- Research
- Open access
- Published:
Common fixed point results on an extended b-metric space
Journal of Inequalities and Applications volume 2018, Article number: 158 (2018)
Abstract
In this paper, we investigate the existence of common fixed points of a certain mapping in the frame of an extended b-metric space. The given results cover a number of well-known fixed point theorems in the literature. We state some examples to illustrate our results.
1 Introduction and preliminaries
Throughout the manuscript, we denote \(\mathbb{N}_{0}:=\mathbb{N}\cup\{0\} \), where \(\mathbb{N}\) is the positive integers. Further, \(\mathbb{R}\) represents the real numbers and \(\mathbb{R}_{0}^{+}:=[0,\infty)\).
Following this pioneering result on b-metric, a number of authors have reported several interesting results in this direction (see, e.g., [1, 2, 4–12, 14, 16, 18] and the related references therein).
Definition 1.1
(Czerwik [11])
Let X be a nonempty set and \(d:X\times X\to[0,\infty)\) be a function satisfying the following conditions:
- \((b1)\) :
-
\(d(x,y)=0\) if and only if \(x=y\).
- \((b2)\) :
-
\(d(x,y)=d(y,x)\) for all \(x,y \in X\).
- \((b3)\) :
-
\(d(x,y)\leq s[d(x,z)+d(z,y)]\) for all \(x,y,z\in X\), where \(s\geq1\).
The function d is called a b-metric and the space \((X,d)\) is called a b-metric space, in short, bMS.
The immediate example of b-metric is the following.
Example 1.1
Let \(Y= \{ x,y,z \}\) and \(X=Y\cup\mathbb {N} \). Define a mapping \(d:X\times X\rightarrow[0,\infty)\) such that
where \(A \in[2,\infty)\). Then we find that
It is evident that \(( X,d ) \) is a b-metric space. Notice also that if \(A >2\), the standard triangle inequality does not hold and \(( X,d ) \) is not a metric space.
Remark 1.1
It is clear that for \(s=1\), the b-metric becomes a usual metric.
Recently, Kamran [13] introduced a new type of generalized metric space and they proved some fixed point theorems on this space.
Definition 1.2
([13])
Let X be a nonempty set and \(\theta:X\times X\rightarrow[1,\infty)\). A function \(d_{\theta}:X\times X\rightarrow[0,\infty)\) is called an extended b-metric if, for all \(x,y,z\in X\), it satisfies
- \((d_{\theta}1)\) :
-
\(d_{\theta}(x,y)=0 \) iff \(x=y\);
- \((d_{\theta}2)\) :
-
\(d_{\theta}(x,y)=d_{\theta}(y,x)\);
- \((d_{\theta}3)\) :
-
\(d_{\theta}(x,y)\leq\theta(x,y) [d_{\theta}(x,z)+d_{\theta}(z,y) ] \).
The pair \((X,d_{\theta})\) is called an extended b-metric space, in short extended-bMS.
Remark 1.2
If \(\theta(x, y)=s\) for \(s\geq1\), then we obtain the definition of bMS.
Example 1.2
Let \(X=[0,1]\) and \(\theta:X\times X\rightarrow[1,\infty)\), \(\theta (x,y)=\frac{x+y+1}{x+y}\). Define \(d_{\theta}:X\times X\rightarrow[0,\infty )\) as
Obviously, \((d_{\theta}1)\) and \((d_{\theta}2)\) hold. For \((d_{\theta}3)\), we distinguish the following cases:
-
(i)
Let \(x,y\in(0,1]\). For \(z\in(0,1]\), we have
$$d_{\theta}(x, y)\leq\theta(x,y)\bigl[d_{\theta}(x,z)+d_{\theta}(z,y) \bigr] \quad\Leftrightarrow\quad\frac{1}{xy}\leq\frac{1+x+y}{x+y}\cdot \frac{x+y}{xyz} \quad\Leftrightarrow\quad z\leq1+x+y. $$If \(z=0\), then
$$d_{\theta}(x, y)\leq\theta(x,y)\bigl[d_{\theta}(x,0)+d_{\theta}(0,y) \bigr] \quad\Leftrightarrow\quad\frac{1}{xy}\leq\frac{1+x+y}{x+y}\cdot \frac{x+y}{xy} \quad\Leftrightarrow\quad 1\leq1+x+y. $$ -
(ii)
For \(x\in(0,1]\) and \(y=0\), let \(z\in(0,1]\).
$$d_{\theta}(x, 0)\leq\theta(x,0)\bigl[d_{\theta}(x,z)+d_{\theta}(z,0) \bigr] \quad\Leftrightarrow\quad\frac{1}{x}\leq\frac{1+x}{x}\cdot \frac{1+x}{xz} \quad\Leftrightarrow\quad xz\leq(1+x)^{2}. $$In conclusion, for any \(x, y, z\in X\),
$$d_{\theta}(x,z)\leq\theta(x,z) \bigl[d_{\theta}(x,y)+d_{\theta}(y,z) \bigr]. $$Hence, \((X,d_{\theta})\) is an extended b-metric space.
Some fundamental concepts, like convergence, Cauchy sequence, and completeness in a extended-bMS, are defined as follows [13].
Definition 1.3
([13])
Let \((X,d_{\theta})\) be an extended-bMS.
-
(i)
A sequence \({x_{n}}\) in X is said to converge to \(x\in X\) if, for every \(\epsilon>0\), there exists \(N=N(\epsilon)\in\mathbb{N}\) such that \(d_{\theta}(x_{n}, x)<\epsilon\) for all \(n\geq N\). In this case, we write \(\lim_{n\rightarrow\infty} x_{n} = x\).
-
(ii)
A sequence \({x_{n}}\) in X is said to be Cauchy if, for every \(\epsilon>0\), there exists \(N=N(\epsilon)\in\mathbb{N}\) such that \(d_{\theta}(x_{m}, x_{n})<\epsilon\) for all \(m,n\geq N\).
Definition 1.4
An extended-b-metric space \((X,d_{\theta})\) is complete if every Cauchy sequence in X is convergent.
Lemma 1.1
Let \((X,d_{\theta})\) be a complete extended-bMS. If \(d_{\theta}\) is continuous, then every convergent sequence has a unique limit.
Theorem 1.1
([13])
Let \((X,d_{\theta})\) be an extended-bMS such that \(d_{\theta}\) is a continuous functional. Let \(T:X\rightarrow X\) satisfy
for all \(x,y\in X \), where \(k\in [0,1 )\) is such that, for each \(x_{0}\in X\), \(\lim_{n,m\rightarrow\infty} \theta(x_{n}, x_{m})<\frac{1}{k}\), here \(x_{n}=T^{n}x_{0}\), \(n=1,2,\ldots \) . Then T has precisely one fixed point u. Moreover, for each \(y\in X\), \(T^{n}y\rightarrow u\).
For our purposes, we need to recall the following definition which is proposed by Popescu [17].
Definition 1.5
Let \(T:X\rightarrow X\) and \(\alpha:X\times X\rightarrow [0, \infty )\). We say that T is an α-orbital admissible if, for all \(x,y\in X\), we have
Definition 1.6
A set X is regular with respect to mapping \(\alpha:X\times X \to [0,\infty)\) if, whenever \(\{x_{n}\}\) is a sequence in X such that \(\alpha (x_{n},x_{n+1})\geq1\) and \(\alpha(x_{n+1},x_{n})\geq1\) for all n and \(x_{n} \rightarrow x\in X\) as \(n\rightarrow\infty\), then there exists a subsequence \(\{x_{n(k)} \}\) of \(\{ x_{n} \}\) such that \(\alpha(x_{n(k)},x)\geq1\) and \(\alpha(x,x_{n(k)})\geq1\) for all n.
2 (S,T) orbital cyclic
Definition 2.1
Suppose that \(T,S\) are two self-mappings on a complete extended-bMS \((X, d_{\theta})\). Suppose also that there are two functions \(\alpha,\beta :X\times X\rightarrow [0,\infty )\) such that, for any \(x\in X\),
Then we say that the pair \(S,T\) is an \((\alpha,\beta)\)-orbital-cyclic admissible pair.
We start with the following lemma which is essential in our main results.
Lemma 2.1
([3])
Let \((X,d_{\theta})\) be an extended b-metric space.If there exists \(q\in [0,1 )\) such that the sequence \(\{ x_{n} \}\) for an arbitrary \(x_{0}\in X\) satisfies \(\lim_{n,m\rightarrow\infty} \theta(x_{n}, x_{m})<\frac{1}{q}\), and also
for any \(n\in\mathbb {N}\), then the sequence \(\{x_{n}\}\) is Cauchy in X.
Proof
Let \(\{x_{n} \}_{n\in\mathbb {N}}\) be a given sequence. By employing inequality (4) recursively, we derive that
Since \(q\in [0,1 )\), we find that
On the other hand, by \((d_{\theta}3)\), together with triangular inequality, for \(p\geq1\), we derive that
Notice the inequality above is dominated by \(\sum_{i=1}^{n+p-1} q^{i} \prod_{j=1}^{i} \theta(x_{n+j}, x_{n+p}) \leq \sum_{i=1}^{n+p-1} q^{i}\times \prod_{j=1}^{i} \theta(x_{j}, x_{n+p}) \).
On the other hand, by employing the ratio test, we conclude that the series \(\sum_{i=1}^{\infty}a_{i}\), where \(a_{i}=q^{i} \prod_{j=1}^{i} \theta (x_{j}, x_{n+p})\) converges to some \(S \in(0,\infty)\). Indeed, \(\lim_{i\rightarrow\infty}\frac{a_{i+1}}{a_{i}} = \lim_{i\rightarrow\infty} q \theta(x_{i}, x_{i+p})<1\), and hence we get the desired result. Thus, we have
Consequently, we observe, for \(n\leq1, p\leq1\), that
Letting \(n\rightarrow\infty\) in (8), we conclude that the constructive sequence \(\{x_{n}\}\) is Cauchy in the extended b-metric space \((X, d_{\theta})\). □
Theorem 2.1
Let \(T,S\) be two self-mappings on a complete extended-bMS \((X,d_{\theta})\) such that the pair \(T,S\) forms an \((\alpha,\beta)\)-orbital-cyclic admissible pair. Suppose that
-
(i)
for each \(x_{0}\in X\), \(\lim_{n,m\rightarrow\infty}\theta (x_{n},x_{m})<\frac{1-k}{k}\), where \(x_{2n}=Sx_{2n-1}\textit{ and }x_{2n+1}=Tx_{2n}\) for each \(n \in\mathbb {N}\);
-
(ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
-
(iii)
either S and T are continuous, or
-
(iii*)
if \({x_{n}}\) is a sequence in X such that \(x_{n}\rightarrow u\), then \(\alpha(u, Tu)\geq1\) and \(\beta(u,Su)\geq1\).
Moreover, if for all \(x,y\in X\) and \(k\in [0,\frac{1}{2} )\)
then the pair of the mappings \(T,S\) possesses a common fixed point u, that is, \(Tu=u=Su\).
Proof
By assumption (ii), there exists a point \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\). Take \(x_{1}=Tx_{0}\) and \(x_{2}=Sx_{1}\). By induction, we construct a sequence \(\{x_{n}\}\) such that
We have \(\alpha(x_{0},x_{1})\geq1\), and since \((S,T)\) is an \(\alpha,\beta \)-orbital-cyclic admissible pair, we get
and
Applying again (3),
and
Recursively, we obtain
and
Without loss of generality, we assume that \(x_{n}\neq x_{n+1}\) for each \(n\in\mathbb{N}_{0}\). Indeed, if \(x_{n_{0}}=x_{n_{0}+1}\) for some \(n_{0}\in \mathbb{N}_{0}\), then \(u=x_{n_{0}}\) forms a common fixed point for S and T, which finalizes the proof. More precisely, to see that u is the common fixed point of S and T, we shall examine the following two cases. First, we assume that \(n_{0}\) is even, that is, \(n_{0}=2k\). In this case, we have \(x_{2k}=x_{2k+1}=Tx_{2k}\), that is, \(x_{2k}\) is a fixed point of T. Now we shall prove that \(x_{2k}=x_{2k+1}=Tx_{2k}=Sx_{2k+1}\). Suppose on the contrary that \(d_{\theta}(Tx_{2k}, Sx_{2k+1})>0\). By letting \(x=x_{2k}\) and \(y=x_{2k+1}\) in (9) and keeping in mind (11) and (12), we get that
a contradiction. Hence, we conclude that \(d_{\theta}(Tx_{2k}, Sx_{2k+1})=0\) and \(x_{2k}=x_{2k+1}=Tx_{2k}=Sx_{2k+1}\), that is, \(x_{2k}=x_{2k+1}=u\) is a common fixed point of T and S. Analogously, one can derive the same conclusion for the case \(n_{0}\) is odd, that is, \(n_{0}=2k-1\).
Thus, throughout the proof, we suppose that
In what follows, we shall prove that the sequence \(\{x_{n}\}\) is Cauchy. For this purpose, it is sufficient to examine the following two cases.
Case (a): Let \(x=x_{2n}\) and \(y=x_{2n+1}\). Then, by inequality (9) and using (11), (12), we get
and from here
for each \(n\in\mathbb{N}_{0}\), where \(q=\frac{k}{1-k}<1\) with \(k\in [0, \frac{1}{2} )\).
Case (b): Let \(x=x_{2n}\) and \(y=x_{2n-1}\). Then, by inequality (9) and using (11), (12), we get
and
for each \(n\in\mathbb{N}_{0}\), where \(q=\frac{k}{1-k}<1\) with \(k\in [0, \frac{1}{2} )\).
Combining (15) and (17), we can conclude that
for all \(m\in\mathbb{N}\). From Lemma 2.1, taking into account (i), \(\lim_{n,m\rightarrow \infty}\theta(x_{n},x_{m})<\frac{1-k}{k}=\frac{1}{q}\), we obtain that \(\{x_{m} \}\) is a Cauchy sequence. By completeness of \((X,d_{\theta})\), there is some point \(u\in X\) such that
Naturally, we also have
Due to the continuity of the mappings T and S, we get
and
Let us consider now the alternative hypothesis (iii*). Taking \(x=u\) and \(y=x_{2n+1}\) in (9) and taking into account (12), we get
Letting \(n\rightarrow\infty\), we obtain
which implies \(d_{\theta}(Tu,u)=0\). Hence, we get that \(Tu=u\). Analogously, regarding (11) and (22), we observe that
Now, letting \(n\rightarrow\infty\) in the inequality above, we derive that
Hence, we find that \(Su=u\). Accordingly, we conclude that T and S have a common fixed point u. □
Example 2.1
Let \(X=[0,1]\) and \(d_{\theta}:X\times X\rightarrow [0,\infty )\) defined by
when
Then \((X, d_{\theta})\) is an extended-bMS (see Example 1.2).
Let \(T:X\rightarrow X\), \(S:X\rightarrow X\), defined as
respectively
and two functions \(\alpha, \beta:X\times X\rightarrow [0,\infty )\) defined by
and
We show that the pair \(T,S\) forms an \((\alpha,\beta)\)-orbital-cyclic admissible pair. Indeed, for \(x=1\),
and
For \(x=\frac{1}{2}\):
and
For \(x=\frac{1}{4}\):
and
We have thus proved that T is α orbital admissible, and sure, because \(\alpha(\frac{1}{4},T\frac{1}{4})\geq1\), assumption (ii) is satisfied.
If \(x_{0}\in \{\frac{1}{4}, \frac{1}{2}, 1 \}\), then \(x_{n}=T^{n}x_{0}=1\), so
where we choose \(k=\frac{1}{4}<\frac{1}{2}\). Otherwise, for each \(x_{0}\in X - \{\frac{1}{4}, \frac{1}{2}, 1 \}\), we have \(x_{2n-1}=\sum_{k=1}^{n} (\frac{1}{2} )^{n}+\frac{x_{0}}{2^{n}}\), \(x_{2n}=x_{2n-1}\) and \(\lim_{n\rightarrow\infty}x_{n}=1\). So,
Hence, (i) is also verified.
We have
and
Because in the other cases \(\alpha(x,y)=0\) and \(\beta(x,y)=0\), it is enough to investigate the following situations:
Case (a): For \(x\in \{1,\frac{1}{2} \}\) and \(y\in \{1, \frac{1}{2}, \frac{1}{4} \}\),
so inequality (9) is satisfied.
Case (b): Let \(x=\frac{1}{4}, y=1\). Then
Case (c): Let \(x=\frac{1}{4}, y=\frac{1}{2}\). Then
Case (d): Let \(x=\frac{1}{4}, y=\frac{1}{4}\). Then
Therefore, all the conditions of Theorem 2.2 are satisfied and T has a unique fixed point, \(x=1\).
2.1 \((\alpha,\beta)\)-orbital-cyclic
Definition 2.2
Let X be a nonempty set, \(T:X\rightarrow X\), and \(\alpha,\beta :X\times X\rightarrow [0,\infty )\). We say that T is an \((\alpha,\beta)\)-orbital-cyclic admissible mapping if
for all \(x\in X\).
Corollary 2.1
Let T be a self-mapping on a complete extended-bMS \((X,d_{\theta})\) such that the mapping T forms an \((\alpha,\beta)\)-orbital-cyclic admissible mapping. Suppose that
-
(i)
for each \(x_{0}\in X\), \(\lim_{n,m\rightarrow\infty}\theta (x_{n},x_{m})<\frac{1-k}{k}\), where \(x_{n}=T^{n}x_{0}\), \(n=1,2,\ldots \) ;
-
(ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\) and \(\beta(x_{0}, Tx_{0})\geq1\);
-
(iii)
either T is continuous, or
-
(iii*)
if \({x_{n}}\) is a sequence in X such that \(x_{n}\rightarrow u\), then \(\alpha(u, Tu))\geq1\) and \(\beta(u, Tu)\geq1\).
Moreover, if for all \(x,y\in X\) and \(k\in [0,\frac{1}{2} )\)
then the pair of the mappings T possesses a fixed point u, that is, \(Tu=u\).
Proof
It is sufficient to take \(S=T\) in Theorem 2.1. □
Example 2.2
Let \(X=[0,2]\) and define \(d_{\theta}:X\times X\rightarrow [0,\infty )\) and \(\theta:X\times X\rightarrow [1,\infty )\) by
respectively
Let the self-map \(T:X\rightarrow X\) be defined by
Define also \(\alpha, \beta:X\times X\rightarrow [0,\infty )\) by
and
We show that T is \((\alpha,\beta)\)-orbital-cyclic admissible. Let \(x,y\in X\) such that \(\alpha(x,Tx)\geq1\) and \(\beta(x,Tx)\geq1\). Then \(x,y\in[0,1)\). On the other hand, if \(x\in[0,1)\), then \(Tx\leq1\) and \(T^{2}x\leq1\). It follows that \(\alpha(Tx,T^{2}x)\geq1\) and \(\beta (Tx,T^{2}x)\geq1\). Thus, the assertion holds. For \(x=0\), we have \(T0=0\) and \(\alpha (0, T0 )\geq1\), respectively, \(\beta(0, T0)\geq1\), so assumption (ii) is satisfied. Let now \(\{x_{n} \}\) be a sequence in X such that \(x_{n}\rightarrow x\). Then \(\{x_{n} \}\subset [0,1 ]\) and \(x\in[0,1]\). This implies that \(\alpha(x, Tx)\geq1\).
For \(x_{0}\in[0,1)\), we get \(T^{n}x_{0}=\frac{x_{0}}{8^{n}}\) and \(\lim_{n,m\rightarrow\infty}\theta(T^{n}x_{0}, T^{m}x_{0})=1\). If \(x_{0}\in[1,2]\), \(Tx_{0}\leq\frac{1}{4}\), and \(\lim_{n,m\rightarrow\infty}\theta(T^{n}x_{0}, T^{m}x_{0})=1\). So, assumption (i) is satisfied for \(k=\frac{1}{3}\). We have the following cases:
(a): For \(x,y\in[0,1)\), we get
Replaced in (26) we get
or
which is true for any \(x,y\in[0,1)\).
(b): For \(x=1\) and \(y=2\), we know that \(\alpha(1,T1)=\alpha(1,0)\geq 1\) and \(\beta(T1, T^{2}1)=\beta(0,0)\geq1\), and \(\beta(2, T2)=\beta (2,0)\geq1\) and \(\alpha(T2, T^{2}2)=\alpha(0,0)\geq1\). But in this case (26) is obvious, because \(d_{\theta}(T1,T2)=0\).
(c): For \(x\in[0,1)\) and \(y=2\), (26) becomes
or
(d): For all other cases, \(\alpha(x,Tx)=0\) or \(\beta(x,Tx)=0\), and for this reason inequality (26) holds. Therefore, all the conditions of Corollary 2.1 are satisfied and T has a fixed point, \(x=0\).
Corollary 2.2
Let T be a self-mapping on a complete extended-bMS \((X,d_{\theta})\) such that T is an α-orbital admissible mapping. Suppose that
-
(i)
for each \(x_{0}\in X\), \(\lim_{n,m\rightarrow\infty}\theta (x_{n},x_{m})<\frac{1-k}{k}\), where \(x_{n}=T^{n}x_{0}\), \(n=1,2,\ldots \) ;
-
(ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq1\);
-
(iii)
either T is continuous, or
-
(iii*)
if \({x_{n}}\) is a sequence in X such that \(x_{n}\rightarrow u\), then \(\alpha(u, Tu))\geq1\).
Moreover, if for all \(x,y\in X\) and \(k\in [0,\frac{1}{2} )\)
then the pair of the mappings T possesses a fixed point u, that is, \(Tu=u\).
Proof
It is sufficient to take \(\beta(x,y)=\alpha(x,y)\) in Corollary 2.1. □
Example 2.3
Let \(X=[0,2]\) be endowed with extended b-metric \(d_{\theta}:X\times X\rightarrow [0,\infty )\), defined by \(d_{\theta}(x,y)=(x-y)^{2}\), where \(\theta:X\times X\rightarrow [1,\infty )\), \(\theta(x,y)=x+y+1\). Let \(T:X\rightarrow X\) such that
Define also \(\alpha:X\times X\rightarrow (0,\infty )\) as
We prove that Corollary 2.2 can be applied to T for \(k=\frac {1}{4}\), but Theorem 1.1 cannot be applied to T. We show that T is an α-orbital admissible mapping. If \(x,y\in [0,\frac{1}{2} ]\), then \(Tx\leq\frac{1}{2}\) and \(T^{2}x\leq1\). Thus, \(\alpha(x,Tx)\geq1\) implies \(\alpha(Tx, T^{2}x)\geq1\). Similarly, we get that \(\alpha(x,Tx)\geq1\) implies \(\alpha(Tx, T^{2}x)\geq1\) for all \(x,y\in [\frac{1}{2}, 1 ]\), so T is α-orbital admissible. In reason of the above arguments, \(\alpha(0,T0)=\alpha(0, \frac {1}{3})\geq1\). Thus, assertion (ii) holds.
Note that, for each \(x_{0}\in X\), \(T^{n}x_{0}=\sum_{k=1}^{n} (\frac {1}{3} )^{n}+\frac{x_{0}}{3^{n}}\) and \(\lim_{n\rightarrow\infty }T^{n}x_{0}=\frac{1}{2}\). Hence,
So assumption (i) is satisfied, and because \(\alpha (\frac{1}{2}, T\frac{1}{2} )=\alpha (\frac{1}{2},\frac{1}{2} )\geq1\), assumption (iii*) is also satisfied. Let \(x,y\in [0,\frac{1}{2} ]\), or \(x,y\in [\frac {1}{2},1 ]\). We have
and
Replaced in inequality \((28)\), we get
or, equivalently,
Hence, inequality \((28)\) is satisfied. In other cases, inequality \((28)\) is obviously satisfied, because \(\alpha(x,y)=0\). Therefore, all conditions of Corollary 2.2 are satisfied and T has a unique fixed point, \(x=\frac{1}{2}\).
Let \(x=2\) and \(y=3\). Then
for any \(k<1\).
2.2 Uniqueness
Notice that in this section we investigate the existence of (common) fixed points of certain operators. For the uniqueness of a fixed point of the observed results, we will consider the following hypothesis.
-
(H)
For all \(x,y\in\operatorname{CFix}(T)\), we have \(\alpha(x,Tx)\geq1\) and \(\beta(y,Sy)\geq1\).
Here, \(\operatorname{CFix}(T)\) denotes the set of common fixed points of T and S.
Theorem 2.2
Adding condition (H) to the hypotheses of Theorem 2.1, we obtain that u is the unique fixed point of T.
Proof
Suppose, on the contrary, that v is another fixed point of T. From (H), there exists \(v\in X\) such that
Since T satisfies (9), we get that
which yields that
Since the inequality above is possible only if \(d_{\theta}(u,v)=0\), that is, \(u=v\). This is a contradiction. Thus we proved that u is the unique fixed point of T. □
Notice also that instead of hypothesis (H), one can suggest different conditions, see, e.g., [15].
3 Conclusions
It is clear that one can list several consequences from our results. By letting \(\theta(x,y)=s\), constant, with \(1\leq s <\frac{k-1}{k}\) in Theorem 2.1 (analogously, in Corollary 2.1 and Corollary 2.2), we get corresponding fixed point results in the setting of standard b-metric space.
On the other hand, regarding the techniques used in [15], one can derive another set of corollaries, by choosing the admissible mapping in a proper way. In this way, for example, several existing fixed point results in the literature in the setting of partially ordered metric spaces can be derived. Furthermore, the analogs of fixed point results for cyclic contractions can be found.
References
Afshari, H., Aydi, H., Karapınar, E.: Existence of fixed points of set-valued mappings in b-metric spaces. East Asian Math. J. 32(3), 319–332 (2016)
Aksoy, U., Karapınar, E., Erhan, Y.M.: Fixed points of generalized alpha-admissible contractions on b-metric spaces with an application to boundary value problems. J. Nonlinear Convex Anal. 17(6), 1095–1108 (2016)
Alqahtani, B., Fulga, A., Karapınar, E.: Non-unique fixed point results in extended b-metric space. Mathematics 6(5), 68 (2018). https://doi.org/10.3390/math6050068
Aydi, H., Bota, M., Karapınar, E., Mitrović, S.: A fixed point theorem for set-valued quasi-contractions in b-metric spaces. Fixed Point Theory Appl. 2012, Article ID 88 (2012)
Aydi, H., Bota, M., Karapınar, E., Moradi, S.: A common fixed point for weak ϕ-contractions in b-metric spaces. Fixed Point Theory 13(2), 337–346 (2012)
Boriceanu, M.: Strict fixed point theorems for multivalued operators in b-metric spaces. Int. J. Mod. Math. 4(3), 285–301 (2009)
Bota, M., Chifu, C., Karapinar, E.: Fixed point theorems for generalized (alpha-psi)-Ciric-type contractive multivalued operators in b-metric spaces. J. Nonlinear Sci. Appl. 9(3), 1165–1177 (2016)
Bota, M., Karapınar, E., Mleşniţe, O.: Ulam–Hyers stability for fixed point problems via \(\alpha-\phi \)-contractive mapping in b-metric spaces. Abstr. Appl. Anal. 2013, Article ID 855293 (2013)
Bota, M.-F., Karapinar, E.: A note on “Some results on multi-valued weakly Jungck mappings in b-metric space”. Cent. Eur. J. Math. 11(9), 1711–1712 (2013). https://doi.org/10.2478/s11533-013-0272-2
Bourbaki, N.: Topologie Générale. Herman, Paris (1974)
Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5–11 (1993)
Gulyaz-Ozyurt, S.: On some alpha-admissible contraction mappings on Branciari b-metric spaces. Adv. Theory Nonlinear Anal. Appl. 2017(1), 1–13 (2017)
Kamran, T., Samreen, M., Ain, O.U.: A generalization of b-metric space and some fixed point theorems. Mathematics 5(2), 19 (2017). https://doi.org/10.3390/math5020019
Karapinar, E., Piri, H., AlSulami, H.: Fixed points of generalized F-Suzuki type contraction in complete b-metric spaces. Discrete Dyn. Nat. Soc. 2015, Article ID 969726 (2015)
Karapinar, E., Samet, B.: Generalized alpha-psi contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012, Article ID 793486 (2012)
Kutbi, M.A., Karapinar, E., Ahmed, J., Azam, A.: Some fixed point results for multi-valued mappings in b-metric spaces. J. Inequal. Appl. 2014, 126 (2014)
Popescu, O.: Some new fixed point theorems for α-Geraghty-contraction type maps in metric spaces. Fixed Point Theory Appl. 2014, 190 (2014)
Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)
Acknowledgements
The first and third authors extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia). The authors thank anonymous referees for their remarkable comments, suggestion, and ideas that helped to improve this paper.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
All 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
Alqahtani, B., Fulga, A. & Karapınar, E. Common fixed point results on an extended b-metric space. J Inequal Appl 2018, 158 (2018). https://doi.org/10.1186/s13660-018-1745-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1745-4