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Fixed point theorems for a class of generalized weak cyclic compatible contractions
Fixed Point Theory and Applicationsvolume 2018, Article number: 13 (2018)
Abstract
In this manuscript, we establish a coincidence point and a unique common fixed point theorem for \((\psi ,\varphi )\)weak cyclic compatible contractions. We also present a fixed point theorem for a class of Λweak cyclic compatible contractions via altering distance functions. Our results extend and improve some wellknown results in the literature. We provide examples to analyze and illustrate our main results.
Introduction and preliminaries
The Banach contraction principle [1] is one of the most powerful and useful tools in modern analysis. Over time, this principle has been extended and improved in many ways and a variety of fixed point theorems have been obtained. In 2003, one of the more notable generalizations of the Banach contraction principle was introduced via cyclic contraction by Kirk et al. [2]. Following the publication of [2], many fixed point theorems for cyclic contractive mappings have been obtained. For more results on cyclic maps, we refer the reader to [3–9] and the references therein. Here, we present some essential definitions.
Definition 1.1
(see [2])
Let A, B be nonempty subsets of a set X and let \(\mathscr{U} : A \cup B \to A \cup B\). \(\mathscr{U}\) is called a cyclic map, if \(\mathscr{U}(A)\subseteq B\) and \(\mathscr{U}(B) \subseteq A\).
Throughout this manuscript, we assume that \(\mathbb{R}^{+}=[0,\infty )\), \(\mathbb{N}\) = the set of all positive integers.
Definition 1.2
(see [10])
Let \((X,d)\) be a complete metric space and let \(\mathscr{U},\mathscr{V}:X\rightarrow X\) be selfmappings. Then \(\mathscr{U}\) and \(\mathscr{V}\) are said to be weakly compatible if \(\mathscr{U}x = \mathscr{V}x\) implies \(\mathscr{U}\mathscr{V}x = \mathscr{V}\mathscr{U}x\).
Definition 1.3
(see [11])
A function \(\eta :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+} \) is called an altering distance function if the following conditions are satisfied:

1.
\(\eta (0)=0\);

2.
η is monotonically nondecreasing;

3.
η is continuous.
We will denote the set of all altering distance functions by Λ.
The concepts of \((\psi ,\varphi )\)weak contractions, weakly compatible maps and altering distance functions is interesting, though brief. These concepts were widely used in the construction of existence theorems and many results, a number of applications have been obtained; see [12–17] for examples.
Below, we provide necessary definitions.
Definition 1.4
(see [6])
Let X be a nonempty set and \(\{d_{\alpha }:\alpha \in (0,1]\}\) a family of the mapping \(d_{\alpha }\) of \(X\times X\) into \(\mathbb{R}^{+}\). Then \((X,d_{\alpha })\) is called a generating space of a bquasimetric family (abbreviated as \(G_{bq}\)family), if it satisfies the following conditions, for any \(x,y,z \in X\) and \(s \ge 1\):

(a)
\(d_{\alpha }(x,y)=0\) if and only if \(x=y\).

(b)
\(d_{\alpha }(x,y)=d_{\alpha }(y,x)\).

(c)
For any \(\alpha \in (0,1]\) there exists \(\beta \in (0,\alpha ]\) such that \(d_{\alpha }(x,z)\leq s[d_{\beta }(x,y)+d_{\beta }(y,z)]\).

(d)
For any \(x,y\in X, d_{\alpha }(x,y)\) is nonincreasing and left continuous in α.
Definition 1.5
(see [6])
Let X be a nonempty set and \(\{d_{\alpha }:\alpha \in (0,1]\}\) a family of the mapping \(d_{\alpha }\) of \(X\times X\) into \(\mathbb{R}^{+}\). Then \((X,d_{\alpha })\) is called a generating space of bdislocated metric family (abbreviated as \(G_{bd}\)family), if it satisfies the following conditions, for any \(x,y,z \in X\) and \(s \ge 1\):

(a)
\(d_{\alpha }(x,y)=0\) implies \(x=y\).

(b)
\(d_{\alpha }(x,y)=d_{\alpha }(y,x)\).

(c)
For any \(\alpha \in (0,1]\) there exists \(\beta \in (0,\alpha ]\) such that \(d_{\alpha }(x,z)\leq s[d_{\beta }(x,y)+d_{\beta }(y,z)]\).

(d)
For any \(x,y\in X, d_{\alpha }(x,y)\) is nonincreasing and left continuous in α.
The construction of topological concepts of the above spaces can be found in [6, 18]. Recently, Kumari and Panthi [7] introduced cyclic compatible contractions and established fixed point theorems in the generating space of a bquasimetric family \((X,d_{\alpha })\).
Definition 1.6
(see [7])
Let A, B be nonempty subsets of a \(G_{bq}\)family \((X,d_{\alpha })\) and let \(\mathscr{U}, \mathscr{V}: A\cup B \to A\cup B\) be cyclic mappings such that \(\mathscr{U}(X)\subset \mathscr{V}(X)\). Then \(\mathscr{U}, \mathscr{V}\) are said to be cyclic compatible contraction, if for some \(x \in A\), there exists a \(\gamma \in (0,1)\) such that
for all \(n\in \mathbb{N}\) and \(y\in B\).
In this paper, motivated and inspired by the above definitions, we investigate the weak cyclic compatible contractions via \((\psi ,\varphi )\)weak contractions and altering distance functions.
Main results
We begin this section by introducing the following definition.
Definition 2.1
Let A and B be nonempty subsets of a \(G_{bd}\)family \((X,d_{\alpha })\). Suppose \(\mathscr{U},\mathscr{V}:A\cup B\rightarrow A\cup B\) are cyclic mappings with \(\mathscr{U}(X)\subset \mathscr{V}(X)\). Then \(\mathscr{U}\), \(\mathscr{V}\) are called \((\psi ,\varphi )\)weak cyclic compatible contractions, if for some \(x\in A\)
where \(\psi ,\varphi :[0,\infty )\rightarrow [0,\infty )\) are both continuous and monotone nondecreasing functions with \(\psi (t)=\varphi (t)=0\) if and only if \(t=0\) and \(n\in \mathbb{N}, y\in B\).
We state and prove our main results.
Theorem 2.2
Let A and B be nonempty closed subsets of a complete \(G_{bd}\)family \((X,d_{\alpha })\) and let \(\mathscr{U},\mathscr{V}:A\cup B\rightarrow A\cup B\) be cyclic mappings with \(\mathscr{U}(X)\subset \mathscr{V}(X)\) and \(\mathscr{V}(X)\) closed in X. Suppose \(\mathscr{U}\), \(\mathscr{V}\) are \((\psi ,\varphi )\)weak cyclic compatible contractions, then \(\mathscr{U}\) and \(\mathscr{V}\) have a point of coincidence and a unique common fixed point in \(A\cap B\).
Proof
Let \(x_{0}\in X\) be fixed. As \(\mathscr{U}(X)\subset \mathscr{V}(X)\), we may choose \(x_{1}\in X\) such that
Hence we can define the sequence \(\{x_{n}\}\) in X by \(\mathscr{U}^{n}x_{0}=\mathscr{U}x_{n}=\mathscr{V}x_{n+1}=x_{n}\) for \(n\in \mathbb{N}\cup \{0\}\).
Now consider
This implies that
Similarly, we have
Inductively, we have
Thus the sequence \(\{d_{\alpha }(x_{n},x_{n+1})\}\) is nonincreasing and hence it is convergent. So, there exists \(\kappa \geq 0\) such that \(\lim _{n\rightarrow \infty }d_{\alpha }(x_{n},x_{n+1})=\kappa \).
From (2), we have
Now taking the limit as \(n\rightarrow \infty \), we get
This is a contradiction, unless \(\kappa =0\). Thus
For \(n,m\in \mathbb{N}, m>n\), consider
Letting \(n,m\rightarrow \infty \), we get
Therefore \(\{\mathscr{U}^{n}x_{0}\}\) is a Cauchy sequence. Since \((X,d_{\alpha })\) is a complete \(G_{bd}\)family, there exist sequences \(\{\mathscr{U}^{2n}x_{0}\}\) in A and \(\{\mathscr{U}^{2n1}x_{0}\}\) in B such that \(\lim _{n\rightarrow \infty }\mathscr{U}^{2n}x_{0}\rightarrow u\) and \(\lim _{n\rightarrow \infty }\mathscr{U}^{2n1}x_{0}\rightarrow u\). Since A and B are closed in X, \(u\in A\cap B\). Since \(\mathscr{V}(X)\) is closed in X, there exists z in X such that \(\mathscr{V}z=u\). Now we shall prove that \(\mathscr{U}z=u\). Consider
By taking the limit as \(n\rightarrow \infty \), we get \(\psi (d_{\alpha }(u,\mathscr{U}z))=0\). Thus \(d_{\alpha }(u,\mathscr{U}z)=0\). Then \(\mathscr{U}z=u\). Hence \(\mathscr{U}z=\mathscr{V}z=u\). So u is a coincidence point of \(\mathscr{U}\) and \(\mathscr{V}\). From weak compatibility, we get
Now we prove that \(\mathscr{V}u=u=\mathscr{U}u\). We assume \(u\neq \mathscr{U}u\), then
That is a contradiction. Therefore
From (7) and (9), we have \(\mathscr{U}u=\mathscr{V}u=u\). Therefore u is a common fixed point of \(\mathscr{U}\) and \(\mathscr{V}\). To prove uniqueness, suppose v is another fixed point of \(\mathscr{U}\) and \(\mathscr{V}\) such that \(u\neq v\), then
This is a contradiction. Hence \(u=v\). This completes our proof. □
Remark 2.3
We will obtain special cases of Theorem 2.2, if we

1.
replace a \(G_{bq}\)family \((X,d_{\alpha })\) by a \(G_{q}\)family, according to Definition 1.4 above, by putting \(s=1\);

2.
replace a \(G_{bq}\)family \((X,d_{\alpha })\) by a \(b_{d}\)metric space, and taking d instead of \(d_{\alpha }\);

3.
replace a \(G_{bq}\)family \((X,d_{\alpha })\) by a complete dislocated metric space, by taking d instead of \(d_{\alpha }\) and letting \(s=1\).
For more details of \(G_{q}\)family, \(b_{d}\)metric space and a complete dislocated metric space, we refer the reader to [6].
Example 2.4
Let \(A=B=X=[0,1]\). Define \(b_{d}:X\times X\rightarrow \mathbb{R}^{+}\) by \(b_{d}(x,y)=(x+y)^{2}\). Then \((X,b_{d})\) is a \(b_{d}\)metric space with \(s=2\), but not a dislocated metric space. Define
and
Clearly \(\mathscr{U}(X)\subset \mathscr{V}(X)\). Define \(\psi (t)=2t\) and \(\varphi (t)=t\).
Fix any \(x\in [0,1]\), we have
For any \(y\in [0,1]\),
Consider
Case (i). If \(0\leq y<\frac{1}{2}\), \(\mathscr{V}y=0\), we get
Case (ii). If \(\frac{1}{2}\leq y\leq 1, \mathscr{U}y=0, \mathscr{V}y=\frac{1}{3}\), we get
and
From both cases, we obtain
Thus \(\mathscr{U}\), \(\mathscr{V}\) are \((\psi ,\varphi )\)weak cyclic compatible contractions. All the conditions of Theorem 2.2 hold true, and \(\mathscr{U}\) and \(\mathscr{V}\) have a unique common fixed point. Here \(u=0\) is the unique common fixed point of \(\mathscr{U}\) and \(\mathscr{V}\).
Example 2.5
Let \(A=B=X=[0,20]\). Define \(d:X\times X\rightarrow \mathbb{R}^{+}\) by \(d(x,y)=x+y\). Then \((X,d)\) is a complete dislocated metric space.
Define
and
Clearly, \(\mathscr{U}(X)\subset \mathscr{V}(X)\). Define \(\psi (t)=2t\) and \(\varphi (t)=t\).
Fix any \(x\in (3,20]\), we have
For any \(y\in [0,20]\).
Consider
Case (i). If \(y=0 \), \(\mathscr{U}y=0, \mathscr{V}y=0\), we have
and
Hence
Case (ii). If \(0< y\leq 3\), \(\mathscr{U}y=2 \) and \(\mathscr{V}y=y+10\). Then we have
and
Therefore, for \(0< y\leq 3\),
Case (iii). If \(3< y\leq 20\), \(\mathscr{U}y=0\) and \(\mathscr{V}y=y1\). Then we have
and
Hence
Therefore, from all cases, we have
Thus \(\mathscr{U}\), \(\mathscr{V}\) are \((\psi ,\varphi )\)weak cyclic compatible contractions. All the conditions of Theorem 2.2 hold true, and \(\mathscr{U}\) and \(\mathscr{V}\) have a unique common fixed point. Here \(u=0\) is the unique common fixed point of \(\mathscr{U}\) and \(\mathscr{V}\).
If we take \(\mathscr{V}=\mathscr{U}\) and \(\mathscr{V}=I\) in Definition 1.6 and Definition 2.1, we obtain the following definition.
Definition 2.6
Let \(\mathscr{U}\) be a cyclic mapping; then

(1)
\(\mathscr{U}\) is called a cyclic idle contraction \(\Leftrightarrow d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y)\leq \gamma d_{\alpha }(\mathscr{U}^{2n1}x,\mathscr{U}y)\);

(2)
\(\mathscr{U}\) is called a \((\psi ,\varphi )\)weak cyclic idle contraction \(\Leftrightarrow \psi (d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y))\leq \psi (d_{\alpha }(\mathscr{U}^{2n1}x,\mathscr{U}y))\varphi (d_{\alpha }(\mathscr{U}^{2n1}x,\mathscr{U}y))\);

(3)
\(\mathscr{U}\) is called a \((\psi ,\varphi )\)weak cyclic orbital contraction \(\Leftrightarrow \psi (d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y))\leq \psi (d_{\alpha }(\mathscr{U}^{2n1}x,y))\varphi (d_{\alpha }(\mathscr{U}^{2n1}x,y))\),
where \(\psi ,\varphi :[0,\infty )\rightarrow [0,\infty )\) are both continuous and monotone nondecreasing functions with \(\psi (t)=\varphi (t)=0\) if and only if \(t=0\).
Theorem 2.7
Let A and B be nonempty closed subsets of a complete \(G_{bd}\)family \((X,d_{\alpha })\) and let \(\mathscr{U}:A\cup B\rightarrow A\cup B\) be a \((\psi\mbox{}\varphi )\)weak cyclic idle contraction. Assume that \(\mathscr{U}(X)\) is closed in X. Then \(\mathscr{U}\) has a unique fixed point in \(A\cap B\).
Proof
Take \(\mathscr{U}=\mathscr{V}\) in Theorem 2.2. □
Theorem 2.8
Let A and B be nonempty closed subsets of a complete \(G_{bd}\)family \((X,d_{\alpha })\) and let \(\mathscr{U}:A\cup B\rightarrow A\cup B\) be a \((\psi\mbox{}\varphi )\)weak cyclic orbital contraction. Assume that \(\mathscr{U}(X)\) is closed in X. Then \(\mathscr{U}\) has a unique fixed point in \(A\cap B\).
Proof
Take \(\mathscr{V}=I\) in Theorem 2.2. □
Next we will state and prove a fixed point theorem via altering distance functions. We introduce the following definition.
Definition 2.9
Let A and B be nonempty closed subsets of the generating space of a bquasimetric family \((X,d_{\alpha })\). Suppose \(\mathscr{U}, \mathscr{V} : A\cup B \to A\cup B\) are cyclic mappings such that \(\mathscr{U}(X)\subset \mathscr{V}(X)\). Then \(\mathscr{U}\), \(\mathscr{V}\) are said to be Λcyclic compatible contractions, if for some \(x\in A\), there exists \(\gamma \in (0,1)\) such that
for all \(n\in \mathbb{N},y\in B\) and \(\eta \in \Lambda \).
Theorem 2.10
Let A and B be nonempty closed subsets of the generating space of a complete bquasimetric family \((X,d_{\alpha })\). Let \(\mathscr{U},\mathscr{V}:A \cup B \to A \cup B\) be cyclic mappings such that \(\mathscr{U}(X)\subset \mathscr{V}(X)\). Suppose

1.
\(\mathscr{U}\), \(\mathscr{V}\) are Λcyclic compatible contractions,

2.
\(\mathscr{U}\), \(\mathscr{V}\) are weakly compatible,

3.
\(\mathscr{V}(X)\) is a closed subset of X.
Then \(\mathscr{U}\) and \(\mathscr{V}\) have a point of coincidence in \(A\cap B\). Moreover, \(\mathscr{U}\) and \(\mathscr{V}\) have a unique common fixed point in \(A\cap B\).
Proof
Take \(x=x_{0}\in A\). Since \(\mathscr{U}(X)\subset \mathscr{V}(X)\), we may choose \(x_{1}\in X\) such that \(\mathscr{U}x_{0}=\mathscr{V}x_{1}\).
Hence we can define the sequence \(\{x_{n}\}\) in X by \(\mathscr{U}^{n}x_{0}=\mathscr{U}x_{n}=\mathscr{V}x_{n+1}=x_{n+1}\) for \(n\in \mathbb{N}\cup \{0\}\). Then \(\{x_{2n}\}\in A\) and \(\{x_{2n1}\}\in B\). Thus we have
Similarly,
In general, we have
Inductively, for each \(n\in \mathbb{N}\), we get
Since \(0\leq \gamma <1\), letting \(n\rightarrow \infty \), we get \(\lim _{n\rightarrow \infty }\eta (d_{\alpha }(x_{n},x_{n+1}))=0\). From the definition of altering distance functions, we get
Therefore
or
Now we claim that \(\{x_{n}\}\) is a Cauchy sequence. According to the definition of generating space of a bquasimetric family \((X,d_{\alpha })\), and for \(n,m \in \mathbb{N}, m >n\), we have
Letting \(n,m\rightarrow \infty \), and from (14), we have \(\lim _{n,m\rightarrow \infty }d_{\alpha }(x_{n},x_{m})=0\) for all \(\alpha \in (0,1]\). Therefore \(\{x_{n}\}\) is a Cauchy sequence in X. Since X is complete, there exist sequences \(\{\mathscr{V}^{2n}x_{0}\}\) in A and \(\{\mathscr{V}^{2n1}x_{0}\}\) in B such that both sequences converge to some ω in X. Since A and B are closed in X, \(\omega \in A\cap B\).
Since \(\mathscr{V}(X)\) is closed in X, there exists z in X such that
Consider
Letting \(n\rightarrow \infty \), we get \(\lim _{n\rightarrow \infty }\eta (d_{\alpha }(\mathscr{U}^{2n}x_{0},\mathscr{U}z))=0\). From the definition of altering distance functions, we get
This implies
Therefore
From (16) and (17), it follows that \(\mathscr{U}z=\mathscr{V}z=\omega \). Thus ω is a point of coincidence of \(\mathscr{U}\) and \(\mathscr{V}\). By weak compatibility, we get \(\mathscr{V}\omega =\mathscr{U}\omega \). Now we prove \(\mathscr{V}\omega =\omega \). Suppose \(\mathscr{V}\omega \neq \omega \), then consider
which is a contradiction. Hence \(\mathscr{V}\omega =\omega \). Therefore \(\mathscr{U}\omega =\mathscr{V}\omega =\omega \). To prove uniqueness, suppose that \(\omega _{1}\) and \(\omega _{2}\) are two common fixed points of \(\mathscr{U}\) and \(\mathscr{V}\). Assume that \(\omega _{1}\neq \omega _{2}\). Then \(\mathscr{U}\omega _{1}=\mathscr{V}\omega _{1}=\omega _{1}\) and \(\mathscr{U}\omega _{2}=\mathscr{V}\omega _{2}=\omega _{2}\).
Consider
which is a contradiction. Hence \(\omega _{1} = \omega _{2}\) and our proof is finished. □
Remark 2.11
We will obtain special cases of Theorem 2.10 above, if we

1.
replace \(G_{bq}\)family \((X,d_{\alpha })\) by \(G_{q}\)family, according to Definition 1.4, by putting \(s=1\);

2.
replace \(G_{bq}\)family \((X,d_{\alpha })\) by bmetric space and taking \(d_{\alpha } = d\);

3.
replace \(G_{bq}\)family \((X,d_{\alpha })\) by complete metric space and taking \(d_{\alpha }=d\) and \(s=1\).
Example 2.12
Let \(A=B=X=[0,1]\). Define \(d:X\times X\rightarrow \mathbb{R}^{+}\) by \(d(x,y)=(xy)^{2}\). This is a bmetric space with \(s=2\), not the usual metric space, since \(d(0,1)\nleq d(0,\frac{1}{2})+d(\frac{1}{2},1)\). Define
and
Clearly \(\mathscr{U}(X)\subset \mathscr{V}(X)\). Define \(\eta (t)=t^{2}\). For any \(x\in [0,1]\), we have
For any \(y\in [0,1]\), we have
Case (i). If \(0\leq y<\frac{1}{2}\), \(\mathscr{U}y=0\), we have
Case (ii). If \(\frac{1}{2}\leq y\leq 1, \mathscr{U}y=0\), we have
and
So
Therefore, from both cases, we get
Thus \(\mathscr{U}\), \(\mathscr{V}\) are Λcyclic compatible contractions. All the conditions of Theorem 2.10 hold true and \(\mathscr{U}\), \(\mathscr{V}\) have a unique common fixed point. Here \(\omega =0\) is the unique common fixed point of \(\mathscr{U}\) and \(\mathscr{V}\).
Example 2.13
Let \(A=B=X=[0,1]\). Define \(d:X\times X\rightarrow \mathbb{R}^{+}\) by \(d(x,y)=\vert xy \vert \). So \((X,d)\) is a complete metric space. Define
and
Clearly \(\mathscr{U}(X)\subset \mathscr{V}(X)\). Define \(\eta (t)=t\).
For any \(x\in [0,1]\), we have
For any \(y\in [0,1]\),
Consider
Case (i). If \(0\leq y<\frac{1}{2}\), \(\mathscr{U}y=0, \mathscr{V}y=0\). Then we have
Case (ii). If \(\frac{1}{2}\leq y\leq 1, \mathscr{U}y=0, \mathscr{V}y=\frac{1}{9}\). Then we have
and
Hence, from both cases we conclude that \(\eta (d(\mathscr{U}^{2n}x,\mathscr{U}y))\leq \gamma \eta (d(\mathscr{U}^{2n1}x,\mathscr{V}y))\), for all \(0\leq \gamma <1\). Thus \(\mathscr{U}\), \(\mathscr{V}\) are Λcyclic compatible contractions. All the conditions of Theorem 2.10 hold true. Here \(\omega =0\) is the unique common fixed point of \(\mathscr{U}\) and \(\mathscr{V}\).
In view of Definition 2.6, we introduce the following definition.
Definition 2.14
Let \(\mathscr{U}\) be a cyclic mapping, then

(1)
\(\mathscr{U}\) is called Λcyclic idle contraction. \(\Leftrightarrow \eta (d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y))\leq \gamma \eta (d_{\alpha }(\mathscr{U}^{2n1}x,\mathscr{U}y))\).

(2)
\(\mathscr{U}\) is called Λcyclic orbital contraction. \(\Leftrightarrow \eta (d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y))\leq \gamma \eta (d_{\alpha }(\mathscr{U}^{2n1}x,y))\).
Theorem 2.15
Let A and B be nonempty closed subsets of the generating space of a complete bquasimetric family \((X,d_{\alpha })\). Suppose \(\mathscr{U} : A \cup B \to A \cup B\) is a Λcyclic idle contraction, then \(\mathscr{U}\) has a unique fixed point in \(A\cap B\).
Proof
Take \(\mathscr{U}=\mathscr{V}\) in Theorem 2.10. □
Theorem 2.16
Let A and B be nonempty closed subsets of the generating space of a complete bquasimetric family \((X,d_{\alpha })\). Suppose \(\mathscr{U} : A \cup B \to A \cup B\) is a Λcyclic orbital contraction, then \(\mathscr{U}\) has a unique fixed point in \(A\cap B\).
Proof
Take \(\mathscr{V}=I\) in Theorem 2.10. □
Discussion
In this manuscript, we establish fixed point theorems in generating space of a \(G_{bd}\)family \((X,d_{\alpha })\), and a bquasimetric family \((X,d_{\alpha })\). One can see that Theorem 2.2 and Theorem 2.10 are generalizations of Theorem 2.2 obtained in [7] in the setting of a complete \(G_{bd}\)family \((X,d_{\alpha })\), and a complete bquasimetric family \((X,d_{\alpha })\), respectively. Besides, we introduce other definition of weak cyclic contractions (see Definition 2.6), and obtain fixed point theorems for those contractions (Theorems 2.7 and 2.8). After Theorem 2.10, we also introduce other definition of Λcyclic contractions (Def. 2.14) and obtain fixed point theorems (Theorems 2.15 and 2.16). Our main results extend and improve some wellknown results in the existing literature. However, in recent remarkable work of Lau and Zhang in [19, 20], the authors studied fixed point properties of semigroups of nonexpansive mappings, nonlinear mappings and amenability. In relation to Theorem 2.2 and Theorem 2.10, we pose the following open problem at the end.
Problem
Can Theorem 2.2 and Theorem 2.10 be generalized to a family or a commuting or amenable semigroup of maps?
Conclusions
In this manuscript, we introduce a new class of generalized \((\psi ,\varphi )\)weak cyclic compatible contractions (Definition 2.1) and prove a coincidence point and a unique common fixed point theorem for these new contractions in a complete \(G_{bd}\)family \((X,d_{\alpha })\). We also introduce a new class of Λcyclic compatible contractions via altering distance functions(Definition 2.9), and prove an existence theorem for this new class of generalized contractions in the generating space of a bquasimetric family \((X,d_{\alpha })\). Our results extend and improve the results obtained in [7] and some wellknown results in the literature. We provide examples to illustrate and support our results and we also pose a problem at the end.
References
 1.
Banach, S.: Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundam. Math. 3, 133–181 (1922)
 2.
Kirk, W.A., Srinivasan, P.S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4(1), 79–89 (2003)
 3.
Sintunavarat, W., Kumam, P.: Common fixed point theorem for cyclic generalized multivalued contraction mappings. Appl. Math. Lett. 25, 1849–1855 (2012)
 4.
Yamaod, O., Sintunavarat, W.: Some fixed point results for generalized contraction mappings with cyclic \((\alpha ,\beta )\)admissible mapping in multiplicative metric spaces. J. Inequal. Appl. 2014, Article ID 488 (2014)
 5.
Zoto, K., Kumari, P.S., Hoxha, E.: Some fixed point theorems and cyclic contractions in dislocated and dislocated quasimetric spaces. Am. J. Numer. Anal. 2(3), 79–84 (2014)
 6.
Kumari, P.S., Panthi, D.: Cyclic contractions and fixed point theorems on various generating spaces. Fixed Point Theory Appl. 2015, Article ID 153 (2015)
 7.
Kumari, P.S., Panthi, D.: Cyclic compatible contraction and related fixed point theorems. Fixed Point Theory Appl. 2016, Article ID 28 (2016)
 8.
Dung, N.V., Radenovic, S.: Remarks on theorems for cyclic quasicontractions in uniformly convex Banach spaces. Kragujev. J. Math. 40(2), 272–279 (2016)
 9.
Han, B.T.N., Hieu, N.T.: A fixed point theorem for generalized cyclic contractive mappings in bmatric spaces. Facta Univ., Ser. Math. Inform. 31(2), 399–415 (2016)
 10.
Jungck, G.: Common fixed points for noncontinuous nonself maps on nonmetric spaces. Far East J. Math. Sci. 4, 199–215 (1996)
 11.
Khan, M.S., Swaleh, M., Sessa, S.: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30(1), 1–9 (1984)
 12.
Altun, I., Turkoglu, D., Rhoades, B.E.: Fixed points of weakly compatible maps satisfying a general contractive condition of integral type. Fixed Point Theory Appl. 2007, Article ID 17301 (2007)
 13.
Erduran, A., Altun, I.: Fixed point theorems for weakly contractive mapping on gmetric spaces and a homotopy result. Int. J. Nonlinear Anal. Appl. 3(1), 1–8 (2012)
 14.
Lee, G.M., Lee, B.S., Jung, J.S., Chang, S.S.: Minimization theorems and fixed point theorems in generating spaces of quasimetric family. Fuzzy Sets Syst. 101, 143–152 (1990)
 15.
Kumari, P.S., Kumar, V.V., Sarma, I.R.: Common fixed point theorems on weakly compatible maps on dislocated metric spaces. Math. Sci. 6, Article ID 71 (2012). https://doi.org/10.1186/22517456671
 16.
Kumari, P.S., Kumar, V.V., Sarma, I.R.: New version for Hardy and Rogers type mapping in dislocated metric space. Int. J. Basic Appl. Sci. 1(4), 609–617 (2012)
 17.
Karapinar, E., Ramaguera, S., Kenan, T.: Fixed points for cyclic orbital generalized contractions on complete metric spaces. Cent. Eur. J. Math. 11(3), 552–560 (2013)
 18.
Kumari, P.S., Sarwar, M.: Some fixed point theorems in generating space of bquasimetric family. SpringerPlus 5, Article ID 268 (2016). https://doi.org/10.1186/s4006401618674
 19.
Lau, A.T., Zhang, Y.: Fixed point properties of semigroups of nonexpansive mappings. J. Funct. Anal. 254(10), 2534–2554 (2008)
 20.
Lau, A.T., Zhang, Y.: Fixed point properties for semigroups of nonlinear mappings and amenability. J. Funct. Anal. 263(10), 2949–2977 (2012)
Acknowledgements
The authors would like to express their sincere gratitude to Mr. Anand Prabhakar for his invaluable suggestions and motivation and to the referees for their helpful comments and suggestions.
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Correspondence to J. Nantadilok.
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MSC
 54H25
 47H10
Keywords
 Cyclic maps
 bquasimetric family
 bdislocated metric family
 \((\psi\mbox{}\varphi)\)weak cyclic compatible contraction
 Λcyclic compatible contraction