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 Open Access
Fixed point theorems in a new type of modular metric spaces
 Duran Turkoglu^{1} and
 Nesrin Manav^{2}Email author
https://doi.org/10.1186/s1366301806503
© The Author(s) 2018
 Received: 30 October 2017
 Accepted: 15 November 2018
 Published: 1 December 2018
Abstract
In this paper, considering both a modular metric space and a generalized metric space in the sense of Jleli and Samet (Fixed Point Theory Appl. 2015:61, 2015), we introduce a new concept of generalized modular metric space. Then we present some examples showing that the generalized modular metric space includes some kind of metric structures. Finally, we provide some fixed point results for both contraction and quasicontraction type mappings on generalized modular metric spaces.
Keywords
 Fixed point
 Fatou property
 Modular metric spaces
 Generalized metric spaces
 Quasicontraction
MSC
 47H10
 54H25
1 Introduction
In 1990, the fixed point theory in modular function spaces was initiated by Khamsi, Kozlowski, and Reich [10]. Modular function spaces are a special case of the theory of modular vector spaces introduced by Nakano [13]. Modular metric spaces were introduced in [2, 3]. Fixed point theory in modular metric spaces was studied by Abdou and Khamsi [1]. Their approach was fundamentally different from the one studied in [2, 3]. In this paper, we follow the same approach as the one used in [1].
Generalizations of standard metric spaces are interesting because they allow for some deep understanding of the classical results obtained in metric spaces. One has always to be careful when coming up with a new generalization. For example, if we relax the triangle inequality, some of the classical known facts in metric spaces may become impossible to obtain. This is the case with the generalized metric distance introduced by Jleli and Samet in [6]. The authors showed that this generalization encompasses metric spaces, bmetric spaces, dislocated metric spaces, and modular vector spaces.
In this paper, considering both a modular metric space and a generalized metric space in the sense of Jleli and Samet [6], we introduce a new concept of generalized modular metric space. Then we proceed to proving the Banach contraction principle (BCP) and Ćirić’s fixed point theorem for quasicontraction mappings in this new space. To prove Ćirić’s fixed point theorem in this new space, we take the contraction constant \(k<\frac{1}{C}\), where C is as given in Definition 1.1. For readers interested in metric fixed point theory, we recommend the book by Khamsi and Kirk [8], and for more details, see [5, 7, 9, 11, 12].
First, we give the definition of generalized modular metric spaces.
Definition 1.1
 (\(\mathit{GMM}_{1}\)):

If \(D_{\lambda}(x,y) = 0\) for some \(\lambda >0\), then \(x=y\) for all \(x,y \in X\);
 (\(\mathit{GMM}_{2}\)):

\(D_{\lambda}(x,y)=D_{\lambda}(y,x)\) for all \(\lambda>0\) and \(x,y \in X\);
 (\(\mathit{GMM}_{3}\)):

There exists \(C > 0 \) such that, if \((x, y) \in X \times X\), \(\{x_{n}\} \subset X\) with \(\lim_{n \to\infty} D_{\lambda}(x_{n},x)=0\) for some \(\lambda>0\), then$$D_{\lambda}(x, y) \leq C \limsup_{n \to\infty} D_{\lambda}(x_{n}, y). $$
It is easy to check that if there exist \(x, y \in X\) such that there exists \(\{x_{n}\} \subset X\) with \(\lim_{n \to\infty} D_{\lambda}(x_{n},x)=0\) for some \(\lambda>0\), and \(D_{\lambda}(x, y) < \infty\), then we must have \(C \geq1\). In fact, throughout this work, we assume \(C \geq1\).
Example 1.1
(Modular vector spaces \((\mathit{MVS})\) [13])
 (1)
\(\rho(x)=0\) if and only if \(x=0\),
 (2)
\(\rho(\alpha x) = \rho(x)\) if \(\alpha = 1\),
 (3)
\(\rho(\alpha x+(1 \alpha)y) \leq\rho(x)+\rho(y)\) for any \(\alpha\in[0,1]\),
 (i)
If \(D_{\lambda}(x,y)=0\) for some \(\lambda> 0\) and any \(x, y \in X\), then \(x=y\);
 (ii)
\(D_{\lambda}(x,y)=D_{\lambda}(y,x)\) for any \(\lambda> 0\) and \(x, y \in X\);
 (iii)If ρ satisfies the FP, then for any \(\lambda>0\) and \(\{x_{n}\}\) such that \(\{x_{n}/ \lambda\}\) ρconverges to \(x/ \lambda \), we havewhich implies$$\rho \biggl(\frac{xy}{\lambda} \biggr) \leq\liminf_{n \to \infty} \rho \biggl(\frac{x_{n}y}{\lambda} \biggr) \leq\limsup_{n \to\infty} \rho \biggl( \frac{x_{n}y}{\lambda} \biggr), $$for any \(x, y,x_{n} \in X_{\rho}\).$$D_{\lambda}(x,y) \leq\liminf_{n \to\infty}D_{\lambda}(x_{n},y) \leq\limsup_{n \to\infty}D_{\lambda}(x_{n},y) $$
In the next example, we discuss the case of modular metric spaces.
Example 1.2
(Modular metric spaces (MMS) [2, 3])
 (i)
\(x=y\) if and only if \(\omega_{ \lambda}(x,y)=0\) for some \(\lambda>0\);
 (ii)
\(\omega_{ \lambda}(x,y)= \omega_{ \lambda}(y,x) \) for all \(\lambda>0\) and \(x,y \in M\);
 (iii)
\(\omega_{ \lambda+ \mu}(x,y) \leq\omega_{ \lambda}(x,z)+ \omega_{\mu}(z,y)\) for all \(\lambda, \mu>0\) and \(x,y,z \in X\).
 (i)
If \(D_{\lambda}(x,y)=0\) for some \(\lambda> 0\) and \(x, y \in X_{\omega}\), then \(x=y\);
 (ii)
\(D_{\lambda}(x,y)=D_{\lambda}(y,x)\) for any \(\lambda> 0\) and \(x, y \in X_{\omega}\);
 (iii)If ω satisfies the FP, then for any \(x \in X_{\omega}\) and \(\{x_{n}\} \subset X_{\omega}\) such that \(\lim_{n \rightarrow\infty} D_{\lambda}(x_{n},x) = 0\) for some \(\lambda>0\), we havefor any \(y \in X_{\omega}\), which implies$$\omega_{\lambda}(x,y) \leq\liminf_{n \to\infty} \omega_{\lambda}(x_{n},y) \leq\limsup_{n \to\infty} \omega_{\lambda}(x_{n},y) $$$$D_{\lambda}(x,y) \leq\liminf_{n \to\infty}D_{\lambda}(x_{n},y) \leq\limsup_{n \to\infty}D_{\lambda}(x_{n},y). $$
Example 1.3
(Generalized metric spaces (GMS) [6])
 (\(\mathcal{D}_{1}\)):

For every \((x,y) \in X \times X\), we have \(\mathcal{D}(x,y)=0 \Rightarrow x=y\);
 (\(\mathcal{D}_{2}\)):

For every \((x,y) \in X \times X\), we have \(\mathcal{D}(x,y)=\mathcal{D}(y,x)\),
 \((\mathcal{D}_{3})\) :

There exists \(C > 0 \) such that, if \((x, y) \in X \times X, \{x_{n}\} \in\mathcal{C}(\mathcal{D},X,x)\), we have$$\mathcal{D}(x,y)\leq C \limsup_{n \to\infty} \mathcal{D}(x_{n},y). $$
 (i)
If \(D_{\lambda}(x,y)=0\) for some \(\lambda> 0\) and \(x, y \in X\), then \(x=y\);
 (ii)
\(D_{\lambda}(x,y)=D_{\lambda}(y,x)\) for any \(\lambda> 0\) and \(x, y \in X\);
 (iii)There exists \(C > 0 \) such that, if \((x, y) \in X \times X, \{x_{n}\} \in\mathcal{C}(D_{\lambda},X,x)\) for some \(\lambda>0\), we have$$D_{\lambda}(x,y)\leq C\limsup_{n \to\infty} D_{\lambda}(x_{n},y). $$
These properties show that \((X,D)\) is a GMMS.
2 Fixed point theorems (FPT) in GMMS
The following definition is useful to set new fixed point theory on GMMS.
Definition 2.1
 (1)
The sequence \(\{x_{n}\}_{n\in \mathbb {N}}\) in \(X_{D}\) is said to be Dconvergent to \(x\in X_{D}\) if and only if \(D_{\lambda }(x_{n},x)\rightarrow0\), as \(n\rightarrow\infty\), for some \(\lambda>0\).
 (2)
The sequence \(\{x_{n}\}_{n\in \mathbb {N}}\) in \(X_{D}\) is said to be DCauchy if \(D_{\lambda}(x_{m},x_{n})\rightarrow0\), as \(m,n\rightarrow \infty\), for some \(\lambda>0\).
 (3)
A subset C of \(X_{D}\) is said to be Dclosed if for any \(\{x_{n}\}\) from C which Dconverges to x, \(x \in C\).
 (4)
A subset C of \(X_{D}\) is said to be Dcomplete if for any \(\{x_{n}\}\) DCauchy sequence in C such that \(\lim_{n,m \to \infty} D_{\lambda}(x_{n},x_{m})=0\) for some λ, there exists a point \(x \in C\) such that \(\lim_{n,m \to\infty} D_{\lambda}(x_{n},x)=0\).
 (5)A subset C of \(X_{D}\) is said to be Dbounded if, for some \(\lambda>0\), we have$$\delta_{D, \lambda}(C)= \sup\bigl\{ D_{\lambda}(x,y);x,y\in C\bigr\} < \infty. $$
In general, if \(\lim_{n \rightarrow\infty} D_{\lambda}(x_{n},x) = 0\) for some \(\lambda>0\), then we may not have \(\lim_{n \rightarrow\infty} D_{\lambda}(x_{n},x) = 0\) for all \(\lambda>0\). Therefore, as it is done in modular function spaces, we will say that D satisfies \(\Delta_{2}\)condition if and only if \(\lim_{n \rightarrow\infty} D_{\lambda}(x_{n},x) = 0\) for some \(\lambda>0\) implies \(\lim_{n \rightarrow\infty} D_{\lambda}(x_{n},x) = 0\) for all \(\lambda>0\).
Another question that comes into this setting is the concept of Dlimit and its uniqueness.
Proposition 2.1
Let \((X_{D},D)\) be a GMMS. Let \(\{x_{n}\}\) be a sequence in \(X_{D}\). Let \((x,y)\in X_{D} \times X_{D}\) such that \(D_{\lambda}(x_{n},x)\rightarrow0\) and \(D_{\lambda }(x_{n},y)\rightarrow0\) as \(n \to\infty\) for some \(\lambda> 0\). Then \(x = y\).
Proof
3 The main results
3.1 The Banach contraction principle (BCP) in GMMS
Now, we show an extension of the BCP to the setting of GMMS presented formerly. From now on, we mean 1 instead of λ for the same reason Abdou and Khmasi used in their work [1].
Definition 3.1
Proposition 3.1
Let \((X_{D},D) \) be a GMMS. Let \(f:X_{D} \to X_{D}\) be a Dcontraction mapping. If \(\omega_{1}\) and \(\omega_{2}\) are fixed points of f and \(D_{1}(\omega_{1},\omega_{2}) < \infty\), then we have \(\omega_{1} = \omega_{2}\).
Proof
Theorem 3.1
Let \((X_{D},D)\) be a GMMS. Assume that \(X_{D}\) is Dcomplete. Let \(f :X_{D} \to X_{D}\) be a Dcontraction mapping. Assume that \(\delta_{D,1}(x_{0})\) is finite for some \(x_{0} \in X_{D}\). Then \(\{ f^{n}(x_{0})\}\) Dconverges to a fixed point ω of f. Moreover, if \(D_{1}(x,\omega)<\infty\) for \(x \in X_{D}\), then \(\{f^{n}(x)\}\) Dconverges to ω.
Proof
Next, we investigate the extension of Ćirić’s FPT [4] for quasicontraction type mappings in GMMS and give a correct version of Theorem 4.3 in [6] since its proof is wrong [7].
3.2 Ćirić quasicontraction in generalized modular metric spaces
First, let us introduce the concept of quasicontraction mappings in the setting of GMMS.
Definition 3.2
Proposition 3.2
Let \((X_{D},D) \) be a GMMS. Let \(f:X_{D} \to X_{D}\) be a Dquasicontraction mapping. If ω is a fixed point of f such that \(D_{1}(\omega, \omega) < \infty\), then we have \(D_{1}(\omega, \omega) = 0\). Moreover, if \(\omega_{1}\) and \(\omega_{2}\) are two fixed points of f such that \(D_{1}(\omega_{1},\omega_{2}) < \infty, D_{1}(\omega _{1},\omega_{1}) < \infty\), and \(D_{1}(\omega_{2},\omega_{2}) < \infty\), then we have \(\omega_{1} = \omega_{2}\).
Proof
Since \(D_{1}(\omega_{1},\omega_{2})<\infty\) and \(k<1\), then \(D_{1}(\omega _{1},\omega_{2})=0\). □
The following result may be seen as an extension of Ćirić’s FPT [4] for quasicontraction type mappings in GMMS.
Theorem 3.2
Let \((X_{D},D)\) be a Dcomplete GMMS. Let \(f :X_{D} \to X_{D}\) be a Dquasicontraction mapping. Assume that \(k<\frac{1}{C}\), where C is the constant from \((\mathit{GMM}_{3})\), and there exists \(x_{0} \in X_{D}\) such that \(\delta_{D,1}(x_{0})<\infty\). Then \(\{f^{n}(x_{0})\}\) Dconverges to some \(\omega\in X_{D}\). If \(D_{1}(x_{0},f(\omega))< \infty\) and \(D_{1}(\omega,f(\omega))<\infty\), then ω is a fixed point of f.
Proof
Declarations
Acknowledgements
The authors would like to thank Professor M.A. Khamsi for his helpful and constructive comments that greatly contributed to improving the final version of this paper. This work was supported by the TUBITAK (The Scientific and Technological Research Council of Turkey).
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Funding
We have no funding for this article.
Authors’ contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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