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Fixed points of αadmissible MeirKeeler contraction mappings on quasimetric spaces
Journal of Inequalities and Applicationsvolume 2015, Article number: 84 (2015)
Abstract
We introduce αadmissible MeirKeller and generalized αadmissible MeirKeller contractions on quasimetric spaces and discuss the existence of fixed points of such contractions. We apply our results to Gmetric spaces and express some fixed point theorems in Gmetric spaces as consequences of the results in quasimetric spaces.
Introduction and preliminaries
One of the generalizations of the metric spaces are the socalled quasimetric spaces in which the commutativity condition does not hold in general. Recently, Jleli and Samet [1] obtained a relation between Gmetric spaces introduced by Mustafa and Sims [2] and quasimetric spaces. This work increased the interest to quasimetric spaces (see [3, 4] for details).
In this paper, we investigate the existence of fixed points of MeirKeeler type contractions defined on quasimetric spaces and apply our results to Gmetric spaces.
First, we recall the definition of a quasimetric and quasimetric space and some topological concepts on these spaces.
Definition 1
Let X be a nonempty set and \(d:X\times X\rightarrow[0,\infty)\) be a function which satisfies:

(d1)
\(d(x,y)=0\) if and only if \(x=y\);

(d2)
\(d(x,y)\leq d(x,z)+d(z,y)\).
Then d called a quasimetric and the pair \((X,d)\) is called a quasimetric space.
Remark 2
Any metric space is a quasimetric space, but the converse is not true in general.
Definition 3
Let \((X,d)\) be a quasimetric space.

(1)
A sequence \(\{x_{n}\}\) in X is said to be convergent to x if \(\lim_{n\rightarrow\infty}d(x_{n},x)= \lim_{n\rightarrow\infty}d(x,x_{n})=0\).

(2)
A sequence \(\{x_{n}\}\) in X is called leftCauchy if for every \(\varepsilon>0\) there exists a positive integer \(N=N(\varepsilon)\) such that \(d(x_{n},x_{m})<\varepsilon\) for all \(n\geq m >N \).

(3)
A sequence \(\{x_{n}\}\) in X is called rightCauchy if for every \(\varepsilon>0\) there exists a positive integer \(N=N(\varepsilon)\) such that \(d(x_{n},x_{m})<\varepsilon\) for all \(m\geq n >N \).

(4)
A sequence \(\{x_{n}\}\) in X is called Cauchy sequence if for every \(\varepsilon>0\) there exists a positive integer \(N=N(\varepsilon)\) such that \(d(x_{n},x_{m})<\varepsilon\) for all \(m, n >N \).
Remark 4
From Definition 3 it is obvious that a sequence \(\{x_{n}\}\) in a quasimetric space is Cauchy if and only if it is both leftCauchy and rightCauchy.
Definition 5
Let \((X,d)\) be a quasimetric space. Then:

(1)
\((X,d)\) is said to be leftcomplete if every leftCauchy sequence in X is convergent.

(2)
\((X,d)\) is said to be rightcomplete if every rightCauchy sequence in X is convergent.

(3)
\((X,d)\) is said to be complete if every Cauchy sequence in X is convergent.
In the sequel, we shall denote by ℕ the set of nonnegative integers, that is, \(\mathbb{N}=\{0,1,2,\ldots\}\). We next define the concept of αadmissible mappings which have been recently introduced by Samet [5] and used by many authors to generalize contraction mappings of various types; see [6–8] for details.
Definition 6
A mapping \(T:X\to X\) is called αadmissible if for all \(x,y\in X\) we have
where \(\alpha:X\times X\to[0,\infty)\) is a given function.
In the existence and uniqueness proofs of fixed points of αadmissible maps, an additional property is required. This property is given below.
Definition 7
A mapping \(T:X\to X\) is called triangular αadmissible if it is αadmissible and satisfies
where \(x,y,z\in X\) and \(\alpha:X\times X\to[0,\infty)\) is a given function.
The following auxiliary result is going to be used in the proof of existence theorems.
Lemma 8
[7]
Let \(T:X\to X\) be a triangular αadmissible mapping. Assume that there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\). If \(x_{n}=T^{n}x_{0}\), then \(\alpha(x_{m},x_{n})\geq1\) for all \(m,n\in\mathbb{N}\).
Proof
Let \(x_{0}\in X\) satisfies \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha (Tx_{0},x_{0})\geq1\). Define the sequence \(\{x_{n}\}\) in X as \(x_{n+1}=Tx_{n}\) for \(n\in\mathbb{N} \). Since T is αadmissible, we have
for all \(n=0,1,\ldots\) . On the other hand, since T is triangular αadmissible, we get
and
Similarly,
and
Continuing in this way, we obtain, for all \(m,n \in\mathbb{N}\),
□
In this paper we study αadmissible MeirKeeler (or shortly αMeirKeeler) contractions which can be regarded as generalizations of the MeirKeeler contractions defined in [9]. In fact, we insert αadmissibility into the definition of the original MeirKeeler contraction.
Definition 9
Let \((X,d)\) be a quasimetric space. Let \(T:X\rightarrow X \) be a triangular αadmissible mapping. Suppose that for every \(\varepsilon> 0\) there exists \(\delta>0 \) such that
for all \(x, y \in X \). Then T is called αMeirKeeler contraction.
Remark 10
Let T be an αMeirKeeler contractive mapping. Then
for all \(x,y\in X\) when \(x\neq y\). Also, if \(x=y\) then \(d(Tx,Ty)=0\), i.e.,
for all \(x,y\in X\).
We also generalize the αMeirKeeler contraction by using a more general expression in the contractive condition. Specifically, we define two types of generalized αMeirKeeler contraction, say type (I) and type (II) as follows.
Definition 11
Let \((X,d)\) be a quasimetric space. Let \(T:X\rightarrow X \) be a triangular αadmissible mapping. Assume that for every \(\varepsilon> 0\) there exists \(\delta>0 \) such that
where
for all \(x, y \in X \). Then T is called a generalized αMeirKeeler contraction of type (I).
Definition 12
Let \((X,d)\) be a quasimetric space. Let \(T:X\rightarrow X \) be a triangular αadmissible mapping. Assume that for every \(\varepsilon> 0\) there exists \(\delta>0 \) such that
where
for all \(x, y \in X \). Then T is called a generalized αMeirKeeler contraction of type (II).
Remark 13
Let \(T:X\to X\) be a generalized αMeirKeeler contraction of type (I) (respectively, type (II)). Then
for all \(x,y\in X\) when \(M(x,y)>0\) (respectively, \(N(x,y)>0\)). Also, if \(M(x,y)=0\) (respectively, \(N(x,y)=0\)), then \(x=y\), which implies \(d(x,y)=0\), i.e.,
for all \(x,y\in X\).
Remark 14
It is obvious that \(N(x,y)\leq M(x,y)\) for all \(x,y\in X\), where \(M(x,y)\) and \(N(x,y)\) are defined in (1.5) and (1.7), respectively.
Main results
Our first result is a fixed point theorem for generalized αMeirKeeler contractions of type (I) on quasimetric spaces.
Theorem 15
Let \((X,d)\) be a complete quasimetric space and \(T:X\to X\) be a continuous generalized αMeirKeeler contraction of type (I). If \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\) for some \(x_{0}\in X\), then T has a fixed point in X.
Proof
Let \(x_{0}\in X\) satisfy \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha (Tx_{0},x_{0})\geq1\). Define the sequence \(\{x_{n}\}\) in X as
Notice that if \(x_{n_{0}}=x_{n_{0}+1}\) for some \(n_{0}>0\), then \(x_{n_{0}}\) is a fixed point of T and the proof is done. Assume that \(x_{n}\neq x_{n+1}\) for all \(n\geq0\). Since T is αadmissible,
and continuing we obtain
Upon substituting \(x=x_{n}\) and \(y=x_{n+1}\) in (1.4) we find that for every \(\varepsilon>0\) there exists \(\delta>0\) such that
where
In what follows, we examine three cases.
Case 1. Assume that \(M(x_{n},x_{n+1})=d(x_{n},x_{n+1})\). Then (2.3) becomes
Therefore, we deduce that
for all n. That is, \(\{d(x_{n},x_{n+1})\}\) is a decreasing positive sequence in \(\mathbb{R_{+}}\) and it converges to some \(r\geq0\). To show that \(r=0\), we assume the contrary, that is, \(r>0\). Then we must have
Since the condition (2.3) holds for every \(\varepsilon>0\), we may choose \(\varepsilon=r\). For this ε, there exists \(\delta (\varepsilon)>0\) satisfying (2.3). In other words,
However, this implies
which contradicts (2.4). Thus, \(r=0\), that is,
Case 2. Assume that \(M(x_{n},x_{n+1})=d(x_{n+1},x_{n})\). In this case (2.3) becomes
from which it follows that
Therefore, we obtain
for all \(n\in\mathbb{N}\). Note that by Remark 13, since
we get
where
Then (2.8) becomes
for all \(n\in\mathbb{N}\).
Clearly, the case \(\max \{d(x_{n},x_{n1}),d(x_{n+1},x_{n}) \} =d(x_{n+1},x_{n})\) is not possible. Indeed, in this case we would get
Therefore, we should have \(\max \{ d(x_{n},x_{n1}),d(x_{n+1},x_{n}) \}=d(x_{n},x_{n1})\), which implies
for all \(n\in\mathbb{N}\). That is, the sequence \(\{ d(x_{n+1},x_{n})\} \) is decreasing and positive sequence and hence it converges to \(L\geq0\). In fact, the limit L of this sequence is 0, which can be shown by mimicking the proof of (2.6) done above. In other words, we get
Finally, taking the limit as \(n\to\infty\) in (2.7) and using (2.11) we obtain
Case 3. Assume that \(M(x_{n},x_{n+1})=d(x_{n+2},x_{n+1})\). In this case (2.3) becomes
or
Therefore, we deduce
for all \(n\in\mathbb{N}\). By Remark 13, we have
where
is clearly positive. Then (2.14) becomes
for all \(n\in\mathbb{N}\).
The case \(\max \{d(x_{n+1},x_{n}),d(x_{n+2},x_{n+1}) \} =d(x_{n+2},x_{n+1})\) is impossible, since it yields
The other case, that is, \(\max \{ d(x_{n+1},x_{n}),d(x_{n+2},x_{n+1}) \}=d(x_{n+1},x_{n})\) implies
for all \(n\in\mathbb{N}\). As in Case 2, the sequence \(\{ d(x_{n+1},x_{n})\}\) is decreasing and positive sequence and hence it converges to \(L=0\). Finally, taking the limit as \(n\to\infty\) in (2.13) we end up with
As a result, we see that in all three cases, the sequence \(\{d_{n}\}\) defined by \(d_{n}:=d(x_{n},x_{n+1})\) converges to 0 as \(n\to\infty\). Using similar arguments, it can be shown that the sequence \(\{f_{n}\}\) where \(f_{n}:=d(x_{n+1},x_{n})\) also converges to 0. We first note that
and continuing in this way, we obtain
Substituting \(x=x_{n+1}\) and \(y=x_{n}\) in (1.4) we find that for every \(\varepsilon>0\) there exists \(\delta>0\) such that
where
We need to examine two cases.
Case 1. Assume that \(M(x_{n+1},x_{n})=d(x_{n+1},x_{n})\). Then (2.20) becomes
Then we have
for all n. That is, \(\{d(x_{n+1},x_{n})\}\) is a decreasing positive sequence in \(\mathbb{R_{+}}\) and it converges to \(L\geq0\). As above, it can be shown that \(L=0\).
Case 2. Assume that \(M(x_{n+1},x_{n})=d(x_{n+2},x_{n+1})\). In this case (2.20) becomes
or
which results in
for all \(n\in\mathbb{N}\) and is not possible.
Thus, we have only one possibility, \(M(x_{n+1},x_{n})=d(x_{n+1},x_{n})\), which leads to the fact that the sequence \(\{f_{n}\}=\{d(x_{n+1},n_{n})\}\) converges to 0.
We next show that the sequence \(\{x_{n}\}\) is both right and left Cauchy. First, we show that \(\{x_{n}\}\) is a rightCauchy sequence in \((X,d)\). We will prove that for every \(\varepsilon>0\) there exists \(N\in\mathbb{N}\) such that
for all \(l\geq N\) and \(k\in\mathbb{N}\). Since the sequences \(\{d_{n}\}\) and \(\{f_{n}\}\) both converge to 0 as \(n\to\infty\), for every \(\delta>0\) there exist \(N_{1},N_{2} \in\mathbb {N}\) such that
Choose δ such as \(\delta<\varepsilon\). We will prove (2.22) by using induction on k. For \(k=1\), (2.22) becomes
and clearly holds for all \(l\geq N=\max\{N_{1},N_{2}\}\) due to (2.23) and the choice of δ. Assume that the inequality (2.22) holds for some \(k=m\), that is,
For \(k=m+1\) we have to show that \(d(x_{l},x_{l+m+1})< \varepsilon\) for all \(l\geq N\). From the triangle inequality, we have
for all \(l\geq N\). If \(d (x_{l1},x_{l+m})\geq\varepsilon\), then for
we have
and because of Lemma 8, the contractive condition (1.4) with \(x=x_{l1}\) and \(y=x_{l+m}\) yields
and hence (2.22) holds for \(k=m+1\).
If \(d (x_{l1},x_{l+m})< \varepsilon\), then
Regarding Remark 13, we deduce
that is, inequality (2.22) holds for \(k=m+1\). Hence \(d(x_{l},x_{l+k})<\varepsilon\) for all \(l\geq N\) and \(k\geq1\), which means
Consequently, \(\{x_{n}\}\) is a rightCauchy sequence in \((X, d)\). Due to the similarity, the proof that \(\{x_{n}\}\) is a leftCauchy sequence in \((X, d)\) is omitted. By Remark 4, we deduce that \(\{x_{n}\}\) is a Cauchy sequence in complete quasimetric space \((X, d)\). Therefore, there exists \(z\in X\) such that
Employing the property (d1) and the continuity of T we get
and
Combining (2.29) and (2.30), we deduce
From (2.28) and (2.31), due to the uniqueness of the limit, we conclude that \(z=Tz\), that is, z is a fixed point of T. □
Below we state an existence theorem for fixed point of generalized αMeirKeeler contraction of type (II). Taking Remark 14 into account, we observe that the proof of this theorem is similar to the proof of Theorem 15.
Theorem 16
Let \((X,d)\) be a complete quasimetric space and \(T:X\to X\) be a continuous generalized αMeirKeeler contraction of type (II). If \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\) for some \(x_{0}\in X\), then T has a fixed point in X.
One advantage of αadmissibility is that the continuity of the contraction is not required whenever the following condition is satisfied.

(A)
If \(\{x_{n}\}\) is a sequence in X which converges to x and satisfies \(\alpha(x_{n+1},x_{n})\geq1\) and \(\alpha(x_{n},x_{n+1})\geq 1\) for all n then there exists a subsequence \(\{x_{n(k)}\}\) of \(\{ x_{n}\}\) such that \(\alpha(x,x_{n(k)})\geq1\) and \(\alpha (x_{n(k)},x)\geq1\) for all k.
Replacing the continuity of the contraction in Theorem 16 by the condition (A) on the space \((X,d)\) we deduce another existence theorem.
Theorem 17
Let \((X,d)\) be a complete quasimetric space and \(T:X\to X\) be a generalized αMeirKeeler contraction of type (II) and let \((X,d)\) satisfies the condition (A). If \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\) for some \(x_{0}\in X\), then T has a fixed point in X.
Proof
Following the lines of the proof of Theorem 15, we know that the sequence \(\{x_{n}\}\) defined by \(x_{n+1} = Tx_{n}\), where \(x_{0}\in X\) satisfies \(\alpha(x_{0},Tx_{0})\geq 1\) and \(\alpha(Tx_{0},x_{0})\geq1\), converges to some \(z\in X\). From (2.2) and condition (A), there exists a subsequence \(\{ x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(z, x_{n(k)})\geq1\) and \(\alpha(x_{n(k)},z)\geq1\) for all k. Regarding Remark 13 we have for all \(k\in\mathbb{N}\)
where
Letting \(k \rightarrow\infty\) in (2.33) we obtain
Thus, upon taking the limit in (2.32) as \(k\to\infty\), we conclude
The first inequality implies \(d(Tz,z)=0\), and hence \(Tz=z\), which completes the proof. □
We next consider a particular case of the main theorems in which the mapping is an αMeirKeeler contractive mapping, that is, it satisfies Definition 9.
Corollary 18
Let \((X,d)\) be a complete quasimetric space and \(T:X\to X\) be a continuous, αMeirKeeler contraction, that is, for every \(\varepsilon>0\) there exists \(\delta>0\) such that
holds for all \(x,y\in X\). If \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha (Tx_{0},x_{0})\geq1\) for some \(x_{0}\in X\), then T has a fixed point in X.
Proof
It is obvious that if (2.35) holds, then using the fact that
we conclude
for all \(x,y\in X\). In other words, T satisfies the conditions in the statement of Theorem 15 and hence has a fixed point. □
Finally, we replace the continuity of the contraction in Corollary 18 by the condition (A) on the space \((X,d)\), which results in the following existence theorem the proof of which is identical to the proof of Theorem 17.
Corollary 19
Let \((X,d)\) be a complete quasimetric space satisfying the condition (A) and let \(T:X\to X\) be an αMeirKeeler contractive mapping. If there exists \(x_{0} \in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\), then T has a fixed point.
We end this section with an example of an αMeirKeeler contraction defined on a quasimetric space.
Example 20
Let \(X=[0,\infty)\). Define
The function \(d(x,y)\) is a quasimetric but not a metric on X. Indeed, note that \(d(1,3)=1\neq d(3,1)=2\). The space \((X,d)\) is a complete quasimetric space. Define the mappings \(T: X\to X\) and \(\alpha: X\times X\to[0,\infty)\) as follows:
It is easy to see that T is triangular αadmissible. Note that if \(\alpha(x,y)\geq1\), then \(x,y\in [0,1]\) and hence both Tx and Ty are also in \([0,1]\). Thus, \(\alpha(Tx,Ty)=\alpha (\frac{x^{2}}{8},\frac{y^{2}}{8})=1\geq1\). Also, if \(\alpha(x,z)\geq1\) and \(\alpha(z,y)\geq1\), then \(x,y,z\in[0,1]\) and thus, \(\alpha(x,y)=1\geq1\). The map T is not continuous, however, the condition (A) holds on X. More precisely, if the sequence \(\{x_{n}\}\subset X\) satisfies \(\alpha(x_{n},x_{n+1})\geq1\) and \(\alpha(x_{n+1},x_{n})\geq1\), and if \(\lim_{n\to\infty}x_{n}=x\), then \(\{x_{n}\}\subset[0,1]\), and hence \(x\in[0,1]\). Then \(\alpha(x_{n},x)\geq1\).
Note that for \(x,y\in[0,1]\), we have \(x+y\leq2\).
If \(x\geq y \) then for \(\varepsilon>0\) we choose \(\delta=3\varepsilon\) so that \(\varepsilon\leq d(x,y)=xy <\varepsilon+\delta\) implies \(\alpha(x,y)d(Tx,Ty)= \frac {x^{2}y^{2}}{8}= \frac{(xy)(x+y)}{8}< \frac {2(\varepsilon+\delta)}{8}=\varepsilon\).
If \(x< y \) then for \(\varepsilon>0\) we choose again \(\delta=3\varepsilon \) so that \(\varepsilon\leq d(x,y)=\frac{yx}{2} <\varepsilon+\delta\) implies \(\alpha(x,y)d(Tx,Ty)= \frac {x^{2}/8y^{2}/8}{2}= \frac{(xy)(x+y)}{16}< \frac {2(\varepsilon+\delta)}{16}= \frac{\varepsilon}{2}\).
In other words, for every \(\varepsilon>0\), there exists δ which is actually \(\delta=3\varepsilon\). Therefore, the map T is an αMeirKeeler contraction. Finally, note that \(\alpha(0,T0)\geq1\) and \(\alpha(T0,0)\geq1\). All conditions of Corollary 19 are satisfied and T has a fixed point \(x=0\).
Consequences: Gmetric spaces
In this section, we present some results which show that several fixed point theorems on Gmetric spaces are in fact direct consequences of the existence theorems given in the previous section.
First, we briefly recollect some basic notions of Gmetric and Gmetric space [2].
Definition 21
Let X be a nonempty set, \(G:X\times X\times X\rightarrow{}[ 0,\infty )\) be a function satisfying the following conditions:

(G1)
\(G(x,y,z)=0\) if \(x=y=z\),

(G2)
\(0< G(x,x,y)\) for all \(x,y\in X\) with \(x\neq y\),

(G3)
\(G(x,x,y)\leq G(x,y,z)\) for all \(x,y,z\in X\) with \(z\neq y\),

(G4)
\(G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots\) (symmetry in all variables),

(G5)
\(G(x,y,z)\leq G(x,a,a)+G(a,y,z)\) for all \(x,y,z,a\in X\) (rectangle inequality).
Then the function G is called a Gmetric on X and the pair \(( X,G ) \) is called a Gmetric space.
It is obvious that for every Gmetric on the set X, the expression
is a standard metric on X.
Definition 22
(see [2])
Let \((X,G)\) be a Gmetric space and let \(\{x_{n}\}\) be a sequence in X.

(1)
A point \(x\in X\) is said to be the limit of the sequence \(\{x_{n}\}\) if
$$ \lim_{n,m\rightarrow\infty}G(x,x_{n},x_{m})=0 $$and the sequence \(\{x_{n}\}\) is said to be Gconvergent to x.

(2)
A sequence \(\{x_{n}\}\) is called a GCauchy sequence if for every \(\varepsilon>0\), there is a positive integer N such that \(G(x_{n},x_{m},x_{l})<\varepsilon\) for all \(n,m,l\geq\) N; that is, if \(G(x_{n},x_{m},x_{l})\rightarrow0\) as \(n,m,l\rightarrow\infty\).

(3)
\((X,G)\) is said to be Gcomplete (or a complete Gmetric space) if every GCauchy sequence in \((X,G)\) is Gconvergent in X.
Theorem 23
[1]
Let \((X,G)\) be a Gmetric space. Let \(d:X\times X \rightarrow [0,\infty)\) be the function defined by \(d(x,y)=G(x,y,y)\). Then

(1)
\((X,d)\) is a quasimetric space;

(2)
\(\{x_{n}\}\subset X\) is Gconvergent to \(x\in X\) if and only if \(\{x_{n}\}\) is convergent to x in \((X,d)\);

(3)
\(\{x_{n}\}\subset X\) is GCauchy if and only if \(\{ x_{n}\}\) is Cauchy in \((X,d)\);

(4)
\((X,G)\) is Gcomplete if and only if \((X,d)\) is complete.
Admissible mappings in the context of Gmetric spaces can be defined as follows [10].
Definition 24
A mapping \(T:X\to X\) is called βadmissible if for all \(x,y\in X\) we have
where \(\beta:X\times X\times X\to[0,\infty)\) is a given function. If in addition,
for all \(x,y,z\in X\), then T is called triangular βadmissible.
Definition 25
Let \((X,G)\) be a Gmetric space. Let \(T:X\rightarrow X \) be a triangular βadmissible mapping. Suppose that for every \(\varepsilon> 0\) there exists \(\delta>0 \) such that
for all \(x, y \in X \). Then T is called βMeirKeeler contraction.
For more details on βadmissible maps on Gmetric spaces we refer the reader to [10].
Definition 26
(I) Let \((X,G)\) be a Gmetric space. Let \(T:X\rightarrow X \) be a triangular βadmissible mapping. Suppose that for every \(\varepsilon> 0\) there exists \(\delta>0 \) such that
for all \(x, y \in X \), where
Then T is called a generalized βMeirKeeler contraction of type (I).
(II) Let \((X,G)\) be a Gmetric space. Let \(T:X\rightarrow X \) be a triangular βadmissible mapping. Suppose that for every \(\varepsilon> 0\) there exists \(\delta>0 \) such that
for all \(x, y \in X \), where
Then T is called a generalized βMeirKeeler contraction of type (II).
Lemma 27
Let \(T:X \rightarrow X\) where X is nonempty set. Then T is βadmissible on \((X, G)\) if and only if T is αadmissible on \((X,d)\).
Proof
The proof is obvious by taking \(\alpha(x,y)=\beta(x,y,y)\). □
The next theorem is a consequence of Corollary 18.
Theorem 28
Let \((X,G)\) be a complete Gmetric space and \(T:X\to X\) be a continuous, βMeirKeeler contraction. If \(\beta(x_{0},Tx_{0}, Tx_{0})\geq1\) and \(\beta(Tx_{0},Tx_{0},x_{0})\geq1\) for some \(x_{0}\in X\), then T has a fixed point in X.
Proof
Consider the quasimetric \(d(x,y)=G(x,y,y)\) for all \(x,y\in X\). Due to Lemma 27 and (3.3), we find that for every \(\varepsilon> 0\) there exists \(\delta>0 \) such that
for all \(x, y \in X \). Then the proof follows from Corollary 18. □
Our last theorem is a consequence of Theorems 15 and 16.
Theorem 29
Let \((X,G)\) be a complete Gmetric space and \(T:X\to X\) be a continuous, generalized βMeirKeeler contraction of type (I) or (II). If \(\beta(x_{0},Tx_{0}, Tx_{0})\geq1\) and \(\alpha(Tx_{0},Tx_{0},x_{0})\geq1\) for some \(x_{0}\in X\), then T has a fixed point in X.
Proof
Since the function \(d(x,y)=G(x,y,y)\) is a quasimetric on X, employing Lemma 27 and (3.4) (respectively (3.6)), we see that for every \(\varepsilon> 0\) there exists \(\delta>0 \) such that
for all \(x, y \in X \) for the generalized αMeirKeeler mappings of types (I) or (II), respectively. Then the proof follows from Theorem 15 (respectively, Theorem 16). □
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MSC
 47H10
 54C60
 54H25
 55M20
Keywords
 quasimetric
 MeirKeeler contraction
 admissible mapping