Abstract
In this paper, we establish dislocated quasi-b-metric spaces and introduce the notions of Geraghty type dqb-cyclic-Banach contraction and dqb-cyclic-Kannan mapping and derive the existence of fixed point theorems for such spaces. Our main theorem extends and unifies existing results in the recent literature.
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1 Introduction and preliminaries
Fixed point theory has been studied extensively, which can be seen from the works of many authors [1–6]. Banach contraction principle was introduced in 1922 by Banach [7] as follows:
-
(i)
Let \((X,d)\) be a metric space and let \(T:X\to X\). Then T is called a Banach contraction mapping if there exists \(k\in[0, 1)\) such that
$$d(Tx,Ty)\leq k d(x,y) $$for all \(x, y \in X\).
The concept of Kannan mapping was introduced in 1969 by Kannan [8] as follows:
-
(ii)
T is called a Kannan mapping if there exists \(r\in[0, \frac{1}{2})\) such that
$$d(Tx, Ty) \leq r d(x,Tx) + r d(y, Ty) $$for all \(x, y \in X\).
Now, we recall the definition of cyclic map. Let A and B be nonempty subsets of a metric space \((X,d)\) and \(T: A\cup B\to A\cup B\). T is called a cyclic map iff \(T(A)\subseteq B\) and \(T(B)\subseteq A\).
In 2003, Kirk et al. [9] introduced cyclic contraction as follows:
-
(iii)
A cyclic map \(T: A\cup B\to A\cup B\) is said to be a cyclic contraction if there exists \(a\in[0, 1)\) such that
$$d(Tx,Ty)\leq a d(x,y) $$for all \(x\in A\) and \(y\in B\).
In 2010, Karapinar and Erhan [10] introduced Kannan type cyclic contraction as follows:
-
(iv)
A cyclic map \(T: A\cup B\to A\cup B\) is called a Kannan type cyclic contraction if there exists \(b\in[0, \frac{1}{2})\) such that
$$d(Tx, Ty) \leq bd(x,Tx) + b d(y, Ty) $$for all \(x\in A\) and \(y\in B\).
If \((X,d)\) is a complete metric space, at least one of (i), (ii), (iii) and (iv) holds, then it has a unique fixed point [7–10]. Next, we discuss the development of spaces. The concept of quasi-metric spaces was introduced by Wilson [11] in 1931 as a generalization of metric spaces, and in 2000 Hitzler and Seda [12] introduced dislocated metric spaces as a generalization of metric spaces, [13] generalized the result of Hitzler, Seda and Wilson and introduced the concept of dislocated quasi-metric space. Włodarczyk et al. (see [14–21]) created uniform spaces as this is the concept of metric spaces. In 1989, Bakhtin [22] introduced b-metric space as a generalization of metric space. Moreover, Czerwik [23] made the results of Bakhtin known more in 1998. Finally, many other generalized b-metric spaces such as quasi-b-metric spaces [24], b-metric-like spaces [25] and quasi-b-metric-like spaces [26] were introduced.
We begin with the following definition as a recall from [11, 12].
Definition 1.1
Let X be a nonempty set. Suppose that the mapping \(d: X\times X\rightarrow [0,\infty)\) satisfies the following conditions:
- (d1):
-
\(d(x,x)=0\) for all \(x\in X\);
- (d2):
-
\(d(x,y)=d(y,x)=0\) implies \(x=y\) for all \(x,y\in X\);
- (d3):
-
\(d(x,y)=d(y,x)\) for all \(x,y\in X\);
- (d4):
-
\(d(x,y)\leq[d(x,z)+d(z,y)]\) for all \(x,y,z\in X\).
If d satisfies conditions (d1), (d2) and (d4), then d is called a quasi-metric on X. If d satisfies conditions (d2), (d3) and (d4), then d is called a dislocated metric on X. If d satisfies conditions (d1)-(d4), then d is called a metric on X.
In 2005 the concept of dislocated quasi-metric spaces [13], which is a new generalization of quasi-b-metric spaces and dislocated b-metric spaces, was introduced. By Definition 1.1, if setting conditions (d2) and (d4) hold true, then d is called a dislocated quasi-metric on X.
Remark 1.2
It is obvious that metric spaces are quasi-metric spaces and dislocated metric spaces, but the converse is not true.
In 1989, Bakhtin [22] introduced the concept of b-metric spaces and investigated some fixed point theorems in such spaces.
Definition 1.3
[22]
Let X be a nonempty set. Suppose that the mapping \(b: X\times X\rightarrow[0,\infty)\) such that the constant \(s\geq1\) satisfies the following conditions:
- (b1):
-
\(b(x,y)=b(y,x)=0 \Leftrightarrow x=y\) for all \(x,y\in X\);
- (b2):
-
\(b(x,y)=b(y,x)\) for all \(x,y\in X\);
- (b3):
-
\(b(x,y)\leq s[b(x,z)+b(z,y)]\) for all \(x,y,z\in X\).
The pair \((X, b)\) is then called a b-metric space.
Remark 1.4
It is obvious that metric spaces are b-metric spaces, but conversely this is not true.
In 2012, Shah and Hussain [24] introduced the concept of quasi-b-metric spaces and verified some fixed point theorems in quasi-b-metric spaces.
Definition 1.5
[24]
Let X be a nonempty set. Suppose that the mapping \(q: X\times X\rightarrow[0,\infty)\) such that constant \(s\geq1\) satisfies the following conditions:
- (q1):
-
\(q(x,y)=q(y,x)=0 \Leftrightarrow x=y\) for all \(x,y\in X\);
- (q2):
-
\(q(x,y)\leq s[q(x,z)+q(z,y)]\) for all \(x,y,z\in X\).
The pair \((X, q)\) is then called a quasi-b-metric space.
Remark 1.6
It is obvious that b-metric spaces are quasi-b-metric spaces, but conversely this is not true.
Recently, the concept of b-metric-like spaces, which is a new generalization of metric-like spaces, was introduced by Alghamdi et al. [25].
Definition 1.7
[25]
Let X be a nonempty set. Suppose that the mapping \(D: X\times X\rightarrow[0,\infty)\) such that constant \(s\geq1\) satisfies the following conditions:
- (D1):
-
\(D(x,y)=0 \Rightarrow x=y\) for all \(x,y\in X\);
- (D2):
-
\(D(x,y)=D(y,x)\) for all \(x,y\in X\);
- (D3):
-
\(D(x,y)\leq s[D(x,z)+D(z,y)]\) for all \(x,y,z\in X\).
The pair \((X, D)\) is then called a b-metric-like space (or a dislocated b-metric space).
Remark 1.8
It is obvious that b-metric spaces are b-metric-like spaces, but conversely this is not true.
In this paper we introduce dislocated quasi-b-metric spaces which generalize quasi-b-metric spaces and b-metric-like spaces, and we introduce the notions of Geraghty type dqb-cyclic-Banach contraction and dqb-cyclic-Kannan mapping and derive the existence of fixed point theorems for such spaces. Our main theorems extend and unify existing results in the recent literature.
2 Main results
In this section, we begin with introducing the notion of dislocated quasi-b-metric space.
Definition 2.1
Let X be a nonempty set. Suppose that the mapping \(d: X\times X\rightarrow[0,\infty)\) such that constant \(s\geq 1\) satisfies the following conditions:
-
(d1)
\(d(x,y)=d(y,x)=0\) implies \(x=y\) for all \(x,y\in X\);
-
(d2)
\(d(x,y)\leq s[d(x,z)+d(z,y)]\) for all \(x,y,z\in X\).
The pair \((X, d)\) is then called a dislocated quasi-b-metric space (or simply dqb-metric). The number s is called the coefficient of \((X, d)\).
Remark 2.2
It is obvious that b-metric spaces, quasi-b-metric spaces and b-metric-like spaces are dislocated quasi-b-metric spaces, but the converse is not true.
Example 2.3
Let \(X=\mathbb{R}\) and let
where \(n,m \in\mathbb{N}\setminus\{1\}\) with \(n\neq m\).
Then \((X, d)\) is a dislocated quasi-b-metric space with the coefficient \(s =2\), but since \(d(1, 1) \neq 0\), we have \((X, b)\) is not a quasi-b-metric space, and since \(d(1, 2) \neq d(2, 1) \), we have \((X, b)\) is not a b-metric-like space. And \((X, b)\) is not a dislocated quasi-metric space. Indeed, let \(x,y,z\in X\). Suppose that \(d(x,y)=0\).
Then
It implies that \(|x-y|^{2}=0\), and so \(x=y\).
Next, consider
where \(s=2\),
where \(n,m>42\).
Example 2.4
[26]
Let \(X = \{0, 1, 2\}\), and let \(d: X\times X\to\mathbb{R}^{+}\) be defined by
Then \((X, d)\) is a dislocated quasi-b-metric space with the coefficient \(s =2\), but since \(d(1, 1) \neq 0\), we have \((X, b)\) is not a quasi-b-metric space, and since \(d(1, 2) \neq d(2, 1) \), we have \((X, b)\) is not a b-metric-like space. It is obvious that \((X, b)\) is not a dislocated quasi-metric space.
Example 2.5
Let \(X=\mathbb{R}\) and let
Then \((X, d)\) is a dislocated quasi-b-metric space with the coefficient \(s =2\), but since \(d(0, 1) \neq d(1, 0)\), we have \((X, b)\) is not a b-metric-like space, since \(d(1, 1) \neq 0\), we have \((X, b)\) is not a quasi-b-metric space. It is obvious that \((X, b)\) is not a dislocated quasi-metric space.
Example 2.6
Let \(X=\mathbb{R}\) and let
Then \((X, d)\) is a dislocated quasi-b-metric space with the coefficient \(s =2\), but since \(d(1, 1) \neq 0\), we have \((X, b)\) is not a quasi-b-metric space. It is obvious that \((X, b)\) is not a dislocated quasi-metric space.
We will introduce a dislocated quasi-b-convergent sequence, a Cauchy sequence and a complete space according to Zoto et al. [27].
Definition 2.7
-
(1)
A sequence \((\{x_{n}\})\) in a dqb-metric space \((X,d)\) dislocated quasi-b-converges (for short, dqb-converges) to \(x\in X\) if
$$\lim_{n\to\infty}d(x_{n},x)=0=\lim_{n\to\infty}d(x,x_{n}). $$In this case x is called a dqb-limit of \((\{x_{n}\})\), and we write \((x_{n}\to x)\).
-
(2)
A sequence \((\{x_{n}\})\) in a dqb-metric space \((X,d)\) is called Cauchy if
$$\lim_{n,m\to\infty}d(x_{n},x_{m})=0=\lim _{n,m\to\infty}d(x_{m},x_{n}). $$ -
(3)
A dqb-metric space \((X,d)\) is complete if every Cauchy sequence in it is dqb-convergent in X.
Next, we begin with introducing the concept of a dqb-cyclic-Banach contraction.
Definition 2.8
Let A and B be nonempty subsets of a dislocated quasi-b-metric space \((X,d)\). A cyclic map \(T: A\cup B\to A\cup B\) is said to be a dqb-cyclic-Banach contraction if there exists \(k\in [0, 1)\) such that
for all \(x\in A\), \(y\in B\) and \(s\geq1\) and \(sk \leq1\).
Now we prove our main results.
Theorem 2.9
Let A and B be nonempty subsets of a complete dislocated quasi-b-metric space \((X,d)\). Let T be a cyclic mapping that satisfies the condition of a dqb-cyclic-Banach contraction. Then T has a unique fixed point in \(A\cap B \).
Proof
Let \(x\in A (\mathrm{fix})\) and, using the contractive condition of the theorem, we have
and
So,
and
where \(\alpha=\max\{d(Tx,x),d(x,Tx)\}\).
By using (2.2) and (2.3), we have \(d(T^{3}x,T^{2}x)\leq k^{2}\alpha\), and \(d(T^{2}x,T^{3}x)\leq k^{2}\alpha\).
For all \(n\in\mathbb{N}\), we get
and
Let \(n,m\in\mathbb{N}\) with \(m>n\), by using the triangular inequality, we have
for some \(\xi>m-n+1\).
Take \(n\to\infty\), we get \(d(T^{m}x,T^{n}x)\to0\).
Similarly, let \(n,m\in\mathbb{N}\) with \(m>n\), by using the triangular inequality, we have
Take \(n\to\infty\), we get \(d(T^{n}x,T^{m}x)\to0\). Thus \(T^{n}x\) is a Cauchy sequence.
Since \((X,d)\) is complete, we have \(\{T^{n}x\}\) converges to some \(z\in X\).
We note that \(\{T^{2n}x\}\) is a sequence in A and \(\{T^{2n-1}x\}\) is a sequence in B in a way that both sequences tend to the same limit z.
Since A and B are closed, we have \(z\in A\cap B\), and then \(A\cap B\neq{\emptyset}\).
Now, we will show that \(Tz=z\).
By using (2.1), consider
Taking limit as \(n\to\infty\) in the above inequality, we have
And so \(d(z,Tz)=kd(z,Tz)\), where \(0\leq k<1\). This implies that \(d(z,Tz)=0\).
Similarly, considering form (2.1), we get
Taking limit as \(n\to\infty\) in the above inequality, we have
And so \(d(Tz,z)=kd(Tz,z)\), where \(0\leq k<1\). This implies that \(d(Tz,z)=0\).
Hence \(d(z,Tz)=d(Tz,z)=0\), this implies that \(Tz=z\), that is, z is a fixed point of T.
Finally, to prove the uniqueness of a fixed point, let \(z^{*}\in X\) be another fixed point of T such that \(Tz^{*}=z^{*}\).
Then we have
On the other hand,
By forms (2.4) and (2.5), we obtain that \(d(z,z^{*})=d(z^{*},z)=0\), this implies that \(z^{*}=z\).
Therefore z is a unique fixed point of T. This completes the proof. □
Example 2.10
Let \(X=[-1,1]\) and \(T: A\cup B\to A\cup B\) defined by \(Tx=\frac{-x}{5}\). Suppose that \(A=[-1,0]\) and \(B=[0,1]\). Define the function \(d:X^{2}\to[0,\infty)\) by
We see that d is a dislocated quasi-b-metric on X.
Now, let \(x\in A\). Then \(-1\leq x\leq0\). So, \(0\leq\frac{-x}{5}\leq \frac{1}{5}\). Thus, \(Tx\in B\).
On the other hand, let \(x\in B\). Then \(0\leq x\leq1\). So, \(\frac {-1}{5}\leq\frac{-x}{5}\leq0\). Thus, \(Tx\in A\).
Hence the map T is cyclic on X because \(T(A)\subset B\) and \(T(B)\subset A\).
Next, we consider
so for \(\frac{1}{5}\leq k<1\).
Thus T satisfies the dqb-cyclic-Banach contraction of Theorem 2.9 and 0 is the unique fixed point of T.
Finally, we begin with introducing the concept of dqb-cyclic-Kannan mapping.
Definition 2.11
Let A and B be nonempty subsets of a dislocated quasi-b-metric space \((X,d)\). A cyclic map \(T: A\cup B\to A\cup B\) is called a dqb-cyclic-Kannan mapping if there exists \(r\in[0, \frac{1}{2})\) such that
for all \(x\in A\), \(y\in B\) and \(s\geq1\) and \(sr \leq\frac{1}{2}\).
In the next theorem, we will prove the fixed point theorem for a cyclic-Kannan mapping in a dislocated quasi-b-metric space.
Theorem 2.12
Let A and B be nonempty subsets of a complete dislocated quasi-b-metric space \((X,d)\). Let T be a cyclic mapping that satisfies the condition of a dqb-cyclic-Kannan mapping. Then T has a unique fixed point in \(A\cap B \).
Proof
Let \(x\in A (\mathrm{fix})\) and, using the contractive condition of the theorem, we have
so
And from (2.7) we have
so
where \(\beta=2d(x,Tx)\).
By using (2.7) and (2.8), we have
and
For all \(n\in\mathbb{N}\), we get
and
Let \(n,m\in\mathbb{N}\) with \(m>n\), by using the triangular inequality, we have
for some \(\xi>m-n+1\). Take \(n\to\infty\), we get \(d(T^{m}x,T^{n}x)\to0\).
Similarly, let \(n,m\in\mathbb{N}\) with \(m>n\), by using the triangular inequality, we have
Take \(n\to\infty\), we get \(d(T^{n}x,T^{m}x)\to0\). Thus \(T^{n}x\) is a Cauchy sequence.
Since \((X,d)\) is complete, we have \(\{(T^{n}x)\}\) converges to some \(z\in X\).
We note that \(\{T^{2n}x\}\) is a sequence in A and \(\{T^{2n-1}x\}\) is a sequence in B in a way that both sequences tend to the same limit z.
Since A and B are closed, we have \(z\in A\cap B\), and then \(A\cap B\neq{\emptyset}\).
Now, we will show that \(Tz=z\).
By using (2.6), consider
Taking limit as \(n\to\infty\) in the above inequality, we have
Since \(0\leq r<\frac{1}{2}\), we have \(d(z,Tz)=0\).
Similarly, considering form (2.6), we get
Taking limit as \(n\to\infty\) in the above inequality, we have
Since \(d(z,Tz)=0\), we have \(d(z,Tz)=0\).
Hence \(d(z,Tz)=d(Tz,z)=0 \Rightarrow Tz=z\) and z is a fixed point of T.
Finally, to prove the uniqueness of a fixed point, let \(z^{*}\in X\) be another fixed point of T such that \(Tz^{*}=z^{*}\).
Then we have \(d(z,z) =d(z^{*},z^{*})=0\), because by assumption
On the other hand,
By forms (2.9) and (2.10), we obtain that \(d(z,z^{*})=d(z^{*},z)=0 \Rightarrow z^{*}=z\).
Therefore z is a unique fixed point of T. This completes the proof. □
Example 2.13
Let \(X=[-1,1]\) and \(T:X\to X\) defined by \(Tx=\frac{-x}{7}\). Suppose that \(A=[-1,0]\) and \(B=[0,1]\). Define the function \(d:X^{2}\to[0,\infty)\) by
We see that d is a dislocated quasi-b-metric on X.
Now, let \(x\in A\). Then \(-1\leq x\leq0\). So, \(0\leq\frac{-x}{7}\leq\frac{1}{7}\). Thus, \(Tx\in B\).
On the other hand, let \(x\in B\). Then \(0\leq x\leq1\). So, \(\frac {-1}{7}\leq\frac{-x}{7}\leq0\). Thus, \(Tx\in A\).
Hence the map T is cyclic on X because \(T(A)\subset B\) and \(T(B)\subset A\).
Next, we consider
so for \(\frac{2}{23}\leq r<\frac{1}{2}\).
Thus T satisfies the dqb-cyclic-Banach contraction of Theorem 2.12 and 0 is the unique fixed point of T.
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Klin-eam, C., Suanoom, C. Dislocated quasi-b-metric spaces and fixed point theorems for cyclic contractions. Fixed Point Theory Appl 2015, 74 (2015). https://doi.org/10.1186/s13663-015-0325-2
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DOI: https://doi.org/10.1186/s13663-015-0325-2