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 Open Access
Extensions of Ćirić and Wardowski type fixed point theorems in Dgeneralized metric spaces
 Tanusri Senapati^{1},
 Lakshmi Kanta Dey^{1}Email author and
 Diana DolićaninÐekić^{2}
https://doi.org/10.1186/s1366301605227
© Senapati et al. 2016
 Received: 21 December 2015
 Accepted: 3 March 2016
 Published: 15 March 2016
Abstract
In this paper, we study an interesting generalization of standard metric spaces, bmetric spaces, dislocated metric spaces, and modular spaces due to the recent work of Jleli and Samet. Here we modify the result for Ćirić quasicontractiontype mappings and also prove the same result by taking Dadmissible mappings. Moreover, we establish fixed point theorems for two wellknown nonlinear contractions like rational contraction mappings and Wardowski type contraction mappings. Several important results in the literature can be derived from our results. Suitable examples are presented to substantiate our obtained results.
Keywords
 Modular Space
 Contraction Mapping
 Type Mapping
 Fixed Point Theory
 Rational Contraction
1 Introduction
Metric fixed point theory plays a crucial role in the field of functional analysis. It was first introduced by the great Polish mathematician Banach [1]. Over the years, due to its significance and application in different fields of science, a lot of generalizations have been done in different directions by several authors; see, for example, [2–11] and references therein. Recently, Jleli and Samet [12] introduced a very interesting generalization of metric spaces from which we can easily derive different known structures, namely standard metric spaces, bmetric spaces, dislocated metric spaces, et cetera. Also, they established a new version of several wellknown fixed point theorems. Before proceeding further, we recall the definition of a generalized metric space.
Definition 1.1
[12]
 (D1)
\(\forall x,y \in X\), \(D(x,y)=0 \Rightarrow x=y\);
 (D2)
\(\forall x,y \in X\), \(D(x,y)=D(y,x)\);
 (D3)there exists \(c>0\) such that for all \((x,y) \in X\times X\) and \((x_{n})\in C(D,X,x)\),$$ D(x,y)\leq c\limsup_{n\rightarrow\infty} D(x_{n},y). $$(1.2)
Throughout this article, we call such a space \((X,D)\) a Dgeneralized metric space. The class of such metric spaces is always larger than the class of standard metric spaces, bmetric spaces, dislocated metric spaces, dislocated bmetric spaces, et cetera. For details, interested readers are referred to [12].
The purpose of this paper is to modify the Ćirić quasicontractions. In this paper we introduce Dadmissible mappings and establish the fixed point theorem for Ćirić quasicontractions with the help of Dadmissible mappings. This article includes an example of a Dgeneralized metric space to show that a sequence in this setting may be convergent without being a Cauchy sequence. We also investigate the existence and uniqueness of a fixed point for the mappings satisfying nonlinear rational contraction and Wardowski type Fcontraction, where the function F is taken from a more general class of functions than that known in the existing literature.
We organize the paper as follows. Section 2 contains some useful notions and important results that will be needed in the paper. In Section 3, we exhibit an example to show that the Theorem 4.3 in [12] does not give the guarantee of the existence of a fixed point for any arbitrary value of \(k\in(0,1)\). Accordingly, we present a modified version of Theorem 4.3 in [12]. Also, we establish the same result for Dadmissible Ćirić quasicontraction mappings. Moreover, we also prove a fixed point theorem for rational contraction type mappings. Finally, in the last section, we present a new version of fixed point theorem due to Wardowski [13].
2 Auxiliary notions and results
We use the standard notation and terminology of functional analysis. For the organization of the paper, we recall the following:
Definition 2.1
[12]
 (i)
convergent to \(x\iff(x_{n})\in C(D,X,x)\);
 (ii)
Cauchy ⇔ \(\lim_{n,m\rightarrow\infty }D(x_{n},x_{n+m})=0\).
Remark 2.2
 (i)
the limit of a convergent sequence is unique (see Jleli and Samet [12]);
 (ii)
a convergent sequence may not be Cauchy.
We construct an example of a Dgeneralized metric space and show that a convergent sequence may not be Cauchy in this structure.
Example 2.3
 (i)
\(D(x,y)=0 \Rightarrow \mbox{ either } x+y=0 \mbox{ or } 1+x+y=0\). Now \(x+y=0\Rightarrow x=y=0\) and \(1+x+y=0\Rightarrow x=1y\), which is impossible. So \(D(x,y)=0\Rightarrow x=y\).
 (ii)
It is clear that for all \(x,y\in X\), \(D(x,y)=D(y,x)\).
 (iii)If \((x_{n})\) is a sequence converging to a point \(x \in X\), then for every \(x,y \in X\), we can always find a number \(c>0\) such that \(D(x,y)\leq c\limsup_{n\rightarrow\infty}D(x_{n},y)\). Note that for all \(x \in X\), \(C(D,X,x)=\emptyset\) except the point 0. So for any sequence \((x_{n})\) converging to 0 and \(y\in X\), we can find \(c>0\) such that$$D(y,0)=y\leq cy=c\limsup_{n\rightarrow\infty}D(x_{n},y). $$
Note 2.4
The authors of [12] show that every metric space, dislocated metric space, bmetric space, or modular metric space is a Dgeneralized metric space. Here, our example establishes that Dgeneralization is a proper generalization of all these spaces since every convergent sequence in a metric space, dislocated metric space, or bmetric space must be a Cauchy sequence, and every modular convergent sequence is a modular Cauchy sequence in a modular metric space.
Definition 2.5
[12]
Definition 2.6
[12]
The following theorem is an extension of the Banach contraction principle.
Theorem 2.7
[12]
 (i)
T is a kcontraction for some \(k\in(0,1)\),
 (ii)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\),
They also proved that the Banach contraction principle in the setting of different abstract spaces is nothing but an immediate consequence of this theorem in the corresponding structure. Continuing in this way, they extended another important fixed point theorem for Ćirić quasicontraction type mappings in Dgeneralized metric spaces, which again, generalizes the theorems concerning the Ćirić quasicontraction type mappings in different topological spaces. In this regard, we recall the definition of a kquasicontraction.
Definition 2.8
[12]
Proposition 2.9
[12]
Theorem 2.10
[12]
 (i)
T is a kquasicontraction for some \(k\in(0,1)\),
 (ii)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\),
Observe that this theorem does not give the guarantee of the existence of a fixed point of the mapping T for any arbitrary value of \(k\in (0,1)\). Indeed, the existence of a fixed point is guaranteed only when \(k\in(0,1)\cap(0,\frac{1}{c})\), where, \(c>0\) is the least number for which (D3)property is satisfied in Definition 1.1. We illustrate this by presenting an example in the next section.
On the other hand, in 2012, Wardowski [13] introduced the notion of an Fcontraction, which is perceived to be one of the most general nonlinear contractions in the literature. After that, a lot of research works have been done concerning Fcontractions; see, for example, [14–17]. Wardowski introduced the Fcontractions as follows.
Definition 2.11
[13]
 (F1)
F is a strictly increasing function on \(\mathbb{R_{+}}\);
 (F2)
For each sequence \((\alpha_{n}) \) of positive numbers, \(\lim_{n\rightarrow\infty} \alpha_{n}=0 \iff\lim_{n\rightarrow\infty} F(\alpha_{n})=\infty\);
 (F3)
\(\exists k\in(0,1)\) such that \(\lim_{\alpha\rightarrow 0^{+}} \alpha^{k} F(\alpha)=0\).
Lemma 2.12
[14]
 (1)
If \(F(\alpha_{n})\rightarrow\infty\), then \(\alpha_{n}\rightarrow0\);
 (2)
If \(\inf F=\infty\) and \(\alpha_{n}\rightarrow0\) then \(F(\alpha _{n})\rightarrow\infty\).
 (F1′):

F is a strictly increasing function on \(\mathbb{R_{+}}\);
 (F2′):

\(\inf F=\infty\);
 (F3′):

F is continuous.
 (F1″):

F is a strictly increasing function, that is, for \(x,y\in\overline{\mathbb{R}}_{+}\) such that \(x< y\), \(F(x)< F(y)\);
 (F2″):

\(\inf F=\infty\).
Example 2.13
Note that \(F\in\mathfrak{F}\), whereas F belongs neither to F nor to \(\mathcal{F}\) since F does not satisfy conditions (F3) and (F3′). Therefore, \(\mathbf{F}\subset\mathfrak{F} \) and \(\mathcal {F}\subset\mathfrak{F}\), but the converse is not true.
Considering the new family \(\mathfrak{F}\) of functions, we prove the result of Wardowski in the setting of the newly defined complete Dgeneralized metric space.
3 Ćirić quasicontraction
We start this section by presenting an example.
Example 3.1
 (i)
It is clear that \(D(x,y)=0\Rightarrow x=y\).
 (ii)
\(D(x,y)=D(y,x)\) for all \(x,y\in X\).
 (iii)
For all \(x \neq0\), we have \(C(D,X,x)=\emptyset\). If \(x=0\), then we can always find a sequence \((x_{n})\) converging to 0. So for any \(y \in X\), there exists a number \(c\geq3\) such that \(D(0,y)=y\leq c\frac{y}{3}=c\limsup_{n\rightarrow\infty}D(x_{n},y)\). Furthermore, if \((x_{n})\) is a zero sequence, then for all \(c\geq1\), we have \(D(0,y)=y\leq cy=c\limsup_{n\rightarrow\infty}D(x_{n},y)\).
Such a problem occurs due to the choice of arbitrary value of \(k\in (0,1)\). We can avoid this problem by taking \(k \in(0,1)\cap(0,\frac {1}{c})\), where c is the least positive number for which (D3)property is satisfied. Here we give a modified version of Theorem 2.10.
Theorem 3.2
 (i)
T is a kquasicontraction for some \(k\in(0,1)\cap (0,\frac{1}{c})\),
 (ii)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\),
Proof
Next, we introduce the concept of a Dadmissible mapping.
Definition 3.3
Lemma 3.4
Suppose that \((X,D)\) is a Dgeneralized metric space and T is a Dadmissible mapping on X. Then for every sequence \((x_{n})\) converging to a point \(w\in X\), we have \(D(w,Tw)<\infty\).
Proof
Using the concept of a Dadmissible mapping, we can establish the fixed point result for Ćirić quasicontraction mappings in a different way.
Theorem 3.5
 (i)
T is a kquasicontraction for some \(k\in(0,1)\cap (0,\frac{1}{c})\),
 (ii)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\),
Proof
Definition 3.6
Theorem 3.7
Suppose that T is a Dadmissible mapping defined on a complete Dgeneralized metric space \((X,D)\) satisfying the rational inequality. If there exists \(x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty \), then the sequence \((T^{n}x_{0})\) converges to some point \(w\in X\), and this w is a fixed point of T. Again, if \(w'\) is another fixed point of T with \(D(w,w')<\infty\) and \(D(w',w')<\infty\), then \(w=w'\).
Proof
The following result is an immediate consequence of Theorem 3.7.
Corollary 3.8
 (i)for all \(x,y\in X\), there exists \(k\in(0,1)\) such that$$D(Tx,Ty) \leq k \max\biggl\{ D(x,y), D(x,Tx), D(y,Ty),\frac {D(x,Ty)+D(Tx,y)}{2}\biggr\} ; $$
 (ii)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\).
4 Results of Fcontraction
Definition 4.1
Now, we state the theorem that establishes the existence and uniqueness of a fixed point for the mappings satisfying the Fcontraction principle.
Theorem 4.2
 (1)
\((X,D)\) is complete;
 (2)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})=c\) for some finite \(c\neq 0\).
Proof
Notice that every standard metric space is a Dgeneralized metric space. Consequently, the result of Wardowski [13] can be presented as an immediate consequence of Theorem 4.2. We state this as follows.
Corollary 4.3
Obviously, it is notable that the domain space of a function F is much wider than that of the existing literature since F does not satisfy conditions (F3) and (F3′) mentioned in Section 2.
Declarations
Acknowledgements
The first named author would like to express her sincere thanks to DSTINSPIRE, New Delhi, India, for their financial support under INSPIRE fellowship scheme. The authors are thankful to the editors and the anonymous referees for their valuable comments, which reasonably improve the presentation of the manuscript.
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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