- Research
- Open Access
A note on recent cyclic fixed point results in dislocated quasi-b-metric spaces
- Diana Dolićanin-Ðekić^{1},
- Tatjana Došenović^{2},
- Huaping Huang^{3} and
- Stojan Radenović^{4, 5}Email author
https://doi.org/10.1186/s13663-016-0565-9
© Dolićanin-Ðekić et al. 2016
- Received: 5 January 2016
- Accepted: 20 June 2016
- Published: 30 June 2016
Abstract
The purpose of this paper is to establish some fixed point results for cyclic contractions in the setting of dislocated quasi-b-metric spaces. We verify that some previous cyclic contraction results in dislocated quasi-b-metric spaces are just equivalent to the non-cyclic ones in the same spaces. Moreover, by using two examples, we highlight the superiority of the results obtained.
Keywords
- fixed point
- dislocated quasi-b-metric space
- Banach contraction
- Kannan contraction
MSC
- 54H25
- 47H10
1 Introduction and preliminaries
French mathematician Poinćare was first to use the concept of fixed point in ‘Poinćare’s final theorem’ during the period of 1895 to 1900, from restricting the existence of periodic solution for three body problem to the existence of fixed point under some conditions of planar continuous transformations. In 1910, Brouwer proved that there exists at least one fixed point for the polyhedron continuous map in finite dimensional space, and this opened the situation of fixed point theory research. Particularly in 1922, Polish mathematician Banach innovated Banach contraction mapping principle by using Picard iteration method. Due to its beautiful assertion and successful way of solving the implicit function existence theorem, the existence of a solution for a differential equation with initial value condition, fixed point theory caught the eyes of scholars and it sparkles people’s inspirations towards in-depth and extensive research. Especially in recent decades, with the development of the computer, many people have coped with numerous applications by utilizing a variety of iteration methods to approach the fixed point and hence they made a breakthrough and brought this subject gradually to perfection. Nowadays fixed point theory plays a crucial role in nonlinear functional analysis. Just because of this, in this paper, we start our fixed point investigation based on some previous work.
To start this article, we first of all recall some basic knowledge.
In 1922, Banach [1] introduced the Banach contraction mapping principle as follows:
After that, based on this finding, a large number of fixed point results have appeared in recent years. Generally speaking, there usually are two generalizations on them. One is from mappings. The other is from spaces.
Concretely, for one thing, from mappings, for example, the concept of a Kannan contraction mapping was introduced in 1969 by Kannan [2] as follows:
Recently, the cyclic contraction mapping has become popular for research activities (see [3–11]). Let A and B be nonempty subsets of a metric space \(( X,d ) \) and let \(T:A\cup B\rightarrow A\cup B\) be a mapping. T is called a cyclic map if and only if \(T ( A ) \subseteq B\) and \(T ( B ) \subseteq A\). In 2003, Kirk et al. [4] introduced a cyclic contraction mapping as follows:
In 2010, Karapınar and Erhan [12] introduced a Kannan type cyclic contraction mapping as follows:
For another thing, from spaces, there are too many generalizations of metric spaces. For instance, we have the b-metric space, quasi-metric space, quasi-b-metric space, dislocated metric space (or metric-like space), dislocated b-metric space (or b-metric-like space), dislocated quasi-metric space (or quasi-metric-like space), and the dislocated quasi-b-metric space (or quasi-b-metric-like space) (see [3, 5, 9–11, 13–18]). Their definitions are as follows:
- (d1)
\(d(x,y)=0\Leftrightarrow x=y\);
- (d2)
\(d(x,y)=0\Rightarrow x=y\);
- (d3)
\(d(x,y)=0=d(y,x)\Rightarrow x=y\);
- (d4)
\(d(x,y)=d(y,x)\);
- (d5)
\(d(x,z)\leq d(x,y)+d(y,z)\);
- (d6)
\(d(x,z)\leq s[d(x,y)+d(y,z)]\), \(s \geq1\).
- (1)
\((X,d)\) is called a metric space if (d1), (d4), and (d5) hold;
- (2)
\((X,d)\) is called a b-metric space if (d1), (d4), and (d6) hold;
- (3)
\((X,d)\) is called a quasi-metric space if (d1) and (d5) hold;
- (4)
\((X,d)\) is called a quasi-b-metric space if (d1) and (d6) hold;
- (5)
\((X,d)\) is called a dislocated metric space if (d2), (d4), and (d5) hold;
- (6)
\((X,d)\) is called a dislocated b-metric space if (d2), (d4), and (d6) hold;
- (7)
\((X,d)\) is called a dislocated quasi-metric space if (d3) and (d5) hold;
- (8)
\((X,d)\) is called a dislocated quasi-b-metric space if (d3) and (d6) hold.
Despite the fact that the given examples were previously known, we thought it is useful for easy reference to give a full review.
Example 1.1
- (a)Let \(X=\mathbb{R}\) and \(d:X \times X \to[0, \infty)\) be defined asThen \((X,d)\) is a quasi-metric space, but it is not a metric space.$$d(x,y)= \left \{ \textstyle\begin{array}{l@{\quad}l} x-y, & x \geq y, \\ 1, & \mbox{otherwise}. \end{array}\displaystyle \right . $$
- (b)Let \(X=[0, \infty)\) and \(d: X \times X \to[0, \infty)\) be defined asThen \((X,d)\) is a b-metric space, but it is not a metric space.$$d(x,y)= \left \{ \textstyle\begin{array}{l@{\quad}l} 0, & x=y, \\ (x+y)^{2}, & \mbox{otherwise}. \end{array}\displaystyle \right . $$
- (c)Let \(X=C([0,1], \mathbb{R})\) with the usual partial ordering, and let \(d:X \times X \to\mathbb{R}^{+}\) be defined asThen \((X,d)\) is a quasi-b-metric space, but it is not a quasi-metric space and b-metric space.$$d(f,g)= \left \{ \textstyle\begin{array}{l@{\quad}l} \int_{0}^{1}(g(t)-f(t))^{3}\, dt, & f \leq g, \\ \int_{0}^{1}(f(t)-g(t))^{3}\, dt, & f \geq g. \end{array}\displaystyle \right . $$
- (d)
Let \(X=\mathbb{R}^{+}\) and \(d:X \times X \to\mathbb{R}^{+}\) be defined as \(d(x,y)=\max\{x,y\}\). Then \((X,d)\) is a dislocated metric space, but it is not a metric space.
- (e)
Let \(X=[0,1]\) and \(d: X \times X\to\mathbb{R}^{+}\) be defined as \(d(x,y)=|x-y|+x\). Then \((X,d)\) is a dislocated quasi-metric space, but it is not a dislocated metric space, and it is not a quasi-metric space.
- (f)
Let \(X=[0, \infty)\) and \(d: X \times X \to[0, \infty)\) be defined as \(d(x,y)=(x+y)^{2}\). Then \((X,d)\) is a dislocated b-metric space, but it is not a b-metric space.
- (g)
Let \(X=\mathbb{R}\) and \(d:X \times X \to[0, \infty)\) be defined as \(d(x,y)=|x-y|^{2}+\frac{|x|}{n}+\frac{|y|}{m}\), where \(n, m \in \mathbb{N}\setminus\{1\}\), \(n \neq m\). Then \((X,d)\) is a dislocated quasi-b-metric space, but it is not a quasi-b-metric space, dislocated b-metric space and dislocated quasi-metric space.
Also, scholars are interested in dislocated quasi-b-metric spaces since they are more general spaces. Based on this fact, we consider fixed point results in such spaces.
For the sake of reader, we recall the following concepts and results.
Definition 1.2
([5])
- (i)\(\{ x_{n} \} _{n\in\mathbb{N}}\) converges to \(x\in X\) ifIn this case x is called a \(dqb\)-limit of \(\{ x_{n} \}\), and we write it as \(x_{n}\rightarrow x\) (\(n\rightarrow\infty\)).$$\lim_{n\rightarrow\infty}d ( x_{n},x ) =0=\lim_{n\rightarrow\infty }d ( x,x_{n} ). $$
- (ii)\(\{ x_{n} \} _{n\in\mathbb{N}}\) is a \(dqb\)-Cauchy sequence if$$\lim_{n,m\rightarrow\infty}d ( x_{n},x_{m} ) =0=\lim _{n,m\rightarrow \infty}d ( x_{m},x_{n} ). $$
- (iii)
\(( X,d ) \) is \(dqb\)-complete if every \(dqb\)-Cauchy sequence is convergent in X.
Definition 1.3
([5], Definition 2.8)
Definition 1.4
([5], Definition 2.11)
In [5], the authors proved the following main results.
Theorem 1.5
([5], Theorem 2.9)
Let A and B be nonempty closed subsets of a \(dqb\)-complete dislocated quasi-b-metric space \(( X,d )\). Let T be a cyclic mapping that satisfies the condition of a dislocated quasi-b-metric-cyclic-Banach contraction. Then T has a unique fixed point in \(A\cap B\).
Theorem 1.6
([5], Theorem 2.12)
Let A and B be nonempty closed subsets of a \(dqb\)-complete dislocated quasi-b-metric space \(( X,d )\). Let T be a cyclic mapping that satisfies the condition of a dislocated quasi-b-metric-cyclic-Kannan contraction. Then T has a unique fixed point in \(A\cap B\).
2 Main results
In this section, we consider and generalize some previous results. We also prove that the results from Theorem 1.5 and Theorem 1.6 are just equivalent to the respective ordinary fixed point results in the same framework.
First, we recall the following lemma ([8], Remark 2.13).
Lemma 2.1
If some ordinary fixed point theorem in the framework of metric (resp. b-metric) spaces has a true cyclic-type extension, then these two theorems are equivalent.
Now we announce the following result.
Claim 1
Claim 2
In order to prove the above two theorems, we use the following crucial lemma.
Lemma 2.4
- (1)
\(T ( A_{i} ) \subseteq A_{i+1}\) for \(1\leq i\leq p\) where \(A_{p+1}=A_{1}\);
- (2)there exists \(k\in[0,\frac{1}{s})\) such that for all \(x\in\bigcup_{i=1}^{p}A_{i}\),$$ d \bigl( T^{2}x,Tx \bigr) \leq kd ( Tx,x ), \qquad d \bigl( Tx,T^{2}x \bigr) \leq kd ( x,Tx ). $$(2.3)
Proof
If \(k=0\), then for all \(x\in\bigcup_{i=1}^{p}A_{i}\), by (2.3) and (d3), we have \(T ( Tx ) =Tx\), i.e., Tx is a fixed point of T. Thus by (1), it is not hard to verify that \(Tx\in\bigcap_{i=1}^{p}A_{i}\). That is, \(\bigcap_{i=1}^{p}A_{i}\neq\emptyset\).
Proof of Theorem 2.2
Putting \(A_{i}=X\) for \(i\in \{ 1,2,\ldots,p \} \) in Theorem 1.5, we obtain Claim 1. Conversely, let Claim 1 hold. We shall prove that Theorem 1.5 also holds. Indeed, if \(x\in\bigcup_{i=1}^{m}A_{i}\), then by virtue of Lemma 2.4, one establishes that \(\{ T^{n}x \} \) is a \(dqb\)-Cauchy sequence, further, \(\bigcap_{i=1}^{p}A_{i}\neq \emptyset\). Now that \(( \bigcap_{i=1}^{p}A_{i},d ) \) is a \(dqb\)-complete dislocated quasi-b-metric space, and we restrict T to \(( \bigcap_{i=1}^{p}A_{i},d ) \) and hence the condition (2.1) holds for all \(x,y\in\bigcap_{i=1}^{p}A_{i}\), then Claim 1 implies that T has a unique fixed point in \(\bigcap_{i=1}^{p}A_{i}\). Accordingly, Theorem 1.5 is satisfied. □
Proof of Theorem 2.3
We could use the same method as in the proof of Theorem 2.2 and hence the proof is omitted. □
Further, we announce the result for the existence of fixed point under cyclical consideration in the framework following dislocated quasi-b-metric spaces.
Lemma 2.5
Let \(( X=\bigcup_{i=1}^{p}A_{i},d ) \) be a \(dqb\)-complete dislocated quasi-b-metric space. If \(T:X\rightarrow X\) satisfies (1) of Lemma 2.4, and for all \(x\in X=\bigcup_{i=1}^{p}A_{i}\), the corresponding Picard sequence \(\{ T^{n}x \} \) is a \(dqb\)-Cauchy sequence, then \(\bigcap_{i=1}^{p}A_{i}\neq\emptyset\).
Proof
Since \(( X,d ) \) is \(dqb\)-complete, then \(dqb\)-Cauchy sequence \(\{ T^{n}x \} \) converges to some \(z\in X\). We shall prove that \(z\in\bigcap_{i=1}^{p}A_{i}\). Actually, observing that \(T (A_{i} ) \subseteq A_{i+1}\) for all \(i\in \{ 1,2,\ldots,p \} \) and \(T ( A_{p+1} ) \subseteq A_{1}\), we conclude that \(\{ T^{n}x \}\) has infinite terms in \(A_{i}\) for all \(i\in \{ 1,2,\ldots,p \}\). As \(A_{i}\) is closed for all \(i\in \{ 1,2,\ldots,p \}\), we claim that \(z\in\bigcap_{i=1}^{p}A_{i}\). Consequently, \(\bigcap_{i=1}^{p}A_{i}\neq\emptyset\). □
The following two examples support our Theorem 2.2 and Theorem 2.3.
Example 2.6
([5], Example 2.10)
Example 2.7
([5], Example 2.13)
The following lemmas are useful in proving of all main results in the framework of dislocated quasi-b-metric spaces. The proofs are almost the same as in [19] for the case of b-metric spaces and hence we omit them.
Lemma 2.8
Lemma 2.9
An example of such a β is given by \(\beta (t ) =\frac{1}{s}e^{-t}\) for \(t>0\) and \(\beta ( 0 ) \in[0,\frac{1}{s})\).
The following result is a generalization of [15], Theorem 2.1, since we do not assume that the dislocated quasi-b-metric d is continuous. With regard to its proof, we omit it because it is almost the same as the counterpart in [15], Theorem 2.1, for b-metric spaces without using Lemma 2.8 and Lemma 2.9, however, many other papers have to use them.
Theorem 2.10
Taking \(g=I_{X}\) (identity mapping of X) in Theorem 2.10, we see that the following Geraghty type theorem in dislocated quasi-b-metric spaces.
Corollary 2.11
Declarations
Acknowledgements
The first and second authors are thankful to the Ministry of Education, Science and Technological Development of Serbia. The third author is thankful to the science and technology research project of education department in Hubei province of China (B2015137).
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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