 Research
 Open Access
Fixed points for cyclic φcontractions in generalized metric spaces
 Fei He^{1}Email author and
 Alatancang Chen^{2}
https://doi.org/10.1186/s1366301605588
© He and Chen 2016
 Received: 1 April 2016
 Accepted: 27 May 2016
 Published: 4 June 2016
Abstract
In this paper, we obtain a fixed point theorem for mappings satisfying cyclic φcontractive conditions in complete metric spaces, which gives a positive answer to the question raised by Radenović (Fixed Point Theory Appl. 2015:189, 2015). We also find that this result and the fixed point result satisfying cyclic weak ϕcontractions given by Karapınar (Appl. Math. Lett. 24:822825, 2011) are independent of each other. Furthermore, when the number of cyclic sets is odd, we obtain fixed point theorems satisfying cyclic weak ϕcontractions and cyclic φcontractions in the setting of generalized metric spaces.
Keywords
 fixed point
 comparison function
 generalized metric space
 cyclic φcontraction
 cyclic weak ϕcontraction
MSC
 47H10
 54H25
1 Introduction and preliminaries
The main purpose of this paper is to answer an open question raised by Radenović in [1]. In order to go further, we attempt to extend our result and the result established by Karapınar [2, 3] to the setting of generalized metric spaces. We show these results are valid in generalized metric spaces when the number of cyclic sets is odd.
Let us recall the definition of a comparison function.
Definition 1.1
[4]
 (i)_{ φ } :

φ is increasing;
 (ii)_{ φ } :

\((\varphi^{n}(t))_{n\in\mathbb{{N}}}\) converges to 0 as \(n\to\infty\), for all \(t\in(0,\infty)\).
 (iii)_{ φ } :

\(\sum_{k=0}^{\infty}\varphi^{k}(t)<\infty\), for all \(t\in(0,\infty)\),
It is clear that a strong comparison function is a comparison function, but the converse is not true.
Example 1.2
Many authors considered fixed point results about cyclic φcontractions in setting of different type of spaces; see, for example, [1–11]. Particularly, in [1], Radenović obtained a fixed point theorem for noncyclic φcontraction, where φ is comparison function, and raised the following question.
Question 1.3
Prove or disprove the following.
 (i)
\(f(A_{i})\subset A_{i+1}\) for \(1\leq i\leq p\);
 (ii)there exists a comparison function \(\varphi:[0,\infty)\to[0,\infty )\) such thatfor any \(x\in A_{i}\), \(y\in A_{i+1}\), \(1\leq i\leq p\).$$d(fx,fy)\leq\varphi\bigl(d(x,y)\bigr), $$
In Section 2, we give an answer to Question 1.3. In Section 3, we obtain a fixed point theorem for a mapping satisfying cyclic weak ϕcontractions and cyclic φcontractions in complete generalized metric spaces, where the number of cyclic sets is odd.
2 Answer of Question 1.3
We start this section by presenting the notion of cyclic φcontraction.
Definition 2.1
 (i)
\(\bigcup_{i=1}^{p} A_{i}\) is a cyclic representation of Y with respect to f;
 (ii)there exists a comparison function \(\varphi:[0,\infty)\to[0,\infty )\) such thatfor any \(x\in A_{i}\), \(y\in A_{i+1}\), where \(A_{p+1}=A_{1}\).$$ d(fx,fy)\leq\varphi\bigl(d(x,y)\bigr), $$(2.1)
Theorem 2.2
Let \((X,d)\) be a complete metric space, \(p\in\mathbb{N}\), \(A_{1},\ldots ,A_{p}\) nonempty closed subsets of X, and \(Y:=\bigcup_{i=1}^{p} A_{i}\). Assume that \(f: Y\to Y\) is a cyclic φcontraction. Then f has a unique fixed point \(x^{*}\in\bigcap_{i=1}^{p} A_{i}\) and a Picard iteration \(\{x_{n}\}_{n\geq1}\) given by \(x_{n}=fx_{n1}\) converging to \(x^{*}\) for any starting point \(x_{0} \in\bigcup_{i=1}^{p} A_{i}\).
Proof
Let \(x_{0}\) be an arbitrary point in Y. Define the sequence \(\{x_{n}\}\) in Y by \(x_{n}=fx_{n1}\), \(n=1,2,\ldots\) . If there exists \(n_{0}\) such that \(x_{n_{0}+1}=x_{n_{0}}\) then \(fx_{n_{0}}=x_{n_{0}+1}=x_{n_{0}}\) and the existence of the fixed point is proved. Consequently, we always assume that \(x_{n}\neq x_{n+1}\) for all \(n\in\mathbb{N}\).
Step 2. We will prove the following claim.
Claim
For every \(\varepsilon>0\), there exists \(N\in\mathbb{N}\) such that if \(n>m>N\) with \(nm\equiv1 \operatorname{mod} p\) then \(d(x_{n},x_{m})<\varepsilon\).
Step 3. We will prove \(\{x_{n}\}\) is a Cauchy sequence in X.
Step 4. We will prove f has a unique fixed point \(x^{*}\in \bigcap_{i=1}^{p} A_{i}\).
As X is a complete metric space, there exists \(x\in X\) such that \(\lim_{n\to\infty} x_{n}=x\). Using the cyclic character of f, there exists a subsequence of \(\{x_{n}\}\) for which belongs to \(A_{i}\) for \(i\in\{1,2,\ldots,p\}\). Hence, from \(A_{i}\) is closed, we see that \(x\in\bigcap_{i=1}^{p} A_{i}\). Now, we consider the restriction \(f_{\bigcap_{i=1}^{p} A_{i}}\) of f on \(\bigcap_{i=1}^{p} A_{i}\). Since \(\bigcap_{i=1}^{p} A_{i}\) is also complete, by Theorem 2.3 in [1], we see that f has a unique fixed point \(x^{*}\) in \(\bigcap_{i=1}^{p} A_{i}\).
Step 5. We prove that the Picard iteration converges to \(x^{*}\) for any initial point \(x_{0}\in\bigcup_{i=1}^{p} A_{i}\).
This completes the proof. □
3 Cyclic weak ϕcontractions and cyclic φcontractions in generalized metric spaces
In 2000, Branciari [12] introduced the notion of generalized metric space and proved the Banach fixed point theorem in such spaces. For more information, the reader can refer to [13–17]. For some notions and facts about generalized metric spaces, one may wish to see [12].
In [3], Karapınar gave a fixed point results satisfying cyclic weak ϕcontractions. For convenience, we rewrite his theorem (i.e., [3], Theorem 2) as the following equivalent statement.
Theorem 3.1
 (i)
\(\bigcup_{i=1}^{p} A_{i}\) is a cyclic representation of Y with respect to f;
 (ii)there exists a function \(\phi:[0,\infty)\to[0,\infty)\) with \(\phi (t)< t\) and \(t\phi(t)\) is nondecreasing for \(t\in(0,\infty)\) and \(\phi(0)=0\) such thatfor any \(x\in A_{i}\), \(y\in A_{i+1}\), where \(A_{p+1}=A_{1}\).$$d(fx,fy)\leq\phi\bigl(d(x,y)\bigr), $$
Based on the concept of cyclic weak ϕcontraction, we can introduce the following notion.
Definition 3.2
 (i)_{ ϕ } :

\(\phi(0)=0\);
 (ii)_{ ϕ } :

\(\phi(t)< t\), for all \(t\in(0,\infty)\);
 (iii)_{ ϕ } :

the function \(\psi(t):=t\phi(t)\) is increasing, i.e., \(t_{1}\leq t_{2}\) implies \(\psi(t_{1})\leq\psi(t_{2})\), for \(t_{1},t_{2}\in [0,\infty)\).
Lemma 3.3
 (1)
\(\phi(t)\leq t\), for any \(t\in[0,\infty)\);
 (2)
for \(k\geq1\), \(\phi^{k}(t)< t\), for any \(t\in(0,\infty)\);
 (3)
\((\phi^{n}(t))_{n\in\mathbb{{N}}}\) converges to 0 as \(n\to\infty\), for all \(t\in(0,\infty)\).
Proof
The next are two basic examples of the comparison function and the \((w)\)comparison function.
Example 3.4
Example 3.5
Remark 3.6
From Example 3.4 and Example 3.5, we see that the comparison function and the \((w)\)comparison function do not imply each other. Consequently, Theorem 2 in [3] and Theorem 2.2 are independent of each other.
Now we carry over the concept of cyclic weak ϕcontraction to generalized metric space.
Definition 3.7
 (i)
\(\bigcup_{i=1}^{p} A_{i}\) is a cyclic representation of Y with respect to f;
 (ii)there exists a \((w)\)comparison function \(\phi:[0,\infty)\to[0,\infty )\) such thatfor any \(x\in A_{i}\), \(y\in A_{i+1}\), where \(A_{p+1}=A_{1}\).$$ d(fx,fy)\leq\phi\bigl(d(x,y)\bigr), $$(3.1)
Theorem 3.8
Let \((X,d)\) be a complete generalized metric space, p an odd number, \(A_{1},\ldots,A_{p}\) nonempty closed subsets of X and \(Y:=\bigcup_{i=1}^{p} A_{i}\). Assume that \(f: Y\to Y\) is a cyclic weak ϕcontraction. Then f has a unique fixed point \(x^{*}\in\bigcap_{i=1}^{p} A_{i}\) and a Picard iteration \(\{x_{n}\}_{n\geq1}\) given by \(x_{n}=fx_{n1}\) converging to \(x^{*}\) for any starting point \(x_{0} \in\bigcup_{i=1}^{p} A_{i}\).
Proof
Let \(x_{0}\in Y\), and \(x_{n}=fx_{n1}\), \(n=1,2,\ldots\) . If there exists \(n_{0}\) such that \(x_{n_{0}+1}=x_{n_{0}}\) then \(fx_{n_{0}}=x_{n_{0}+1}=x_{n_{0}}\) and the existence of the fixed point is proved. Consequently, we will assume that \(x_{n}\neq x_{n+1}\) for all \(n\in\mathbb{N}\).
Step 1. We will prove that \(x_{n}\neq x_{m}\) for all \(n\neq m\).
Step 3. We will prove the following claim.
Claim
For every \(\varepsilon>0\), there exists \(N\in\mathbb{N}\) such that if \(n>m>N\) with \(nm\equiv1 \operatorname{mod} p\) then \(d(x_{n},x_{m})<\varepsilon\).
Step 4. We will prove \(\{x_{n}\}\) is a Cauchy sequence in X.
Step 5. We will prove f has a unique fixed point \(x^{*}\in \bigcap_{i=1}^{p} A_{i}\) and the Picard iteration \(\{x_{n}\}\) converges to \(x^{*}\).
Theorem 3.9
Let \((X,d)\) be a complete generalized metric space, p an odd number, \(A_{1},\ldots,A_{p}\) nonempty closed subsets of X and \(Y:=\bigcup_{i=1}^{p} A_{i}\). Assume that \(f: Y\to Y\) is a cyclic φcontraction. Then f has a unique fixed point \(x^{*}\in\bigcap_{i=1}^{p} A_{i}\) and a Picard iteration \(\{x_{n}\}_{n\geq1}\) given by \(x_{n}=fx_{n1}\) converging to \(x^{*}\) for any starting point \(x_{0} \in\bigcup_{i=1}^{p} A_{i}\).
Proof
Let \(x_{0}\in Y\), and \(x_{n}=fx_{n1}\), \(n=1,2,\ldots\) .
Now, we will prove the following claim.
Claim
For every \(\varepsilon>0\), there exists \(N\in\mathbb{N}\) such that if \(n>m>N\) with \(nm\equiv1 \operatorname{mod} p\) then \(d(x_{n},x_{m})<\varepsilon\).
Similar to Step 4 and Step 5 in the proof of Theorem 3.8, we can finish the proof. □
Example 3.10
Finally, a natural question arises.
Declarations
Acknowledgements
The first author is supported by the National Natural Science Foundation of China (11471236, 11570049). The second author is supported by the National Natural Science Foundation of China (11371185).
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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