Fixed point and periodic point results for α-type F-contractions in modular metric spaces
- Anantachai Padcharoen^{1},
- Dhananjay Gopal^{2},
- Parin Chaipunya^{1} and
- Poom Kumam^{1, 3}Email author
https://doi.org/10.1186/s13663-016-0525-4
© Padcharoen et al. 2016
Received: 4 November 2015
Accepted: 8 March 2016
Published: 22 March 2016
Abstract
Motivated by Gopal et al. (Acta Math. Sci. 36B(3):1-14, 2016). We introduce the notion of α-type F-contraction in the setting of modular metric spaces which is independent from one given in (Hussain et al. in Fixed Point Theory Appl. 2015:158, 2015). Further, we establish some fixed point and periodic point results for such contraction. The obtained results encompass various generalizations of the Banach contraction principle and others.
Keywords
MSC
1 Introduction and preliminaries
The fixed point technique is one of the important tools with respect to studying the existence and uniqueness of the solution of various mathematical methods appearing in the practical problems. In particular, the Banach contraction principle provides a constructive method of finding a unique solution for models involving various types of differential and integral equations. This principle is generalized by several authors in various directions; see [3–8]. Recently, Gopal et al. [1] introduced the concept of α-type F-contraction in metric space by combining the ideas given in [8] and obtained some fixed point results.
On the other hand, to deal with the problems of description of superposition operators, Chistyakov [9] introduced the notion of modular metric spaces and gave some fundamental results on this topic, whereas some authors introduced the analog of the Banach contraction theorem in modular metric spaces and described the important aspects of applications of fixed point of mappings in modular metric spaces. Some recent results in this direction can be found in [2, 10–13]. In this paper, we introduce the concept of α-type F-contraction in the setting of modular metric spaces and establish fixed point and periodic point results for such a contraction. Consequently, our results generalize and improve some known results from the literature.
Consistent with Chistyakov [9], we begin with some basic definitions and results which will be used in the sequel.
Throughout this paper \(\mathbb{N}\), \(\mathbb{R}^{+} \), and \(\mathbb{R}\) will denote the set of natural numbers, positive real numbers, and real numbers, respectively.
Definition 1.1
[9]
- (i)
\(w_{\lambda}(x,y) = 0\) for all \(\lambda> 0 \) if and only if \(x = y \);
- (ii)
\(w_{\lambda}(x,y) = \omega_{\lambda}(y,x)\) for all \(\lambda> 0 \);
- (iii)
\(w_{\lambda+ \mu}(x,y) \leq w_{\lambda}(x,z) + w_{ \mu }(z,y)\) for all \(\lambda, \mu> 0 \).
Definition 1.2
[9]
Definition 1.3
- (i)
The sequence \((x_{n})_{n\in\mathbb{N}}\) in \(X_{w}\) is said to be w-convergent to \(x \in X_{\omega}\) if and only if \(w_{\lambda}(x_{n},x)\rightarrow0\), as \(n \rightarrow \infty\) for some \(\lambda> 0 \).
- (ii)
The sequence \((x_{n})_{n\in\mathbb{N}}\) in \(X_{w}\) is said to be w-Cauchy if \(w_{\lambda}(x_{m},x_{n})\rightarrow0\), as \(m,n \rightarrow\infty\) for some \(\lambda> 0 \).
- (iii)
A subset C of \(X_{w}\) is said to be w-complete if any w-Cauchy sequence in C is a convergent sequence and its limit is in C.
- (iv)
A subset C of \(X_{w}\) is said to be w-bounded if for some \(\lambda> 0\), we have \(\delta_{w} (C) = \sup\{ w_{\lambda}(x,y); x,y \in C \} < \infty\).
- (F1)
F is strictly increasing on \(\mathbb{R}^{+}\),
- (F2)
for every sequence \(\{s_{n}\}\) in \(\mathbb{R}^{+}\), we have \(\lim_{n\rightarrow\infty}s_{n} = 0\) if and only if \(\lim_{n\rightarrow\infty} F(s_{n}) = -\infty\),
- (F3)
there exists a number \(k \in(0, 1)\) such that \(\lim_{s\rightarrow0^{+}}s^{k}F(s) = 0\).
Example 1.4
- (i)
\(F(t) = \ln t\), with \(t > 0\),
- (ii)
\(F(t) = \ln t + t\), with \(t > 0\).
Definition 1.5
[8]
Definition 1.6
[2]
Let \(\Delta_{G}\) denote the set of all functions \(G:(\mathbb{R}^{+})^{4}\rightarrow \mathbb{R}^{+}\) satisfying the condition \((G)\) for all \(t_{1}, t_{2}, t_{3}, t_{4} \in \mathbb{R}^{+}\) with \(t_{1} t_{2} t_{3} t_{4} = 0\), there exists \(\tau>0\) such that \(G(t_{1}, t_{2}, t_{3}, t_{4}) =\tau\).
Example 1.7
- (i)
\(G(t_{1}, t_{2}, t_{3}, t_{4}) = L \min(t_{1}, t_{2}, t_{3}, t_{4}) +\tau\),
- (ii)
\(G(t_{1}, t_{2}, t_{3}, t_{4}) =\tau e^{L \min(t_{1}, t_{2}, t_{3}, t_{4})}\), where \(L \in\mathbb{R}^{+}\).
Definition 1.8
[2]
2 Fixed point results for α-type F-contractions
We begin with the following definitions.
Definition 2.1
Definition 2.2
Remark 2.3
Every α-type F-contraction is an α-type F-weak contraction, but the converse is not necessarily true.
Example 2.4
Remark 2.5
Definition 2.1 (respectively, Definition 2.2) reduces to an F-contraction (respectively, an F-weak contraction) for \(\alpha(x, y) = 1\).
The next two examples demonstrate that α-type F-contractions (defined above) and α-η-GF-contractions [2] are independent.
Example 2.6
Example 2.7
The motivation of the following definition is in the last step of the proof of the Cauchy sequence in our theorems.
Definition 2.8
Let \((X,w)\) be a modular metric space. Then we will say that w satisfies the \(\Delta_{M}\)-condition if it is the case that \(\lim_{m,n\rightarrow\infty}w_{\lambda}(x_{n},x_{m})=0\), for \(\lambda=m\) implies \(\lim_{m,n\rightarrow\infty}w_{\lambda}(x_{n},x_{m})=0\) (\(m,n\in\mathbb {N}\), \(m\geq n\)), for some \(\lambda>0\).
Now, we are ready to state our first theorem which generalizes the main theorem of Gopal et al. [1] for modular metric spaces.
Theorem 2.9
- (i)
T is α-admissible,
- (ii)
there exists \(x_{0} \in C\) such that \(\alpha(x_{0}, Tx_{0})\geq1\),
- (iii)
T is continuous.
Proof
Let \(x_{0}\in C\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and define a sequence \(\{x_{n}\}\) in C by \(x_{n+1} = Tx_{n}\) for all \(n \in\mathbb{N}\).
Theorem 2.10
- (i)
there exists \(x_{0} \in C\) such that \(\alpha(x_{0}, Tx_{0})\geq1\),
- (ii)
T is α-admissible,
- (iii)
if \(\{x_{n}\}\) is a sequence in \(X_{w}\) such that \(\alpha (x_{n}, x_{n+1})\geq1\) for all \(n \in\mathbb{N}\) and \(x_{n}\rightarrow x\) as \(n\rightarrow \infty\), then \(\alpha (x_{n}, x)\geq1\) for all \(n \in \mathbb{N}\),
- (iv)
F is continuous.
Proof
- (H):
for all \(x, y \in \operatorname{Fix}(T)\), \(\alpha(x, y) \geq1\).
Theorem 2.11
Adding condition (H) to the hypotheses of Theorem 2.9 (respectively, Theorem 2.10) the uniqueness of the fixed point is obtained.
Proof
Example 2.6 satisfies all the hypotheses of Theorem 2.10, hence T has a unique fixed point \(x=\frac{3}{2}\).
The following result improves the main theorem of the F-contraction for a modular metric space.
Corollary 2.12
Let \((X,w)\) be a modular metric space. Assume that w is regular and satisfies the \(\Delta_{M}\)-condition. Let C be a nonempty subset of \(X_{w}\). Assume that C is w-complete and w-bounded, i.e., \(\delta_{w}(C)=\sup\{w_{1}(x,y) : x,y\in C\}<\infty\). Let \(T : C \rightarrow C\) be an α-type F-contraction satisfying the hypotheses of Theorem 2.11, then T has unique fixed point.
From Example 1.4(i) and Corollary 2.12, we obtain the following result.
Theorem 2.13
Let \((X,w)\) be a modular metric space. Assume that w is regular. Let C be a nonempty subset of \(X_{w}\). Assume that C is w-complete and w-bounded, i.e., \(\delta_{w}(C)=\sup\{w_{1}(x,y); x,y\in C\} <\infty\). Let \(T : C\rightarrow C\) be a contraction. Then T has a unique fixed point \(x_{0}\). Moreover, the orbit \(\{T^{n}(x)\}\) w-converges to \(x_{0}\) for \(x\in C\).
3 Periodic point results
In this section, we prove some periodic point results for self-mappings on a modular metric space. In the sequel, we need the following definition.
Definition 3.1
[14]
A mapping \(T : C \rightarrow C\) is said to have the property \((P)\) if \(\operatorname{Fix}(T^{n}) = \operatorname{Fix}(T)\) for every \(n \in\mathbb{N}\), where \(\operatorname{Fix}(T) := \{x \in X_{w} : Tx = x\}\).
Theorem 3.2
- (i)there exists \(\tau>0\) and two functions \(F \in\mathcal {F}\) and \(\alpha:C\times C\rightarrow (0,\infty)\) such thatholds for all \(x \in C\) with \(w_{1}(Tx, T^{2}x) > 0\),$$\tau+ \alpha(x, Tx)F\bigl(w_{1}\bigl(Tx, T^{2}x\bigr)\bigr) \leq F\bigl(w_{1}(x, Tx)\bigr) $$
- (ii)
there exists \(x_{0} \in C\) such that \(\alpha(x_{0}, Tx_{0})\geq1\),
- (iii)
T is α-admissible,
- (iv)
if \(\{x_{n}\}\) is a sequence in C such that \(\alpha(x_{n}, x_{n+1})\leq1\) for all \(n \in\mathbb{N}\) and \(w_{1}(x_{n},x)\rightarrow 0\), as \(n \rightarrow \infty\), then \(w_{1}(Tx_{n},Tx) \rightarrow 0\) as \(n \rightarrow \infty\),
- (v)
if \(z \in \operatorname{Fix}(T^{n})\) and \(z \notin \operatorname{Fix}(T)\), then \(\alpha (T^{n-1}z, T^{n}z)\geq1\). Then T has the property \((P)\).
Proof
By using a similar reasoning to the proof of Theorem 2.9, we see that the sequence \(\{x_{n}\}\) is a w-Cauchy sequence and thus, by w-completeness, there exists \(x^{*}\in X_{w}\) such that \(x_{n}\rightarrow x^{*} \) as \(n \rightarrow \infty\).
By taking the limit as \(n\rightarrow \infty\) in the above inequality, we have \(F(w_{1}(z,Tz))=-\infty\), which is a contradiction until \(w_{1}(z,Tz)= 0\) and by the regularity of w, we set \(z=Tz\). Hence, \(\operatorname{Fix}(T^{n}) = \operatorname{Fix}(T)\) for all \(n \in\mathbb{N}\). □
Taking \(\alpha(x, y) = 1\) for all \(x, y \in C\) in Theorem 3.2, we get the following result, which is a generalization of Theorem 4 of Abbas et al. [14] in the setting of a modular metric.
Corollary 3.3
Declarations
Acknowledgements
The first author thanks for the support of Petchra Pra Jom Klao Doctoral Scholarship Academic. This work was completed while the second author (Dr. Gopal) was visiting Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, Thailand, during 15 October-8 November, 2015. He thanks Professor Poom Kumam and the University for their hospitality and support.
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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