- Research
- Open Access
Fixed points of α-admissible Meir-Keeler contraction mappings on quasi-metric spaces
- Hamed H Alsulami^{1},
- Selma Gülyaz^{2} and
- İnci M Erhan^{3}Email author
https://doi.org/10.1186/s13660-015-0604-9
© Alsulami et al.; licensee Springer. 2015
- Received: 28 October 2014
- Accepted: 18 February 2015
- Published: 5 March 2015
Abstract
We introduce α-admissible Meir-Keller and generalized α-admissible Meir-Keller contractions on quasi-metric spaces and discuss the existence of fixed points of such contractions. We apply our results to G-metric spaces and express some fixed point theorems in G-metric spaces as consequences of the results in quasi-metric spaces.
Keywords
- quasi-metric
- Meir-Keeler contraction
- admissible mapping
MSC
- 47H10
- 54C60
- 54H25
- 55M20
1 Introduction and preliminaries
One of the generalizations of the metric spaces are the so-called quasi-metric spaces in which the commutativity condition does not hold in general. Recently, Jleli and Samet [1] obtained a relation between G-metric spaces introduced by Mustafa and Sims [2] and quasi-metric spaces. This work increased the interest to quasi-metric spaces (see [3, 4] for details).
In this paper, we investigate the existence of fixed points of Meir-Keeler type contractions defined on quasi-metric spaces and apply our results to G-metric spaces.
First, we recall the definition of a quasi-metric and quasi-metric space and some topological concepts on these spaces.
Definition 1
- (d1)
\(d(x,y)=0\) if and only if \(x=y\);
- (d2)
\(d(x,y)\leq d(x,z)+d(z,y)\).
Remark 2
Any metric space is a quasi-metric space, but the converse is not true in general.
Definition 3
- (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 left-Cauchy 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 right-Cauchy 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 quasi-metric space is Cauchy if and only if it is both left-Cauchy and right-Cauchy.
Definition 5
- (1)
\((X,d)\) is said to be left-complete if every left-Cauchy sequence in X is convergent.
- (2)
\((X,d)\) is said to be right-complete if every right-Cauchy 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 α-admissible 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
In the existence and uniqueness proofs of fixed points of α-admissible maps, an additional property is required. This property is given below.
Definition 7
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 α-admissible 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
In this paper we study α-admissible Meir-Keeler (or shortly α-Meir-Keeler) contractions which can be regarded as generalizations of the Meir-Keeler contractions defined in [9]. In fact, we insert α-admissibility into the definition of the original Meir-Keeler contraction.
Definition 9
Remark 10
We also generalize the α-Meir-Keeler contraction by using a more general expression in the contractive condition. Specifically, we define two types of generalized α-Meir-Keeler contraction, say type (I) and type (II) as follows.
Definition 11
Definition 12
Remark 13
2 Main results
Our first result is a fixed point theorem for generalized α-Meir-Keeler contractions of type (I) on quasi-metric spaces.
Theorem 15
Let \((X,d)\) be a complete quasi-metric space and \(T:X\to X\) be a continuous generalized α-Meir-Keeler 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
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.
Below we state an existence theorem for fixed point of generalized α-Meir-Keeler 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 quasi-metric space and \(T:X\to X\) be a continuous generalized α-Meir-Keeler 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.
- (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 quasi-metric space and \(T:X\to X\) be a generalized α-Meir-Keeler 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
We next consider a particular case of the main theorems in which the mapping is an α-Meir-Keeler contractive mapping, that is, it satisfies Definition 9.
Corollary 18
Proof
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 quasi-metric space satisfying the condition (A) and let \(T:X\to X\) be an α-Meir-Keeler 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 α-Meir-Keeler contraction defined on a quasi-metric space.
Example 20
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 \(\varepsilon\leq d(x,y)=x-y <\varepsilon+\delta\) implies \(\alpha(x,y)d(Tx,Ty)= \frac {x^{2}-y^{2}}{8}= \frac{(x-y)(x+y)}{8}< \frac {2(\varepsilon+\delta)}{8}=\varepsilon\).
If \(x< y \) then for \(\varepsilon>0\) we choose again \(\delta=3\varepsilon \) so that \(\varepsilon\leq d(x,y)=\frac{y-x}{2} <\varepsilon+\delta\) implies \(\alpha(x,y)d(Tx,Ty)= \frac {x^{2}/8-y^{2}/8}{2}= \frac{(x-y)(x+y)}{16}< \frac {2(\varepsilon+\delta)}{16}= \frac{\varepsilon}{2}\).
In other words, for every \(\varepsilon>0\), there exists δ which is actually \(\delta=3\varepsilon\). Therefore, the map T is an α-Meir-Keeler 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 \(x=0\).
3 Consequences: G-metric spaces
In this section, we present some results which show that several fixed point theorems on G-metric spaces are in fact direct consequences of the existence theorems given in the previous section.
First, we briefly recollect some basic notions of G-metric and G-metric space [2].
Definition 21
- (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 G-metric on X and the pair \(( X,G ) \) is called a G-metric space.
Definition 22
(see [2])
- (1)A point \(x\in X\) is said to be the limit of the sequence \(\{x_{n}\}\) ifand the sequence \(\{x_{n}\}\) is said to be G-convergent to x.$$ \lim_{n,m\rightarrow\infty}G(x,x_{n},x_{m})=0 $$
- (2)
A sequence \(\{x_{n}\}\) is called a G-Cauchy 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 G-complete (or a complete G-metric space) if every G-Cauchy sequence in \((X,G)\) is G-convergent in X.
Theorem 23
[1]
- (1)
\((X,d)\) is a quasi-metric space;
- (2)
\(\{x_{n}\}\subset X\) is G-convergent to \(x\in X\) if and only if \(\{x_{n}\}\) is convergent to x in \((X,d)\);
- (3)
\(\{x_{n}\}\subset X\) is G-Cauchy if and only if \(\{ x_{n}\}\) is Cauchy in \((X,d)\);
- (4)
\((X,G)\) is G-complete if and only if \((X,d)\) is complete.
Admissible mappings in the context of G-metric spaces can be defined as follows [10].
Definition 24
Definition 25
For more details on β-admissible maps on G-metric spaces we refer the reader to [10].
Definition 26
Lemma 27
Let \(T:X \rightarrow X\) where X is nonempty set. Then T is β-admissible on \((X, G)\) if and only if T is α-admissible 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 G-metric space and \(T:X\to X\) be a continuous, β-Meir-Keeler 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
Our last theorem is a consequence of Theorems 15 and 16.
Theorem 29
Let \((X,G)\) be a complete G-metric space and \(T:X\to X\) be a continuous, generalized β-Meir-Keeler 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
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions, which helped in the improvement of the paper.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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