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Fixed point theorems for a class of generalized weak cyclic compatible contractions
- P. S. Kumari^{1},
- J. Nantadilok^{2}Email authorView ORCID ID profile and
- M. Sarwar^{3}
https://doi.org/10.1186/s13663-018-0638-z
© The Author(s) 2018
Received: 3 January 2018
Accepted: 19 April 2018
Published: 7 May 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 Λ-weak cyclic compatible contractions via altering distance functions. Our results extend and improve some well-known results in the literature. We provide examples to analyze and illustrate our main results.
Keywords
- Cyclic maps
- b-quasi-metric family
- b-dislocated metric family
- \((\psi\mbox{-}\varphi)\)-weak cyclic compatible contraction
- Λ-cyclic compatible contraction
MSC
- 54H25
- 47H10
1 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 non-empty 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 self-mappings. 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])
- 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])
- (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 non-increasing and left continuous in α.
Definition 1.5
(see [6])
- (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 non-increasing 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 b-quasi-metric family \((X,d_{\alpha })\).
Definition 1.6
(see [7])
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.
2 Main results
We begin this section by introducing the following definition.
Definition 2.1
We state and prove our main results.
Theorem 2.2
Let A and B be non-empty 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
Remark 2.3
- 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
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.
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
- (1)
\(\mathscr{U}\) is called a cyclic idle contraction \(\Leftrightarrow d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y)\leq \gamma d_{\alpha }(\mathscr{U}^{2n-1}x,\mathscr{U}y)\);
- (2)
\(\mathscr{U}\) is called a \((\psi ,\varphi )\)-weak cyclic idle contraction \(\Leftrightarrow \psi (d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y))\leq \psi (d_{\alpha }(\mathscr{U}^{2n-1}x,\mathscr{U}y))-\varphi (d_{\alpha }(\mathscr{U}^{2n-1}x,\mathscr{U}y))\);
- (3)
\(\mathscr{U}\) is called a \((\psi ,\varphi )\)-weak cyclic orbital contraction \(\Leftrightarrow \psi (d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y))\leq \psi (d_{\alpha }(\mathscr{U}^{2n-1}x,y))-\varphi (d_{\alpha }(\mathscr{U}^{2n-1}x,y))\),
Theorem 2.7
Let A and B be non-empty 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 non-empty 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
Theorem 2.10
- 1.
\(\mathscr{U}\), \(\mathscr{V}\) are Λ-cyclic compatible contractions,
- 2.
\(\mathscr{U}\), \(\mathscr{V}\) are weakly compatible,
- 3.
\(\mathscr{V}(X)\) is a closed subset of X.
Proof
Take \(x=x_{0}\in A\). Since \(\mathscr{U}(X)\subset \mathscr{V}(X)\), we may choose \(x_{1}\in X\) such that \(\mathscr{U}x_{0}=\mathscr{V}x_{1}\).
Remark 2.11
- 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 b-metric 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
Example 2.13
In view of Definition 2.6, we introduce the following definition.
Definition 2.14
- (1)
\(\mathscr{U}\) is called Λ-cyclic idle contraction. \(\Leftrightarrow \eta (d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y))\leq \gamma \eta (d_{\alpha }(\mathscr{U}^{2n-1}x,\mathscr{U}y))\).
- (2)
\(\mathscr{U}\) is called Λ-cyclic orbital contraction. \(\Leftrightarrow \eta (d_{\alpha }(\mathscr{U}^{2n}x,\mathscr{U}y))\leq \gamma \eta (d_{\alpha }(\mathscr{U}^{2n-1}x,y))\).
Theorem 2.15
Let A and B be non-empty closed subsets of the generating space of a complete b-quasi-metric family \((X,d_{\alpha })\). Suppose \(\mathscr{U} : A \cup B \to A \cup B\) is a Λ-cyclic 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 non-empty closed subsets of the generating space of a complete b-quasi-metric family \((X,d_{\alpha })\). Suppose \(\mathscr{U} : A \cup B \to A \cup B\) is a Λ-cyclic orbital contraction, then \(\mathscr{U}\) has a unique fixed point in \(A\cap B\).
Proof
Take \(\mathscr{V}=I\) in Theorem 2.10. □
3 Discussion
In this manuscript, we establish fixed point theorems in generating space of a \(G_{bd}\)-family \((X,d_{\alpha })\), and a b-quasi-metric 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 b-quasi-metric 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 Λ-cyclic contractions (Def. 2.14) and obtain fixed point theorems (Theorems 2.15 and 2.16). Our main results extend and improve some well-known 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 non-expansive mappings, nonlinear mappings and amenability. In relation to Theorem 2.2 and Theorem 2.10, we pose the following open problem at the end.
4 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 Λ-cyclic 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 b-quasi-metric family \((X,d_{\alpha })\). Our results extend and improve the results obtained in [7] and some well-known results in the literature. We provide examples to illustrate and support our results and we also pose a problem at the end.
Declarations
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.
Authors’ contributions
Each author equally contributed to this paper, read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interest regarding the publication of this article.
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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