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 Open Access
Cyclic compatible contraction and related fixed point theorems
 Panda Sumati Kumari^{1}Email author and
 Dinesh Panthi^{2}
https://doi.org/10.1186/s1366301605218
© Kumari and Panthi 2016
 Received: 18 November 2015
 Accepted: 3 March 2016
 Published: 11 March 2016
Abstract
In this work, we introduce the concept of a cyclic compatible contraction and prove related fixed point theorems in the generating space of a bquasimetric family.
Keywords
 cyclic compatible contraction
 generating space of bquasimetric family
 generating space of quasimetric family
 bmetric space
 weakly compatible
 point of coincidence and fixed point
MSC
 47H10
 54H25
1 Introduction and preliminaries
Throughout this paper \(\mathbb{R}\), \(\mathbb{R}^{+}\), and \(\mathbb{N}\) represent the set of real numbers, the set of positive real numbers, and the set of positive integers, respectively.
In 1922, Banach introduced the contraction mapping theorem which is famously known as the Banach contraction principle. It is also known that Banach’s contraction principle is one of the pivotal result of metric fixed point theory.
This theorem ensures the existence and uniqueness of fixed points of certain selfmaps of metric spaces, and it gives a useful constructive method to find those fixed points.
The traditional Banach contraction principle has been extended and generalized in wide directions.

In 1968, Kannan fixed point theorem [2]: If \((X,d)\) is a complete metric space and \(T:X\rightarrow X\) is a selfmapping such thatfor all \(x,y\in X\), where \(0\leq\beta<\frac{1}{2}\), then T has a unique fixed point.$$d(Tx,Ty)\leq\beta\bigl[d(x,Tx)+d(y,Ty)\bigr], $$

In 1971, Reich fixed point theorem [3]: If \((X,d)\) is a complete metric space and \(T:X\rightarrow X\) is a selfmapping such thatfor all \(x,y\in X\), where α, β, γ are nonnegative constants with \(\alpha+\beta+\gamma<1\), then T has a unique fixed point.$$d(Tx,Ty)\leq\alpha d(x,y)+\beta d(x,Tx)+\gamma d(y,Ty), $$

In 1971, Ciric fixed point theorem [4]: If \((X,d)\) is a complete metric space and \(T:X\rightarrow X\) is a selfmapping such thatfor all \(x,y\in X\), where α, β, γ, δ are nonnegative constants with \(\alpha+\beta+\gamma+2\delta<1\), then T has a unique fixed point.$$d(Tx,Ty)\leq\alpha d(x,y)+\beta d(x,Tx)+\gamma d(y,Ty)+\delta \bigl[d(x,Ty)+d(y,Tx)\bigr], $$
The above named fixed point theorems are undoubtedly the most valuable theorems in nonlinear phenomena. Many fixed point theorems concerning the above named theorems and their generalizations have been given by several authors (for example, see [5–12]).
The Banach contraction principle appears everywhere in mathematics: Analysis, geometry, statistics, graph theory, and logic programming are some of the fields in which the Banach contraction principle and/or generalizations play an important role. In the literature, we can say that the elegant generalizations below are the standard generalizations of the Banach contraction principle in 20th century.
In 2003, Kirk et al., generalized the Banach contraction principle by using cyclic map and proved below fixed point theorem.
Theorem 1.1
[13]
In 2012, Wardowski [14] introduced the Fcontraction and generalized the Banach contraction principle in a new way.
 (F1)
F is strictly increasing, i.e. for all \(\alpha ,\beta\in \mathbb{R}^{+}\) such that \(\alpha<\beta\), \(F(\alpha)< F(\beta)\);
 (F2)
for each sequence \(\{\alpha_{n}\}_{n\in\mathbb{N}}\) of positive numbers \(\lim_{n\rightarrow\infty}\alpha_{n}=0\) iff \(\lim_{n\rightarrow\infty}F(\alpha_{n})= \infty\);
 (F3)
there exists \(k\in(0,1)\) such that \(\lim_{\alpha \rightarrow0^{+}}\alpha^{k}F(\alpha)=0\).
Theorem 1.2
[14]
Let \((X,d)\) be a complete metric space and let \(T:X\rightarrow X\) be an Fcontraction, then T has a unique fixed point \(x^{\ast}\in X\) and for every \(x_{0}\in X\) a sequence \(\{T^{n}x_{0}\}_{n\in\mathbb{N}}\) is convergent to \(x^{\ast}\).
 (F3′):

F is continuous on \((0,\infty)\).
The authors of [15] generalized the standard Fcontraction and proved a fixed point result with the above new setup.
The concept of weakly compatible maps was introduced by Jungck [16].
Definition 1.3
[16]
Let \((X,d)\) be a complete metric space and \(T,S\) be two mappings. Then T and S are said to be weakly compatible if they commute at their coincidence point x, that is, \(Tx = Sx\) implies \(TSx = STx\).
The above concept is used to prove existence theorems in common fixed point theory. However, the study of common fixed points of weakly compatible maps is very impressive. In the literature one can find some interesting papers concerning cyclic contraction, Fcontraction and weakly compatible mapping (see for example [17–26]).
Definition 1.4
[18]
 (d_{1}):

the family of self distances are zero: \(d_{\alpha}(x,x)=0\);
 (d_{2}):

the family of distances are symmetric: \(d_{\alpha}(x,y)=d_{\alpha}(y,x)\);
 (d_{3}):

the family of positive distances between distinct points: \(d_{\alpha}(x,y)=d_{\alpha}(y,x)=0\) implies \(x=y\);
 (d_{4}):

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_{5}):

for any \(x,y\in X\), \(d_{\alpha}(x,y)\) is nonincreasing and left continuous in α.
 (i)
the generating space of the bquasimetric family (shortly, the \(G_{bq}\)family) if \(d_{\alpha}\) satisfies (d_{1}) through (d_{5});
 (ii)
the generating space of the bdislocated metric family (shortly, the \(G_{bd}\)family) if \(d_{\alpha}\) satisfies (d_{2}) through (d_{5});
 (iii)
the generating space of the bdislocatedquasimetric family (shortly, the \(G_{bdq}\)family) if \(d_{\alpha}\) satisfies (d_{3}) through (d_{5}).
Definition 1.5
 1.
Let \((X,d_{\alpha})\) be a \(G_{bq}\)family and let \(\{x_{n}\}\) be a sequence in X. We say that \(\{x_{n}\}\) \(G_{bq}\)converges to x in \((X,d_{\alpha})\) if \(\lim_{n\rightarrow\infty}d_{\alpha}(x_{n},x)=0\) for all \(\alpha\in(0,1]\).
In this case we write \(x_{n}\rightarrow x\).
 2.
Let \((X,d_{\alpha})\) be a \(G_{bq}\)family and let \(A\subseteq X\), \(x\in X\). We say that x is a \(G_{bq}\)limit point of A if there exists a sequence \(\{x_{n}\}\) in \(A\{x\}\) such that \(\lim_{n\rightarrow\infty}x_{n}=x\).
 3.
A sequence \(\{x_{n}\}\) in a \(G_{bq}\)family is called a \(G_{bq}\)Cauchy sequence if given \(\epsilon>0\), there exists \(n_{0}\in\mathbb{N}\) such that for all \(n,m\geq n_{0}\), we have \(d_{\alpha}(x_{n},x_{m})<\epsilon\) or \(\lim_{ n,m\rightarrow \infty} d_{\alpha}(x_{n},x_{m})=0 \) for all \(\alpha\in(0,1]\).
 4.
A \(G_{bq}\)family \((X,d_{\alpha})\) is called complete if every \(G_{bq}\)Cauchy sequence in X is \(G_{bq}\)Convergent.
Remark 1.6
Every \(G_{bq}\)convergent sequence in a \(G_{bq}\)family is \(G_{bq}\)Cauchy.
If we take \(s=1\) then generating space of bquasimetric family becomes generating space of quasimetric family as defined by Chang et al. [36].
Example 1.7
Let \((X, d)\) be a metric space. If we put \(d_{\alpha}\) instead of d for all \(\alpha\in(0,1]\) and \(x,y\in X\), then \((X, d_{\alpha})\) is a generating space of quasimetric family.
In [34], the author proved that each generating space of quasimetric family generates a topology \(\Im_{d_{\alpha}}\) whose base is the family of open balls. The ‘\(G_{bq}\)family’ will play a very predominant role in fixed point theory because the class of \(G_{bq}\)family is larger than the generating space of quasimetric family.
Motivated by the above facts, in this paper, we introduce the concept of a cyclic compatible contraction and prove related fixed point theorems in the generating space of a bquasimetric family.
2 Main results
Definition 2.1
Theorem 2.2
 1.
\(S,T:A\cup B \rightarrow A\cup B\) be a cyclic compatible contraction.
 2.
TX is a closed subset of X.
Proof
Consider \(d_{\alpha}(S^{2n1}x_{0},Su)\leq\gamma d_{\alpha}(S^{2n2}x_{0},Tu)\).
By letting \(n\rightarrow\infty\), \(d_{\alpha}(\eta,Su)\leq\gamma d_{\alpha}(\eta,Tu)\).
Therefore \(d_{\alpha}(\eta,T\eta)=0\).
Thus \(\eta=T\eta\).
From (4), we get \(S\eta=T\eta=\eta\).
Hence η is a common fixed point of S and T.
To prove uniqueness, let us suppose that \(\eta_{1}\) and \(\eta_{2}\) are two fixed points of S and T.
Hence \(\eta_{1}=\eta_{2}\), since \(0<\gamma<1\).
If we put \(s=1\) in the above theorem, we obtain the following corollary in the generating space of a quasimetric family. □
Corollary 2.3
 1.
\(S,T:A\cup B \rightarrow A\cup B\) are cyclic compatible contractions.
 2.
TX is a closed subset of X.
If we write d instead of \(d_{\alpha}\) in the above theorem, we obtain the following corollary in a complete bmetric space.
Corollary 2.4
 1.
\(S,T:A\cup B \rightarrow A\cup B\) are cyclic compatible contractions.
 2.
TX is a closed subset of X.
Example 2.5
If we put \(s=1\) and d instead of \(d_{\alpha}\) in the above theorem, we obtain the following corollary in a complete metric space.
Corollary 2.6
 1.
\(S,T:A\cup B \rightarrow A\cup B\) are cyclic compatible contractions.
 2.
TX is a closed subset of X.
Example 2.7
Theorem 2.8
Proof
Now we prove that \(\{x_{n}\}\) is a Cauchy sequence.
In order to prove uniqueness, suppose that \(\eta_{1}\) and \(\eta_{2}\) are two common fixed points of S and T.
If we put \(s=1\) in the above theorem, we obtain the following corollary in the generating space of a quasimetric family.
Corollary 2.9
If we write d instead of \(d_{\alpha}\) in the above theorem, we obtain the following corollary in a complete bmetric space.
Corollary 2.10
If we put \(s=1\) and d instead of \(d_{\alpha}\) in the above theorem, we obtain the following corollary in a complete metric space.
Corollary 2.11
Theorem 2.12
 (F_{1}):

\(\mathcal{F}\) is strictly increasing,
 (F_{2}):

\(\operatorname{Inf}\mathcal{F}=\infty\),
 (F_{3}):

\(\mathcal{F}\) is continuous on \((0,\infty)\),
 (F_{4}):

for some \(x\in A\) there exists \(\tau>0\) such thatfor \(n\in\mathbb{N}\), \(y\in A\).$$d_{\alpha}(Tx,Ty)>0\quad\Rightarrow\quad\tau+\mathcal{F}\bigl(d_{\alpha } \bigl(S^{n}x,Sy\bigr)\bigr)\leq\mathcal{F}\bigl(d_{\alpha} \bigl(S^{n1}x,Ty\bigr)\bigr), $$
Then S and T have a point of coincidence in \(A\cap B\). Moreover, if S and T are weakly compatible then S and T have a unique common fixed point in \(A\cap B\).
Proof
Fix \(x\in A\). Since \(SX\subset TX\), we may choose \(x_{0}=x\in X\) such that \(Sx_{0}=Tx_{1}\).
Hence define the sequence \(\{x_{n}\}\) in X by \(Sx_{n}=Tx_{n+1}=T^{n+1}x_{0}=x_{n+1}\) for \(n\in \mathbb{N}\cup\{0\}\).
If \(x_{0}=Tx_{0}\), the proof is complete. So we assume that \(x_{0}\neq Tx_{0}\). This yields \(d_{\alpha}(x_{0},Tx_{0})>0\).
Now, we claim that \(d_{\alpha}(Tx_{\eta(n)},Tx_{\psi(n)})>0\).
This is a contradiction. Hence \(\{x_{n}\}_{n=1}^{\infty}\) is a Cauchy sequence. By completeness of \((X,d_{\alpha})\), there is a sequence \(\{T^{2n}x_{0}\}\) in A and \(\{T^{2n1}x_{0}\}\) in B such that both converge to some u in X for all \(\alpha\in(0,1]\). Since A and B are closed subsets of X, \(u\in A\cup B\).
Hence u is a common fixed point of S and T.
Now we prove the uniqueness of the common fixed point.
Let us assume that u and v are two common fixed points of S and T such that \(Su=Tu=u\) and \(Sv=Tv=v\) but \(u\neq v\).
Hence \(u=v\). This completes the proof of the theorem. □
If we put \(s=1\) in the above theorem, we obtain the following corollary in the generating space of a quasimetric family.
Corollary 2.13
 (F_{1}):

\(\mathcal{F}\) is strictly increasing,
 (F_{2}):

\(\operatorname{Inf}\mathcal{F}=\infty\),
 (F_{3}):

\(\mathcal{F}\) is continuous on \((0,\infty)\),
 (F_{4}):

for some \(x\in A\) there exists \(\tau>0\) such thatfor \(n\in\mathbb{N}\), \(y\in A\).$$d_{\alpha}(Tx,Ty)>0\quad\Rightarrow\quad\tau+\mathcal{F}\bigl(d_{\alpha } \bigl(S^{n}x,Sy\bigr)\bigr)\leq\mathcal{F}\bigl(d_{\alpha} \bigl(S^{n1}x,Ty\bigr)\bigr), $$
Then S and T have a point of coincidence in \(A\cap B\). Moreover, if S and T are weakly compatible then S and T have a unique common fixed point in \(A\cap B\).
If we write d instead of \(d_{\alpha}\) in the above theorem, we obtain the following corollary in complete bmetric space.
Corollary 2.14
 (F_{1}):

\(\mathcal{F}\) is strictly increasing,
 (F_{2}):

\(\operatorname{Inf}\mathcal{F}=\infty\),
 (F_{3}):

\(\mathcal{F}\) is continuous on \((0,\infty)\),
 (F_{4}):

for some \(x\in A\) there exists \(\tau>0\) such thatfor \(n\in\mathbb{N}\), \(y\in A\).$$d(Tx,Ty)>0\quad\Rightarrow\quad\tau+\mathcal{F}\bigl(d\bigl(S^{n}x,Sy\bigr) \bigr)\leq\mathcal {F}\bigl(d\bigl(S^{n1}x,Ty\bigr)\bigr), $$
Then S and T have a point of coincidence in \(A\cap B\). Moreover, if S and T are weakly compatible then S and T have a unique common fixed point in \(A\cap B\).
If we put \(s=1\) and d instead of \(d_{\alpha}\) in the above theorem, we obtain the following corollary in a complete metric space.
Corollary 2.15
 (F_{1}):

\(\mathcal{F}\) is strictly increasing,
 (F_{2}):

\(\operatorname{Inf}\mathcal{F}=\infty\),
 (F_{3}):

\(\mathcal{F}\) is continuous on \((0,\infty)\),
 (F_{4}):

for some \(x\in A\) there exists \(\tau>0\) such thatfor \(n\in\mathbb{N}\), \(y\in A\).$$d(Tx,Ty)>0\quad\Rightarrow\quad\tau+\mathcal{F}\bigl(d\bigl(S^{n}x,Sy\bigr) \bigr)\leq\mathcal {F}\bigl(d\bigl(S^{n1}x,Ty\bigr)\bigr), $$
Then S and T have a point of coincidence in \(A\cap B\). Moreover, if S and T are weakly compatible then S and T have a unique common fixed point in \(A\cap B\).
Declarations
Acknowledgements
The first author would like to express her sincere gratitude to Mr. Anand Prabhakar for his invaluable support and motivation. The authors would like to express their thanks to the referees for their helpful comments and 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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