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Extensions of Ćirić and Wardowski type fixed point theorems in Dgeneralized metric spaces
Fixed Point Theory and Applications volume 2016, Article number: 33 (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.
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.
Let X be a nonempty set, and \(D:X\times X\rightarrow[0,\infty]\) be a mapping. For every \(x\in X\), we define the set \(C(D,X,x)\) as follows:
Definition 1.1
[12]
Let X be a nonempty set, and \(D:X\times X\rightarrow[0,\infty]\) be a mapping. Then \((X,D)\) is said to be a generalized metric space if the following conditions are satisfied:

(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].
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]
Let \((X,D)\) be a Dgeneralized metric space. Then a sequence \((x_{n})\) in X is said to be:

(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
In the Dgeneralized metric space \((X,D)\), the following results hold:

(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
Let \(X=\mathbb{R^{+}} \cup\{0,\infty\}\), and let \(D:X\times X\rightarrow [0,\infty]\) be defined as follows:
Now we check the axioms of a Dgeneralized metric space:

(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
$$D(y,0)=y\leq cy=c\limsup_{n\rightarrow\infty}D(x_{n},y). $$
Therefore, all conditions (D1)(D3) are satisfied. So \((X,D)\) is a Dgeneralized metric space.
Now, in this structure, we show that every convergent sequence may not be a Cauchy sequence. Let us consider the sequence \((x_{n})\) where \(x_{n}=\frac{1}{n}\) for all \(n\in\mathbb{N}\). Then,
But,
This shows that \((x_{n})\) is a convergent sequence but not a Cauchy sequence.
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]
Let \((X,D)\) be a Dgeneralized metric space, and \(T:X \rightarrow X\) be a mapping. For any \(k \in(0,1)\), T is said to be a kcontraction if
Definition 2.6
[12]
For every \(x_{0}\in X\), we define
The following theorem is an extension of the Banach contraction principle.
Theorem 2.7
[12]
Suppose that \((X,D)\) is a complete Dgeneralized metric space and T is a self mapping defined on X. If

(i)
T is a kcontraction for some \(k\in(0,1)\),

(ii)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\),
then \(\{T^{n}(x_{0})\}\) converges to some \(w\in X\), a fixed point of T. If \(w'\) is another fixed point of T with \(D(w,w')<\infty\), then \(w=w'\).
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]
Let \((X,D)\) be a Dgeneralized metric space, and \(T:X\rightarrow X \) be a selfmapping. For any \(k\in(0,1)\), T is a kquasicontraction if for all \(x,y\in X\),
where \(M(x,y)=\max\{D(x,y),D(x,Tx),D(y,Ty),D(x,Ty),D(y,Tx)\}\).
Proposition 2.9
[12]
Suppose that T is a kquasicontraction for some \(k\in(0,1)\). Then any fixed point \(w\in X\) of T satisfies
Theorem 2.10
[12]
Suppose that \((X,D)\) is a complete Dgeneralized metric space and T is a selfmapping defined on X. If

(i)
T is a kquasicontraction for some \(k\in(0,1)\),

(ii)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\),
then \(\{T^{n}(x_{0})\}\) converges to some \(w\in X\). If \(D(x_{0},T(w))<\infty\) and \(D(w,T(w))<\infty\), then w is a fixed point of T. If \(w'\) is another fixed point of T with \(D(w,w')<\infty\) and \(D(w',w')<\infty\), then \(w=w'\).
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]
Let \((X,d)\) be a metric space, and \(T:X\rightarrow X \) be a selfmapping. The function T is said to be an Fcontraction mapping if there exists \(\tau>0\) such that for all \(x,y\in X\),
where F belongs to a family of functions F from \(\mathbb {R_{+}}\) to \(\mathbb{R}\) having the following properties:

(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]
Let \(F:\mathbb{R_{+}}\rightarrow\mathbb{R}\) be an increasing function, and \((\alpha_{n})\) be a sequence of positive real numbers. Then the following assertions hold:

(1)
If \(F(\alpha_{n})\rightarrow\infty\), then \(\alpha_{n}\rightarrow0\);

(2)
If \(\inf F=\infty\) and \(\alpha_{n}\rightarrow0\) then \(F(\alpha _{n})\rightarrow\infty\).
Taking into account this lemma, Piri and Kumam [14] considered a new set \(\mathcal{F}\) of functions \(F:\mathbb{R_{+}}\rightarrow \mathbb{R}\) satisfying the following conditions:
 (F1′):

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

\(\inf F=\infty\);
 (F3′):

F is continuous.
Here, we consider \(\overline{\mathbb{ R}}_{+}=(0,\infty]\). We use the standard arithmetic operations on \(\overline{\mathbb{R}}_{+}\) and suppose that \(a\leq\infty\) for all \(a\in \overline{\mathbb{R}}_{+}\). Now, we consider a new family \(\mathfrak{F}\) of functions having the following properties:
 (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
We consider the function \(F:\overline{\mathbb{R}}_{+}\rightarrow\mathbb {R}\) defined by
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.
Ćirić quasicontraction
We start this section by presenting an example.
Example 3.1
Let \(X=[0,1]\). We define the distance function D on X as follows:
First, we check the axioms of a Dgeneralized metric space.

(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 \(D(0,y)=y\leq cy=c\limsup_{n\rightarrow\infty}D(x_{n},y)\).
Therefore, all conditions (D1)(D3) are satisfied. So \((X,D)\) is a Dgeneralized metric space. Now we define the mapping \(T:X\rightarrow X\) by
Now, we show that this mapping satisfies Ćirić quasicontraction condition.
Case I: For all \(x,y\in(0,1]\), we have
and
Clearly, \(\frac{x+y}{3}\leq M(x,y)\), and hence, for all \(k\in[\frac {1}{2},1)\), we get
Case II: Let \(x=0\), and y be any arbitrary point. Then
and
Let us consider \(y=1\). Then \(D(T0,T1)=\frac{1}{2}\) and \(M(0,1)=1\). Therefore, for all \(k\in[\frac{1}{2},1)\), we get
Similarly, for all \(y\in[0,1]\), we can find some \(k\in(0,1)\) such that
Considering these two cases, we can conclude that, for all \(x,y\in X\),
for some \(k\in(0,1)\), that is, T satisfies the kquasicontraction condition. Let us set \(x_{0}=1\). Then it is clear that \(\delta (D,T,x_{0})<\infty\). Now, \(Tx_{0} = \frac{1}{2},T^{2}x_{0}= \frac{1}{2^{2}},\dots {}, T^{n}{x_{0} } = \frac{1}{2^{n}}\), and so on. Clearly, \((T^{n}x_{0})\) is a Cauchy sequence and converges to \(w=0\). Comparing with the conditions of Theorem 2.10, we have
and
Thus, all the conditions of Theorem 2.10 are satisfied, but still \(w=0\) is not a fixed point of T since \(T0=1\). Also, note that the mapping T does not have any fixed points.
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
Suppose that \((X,D)\) is a complete Dgeneralized metric space and T is a selfmapping defined on X. If

(i)
T is a kquasicontraction for some \(k\in(0,1)\cap (0,\frac{1}{c})\),

(ii)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\),
then \(\{T^{n}(x_{0})\}\) converges to some \(w\in X\). If \(D(x_{0},T(w))<\infty\) and \(D(w,T(w))<\infty\), then w is a fixed point of T. If \(w'\) is another fixed point of T with \(D(w,w')<\infty\) and \(D(w',w')<\infty\), then \(w=w'\).
Proof
Proof of the first part of this theorem follows from that of Theorem 4.3 in [12]. Here, we just give a corrected version of the last part of the proof. Using property (D3) and \(D(w,T(w))<\infty\), we have
that is, w is a fixed point of T. □
Next, we introduce the concept of a Dadmissible mapping.
Definition 3.3
Let \((X,D)\) be a Dgeneralized metric space, and T be a selfmapping on X. Then T is said to be a Dadmissible mapping if for all \(x,y\in X\),
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
Since \(x_{n}\rightarrow w\) as \(n\rightarrow\infty\Rightarrow\lim_{n\rightarrow\infty}D(x_{n},w)=0\), we can find a positive integer \(n_{0}\) such that \(D(x_{n},w)<\infty\) for all \(n>n_{0}\). Again, since T is a Dadmissible mapping, we must have \(D(Tx_{n},Tw)<\infty\) for all \(n>n_{0}\), that is, \(\limsup_{n\rightarrow\infty} D(Tx_{n},Tw)<\infty\). Using the (D3)property, we have,
□
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
Suppose that \((X,D)\) is a complete Dgeneralized metric space and T is a Dadmissible selfmapping defined on X. If

(i)
T is a kquasicontraction for some \(k\in(0,1)\cap (0,\frac{1}{c})\),

(ii)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\),
then \(\{T^{n}(x_{0})\}\) converges to some \(w\in X\), and this w is a fixed point of T. If \(w'\) is another fixed point of T with \(D(w,w')<\infty \) and \(D(w',w')<\infty\), then \(w=w'\).
Proof
Since T is kquasicontraction, for all \(n\geq1 \) and \(i,j \in \mathbb{N}\), we have
Using Definition 2.6, we obtain
From this inequality, for all \(n,m\in\mathbb{N}\), we get
Since \(k\in(0,1)\), we have
which implies that \((T^{n}(x_{0}))\) is a Cauchy sequence. Since \((X,D) \) is complete, we must have some \(w\in X\) such that \((T^{n}(x_{0}))\) is convergent to w. We prove that w is a fixed point of T. Now,
If \(\limsup_{n\rightarrow\infty} D(T^{n}x_{0},Tw)\) is the maximum then from inequality (3.3), then we have
Since T is a Dadmissible mapping, we must have \(\limsup_{n\rightarrow\infty} D(T^{n}x_{0},Tw)<\infty\), which implies that the last inequality is impossible since \(k\in (0,1)\). Therefore, we must have
Using property (D3) and Lemma 3.4, we have
that is, w is a fixed point of T.
Let \(w'\) be another fixed point of T with \(D(w,w')<\infty\) and \(D(w',w')<\infty\). So by Proposition 2.9 we must have \(D(w',w')=0\). Again, by the property of kquasicontraction of T,
since \(k\in(0,1)\) and \(D(w,w')<\infty\). Hence, the proof is completed. □
Definition 3.6
Let \((X,D)\) be a complete Dgeneralized metric space, and T be a selfmapping on X. T is said to satisfy the rational inequality if
for some \(k\in(0,1)\).
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
For all \(i,j,n\in\mathbb{N}\), we have
Taking the limiting value of n, we get
As previously, for all \(n,m\in\mathbb{N}\), we get
which shows that \(T^{n}(x_{0})\) is a Cauchy sequence. Let \(T^{n}(x_{0})\) converge to some \(w\in X\) since \((X,D)\) is complete. Let us show that this w is a fixed point of T. We have
From Equation (3.5) it is clear that
Proceeding as before, we get that w is a fixed point of T. □
The following result is an immediate consequence of Theorem 3.7.
Corollary 3.8
Let \(T:X\rightarrow X\) be a Dadmissible self mapping, and \((X,D)\) be a complete Dgeneralized metric space. Suppose that the following conditions hold:

(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 \(\delta(D,T,x_{0})<\infty\).
Then \((T^{n}(x_{0}))\) converges to some \(w\in X\), and this w is a fixed point of T. Moreover, if \(w'\) is another fixed point of T with \(D(w,w')<\infty\) and \(D(w',w')<\infty\), then \(w=w'\).
Results of Fcontraction
This section is devoted to a fixed point theorem of Wardowski type contraction. It is worth mentioning that our proof of the following theorem is very precise and it is interesting to compare it with the existing proof in the literature. First, we introduce the definition of Fcontraction. Note that in a metric space, \(d(x,x)=0\) for all x, so the condition \(d(Tx,Ty) > 0\Rightarrow d(x,y) > 0\), and hence the condition
for some \(\tau>0\), is appropriate. But in a Dgeneralized metric space, since \(D(x,x)\neq0\) for all x, \(D(Tx,Ty) > 0\) may not give the guarantee that \(D(x,y) > 0\). Hence, we modify the definition of Fcontraction as follows.
Definition 4.1
A selfmapping T defined on X is said to be an Fcontraction mapping if for all \(x,y\in X\),
for some \(\tau>0\).
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
Let \((X,D)\) be a Dgeneralized metric space, and \(T:X\rightarrow X\) be an Fcontraction mapping with \(F\in\mathfrak {F}\). Assume that the following conditions hold:

(1)
\((X,D)\) is complete;

(2)
\(\exists x_{0}\in X\) such that \(\delta(D,T,x_{0})=c\) for some finite \(c\neq 0\).
Then T has a fixed point. If \(w'\) is another fixed point with \(D(w,w')<\infty\), then \(w=w'\).
Proof
By the hypothesis of the theorem, there exists some \(x_{0}\in X\) such that \(\delta(D,T,x_{0})=c\) for some finite \(c\neq0\). Since T is an Fcontraction, for all \(n\geq1\) and \(i,j\in\mathbb {N}\), we get
From Definition 2.6 it is clear that, for all \(i,j\in\mathbb{N}\),
Since F is a strictly increasing function, we must have
So from Equation (4.1) we obtain
Therefore, for all \(n,m \in\mathbb{N} \),
Taking \(n \rightarrow\infty\) in both sides of inequality (4.2) and using property (F2″) and Lemma 2.12, we have
Since \((X,D)\) is complete, \(T^{n}x_{0} \rightarrow w\) for some \(w\in X\) as \(n\rightarrow\infty\). We prove that this w is a fixed point of T. Since F is a strictly increasing function, we have
Using property (D3) and Equation (4.3), we obtain
If possible, let \(w'\) be another fixed point of T with \(D(w,w')<\infty \). We show that \(D(w,w')=0\). If not, let \(D(w,w')=k\) for some positive k. Then by the property of Fcontraction of T we have
So, w is a unique fixed point of T. □
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
Let \((X,d)\) be a complete metric space, and T be a selfmapping defined on X such that
where \(F\in\mathfrak{F}\). Now if there exists \(x_{0}\in X\) such that
for some finite \(c\neq0\), then T has a fixed point. Also, if \(w'\) is another fixed point with \(D(w,w')<\infty\), then \(w=w'\).
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.
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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.
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Senapati, T., Dey, L.K. & DolićaninÐekić, D. Extensions of Ćirić and Wardowski type fixed point theorems in Dgeneralized metric spaces. Fixed Point Theory Appl 2016, 33 (2016). https://doi.org/10.1186/s1366301605227
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Keywords
 Modular Space
 Contraction Mapping
 Type Mapping
 Fixed Point Theory
 Rational Contraction