- Research
- Open Access
- Published:
On some fixed point theorems in generalized metric spaces
Fixed Point Theory and Applications volume 2017, Article number: 23 (2017)
Abstract
In this paper, we obtain some generalizations of fixed point results for Kannan, Chatterjea and Hardy-Rogers contraction mappings in a new class of generalized metric spaces introduced recently by Jleli and Samet (Fixed Point Theory Appl. 2015:33, 2015).
1 Introduction
The concept of a metric space is a very important tool in many scientific fields and particulary in the fixed point theory.
In recent years, this notion has been generalized in several directions and many notions of a metric-type space was introduced (b-metric, dislocated space, generalized metric space, quasi-metric space, symmetric space, etc.).
In 2015, Jleli and Samet [1] introduced a very interesting concept of a generalized metric space, which covers different well-known metric structures including classical metric spaces, b-metric spaces, dislocated metric spaces, modular spaces, and so on.
In this paper, we establish and generalize some well-known fixed point results for nonlinear contractions in this new class of generalized metric spaces.
Let us recall that a mapping T on a metric space \((X,d)\) is called a Kannan contraction if there exists \(\alpha\in[0,\frac{1}{2}[\) such that
for all \(x,y\in X\).
Using this contraction notion, Kannan [2] proved the following result.
Theorem 1.1
([2])
Let \((X,d) \) be a complete metric space, \(\lambda\in [0 ,\frac {1}{2} [ \), and T a self-mapping on X such that
for all \(x,y\in X\). Then T has a unique fixed point.
In 1972, Chatterjea [3] obtained a similar result by considering a constant \(\lambda\in [0 ,\frac{1}{2} [\) and a mapping \({ T:X\longrightarrow X} \) such that
for all \(x,y\in X\).
In this paper, we are interested by Kannan, Chatterjea, and Hardy-Rogers contraction types (see [2–4] and [5]); we establish some results on fixed points in generalized metric spaces. We also give some examples to show the effectiveness of the obtained results.
Our results generalize and improve many fixed point theorems existing in the literature in various metric-type spaces.
2 Definitions and preliminaries
Let X be a nonempty set, and let \(D:X\times X\longrightarrow [ 0,+\infty ] \) be a given mapping. For every \(x \in X \), let us define the set
Definition 2.1
([1])
D is called a generalized metric on X if it satisfies the following conditions:
- \((D_{1})\) :
-
For every \((x,y)\in X\times X \), we have
$$D(x,y)=0 \quad \Rightarrow\quad x=y. $$ - \((D_{2})\) :
-
For all \((x,y)\in X\times X \), we have
$$D(x,y)=D(y,x). $$ - \((D_{3})\) :
-
There exists a real constant \(C>0\) such that, for all \((x,y)\in X\times X \) and \(\{ x_{n}\}\in C(D,X,x)\), we have
$${D(x,y)\leq C \limsup_{n\to\infty} D(x_{n},y)}. $$The pair \((X,D) \) is called a generalized metric space.
Remark 2.2
If the set \(C(D,X,x) \) is empty for every \(x\in X \), then \((X,D) \) is a generalized metric space if and only if \((D_{1})\) and \((D_{2})\) are satisfied.
Definition 2.3
Let \((X,D) \) be a generalized metric space, let \(\{ x_{n}\}\) be a sequence in X, and let \(x\in X \). We say that \(\{ x_{n}\}\) D-converges to x in X if \(\{ x_{n}\}\in C(D,X,x)\).
Remark 2.4
Let \(\lbrace x_{n} \rbrace\) be the sequence defined by \(x_{n}=x \) for all \(n\in\mathbb{N}\). If it D-converges to x, then \(D(x,x)=0\).
Definition 2.5
Let \((X,D) \) be a generalized metric space. A sequence \(\{ x_{n}\}\) in X is called a D-Cauchy sequence if \({\lim_{m,n\to \infty} D(x_{n},x_{n+m})=0}\).
The space \((X,D) \) is said to be D-complete if every Cauchy sequence in X is D-convergent to some element in X.
In the sequel, we use the following definition of a Cauchy sequence.
Definition 2.6
Let \((X,D) \) be a generalized metric space, and let \(\{ x_{n}\}\) be a sequence in X. We say that \(\{ x_{n}\}\) is a D-Cauchy sequence if \({\lim_{m,n\to\infty} D(x_{n},x_{m})=0}\).
Proposition 2.7
\(C(D,X,x) \) is a nonempty set if and only if \(D(x,x)=0 \).
Proof
If \(C(D,X,x)\neq\emptyset\), then there exists a sequence \(\lbrace x_{n} \rbrace\subset X \) such that \(\lim_{n \to\infty} D(x_{n},x)=0\). Using property \((D_{3}) \), we obtain
and thus \(D(x,x)=0\).
Assume now that \(D(x,x)=0 \). The sequence \(\lbrace x_{n} \rbrace \subset X \) defined by \(x_{n}=x\) for all \(n\in\mathbb{N}\) converges to x, which ends the proof. □
3 Main results
Proposition 3.1
Let \((X,D) \) be a generalized metric space, and let \(f:X\longrightarrow X \) be a mapping satisfying inequality (1) for some \(\lambda\in [0,\frac{1}{2} )\). Then any fixed point \(\omega\in X \) of f satisfies
Proof
Let \(\omega\in X \) be a fixed point of f such that \(D(\omega ,\omega) <\infty\). Using (1), we obtain
Since \(2\lambda\in [ 0,1 [\), we obtain \(D(\omega,\omega)=0\). □
For every \(x\in X\), we define
Theorem 3.2
Let \((X,D) \) be a D-complete generalized metric space, and let f be a self-mapping on X satisfying (1) for some constant \(\lambda\in [0 ,\frac{1}{2} [\) such that \(C\lambda<1\).
If there exists an element \(x_{0}\in X \) such that \(\delta (D,f,x_{0})<\infty\), then the sequence \(\{ f^{n}x_{0}\} \) converges to some \(\omega\in X \). Moreover, if \(D(\omega, f\omega) < \infty\), then ω is a fixed point of f. Moreover, for each fixed point \(\omega'\) of f in X such that \(D(\omega',\omega')<\infty\), we have \(\omega=\omega'\).
Proof
Let \(n\in\mathbb{N}\) (\(n\geq1\)). For all \(i,j\in\mathbb{N}\), we have
and then
which gives
Consequently, we obtain
and
for all integer m such that \(m>n\).
Since \(\delta(D,f,x_{0})< \infty\) and \(2\lambda\in [ 0,1 [\), we obtain
It follows that \(\{ f^{n}x_{0}\} \) is a D-Cauchy sequence, and thus there exists \(\omega\in X \) such that
and
By (1) we have
Using (4), we obtain
Since \(C\lambda<1\) and \(D(\omega,f\omega)< \infty\), we deduce that \(D(\omega,f\omega)=0\), which implies that \(f\omega=\omega\).
If \(\omega' \) is any fixed point of f such that \(D(\omega',\omega ')<\infty\), we obtain
which implies that \(\omega'=\omega\). □
Example 3.3
Let \(X= [ 0,1 ] \), and let \(D:X\times X \rightarrow [ 0,\infty [ \) be the mapping defined by
Conditions \((D_{1}) \) and \((D_{2}) \) are trivially satisfied. By Proposition 2.7 we need to verify condition \((D_{3})\) only for elements x of X such that \(D(x,x)=0\), which implies that \(x=0\).
Let \((x_{n})\subset X \) be a sequence such that \(\lim_{n\to\infty} D(x_{n},0) =0\). For all \(n\in\mathbb{N}\) and \(y\in X\), we have:
Then
which implies that
It follows that \((X,D)\) is a generalized metric space that is not a standard metric space since the triangular inequality does not hold: If \(x,y\in X-\lbrace0\rbrace\), then we have \(D(x,y)=x+y \) and \(D(x,0)+D(0,y)=\frac{x+y}{2} \), and thus
Note that \((X,D) \) is D-complete.
Define the mapping T on X by
For any \(x\in X\), we have:
and
Then
For \(x,y\in X- \lbrace0 \rbrace\), we have
and
Then
The hypotheses of Theorem 3.2 are satisfied. Therefore T has a unique fixed point since D is bounded; note that \(T(0)=0\).
Example 3.4
Let \(X=\{a,b,c\}\) and define T on X by \(T(a)=a\), \(T(b)=b\), and \(T(c)=a\). There is no metric for which T is a Kannan contraction on X.
Define \(D: X\times X\longrightarrow[0,+\infty]\) by
Then \((X,D)\) is a complete generalized metric space, T is a Kannan contraction on \((X,D)\) for any \(\lambda\in{]}0,\frac{1}{2}[\), and we can apply Theorem 3.2.
Theorem 3.2 generalizes well-known results for b-metric and metric spaces.
Corollary 3.5
Let \((X,d) \) be a complete b-metric space with constant \(s\geq1 \), and let \(f:X\rightarrow X \) a mapping for which there exists \(\lambda \in [0 ,\frac{1}{s+1} [\) such that
for all \(x,y\in X\). Then f has a unique fixed point.
Proof
Let \(x_{0} \in X \). For all \(n\in\mathbb{N} \), we have
which implies
Let \(k=\frac{\lambda}{1-\lambda}\). By induction we obtain
and then
for all \(n,m\in\mathbb{N}\) such that \(m>n\).
This implies that \(\delta(D,f,x_{0})<\infty\). Then by Theorem 3.2 we conclude that f has a unique fixed point. □
Corollary 3.6
(Kannan fixed point theorem [2])
Let \((X,d) \) be a complete metric space, and let \(f:X\rightarrow X \) a mapping for which there exists \(\lambda\in [0 ,\frac{1}{2} [\) such that
for all \(x,y\in X\). Then f has a unique fixed point.
Corollary 3.7
Let \((X,d) \) be a dislocated metric space, and let \(f:X\rightarrow X \) be a mapping for which there exists \(\lambda\in [0 ,\frac{1}{2} [\) such that
for all \(x,y\in X \). Then f has a unique fixed point.
In the following, we need the basic lemma.
Lemma 3.8
([6])
Let λ is a real number such that \(0\leq\lambda<1\), and let \(\{ b_{n}\}\) be a sequence of positives reals numbers such that \(\lim_{n\to \infty} b_{n} =0\). Then, for any sequence of positives numbers \(\{ a_{n}\}\) satisfying
we have \(\lim_{n\to\infty} a_{n} =0\).
Theorem 3.9
Let \((X,D) \) be a D-complete generalized metric space, \(\lambda\in [0 ,\frac{1}{2} [\), and let f be a self-mapping on X such that
for all \(x,y\in X\). If there exists a point \(x_{0}\in X \) such that \(\delta(D,f,x_{0})<\infty\), then the sequence \(\{ f^{n}x_{0}\} \) converges to some \(\omega\in X \). Moreover, if \(D(x_{0},f\omega )<\infty\), then ω is a fixed point of f, and for any fixed point \(\omega'\) of f such that \(D(\omega,\omega^{\prime})<\infty\), we have \(\omega=\omega'\).
Proof
Let \(n\in\mathbb{N}\) (\(n\geq1\)). For all integers \(i,j\), we have
which implies that
Hence
and consequently
This inequality implies that
for all integers \(n,m\) such that \(m>n\). Since \(\delta(D,f,x_{0})< \infty\) and \(2\lambda\in [ 0,1 )\), we obtain
It follows that \(\{ f^{n}x_{0}\} \) is a D-Cauchy sequence, and thus there exists \(\omega\in X \) such that
By \((D_{3})\) we have
Then
By (7) we have
Since \(D(x_{0},f\omega)<\infty\), we have \(D(f^{n}x_{0},f\omega )<\infty\) for all \(n\in\mathbb{N}\). By Lemma 3.8 we obtain
It follows that \(f\omega=\omega\).
Let \(\omega'\) be any fixed point of X. We have
Since \(D(\omega,\omega^{\prime})<\infty\), we obtain \(D(\omega, \omega ')=0 \), which ends the proof. □
Example 3.10
Let \(X= [ 0,1 ] \), and let \(D:X\times X \rightarrow [ 0,\infty ] \) be defined by
It is easy to see that \((X,D) \) is a D-complete generalized metric space with \(C=1\).
Consider the function \(T:[0,1]\rightarrow[0,1] \) given by
The function T is a Chatterjea contraction with \(\lambda=\frac {1}{3} \) in \((X,D)\). By Theorem 3.9, T has a fixed point \(\omega\in X \).
Note that the mapping T has two different fixed points, so we cannot apply the classical fixed point theorems for Banach, Kannan, and Chatterjea contractions since they give the uniqueness of the fixed point.
Definition 3.11
Let \((X,D) \) be a generalized metric space. A self-mapping f on X is called a Hardy-Rogers contraction if there exists nonnegative real constants \(\lambda_{i}\) for \(i=1,2,3,4,5\) such that \(\lambda= \sum_{i=1}^{i=5} \lambda_{i} \in [0,1 [\) and
for all \(x,y\in X\).
Proposition 3.12
Let \((X,D) \) be a generalized metric space, and let \({f:X\longrightarrow X}\) be a Hardy-Rogers contraction. Then any fixed point \(\omega\in X \) of f satisfies
Proof
Let \(\omega\in X \) be a fixed point of f such that \(D(\omega ,\omega) <\infty\). We have
Since \(\lambda\in [ 0,1 [ \), we have \(D(\omega,\omega)=0\). □
To prove a fixed point result for Hardy-Rogers contraction mappings, we need the following lemma.
Lemma 3.13
([6])
Let \(( a_{n} ) \) be a sequence of nonnegative real numbers, and let \(( \lambda_{n} ) \) be a real sequence in \([ 0,1 ] \) such that
If, for a given \(\varepsilon>0 \), there exists a positive integer \(n_{0} \) such that
then \(0\leq\limsup_{n\rightarrow\infty} a_{n} \leq\varepsilon\).
Theorem 3.14
Let \((X,D) \) be a D-complete generalized metric space, and let f be a self-mapping on X satisfying (8).
Assume that \(C\lambda_{3}+\lambda_{5}<1 \) and that there exists a point \(x_{0}\in X \) such that \(\delta(D,f,x_{0})<\infty\). Then the sequence \(\{ f^{n}x_{0}\} \) converges to some \(\omega\in X \). If \(D(x_{0},f\omega)<\infty\), then ω is a fixed point of f. Moreover, if \(\omega' \in X \) is another fixed point of f such that \(D(\omega,\omega')<\infty\) and \(D(\omega',\omega')<\infty\), then \(\omega=\omega'\).
Proof
Let \(n\in\mathbb{N}\) (\(n\geq1\)). For all \(i,j\in\mathbb{N} \), we have
We obtain
and
which leads to
Since \(\delta(D,f,x_{0})< \infty\) and \(\lambda\in [ 0,1 )\), we obtain
It follows that \(\{ f^{n}x_{0}\} \) is a D-Cauchy sequence, and thus there exists \(\omega\in X \) such that
From (8) we have
Let
By (10) we obtain
Since \(\lim_{n\rightarrow\infty} b_{n}=0\), for every \(\epsilon>0 \) such that \(\epsilon>\frac{K}{1-\lambda_{5}}\), there exists \(N_{\epsilon}\) such that
Then
By Lemma 3.13 we have
Then
By \((D_{3})\) we obtain
and by (11) we have
Since \(C\lambda_{3}+\lambda_{5}<1 \), we have \(\frac{C\lambda _{3}}{1-\lambda_{5}} <1\). Then \(D(f\omega,\omega)=0\), which implies \(f\omega=\omega\).
If \(\omega'\) is any fixed point of f such that \(D(\omega,\omega ')<\infty\) and \(D(\omega',\omega')<\infty\), then (8) implies
Then we obtain
From Proposition 3.12 we have \(D(\omega,\omega)=D(\omega',\omega ')=0\), and then \(D(\omega,\omega')=0 \), which ends the proof. □
By the symmetry of the generalized metric D the theorem is true if either \({C\lambda_{3}+\lambda_{5}<1}\) or \(C\lambda_{2}+\lambda_{4}<1 \).
4 Fixed point results for T-contractions
Beiranvand, Moradi, Omid, and Pazandeh [7] introduced the notion of a T-contraction and established a version of the Banach contraction principle.
Let us introduce the following definitions.
Definition 4.1
Let \((X,D ) \) be a metric space, and let T be a self-mapping on X. We say that
-
(a)
T is continuous if
$$\lim_{n\rightarrow\infty}D(x_{n},x)=0\quad \Rightarrow\quad \lim _{n\rightarrow \infty}D\bigl(T(x_{n}),T(x)\bigr)=0 $$for all \(x\in X\);
-
(b)
T is sequentially convergent if for every sequence \(\{x_{n}\} \) such that \(\{T(x_{n})\} \) converges, \(\{x_{n}\} \) converges;
-
(c)
T is subsequentially convergent if for every sequence \(\{x_{n}\} \) such that \(\{T(x_{n})\} \) converges, \(\{x_{n}\} \) has a convergent subsequence.
Definition 4.2
Let \((X,D ) \) be a metric space, and let T and f be two self-mappings on X. We say that
-
(a)
f is a T-Banach contraction if there exists \(k\in {]}0,1 [ \) such that
$$D\bigl(Tf(x),Tf(y)\bigr)\leq kD(Tx,Ty) $$for all \(x,y \in X \);
-
(b)
f is a T-Kannan contraction if there exists \(k\in {]}0,\frac{1}{2} [ \) such that
$$D\bigl(Tf(x),Tf(y)\bigr)\leq k\bigl[D\bigl(Tx,Tf(x)\bigr)+D\bigl(Ty,Tf(y)\bigr) \bigr] $$for all \(x,y \in X \);
-
(c)
f is a T-Chatterjea contraction if there exists \(k\in {]}0,\frac{1}{2} [ \) such that
$$D\bigl(Tf(x),Tf(y)\bigr)\leq k\bigl[D\bigl(Tx,Tf(y)\bigr)+D\bigl(Ty,Tf(x)\bigr) \bigr] $$for all \(x,y \in X \);
-
(d)
f is a T-Hardy-Rogers contraction if
$$D(Tfx,Tfy)\leq\lambda_{1}D(Tx,Ty)+\lambda_{2}D(Tx,Tfx)+ \lambda _{3}D(Ty,Tfy)+\lambda_{4}D(Ty,Tfx)+ \lambda_{5}D(Tx,Tfy) $$for all \(x,y\in X \), where \(\lambda_{i}\), \(i=1,2,3,4,5 \), are nonnegative constants such that \(\lambda= \sum_{i=0}^{i=5} \lambda_{i} \in [0,1 [ \).
To show new results for T-contractions on a complete generalized metric space \((X,D)\), we consider the mapping \(D_{T}:X\times X\rightarrow [0,\infty]\) defined by
where T is continuous, sequentially convergent, and one-to-one.
Proposition 4.3
We have:
-
1.
For every sequence \({x_{n}} \) in X,
$$\lim_{n} D(x_{n},x)=0\quad \Longleftrightarrow\quad \lim _{n} D_{T}(x_{n},x)=0; $$ -
2.
\(D_{T} \) is a generalized metric on X;
-
3.
If \((X,D)\) is complete, then \((X,D_{T}) \) is complete.
Proof
1. Let \(\{x_{n}\} \) be a sequence such that
By continuity we have
Assume that \(\lim_{n \to\infty} D(Tx_{n},Tx)=0\). Since T is sequentially convergent, there exists \(u\in X\) such that
and by the continuity of T we have \(\lim_{n}D(Tx_{n},Tu)=0\). It follows that \(Tu=Tx\), and since T is one-to-one, we have \(u=x\).
2. Let \(x,y \in X \) be such that \(D_{T}(x,y)=0 \). Then \(D(Tx,Ty)=0\). Since T is one-to-one, we obtain \(x=y\) by \((D_{1})\).
The symmetry is obvious. Let now \(x,y \in X\), and let \(\{x_{n}\}\) be a sequence that converges to x in \((X,D_{T})\). Then \(\{Tx_{n}\}\) converges to Tx in \((X,D)\), and by \((D_{3})\) we have
which is equivalent to \(D_{T}(x,y)\le C\limsup_{n}D_{T}(x_{n},y)\). Hence \((X,D_{T})\) is a generalized metric space with the same constant C.
3. If \(\lim_{n,m \to\infty} D_{T}(x_{n},x_{m})=0\), then \(\lim_{n,m \to\infty} D(Tx_{n},Tx_{m})=0\). So \(\{Tx_{n}\} \) is a Cauchy sequence in \((X,D) \), which is complete. It follows that there exists \(u \in X \) such that
Since T is sequentially convergent, there exists \(x\in X\) such that \(\lim_{n}D(x_{n},x)=0\), which is equivalent to \(\lim_{n}D_{T}(x_{n},x)=0\). □
Remark 4.4
If a mapping f is a T-Banach (resp. T-Kannan, T-Chatterjea, T-Hardy-Rogers) contraction in \((X, D)\), then f is a Banach (resp. Kannan, Chatterjea, Hardy-Rogers) contraction in \((X, D_{T}) \) with the same constants.
For every \(x\in X\), we define
From Remark 4.4 and Theorem 3.3 in [1] we can deduce the following corollaries.
Corollary 4.5
Let \((X,D) \) be a complete metric space, and let \(T,f:X\rightarrow X \) be two mappings such that T is continuous, one-to-one, and sequentially convergent. Assume that f is a T-Banach contraction. If there exists \(x_{0}\in X \) such that \(\delta_{T} (D,f,x_{0})<\infty \), then \(\{f^{n}x_{0}\} \) converges to a fixed point ω of f. Moreover, if \(\omega' \) is any fixed point of f such that \(D(T\omega ,T\omega') < \infty\), then \(\omega= \omega'\).
Corollary 4.6
Let \((X,D) \) be a complete generalized metric space, and let \(T,f:X\rightarrow X \) be two mappings such that T is continuous, one-to-one, and sequentially convergent. Assume that f is a T-Kannan contraction with constant \(k>0 \) such that \(Ck<1\). If there exists \(x_{0}\in X \) such that \(\delta_{T} (D,f,x_{0})<\infty\), then \(\{ f^{n}x_{0}\} \) converges to some \(\omega\in X\). Moreover, if \(D(Tx_{0},Tf\omega)<\infty\), then ω is a fixed point of f, and for every fixed point \(\omega'\) of f such that \(D(T\omega,T\omega ')<\infty\), we have \(\omega=\omega'\).
Corollary 4.7
Let \((X,D) \) be a complete generalized metric space, and let \(T,f:X\rightarrow X \) be two mappings such that T is continuous, one-to-one, sequentially convergent. Assume that f is a T-Chatterjea contraction and that there exists \(x_{0}\in X \) such that \(\delta_{T} (D,f,x_{0})<\infty\). Then \(\{f^{n}x_{0}\} \) converges to some \(\omega\in X\), and if \(D(Tx_{0},Tf\omega)<\infty\), then ω is a fixed point of f. Moreover, if \(\omega' \in X \) is another fixed point of f such that \(D(T\omega,T\omega')<\infty\), then \(\omega=\omega'\).
Corollary 4.8
Let \((X,D) \) be a complete generalized metric space, and let \(T,f:X\rightarrow X \) be two mappings such that T is continuous, one-to-one, and sequentially convergent. Assume that f is a T-Hardy-Rogers contraction with nonnegative constants \(\lambda_{i} ,i=1,2,3,4,5 \), such that \(\lambda= \sum_{i=0}^{i=5} \lambda_{i} \in [0,1 [ \) and \(C\lambda_{3}+\lambda_{5}<1\). Assume that there exists \(x_{0}\in X \) such that \(\delta_{T} (D,f,x_{0})<\infty\). Then \(\{f^{n}x_{0}\} \) converges to some \(\omega\in X\). If \(D(Tx_{0},Tf\omega)<\infty\), then ω is a fixed point of f. Moreover, if \(\omega' \in X \) is another fixed point of f such that \(D(T\omega,T\omega')<\infty\) and \(D(T\omega^{\prime},T\omega')<\infty\), then \(\omega=\omega'\).
5 Some remarks on Senapati et al. results
In this section, we discuss some results by Senapati et al. [8] on extensions of Ciric and Wardowski-type fixed point theorems in generalized metric spaces. The authors introduced the following definition.
Definition 5.1
([8], Definition 3.3)
Let \((X,D)\) be a generalized metric space, and let T be a self-mapping on X. Then T is said to be a D-admissible mapping if for all \(x,y \in X\),
They proved the following lemma.
Lemma 5.2
([8], Lemma 3.4)
Let \((X,D)\) be a generalized metric space, and let T be a D-admissible mapping on X. Then, for every sequence \(\{x_{n}\}\) converging to a point a point \(w\in X\), we have \(D(w,Tw)<\infty\).
The proof of this lemma uses the following implication:
Example 5.3
We consider Example 2.3 in [8] and the mapping \(T: X\longrightarrow X\) defined by
For all \(x, y\in X\), we have \(Tx+Ty\le D(Tx,Ty)\le Tx+Ty+1\). Then \(D(Tx,Ty)<\infty\), which implies that T is admissible.
For \(x_{n}=\frac{1}{n}\), we have \(\lim_{n}D(x_{n},0)=0\) but \(\limsup_{n}D(Tx_{n},T0)=\infty\).
Example 5.4
Let \(X=[0,+\infty[\), and let D be defined by
Let T be the mapping defined by
Then T is admissible. If the sequence \((x_{n})_{n}\) is defined by \(x_{1}=1\) and \(x_{n+1}=Tx_{n}\) for all \(n\ge2\), then we have
The lemma was used to prove Theorems 3.5 and 3.7 of [8], and they deduced the following corollary:
Corollary 5.5
([8], Corollary 3.8)
Let \(T:X\longrightarrow X\) be a D-admissible self-mapping, and let \((X,D)\) be a complete D-generalized metric space. Suppose the following conditions hold:
-
(i)
for all \(x, y\in X\), there exists \(k\in]0,1[\) such that
$$D(Tx,Ty)\le k\max \biggl\{ D(x,y),D(x,Tx),D(y,Ty),\frac {D(x,Ty)+D(y,Tx)}{2} \biggr\} ; $$ -
(ii)
there exists \(x_{0}\in X\) such that \(\delta(D,T,x_{0})<\infty\).
Then \((T^{n}(x_{0}))_{n}\) 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 \(w=w'\).
The inequalities
give the following implications:
and
These implications show that in [8], Theorem 3.7 is a consequence of Corollary 3.8.
Finally, in the proof of Theorem 4.2 in [8], the authors use the F-contraction defined as follows.
Definition 5.6
([8], Definition 4.1)
A self-mapping T defined on X is said to be an F-contraction if, for all \(x, y\in X\),
for some \(\tau>0\).
We suspect that that existence of some \(x_{0}\in X\) such that \(\delta (D,T,x_{0})=c\) does not give \(D(T^{n+i}x_{0},T^{n+j}x_{0})>0\) for all integers n, i, j as the proofs of Theorem 4.2 and Corollary 4.3 of [8] blame.
6 Conclusion
In this paper, we gave a generalized version of Kannan, Chatterjea, and Hardy-Rogers contraction fixed point theorems and some fixed point results for T-contractions in a generalized metric space.
Our examples show that the results can be applied to prove the existence of fixed points in generalized metric spaces, whereas its classical counterpart fails to give positive answers.
References
Jleli, M, Samet, B: A generalized metric space and related fixed point theorems. Fixed Point Theory Appl. 2015, 33 (2015)
Kannan, R: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71-76 (1968)
Chatterjea, SK: Fixed points theorems. C. R. Acad. Bulgare Sci. 25, 727-730 (1972)
Kannan, R: Some results on fixed point theory II. Am. Math. Mon. 76, 405-408 (1969)
Hardy, GE, Rogers, TD: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16, 201-206 (1973)
Berinde, V: Iterative Approximation of Fixed Points. Lecture Notes in Mathematics 2nd Edition. Springer, Berlin (2007)
Beiranvand, A, Moradi, S, Omid, M, Pazandeh, H: Two fixed point theorems for special mappings (2009) arXiv:0903.1504v1
Senapati, T, Dey, LK, Dekic, DD: Extensions of Ciric and Wardowski type fixed point theorems in D-generalized metric spaces. Fixed Point Theory Appl. 2016, 33 (2016). doi:10.1186/s113663-016-0522-7
Acknowledgements
The authors are thankful to the editors and the anonymous referees for their valuable comments, which reasonably improved the presentation of the manuscript.
Author information
Authors and Affiliations
Contributions
Both authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
ElKouch, Y., Marhrani, E.M. On some fixed point theorems in generalized metric spaces. Fixed Point Theory Appl 2017, 23 (2017). https://doi.org/10.1186/s13663-017-0617-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13663-017-0617-9
MSC
- 47H10
- 54H25
Keywords
- fixed point theorem
- Kannan contraction
- Chatterjea contraction
- Hardy-Rogers contraction
- T-contraction