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Fixed point and periodic point results for αtype Fcontractions in modular metric spaces
Fixed Point Theory and Applications volume 2016, Article number: 39 (2016)
Abstract
Motivated by Gopal et al. (Acta Math. Sci. 36B(3):114, 2016). We introduce the notion of αtype Fcontraction 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.
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 αtype Fcontraction 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 αtype Fcontraction 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.
Let X be a nonempty set. Throughout this paper, for a function \(w:(0,\infty) \times X \times X \rightarrow[0,\infty)\), we write
for all \(\lambda>0\) and \(x,y\in X\).
Definition 1.1
[9]
Let X be a nonempty set. A function \(w : (0,\infty) \times X \times X \rightarrow[0,\infty]\) is said to be a metric modular on X if it satisfies, for all \(x, y, z \in X\), the following conditions:

(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 \(\lambda, \mu> 0 \).
If instead of (i) we have only the condition (i′)
then w is said to be a pseudomodular (metric) on X. A modular metric w on X is said to be regular if the following weaker version of (i) is satisfied:
Definition 1.2
[9]
Let w be a pseudomodular on X. Fix \(x_{0}\in X\). The set
is said to be modular space (around \(x_{0}\)).
Definition 1.3
Let \(X_{w}\) be a modular metric space.

(i)
The sequence \((x_{n})_{n\in\mathbb{N}}\) in \(X_{w}\) is said to be wconvergent 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 wCauchy 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 wcomplete if any wCauchy sequence in C is a convergent sequence and its limit is in C.

(iv)
A subset C of \(X_{w}\) is said to be wbounded if for some \(\lambda> 0\), we have \(\delta_{w} (C) = \sup\{ w_{\lambda}(x,y); x,y \in C \} < \infty\).
Next, we denote by \(\mathcal{F}\) the family of all functions \(F : \mathbb{R}^{+} \rightarrow\mathbb{R}\) satisfying the following conditions:

(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
The following functions \(F : \mathbb{R}^{+}\rightarrow \mathbb{R}\) belongs to \(\mathcal{F}\):

(i)
\(F(t) = \ln t\), with \(t > 0\),

(ii)
\(F(t) = \ln t + t\), with \(t > 0\).
Definition 1.5
[8]
A mapping \(T : X \rightarrow X\) is said to be αadmissible if there exists a function \(\alpha: X \times X \rightarrow\mathbb{R}^{+}\) such that
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
The following function \(G : (\mathbb {R}^{+})^{4}\rightarrow \mathbb{R}\) belongs to \(\Delta_{G}\):

(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]
Let \(X_{\omega}\) be a modular metric space and T be a selfmapping on \(X_{\omega}\). Suppose that \(\alpha, \eta: X_{\omega}\times X_{\omega} \rightarrow [0,\infty )\) are two functions. We say T is an αηGFcontraction if for \(x, y \in X_{\omega}\) with \(\eta(x, Tx) \leq\alpha(x, y)\), \(\omega _{\lambda/l}(Tx, Ty) > 0\), and \(\lambda,l> 0\), we have
where \(G \in\Delta_{G}\) and \(F \in\mathcal{F}\).
Fixed point results for αtype Fcontractions
We begin with the following definitions.
Definition 2.1
Let \((X,w)\) be a modular metric space. Let C be a nonempty subset of \(X_{w}\). A mapping \(T:C\rightarrow C\) is said to be an αtype Fcontraction if there exist \(\tau>0\) and two functions \(F\in\mathcal{F}\), \(\alpha:C\times C\rightarrow (0,\infty)\) such that, for all \(x, y \in C\), satisfying \(w_{1}(Tx,Ty)>0\), the following inequality holds:
Definition 2.2
Let \((X,w)\) be a modular metric space. Let C be a nonempty subset of \(X_{w}\). A mapping \(T:C\rightarrow C\) is said to be an αtype Fweak contraction if there exist \(\tau>0\) and two functions \(F\in\mathcal{F}\), \(\alpha:C\times C\rightarrow (0,\infty)\) such that, for all \(x, y \in C\), satisfying \(w_{1}(Tx,Ty)>0\), the following inequality holds:
Remark 2.3
Every αtype Fcontraction is an αtype Fweak contraction, but the converse is not necessarily true.
Example 2.4
Let \(X_{w}=C = [0, \frac{9}{2}]\), \(w_{1}=xy\), and \(w_{2}=xy\). Define \(T : C \rightarrow C\), \(\alpha:C\times C\rightarrow (0,\infty)\), and \(F:\mathbb{R}^{+}\rightarrow \mathbb{R}\) by
Then, for \(x = 0\) and \(y = 1\), by putting \(F(t) = \ln t\) with \(t > 0\), we have
and
Clearly, we have
However, since
T is an αtype Fweak contraction for the choice
and \(\tau>0\) such that \(e^{\tau}=\frac{8}{9}\).
Remark 2.5
Definition 2.1 (respectively, Definition 2.2) reduces to an Fcontraction (respectively, an Fweak contraction) for \(\alpha(x, y) = 1\).
The next two examples demonstrate that αtype Fcontractions (defined above) and αηGFcontractions [2] are independent.
Example 2.6
Let \(X_{w}=C = [0, 3]\), \(w_{1}=xy\), and \(w_{\lambda}=\frac{1}{\lambda }xy\). Define \(T : C \rightarrow C\), \(\alpha:C\times C\rightarrow(0,\infty)\), and \(F:\mathbb{R}^{+}\rightarrow \mathbb{R}\) by
So, define \(F(t) = \ln t\) with \(t > 0\). Then T is an αtype Fweak contraction with \(\alpha(x, y) = 1\) for all \(x, y \in C\) and \(\tau>0\) such that \(e^{\tau}=\frac{3}{2}\). But T is not an αηGFcontraction [2]. To see this, consider \(\eta:C\times C\rightarrow [0,\infty)\) such that
and
Then, for \(x = \frac{3}{2}\) and \(y = 3\), we have
and
Consequently, we have
and thus, T is not an αηGFcontraction.
Example 2.7
Let \(X_{w}=C = [0,1]\), \(w_{1}=xy\), and \(w_{\lambda}=\frac {1}{\lambda}xy\). Define \(T : C \rightarrow C\), \(\alpha,\eta: C \times C \rightarrow [0,\infty)\), \(G : (\mathbb {R}^{+})^{4} \rightarrow \mathbb{R}^{+}\) by
\(\alpha(x,y) = x+y\) and \(\eta(x,y)=\frac{x+y}{2}\), if x and y both are rational or irrational, \(\alpha(x,y)=1\) and \(\eta(x,y)=0\) if x is irrational and y is rational (and vice versa), \(G(t_{1}, t_{2}, t_{3}, t_{4}) = L \min\{t_{1},t_{2},t_{3},t_{4}\}+\tau\) (\(\tau>0\)), and \(F(t)=\ln t\). Then T is an αηGFcontraction. But it is not an αtype Fweak contraction. To see this, consider \(x = 0\) and y is any irrational.
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
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 wcomplete and wbounded, i.e., \(\delta_{w}(C)=\sup\{w_{1}(x,y) : x,y\in C\}<\infty\). Let \(T : C \rightarrow C\) be an αtype Fweak contraction satisfying the following conditions:

(i)
T is αadmissible,

(ii)
there exists \(x_{0} \in C\) such that \(\alpha(x_{0}, Tx_{0})\geq1\),

(iii)
T is continuous.
Then T has a fixed point \(x^{*} \in C\) and for every \(x_{0} \in C\) the sequence \(\{T^{n}x_{0}\}_{n\in\mathbb{N}}\) is convergent to \(x^{*}\).
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 \(n \in\mathbb{N}\).
Obviously, if there exists \(n_{0} \in\mathbb{N}\) such that \(x_{{n_{0}}+1} = x_{n_{0}}\), then \(Tx_{n_{0}} = x_{n_{0}}\) and the proof is finished. Hence, we suppose that \(x_{n+1}\neq x_{n}\) for every \(n \in\mathbb{N}\). Now from conditions (ii) and (i), we have
By induction we have
Since T is an αtype Fweak contraction, for every \(n \in \mathbb{N}\), we have
so
If there exists \(n \in\mathbb{N}\) such that \(\max\{ w_{1}(x_{n1},x_{n}),w_{1}(x_{n},x_{n+1})\}=w_{1}(x_{n},x_{n+1})\), then, from (2.5), we have
a contradiction. Therefore \(\max\{ w_{1}(x_{n1},x_{n}),w_{1}(x_{n},x_{n+1})\}=w_{1}(x_{n1},x_{n})\), for all \(n \in\mathbb{N}\). Hence, from (2.5), we have
This implies that
Taking the limit as \(n\rightarrow\infty\) in (2.6) and since C is wbounded, we have
from (F2), we obtain
From (F3), there exists \(k \in(0, 1)\) such that
From (2.8), for all \(n \in\mathbb{N}\), we deduce that
By using (2.7), (2.8), and taking the limit as \(n\rightarrow \infty\) in (2.9), we have
Then there exists \(n_{1} \in\mathbb{N}\) such that \(n (w_{1}(x_{n+1},x_{n}) )^{k}\leq1\) for all \(n\geq n_{1}\), that is,
For all \(m > n > n_{1}\), we have
Since the series \(\sum_{i=n}^{\infty}\frac{1}{i^{1/k}}\) is convergent, this implies
Since w satisfies the \(\Delta_{M}\)condition. Hence, we have
This shows that \(\{x_{n}\}\) is a wCauchy sequence. Since C is wcomplete, there exists \(x^{*}\in C\) such that \(x_{n}\rightarrow x^{*}\) as \(n\rightarrow \infty\). By the continuity of T and since w is regular, we have
Hence, \(x^{*}\) is a fixed point of T. □
Theorem 2.10
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 wcomplete modular metric space and wbounded, i.e., \(\delta_{w}(C)=\sup\{w_{1}(x,y) : x,y\in C\} <\infty\). Let \(T : C \rightarrow C\) be an αtype Fweak contraction satisfying the following conditions:

(i)
there exists \(x_{0} \in C\) such that \(\alpha(x_{0}, Tx_{0})\geq1\),

(ii)
T is αadmissible,

(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.
Then T has a fixed point \(x^{*} \in C\) and for every \(x_{0} \in C\) the sequence \(\{T^{n}x_{0}\}_{n\in\mathbb{N}}\) is convergent to \(x^{*}\).
Proof
Let \(x_{0} \in C\) be such that \(\alpha(x_{0}, Tx_{0})\geq1\) and let \(x_{n} = Tx_{n1}\) for all \(n \in\mathbb{N}\) Following the proof of Theorem 2.9, we see that \(\{x_{n}\}\) is a wCauchy sequence in the wcomplete modular metric space. Then there exists \(x^{*} \in C\) such that \(x_{n} \rightarrow x^{*}\) as \(n\rightarrow\infty\). From (2.3) and the hypothesis (iii), we have
Case I: Suppose, for every \(n \in\mathbb{N}\), there exists \(i_{n} \in\mathbb{N}\) such that \(x_{i_{n}+1} = Tx^{*}\) and \(i_{n} > i_{n1}\). Then we have
that is, \(x^{*}\) is a fixed point of T.
Case II: Assume there exists \(n_{0} \in\mathbb{N}\) such that \(x_{n+1} \neq Tx^{*}\) for all \(n\geq n_{0}\), i.e., \(w_{1}(Tx_{n}, Tx^{*}) > 0\) for all \(n \geq n_{0}\). It follows from (2.2) and (F1) that
If \(w_{1}(x^{*}, Tx^{*}) > 0\) and by the fact that
there exists \(n_{1} \in\mathbb{N}\) such that, for all \(n \geq n_{1}\), we have
From (2.11), we obtain
for all \(n\geq\max\{n_{0}, n_{1}\}\). Since F is continuous, taking the limit as \(n \rightarrow\infty\) in (2.12), we have
a contradiction. Thus \(w_{1}(x^{*}, Tx^{*}) = 0\) and hence \(x^{*}\) is a fixed point of T. □
Indeed, uniqueness of the fixed point, we will consider the following hypothesis.

(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
Assume that \(y^{*}\in C\) is an another fixed point of T, such that \(w_{1}(x,y)<\infty\) and \(w_{1}(Tx^{*}, Ty^{*}) = w_{1}(x^{*}, y^{*}) > 0\). Then we have
a contradiction. This implies that \(x^{*} = y^{*}\). □
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 Fcontraction 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 wcomplete and wbounded, i.e., \(\delta_{w}(C)=\sup\{w_{1}(x,y) : x,y\in C\}<\infty\). Let \(T : C \rightarrow C\) be an αtype Fcontraction 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 wcomplete and wbounded, 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 \(\{T^{n}(x)\}\) wconverges to \(x_{0}\) for \(x\in C\).
Periodic point results
In this section, we prove some periodic point results for selfmappings 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
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 wcomplete and wbounded, i.e., \(\delta_{w}(C)=\sup\{w_{1}(x,y) : x,y\in C\}<\infty\). Let C be a wcomplete and wbounded subset of X. Let \(T : C \rightarrow C\) be a mapping satisfying the following conditions:

(i)
there exists \(\tau>0\) and two functions \(F \in\mathcal {F}\) and \(\alpha:C\times C\rightarrow (0,\infty)\) such that
$$\tau+ \alpha(x, Tx)F\bigl(w_{1}\bigl(Tx, T^{2}x\bigr)\bigr) \leq F\bigl(w_{1}(x, Tx)\bigr) $$holds for all \(x \in C\) with \(w_{1}(Tx, T^{2}x) > 0\),

(ii)
there exists \(x_{0} \in C\) such that \(\alpha(x_{0}, Tx_{0})\geq1\),

(iii)
T is αadmissible,

(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 \(\alpha (T^{n1}z, T^{n}z)\geq1\). Then T has the property \((P)\).
Proof
Let \(x_{0} \in C\) be such that \(\alpha(x_{0}, Tx_{0})\geq1\). Now, for \(x_{0} \in C\), we define the sequence \(\{x_{n}\}\) by the rule \(x_{n} = T^{n}x_{n} = Tx_{n1}\). By (iii), we have \(\alpha(x_{1}, x_{2}) = \alpha(Tx_{0}, Tx_{1})\geq1\) and by induction we write
If there exists \(n_{0}\in\mathbb{N}\) such that \(x_{n_{0}} = x_{n_{0}+1} = Tx_{n_{0}}\), then \(x_{n_{0}}\) is a fixed point of T and the proof is finished. Thus, we assume \(x_{n} \neq x_{n+1}\) or \(w_{1}(Tx_{n1}, T^{2}x_{n1}) > 0\) for all \(n \in\mathbb{N}\). From (3.1) and (i), we have
or equivalently
By using a similar reasoning to the proof of Theorem 2.9, we see that the sequence \(\{x_{n}\}\) is a wCauchy sequence and thus, by wcompleteness, there exists \(x^{*}\in X_{w}\) such that \(x_{n}\rightarrow x^{*} \) as \(n \rightarrow \infty\).
By (iv), we have \(w_{1}(x_{n+1},Tx^{*})=w_{1}(Tx_{n},Tx^{*})\rightarrow0\) as \(n \rightarrow \infty\), that is, \(x^{*} = Tx^{*}\). Hence, T has a fixed point and \(\operatorname{Fix}(T^{n}) = \operatorname{Fix}(T)\) is true for \(n = 1\). Let \(n > 1\) and assume, by contradiction, that \(z \in \operatorname{Fix}(T^{n})\) and \(z \notin \operatorname{Fix}(T)\), such that \(w_{1}(z, Tz) > 0\). Now, applying (v) and (i), we have
and
Consequently, we have
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
Let \((X,w)\) be a complete 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 wcomplete and wbounded, i.e., \(\delta_{w}(C)=\sup\{w_{1}(x,y) : x,y\in C\}<\infty\). Let \(T : C \rightarrow C\) be a continuous mapping satisfying
for some \(\tau> 0\) and for all \(x \in X_{w}\) such that \(w_{1}(Tx,T^{2}x) > 0\). Then T has property \((P)\).
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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 October8 November, 2015. He thanks Professor Poom Kumam and the University for their hospitality and support.
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Padcharoen, A., Gopal, D., Chaipunya, P. et al. Fixed point and periodic point results for αtype Fcontractions in modular metric spaces. Fixed Point Theory Appl 2016, 39 (2016). https://doi.org/10.1186/s1366301605254
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MSC
 47H09
 47H10
 54H25
 37C25
Keywords
 fixed point
 αtype Fcontraction
 modular metric space