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 Open Access
Fixed points of generalized MeirKeeler contraction mappings in bmetriclike spaces
 Nayereh Gholamian^{1} and
 Mahnaz Khanehgir^{1}Email author
https://doi.org/10.1186/s1366301605076
© Gholamian and Khanehgir 2016
 Received: 17 November 2015
 Accepted: 27 February 2016
 Published: 15 March 2016
Abstract
In this paper, we introduce the notion of generalized MeirKeeler contraction mappings in the setup of bmetriclike spaces. Then we establish some fixed point results for this class of contractions. We also provide some examples to verify the effectiveness and applicability of our main results.
Keywords
 αadmissible
 fixed point
 MeirKeeler contraction
 bmetriclike
MSC
 47H10
 54H25
1 Introduction and preliminaries
In 1969, Meir and Keeler [1] proved the following very attractive fixed point theorem, which is a generalization of the Banach contraction principle [2].
Definition 1.1
[1]
Theorem 1.2
[1]
Let \((X,d)\) be a complete metric space. If \(T:X\rightarrow X \) is a MeirKeeler contraction, then T has a unique fixed point.
Alsulami et al. [3] defined two types of generalized αadmissible [4] MeirKeeler contractions and proved some fixed point theorems for these kinds of mappings. MeirKeeler contraction has many generalizations in the area studied by some scholars (cf. e.g. [5–8]).
On the other hand, AminiHarandi [9] presented a new extension of the concept of the partial metric space [10], called a metriclike space. The concept of bmetriclike space which generalizes the notions of partial metric space, metriclike space and bmetric space [11] was introduced by Alghamdi et al. in [12]. They established some fixed point theorems in partial metric spaces, bmetric spaces and bmetriclike spaces. It is well known that all these spaces are generalization of the usual metric spaces. There are several types of generalized metric spaces [13, 14], introduced by modifying and improving metric axioms. These generalized metric spaces often appear to be metrizable and the contraction conditions may be preserved under special transforms. Hence the fixed point theory in such spaces may be a consequence of the fixed point theory in certain metric spaces. However, it is not true that all generalized fixed point results become evident in this way. More precisely, these results are based on some contractive conditions, and some of these conditions do not remain authentic when one considers the problem in the associated metric space [15].
In the present work, using the concepts of MeirKeeler contractions and bmetriclike spaces, we define a new concept of generalized MeirKeeler contraction on a bmetriclike space. Then we investigate some fixed point results for this class of contractions. We give an example that shows that our results in bmetriclike spaces may not be deduced from certain ones in bmetric spaces. We also provide some examples to support the usability of our results.
It is convenient and, more importantly helpful to recall some basic definitions and facts which will be used further on. Throughout this paper, we denote by \(\mathbb{R}^{+}\) the set of nonnegative real numbers.
Definition 1.3
[11]
 (b1):

\(d(x,y) = 0\) if and only if \(x = y\),
 (b2):

\(d(x,y) = d(y,x)\) for all \(x,y\in X\),
 (b3):

there exists a real number \(s \geq1\) such that \(d(x,z) \leq s[d(x,y) + d(y,z)]\) for all \(x,y,z\in X\).
Definition 1.4
[16]
 (p_{ b }1):

\(x=y\) if and only if \(p_{b}(x,x)=p_{b}(x,y)=p_{b}(y,y)\),
 (p_{ b }2):

\(p_{b}(x,x)\leq p_{b}(x,y)\),
 (p_{ b }3):

\(p_{b}(x,y)=p_{b}(y,x)\),
 (p_{ b }4):

\(p_{b}(x,y)\leq s[p_{b}(x,z)+p_{b}(z,y)]p_{b}(z,z)\).
A pair \((X,p_{b})\) is called a partial bmetric space, if X is a nonempty set and \(p_{b}\) is a partial bmetric on X. The number s is called the coefficient of \((X,p_{b})\).
It is clear that if in Definitions 1.3 and 1.4 \(s=1\), then they are the usual metric and partial metric space, respectively.
Definition 1.5
[9]
 (σ1):

\(\sigma(x,y)=0\) implies \(x=y\),
 (σ2):

\(\sigma(x,y)=\sigma(y,x)\),
 (σ3):

\(\sigma(x,y)\leq\sigma(x,z)+\sigma(z,y)\).
The pair \((X,\sigma)\) is called a metriclike space.
Example 1.6
[17]
Let \(X=[0,1]\). Then the mapping \(\sigma_{1}:X\times X\rightarrow\mathbb{R}^{+}\) defined by \(\sigma_{1}(x,y)=x+yxy\) is a metriclike on X.
Example 1.7
[17]
Definition 1.8
[12]
 (\(\sigma_{b}1\)):

\(\sigma_{b}(x,y)=0\) implies \(x=y\),
 (\(\sigma_{b}2\)):

\(\sigma_{b}(x,y)=\sigma_{b}(y,x)\),
 (\(\sigma_{b}3\)):

\(\sigma_{b}(x,y)\leq s[\sigma_{b}(x,z)+\sigma_{b}(z,y)]\).
A bmetriclike space is a pair \((X,\sigma_{b})\) such that X is a nonempty set and \(\sigma_{b}\) is a bmetriclike on X. The number s is called the coefficient of \((X,\sigma_{b})\).
Some examples of bmetriclike spaces can be constructed with the help of the following proposition.
Proposition 1.9
[18]
Let \((X,\sigma)\) be a metriclike space and \(\sigma_{b}(x,y)=[\sigma(x,y)]^{p}\), where \(p>1\). Then \(\sigma_{b}\) is a bmetriclike with coefficient \(s=2^{p1}\).
Every partial bmetric space is a bmetriclike space with the same coefficient s and every bmetric space is also a bmetriclike space with the same coefficient s. However, the converses of these facts need not hold. For instance, assume that \(p>1\), then \(\sigma_{1}^{p}\) and \(\sigma_{4}^{p}\) are bmetricslike, but \(\sigma_{1}^{p}\) is not bmetric and \(\sigma_{4}^{p}\) is not partial bmetric.
Each bmetriclike \(\sigma_{b}\) on a nonempty set X generates a topology \(\tau_{\sigma_{b}}\) on X whose base is the family of open \(\sigma_{b}\)balls \(\{B_{\sigma_{b}}(x,\varepsilon):x\in X,\varepsilon >0\}\) where \(B_{\sigma_{b}}(x,\varepsilon)=\{y\in X:\sigma_{b}(x,y)\sigma_{b}(x,x)<\varepsilon\}\) for all \(x\in X\) and \(\varepsilon>0\).
Now, we recall the concepts of Cauchy sequence and convergent sequence in the framework of bmetriclike spaces.
Definition 1.10
[12]
 (i)
The sequence \(\{x_{n}\}\) is said to be convergent to x with respect to \(\tau_{\sigma_{b}}\) if \(\lim_{n\rightarrow \infty}\sigma_{b}(x_{n},x)=\sigma_{b}(x,x)\).
 (ii)
The sequence \(\{x_{n}\}\) is said to be a Cauchy sequence in \((X,\sigma_{b})\), if \(\lim_{n,m\rightarrow \infty}\sigma_{b}(x_{n},x_{m})\) exists and is finite.
 (iii)\((X,\sigma_{b})\) is said to be a complete bmetriclike space if for every Cauchy sequence \(\{x_{n}\}\) in X there exists \(x\in X \) such that$$ \lim_{n,m\rightarrow\infty }\sigma_{b}(x_{n},x_{m})= \lim_{n\rightarrow\infty }\sigma_{b}(x_{n},x)= \sigma_{b}(x,x). $$
Note that in a bmetriclike space the limit of a convergent sequence may not be unique (since already partial metric spaces share this property).
Definition 1.11
[19]
Suppose that \((X,\sigma_{b})\) is a bmetriclike space. A mapping \(T:X \rightarrow X\) is said to be continuous at a point \(x\in X\), if for every \(\varepsilon>0\) there exists a \(\delta>0\) such that \(T(B_{\sigma_{b}}(x,\delta))\subseteq B_{\sigma_{b}}(Tx,\varepsilon)\). The mapping T is continuous on X if it is continuous at each point x in X.
Note that if \(T:X \rightarrow X\) is a continuous mapping and \(\{x_{n}\}\) is a sequence in X with \(\lim_{n\rightarrow \infty}\sigma_{b}(x_{n},x)=\sigma_{b}(x,x)\), then \(\lim_{n\rightarrow \infty}\sigma_{b}(Tx_{n},Tx)=\sigma_{b}(Tx,Tx)\).
Samet et al. in [4] introduced the concept of αadmissible mappings and established some new fixed point theorems for these mappings. Thereafter, many researchers improved and generalized fixed point results by using this notion for single valued and multivalued mappings (cf. e.g. [3, 7, 20] for details).
Definition 1.12
[3]
Definition 1.13
[3]
The following lemma is useful in proving our main results, stated and proved according to [7], Lemma 7.
Lemma 1.14
Let \((X,\sigma_{b})\) be a bmetriclike space and \(T:X\rightarrow X\) be a triangular αadmissible mapping. Assume that there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq 1\). If \(x_{n}=T^{n}x_{0}\), then \(\alpha(x_{m},x_{n})\geq1\) for all \(m,n\in\mathbb{N}\).
2 Main results
In this section, first we describe the concept of generalized MeirKeeler contraction on a bmetriclike space which can be regarded as an extension of the MeirKeeler contractions defined in [1]. Then we demonstrate some fixed point results for this class of contractions.
Definition 2.1
Remark 2.2
We are now in a position to define two types of generalized MeirKeeler contractions on bmetriclike spaces, say type (I) and type (II).
Definition 2.3
Definition 2.4
In the following, we illustrate two important properties concerned with these new generalized MeirKeeler contractions, which we will require in our subsequent arguments.
Remark 2.5
Remark 2.6
It is readily verified 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 (3) and (5), respectively.
Next, we establish a fixed point theorem for generalized MeirKeeler type contractions via a rational expression. The presented theorem is a generalization of the result of Samet et al. [8].
Theorem 2.7
 (a)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), \(\alpha(Tx_{0},x_{0})\geq1\),
 (b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\rightarrow z\) as \(n\rightarrow\infty\) and \(\alpha(x_{n},x_{m})\geq1\) for all \(n,m\in\mathbb{N}\), then \(\alpha(x_{n},z)\geq1\) for all \(n\in\mathbb{N}\),
 (c)for each \(\varepsilon>0\), there exists \(\delta>0\) satisfying the following condition:$$ \begin{aligned} &2s\varepsilon\leq \sigma_{b}(y,Ty)\frac{1+\sigma _{b}(x,Tx)}{1+M(x,y)}+N(x,y)< s(2 \varepsilon+\delta)\quad \textit{implies} \\ &\alpha(x,y) \sigma_{b}(Tx,Ty)< \varepsilon. \end{aligned} $$(6)
Proof
The following example reveals the usefulness of Theorem 2.7.
Example 2.8
Note that \(\alpha(1,T1)\geq1\), \(\alpha(T1,1)\geq1\). Now, all conditions of Theorem 2.7 are satisfied and so T has a fixed point.
On the other hand, let \(d_{\sigma_{b}}\) be the bmetric associated to bmetriclike \(\sigma_{b}\) defined by \(d_{\sigma_{b}}(x,y)=0\) if \(x=y\) and \(d_{\sigma_{b}}(x,y)=\sigma_{b}(x,y)\), elsewhere. Then condition (6) does not hold in bmetric space \((X,d_{\sigma_{b}})\). Let \(\varepsilon=\frac{1}{4}\), \(x=0\), and \(y=2\). Then \(1=4\varepsilon\leq d_{\sigma_{b}}(2,T2)\frac{1+d_{\sigma_{b}}(0,T0)}{1+M(0,2)}+N(0,2)=1 <4\varepsilon+2\delta=1+2\delta\), for each \(\delta>0\). But \(\alpha(0,2)d_{\sigma_{b}}(T0,T2)=\frac{1}{3} \nless\frac{1}{4}\).
Theorem 2.9
 (a)
\(\theta(0)=0\) and \(\theta(t)>0\) for every \(t>0\),
 (b)
θ is nondecreasing and right continuous,
 (c)for every \(\varepsilon>0\), there exists \(\delta>0\) such thatfor all \(x,y\in X\), then (6) is satisfied.$$\begin{aligned}& 2\varepsilon\leq\theta \biggl(\frac{1}{s} \sigma_{b}(y,Ty) \frac{1+\sigma_{b}(x,Tx)}{1+M(x,y)}+\frac {1}{s}N(x,y) \biggr)< 2\varepsilon+\delta \quad \textit{implies} \\& \theta\bigl(2\alpha(x,y) \sigma_{b}(Tx,Ty) \bigr)< 2\varepsilon, \end{aligned}$$
Proof
Corollary 2.10
Proof
Now, we establish an existence of fixed point of mapping satisfying generalized MeirKeeler contractions of type (I) in the setup of bmetriclike spaces. For this purpose, we need the following definition.
Definition 2.11
Theorem 2.12
 (a)
T is an orbitally continuous generalized MeirKeeler contraction of type (I),
 (b)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), \(\alpha(Tx_{0},x_{0})\geq1\),
 (c)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\rightarrow z\) as \(n\rightarrow\infty\) and \(\alpha(x_{n},x_{m})\geq1\) for all \(n,m\in\mathbb{N}\), then \(\alpha(z,z)\geq1\),
 (d)
\(s>1\) or β is a continuous function.
Proof
(I) \(r'=0\).
(II) \(r'>0\).
By Remark 2.6 we know \(N(x,y)\leq M(x,y)\), so a slight change in the proof of Theorem 2.12 shows actually the following theorem holds.
Theorem 2.13
 (a)
T is an orbitally continuous generalized MeirKeeler contraction of type (II),
 (b)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), \(\alpha(Tx_{0},x_{0})\geq1\),
 (c)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\rightarrow z\) as \(n\rightarrow\infty\) and \(\alpha(x_{n},x_{m})\geq1\) for all \(n,m\in\mathbb{N}\), then \(\alpha(z,z)\geq1\),
 (d)
\(s>1\) or β is a continuous function.
There is an analogous result for the generalized MeirKeeler contraction. The proof is an easy adaptation of the one given in Theorem 2.12.
Proposition 2.14
Under the hypotheses of Theorem 2.12 consider a particular case, T is a generalized MeirKeeler contraction, then T has a fixed point in X.
 (A)
If \(\{x_{n}\}\) is a sequence in X which converges to z with respect to \(\tau_{\sigma_{b}}\) 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 \(\alpha(z,x_{n_{k}})\geq1\) and \(\alpha (x_{n_{k}},z)\geq 1\) for all k.
Theorem 2.15
 (a)
\(T:X \rightarrow X\) is a generalized MeirKeeler contraction of type (II),
 (b)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), \(\alpha(Tx_{0},x_{0})\geq1\),
 (c)
\(s>1\) or β is a continuous function.
Proof
Proposition 2.16
 (a)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), \(\alpha(Tx_{0},x_{0})\geq1\),
 (b)
\(s>1\) or β is a continuous function.
The usability of these results is demonstrated by the two following examples.
Example 2.17
In order to check the condition (2), we choose \(\delta=\varepsilon\) so that \(\varepsilon\leq\beta(\sigma_{b}(x,y)) M(x,y)<\varepsilon+\delta=2\varepsilon\), which implies \(\alpha (x,y)\sigma_{b}(Tx,Ty)<\varepsilon\).
Therefore, the map T is a generalized MeirKeeler contraction of type (I). Note that T is continuous with respect to \(\tau_{\sigma_{b}}\) and \(\alpha(0,T0)\geq1\), \(\alpha(T0,0)\geq1\). Now, all conditions of Theorem 2.12 are satisfied and so T has a fixed point.
Example 2.18
Next we prove that T is a generalized MeirKeeler contraction. We show this in the three following steps.
Step 1. If \(x\notin[0,1]\) or \(y\notin[0,1]\).
In this case, \(\alpha(x,y)=0\) and evidently (1) holds.
Step 2. Let \(x,y\in[0,1]\) with \(\sigma_{b}(x,y)\in[0,1]\).
Step 3. Let \(x,y\in[0,1]\) with \(\sigma_{b}(x,y)\notin[0,1]\).
Take \(\delta=3\varepsilon\). Then the inequality \(\varepsilon\leq \beta(\sigma_{b}(x,y))\sigma_{b}(x,y)=\frac {(x^{2}+y^{2})^{4}}{3(x^{2}+y^{2})^{2}+1}<4\varepsilon\), implies that \(\alpha (x,y)\sigma_{b}(Tx,Ty)=\frac{1}{16}(x^{2}+y^{2})^{2}<\varepsilon\).
Also, notice that \(\alpha(0,T0)\geq1\) and \(\alpha(T0,0)\geq1\). We conclude that all of the assumptions of Proposition 2.16 are satisfied. Moreover, T has fixed points \(x=0\) and \(x=2\).
A remarkable fact concerning Example 2.18 is that the restriction of T to the interval \([0,1]\) is orbitally continuous and so by the definition of α that example fulfills all conditions of Theorem 2.12, too.
Declarations
Acknowledgements
The authors gratefully acknowledge the anonymous reviewers for their carefully reading of the paper and helpful suggestions.
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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