that \(u \leq r t\) holds for any \((t,u) \in Q\).
 (2)Q is said to be Browder (Bro for short) [14] if there exists a function φ from \((0, \infty)\) into itself satisfying the following:
 (2i)
φ is nondecreasing and rightcontinuous;
 (2ii)
\(\varphi(t) < t\) holds for any \(t \in(0, \infty)\);
 (2iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in Q\).
 (2i)
 (3)Q is said to be BoydWong (BW for short) [5] if there exists a function φ from \((0, \infty)\) into itself satisfying the following:
 (3i)
φ is upper semicontinuous from the right;
 (3ii)
\(\varphi(t) < t\) holds for any \(t \in(0, \infty)\);
 (3iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in Q\).
 (3i)
 (4)
Q is said to be MeirKeeler (MK for short) [6] if, for any \(\varepsilon> 0\), there exists \(\delta> 0\) such that \(u < \varepsilon\) holds for any \((t,u) \in Q\) with \(t < \varepsilon+ \delta\).
 (5)Q is said to be Matkowski (Mat for short) [7] if there exists a function φ from \((0, \infty)\) into itself satisfying the following:
 (5i)
φ is nondecreasing;
 (5ii)
\(\lim_{n} \varphi^{n}(t) = 0\) for any \(t \in(0, \infty)\);
 (5iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in Q\).
 (5i)
 (6)Q is said to be of Newtype (NT for short) if there exists a function φ from \((0, \infty)\) into itself satisfying the following:
 (6i)
\(\varphi(t) < t\) for any \(t \in(0,\infty)\);
 (6ii)
For any \(\varepsilon> 0\), there exists \(\delta> 0\) such that \(\varepsilon< t < \varepsilon+ \delta\) implies \(\varphi(t) \leq\varepsilon\);
 (6iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in Q\).
 (6i)
 (7)
It is well known that the converse implication of (Cont → Bro) does not hold. The following three examples tell us that for each implication except (Cont → Bro), there exists a counterexample for its converse implication. In particular, MK and NT are independent.
Example 9
Remark 3
We note that the converse implication of (BW → NT) does not hold.
Example 10
Remark 4
We note that the converse implication of (Mat → NT) does not hold.
Example 11
Remark 5
We note that the converse implication of (NT → CJM) does not hold.
We will give three mappings such that \(Q_{T}\) for each mapping matches one of Examples 911, respectively.
Lemma 12

\(f(x) > 0\) implies \(Tx \neq x\) and \(f(Tx) \leq f(x)\);

\(f(x) = 0\) implies \(Tx = x\).
 (i)
\((X, d)\) is a metric space;
 (ii)
if either \(\{ x \in X : f(x) = 0 \} \neq\varnothing\) or \(\inf f(X) > 0\) holds, then X is complete;
 (iii)
\(P_{T} = Q_{T}\).
Proof
Example 13
 (i)
\((X,d)\) is a complete metric space;
 (ii)
\(f(x) > 0\) implies \(f(Tx) < f(x)\);
 (iii)
\(f(x) = 0\) implies \(Tx = x\);
 (iv)
\(P_{T} = Q_{T} = \{ (t,1) : 1 < t \}\);
 (v)
\(P_{T}\) and \(Q_{T}\) are Mat;
 (vi)
neither \(P_{T}\) nor \(Q_{T}\) are MK.
Proof
We can prove (i)(iii) easily. Using Lemma 12, we can prove (iv). (v) and (vi) follow from Example 9. □
Example 14
 (iv)
\(P_{T} = Q_{T} = \{ (1+\lambda,2 \lambda) : \lambda\in(0,1) \}\);
 (v)
\(P_{T}\) and \(Q_{T}\) are BW;
 (vi)
neither \(P_{T}\) nor \(Q_{T}\) are Mat.
Proof
We can prove (i)(iii) easily. Using Lemma 12, we can prove (iv). (v) and (vi) follow from Example 10. □
Example 15
 (iv)
\(P_{T} = Q_{T} = \{ (1,u) : 0 < u < 1 \}\);
 (v)
\(P_{T}\) and \(Q_{T}\) are MK;
 (vi)
neither \(P_{T}\) nor \(Q_{T}\) are NT.
Proof
We can prove (i)(iii) easily. Using Lemma 12, we can prove (iv). (v) and (vi) follow from Example 11. □
We finally give the following example, which tells us that we cannot extend Theorem 1 to a MeirKeeler type contractive condition.
Example 16
 (i)
\((X,d)\) is a complete metric space;
 (ii)
\(d(x,y) < 1\) holds for any \(x, y \in X\);
 (iii)
for any \(x \in X\), \(\{ T^{n} x \}\) converges to 1 in the Euclidean space \(\mathbb {R}^{1}\);
 (iv)
\(D_{T}(x) = 1\) holds for any \(x \in X\);
 (v)
\(TX = (0,1)\);
 (vi)
\(Q_{T} = \{ (1,u) : 0 < u < 1 \}\);
 (vii)
\(Q_{T}\) is MK;
 (viii)
\(Q_{T}\) is not NT;
 (ix)
T does not have a fixed point.
Proof
We can easily prove (i)(vi) and (ix). (vii) and (viii) follow from Example 11. □
4 Conclusions
In this paper, introducing a new contractive condition (see Definition 8(6)), we generalize HegedüsSzilágyi’s fixed point theorem (Theorem 1) in complete metric spaces proved in 1980. In Section 3, we discuss the relationship between the new contractive condition and other contractive conditions. We also show that we cannot extend Theorem 1 to MeirKeeler type (see Example 16).
Declarations
Acknowledgements
The author is supported in part by JSPS KAKENHI Grant Number 16K05207 from Japan Society for the Promotion of Science.
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that he has 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
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