Abstract
In this paper, we introduce certain α-admissible mappings which are \(F(\psi,\varphi)\)-contractions on M-metric spaces, and we establish some fixed point results. Our results generalize and extend some well-known results on this topic in the literature.
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1 Introduction and preliminaries
Geraghty in [10] introduced an interesting class of auxiliary functions to refine the Banach contraction mapping principle. Let \(\mathcal{F} \) be the function \(\beta:[0,\infty) \to[0,1)\) which satisfies the condition
By using \(\mathcal{F}\), Geraghty [10] proved the following theorem.
Theorem 1.1
([10])
Let \((X, d)\) be a complete metric space and \(T : X \to X \) be an operator. Suppose that there exists \(\beta\in \mathcal{F}\) satisfying the condition
If T satisfies the following inequality
then T has a unique fixed point.
We now present definitions, lemmas, remarks, and examples that we will use.
Definition 1.2
([4])
Let \(f:X \to X\) and \(\alpha:X\times X\to[0, +\infty)\). We say that f is an α-admissible mapping if \(\alpha (x,y)\geq1\) implies \(\alpha(fx,fy)\geq1\) for all \(x,y \in X\).
Definition 1.3
([12])
Let Ψ denote all functions \(\psi:[0,\infty)\rightarrow[0,\infty)\) satisfying:
-
(i)
ψ is strictly increasing and continuous,
-
(ii)
\(\psi ( t ) =0\) if and only if \(t=0\).
Definition 1.4
([5])
An ultra altering distance function is a continuous, nondecreasing mapping \(\varphi:[0,\infty)\rightarrow{}[0,\infty)\) such that \(\varphi(t)>0\) for \(t>0\).
Remark 1.5
We let Φ denote the class of the ultra altering distance functions.
Definition 1.6
([5])
A mapping \(F:[0,\infty)^{2}\rightarrow\Bbb{R}\) is called a C-class function if it is continuous and satisfies the following axioms:
-
1.
\(F(s,t)\leq s\);
-
2.
\(F(s,t)=s\) implies that either \(s=0\) or \(t=0\) for all \(s,t\in [0,\infty)\).
We denote the C-class functions by \(\mathcal{C}\).
Example 1.7
([5])
The following functions are elements of \(\mathcal{C}\):
-
1.
\(F(s,t)=s-t\).
-
2.
\(F(s,t)=ms\), \(0 < m < 1\).
-
3.
\(F(s,t)=\frac{s}{(1+t)^{r}}\); \(r\in(0,\infty)\).
-
4.
\(F(s,t)=s\beta(s)\), \(\beta:[0,\infty)\rightarrow(0,1)\) and is continuous.
-
5.
\(F(s,t)=s-(\frac{2+t}{1+t})t\).
-
6.
\(F(s,t)=\sqrt[n]{\ln(1+s^{n})}\).
Definition 1.8
A partial metric on a nonempty set X is a function \(p: X \times X \to\mathbb {R}^{+}\) such that, for all \(x, y, z \in X\):
-
(p1)
\(p(x,x)=p(y,y)=p(x,y)\iff x=y\),
-
(p2)
\(p(x,x)\leq p(x,y)\),
-
(p3)
\(p(x, y)=p(y,x)\),
-
(p4)
\(p(x, y)\leq p(x,z)+p(z,y)-p(z,z)\).
A partial metric space is a pair \((X, p)\) such that X is a nonempty set and p is a partial metric on X.
For more details and examples see [14–16].
Definition 1.9
([7])
Let X be a nonempty set. A function \(\mu:X \times X\to\mathbb {R}^{+}\) is called an m-metric if the following conditions are satisfied:
-
(m1)
\(\mu(x,x)=\mu(y,y)=\mu(x,y)\iff x=y\),
-
(m2)
\(m_{xy}\leq\mu(x,y)\),
-
(m3)
\(\mu(x, y)=\mu(y,x)\),
-
(m4)
\((\mu(x, y)-m_{xy} )\leq (\mu (x,z)-m_{xz} )+ (\mu(z,y)-m_{zy} )\),
where
Then the pair \((X,\mu)\) is called an M-metric space. The following notation is useful in the sequel:
Remark 1.10
([7])
For every \(x,y\in X\),
-
1.
\(0\leq M_{xy}+m_{xy} =\mu(x,x)+\mu(y,y)\);
-
2.
\(0\leq M_{xy}-m_{xy} = |\mu(x,x)-\mu(y,y)|\);
-
3.
\(M_{xy}-m_{xy} \leq(M_{xz}-m_{xz}) +(M_{zy}-m_{zy})\).
2 Topology on M-metric space
It is clear that each M-metric m on X generates a \(T_{0}\) topology \(\tau_{m}\) on X. The set
where
for all \(x \in X\) and \(\varepsilon> 0\), forms the base of \(\tau_{m}\).
Definition 2.1
([7])
Let \((X,\mu)\) be an M-metric space. Then:
-
1.
A sequence \(\{x_{n}\}\) in an M-metric space \((X, m)\) converges to a point \(x \in X\) if
$$ \lim_{n\to\infty} \bigl(\mu(x_{n}, x)-m_{x_{n},x}\bigr)=0. $$(2) -
2.
A sequence \(\{x_{n}\}\) in an M-metric space \((X, m)\) is called an m-Cauchy sequence if
$$ \lim_{n,m\to \infty}\bigl(\mu(x_{n}, x_{m})-m_{x_{n},x_{m}}\bigr) \quad \mbox{and}\quad \lim _{n,m\to\infty}(M_{x_{n}, x_{m}}-m_{x_{n},x_{m}}) $$(3)exist (and are finite).
-
3.
An M-metric space \((X, m)\) is said to be complete if every m-Cauchy sequence \(\{x_{n}\}\) in X converges, with respect to \(\tau _{m}\), to a point \(x\in X\) such that
$$\Bigl(\lim_{n\to\infty}\bigl(\mu(x_{n}, x)-m_{x_{n},x}\bigr)=0 \mbox{ and } \lim_{n\to\infty}(M_{x_{n}, x}-m_{x_{n},x})=0 \Bigr). $$
Lemma 2.2
([7])
Assume that \(x_{n}\to x\) and \(y_{n}\to y\) as \(n\to\infty\) in an M-metric space \((X, m)\). Then
Lemma 2.3
([7])
Assume that \(x_{n}\to x\) as \(n\to\infty\) in an M-metric space \((X, m)\). Then
for all \(y\in X\).
Lemma 2.4
([7])
Assume that \(x_{n}\to x\) and \(x_{n}\to y\) as \(n\to\infty\) in an M-metric space \((X, m)\). Then \(\mu(x,y)=m_{xy}\). Further if \(\mu(x,x)=\mu(y,y)\), then \(x=y\).
3 Methods
Many authors studied the class of \(\alpha-\psi\) contractive type mappings and obtained fixed point results for this new class of mappings in metric spaces. Their results contain several well-known fixed point theorems including the Banach contraction principle.
The goal of this article is to introduce the class of \(F(\psi,\varphi )\)-contractions and investigate the existence and uniqueness of fixed points for α-admissible mappings on M-metric spaces.
4 Discussion and main results
We start this section with the following main theorem.
Theorem 4.1
Let \((X,\mu) \) be a complete M-metric space and \(T: X\to X\) be an α-admissible mapping. Suppose that the following condition is satisfied:
for all \(x,y \in X\) and \(l\geq1\), where \(\psi\in\Psi\), \(\varphi\in \Phi\), and \(F\in\mathcal{C}\). Suppose that either
-
(a)
T is continuous,
or
-
(b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\), \(\alpha(x_{n},x_{n+1})\geq1\) for all n, then \(\alpha (x,Tx)\geq1\).
If there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), then T has a fixed point.
Proof
Let \(x_{0} \in X \) be such that \(\alpha(x_{0},Tx_{0})\geq1\). Define a sequence \(\{x_{n}\}\) in X by \(x_{n}=T^{n} x_{0}=Tx_{n-1}\) for all \(n\in\Bbb{N}\). Since T is an α-admissible mapping and \(\alpha(x_{0},Tx_{0})\geq 1\), we deduce that \(\alpha(x_{1},x_{2})=\alpha(Tx_{0},T^{2}x_{0})\geq1\). Continuing this process, we get \(\alpha(x_{n},Tx_{n})\geq1\) for all \(n\in\Bbb{N}\cup\{0\} \). From inequality (4) we have
Then we have
We want to prove that \(\mu(x_{n},x_{n+1})\to0\), as \(n\to\infty\). If \(\mu(x_{n_{0}},x_{n_{0}+1})=0\), for some \(n_{0}\in\Bbb{N}\), then by (5)
hence from the properties of functions F, ψ, and φ we have \(\mu(x_{n_{0}+1},x_{n_{0}+2})=0\) which means
Now let
Inequality (5) implies that \(\mu(x_{n},x_{n+1})\leq \mu(x_{n-1},x_{n})\). It follows that the sequence \(\{\mu(x_{n},x_{n+1})\}\) is decreasing. Thus, there exists \(m\in \mathbb{R_{+}}\) such that
We want to prove that \(m=0\). Let \(m > 0\). From (5) we have
Hence we get
so
Using the properties of functions F, ψ, and φ, we obtain that \(\psi(m)=0\) or \(\varphi(m)=0\), so then \(m=0\), which is a contradiction. Therefore
Now we prove that \(\{x_{n}\}\) is an M-Cauchy sequence in \((X,\mu)\). We have
and
On the other hand,
so
We show
Let
If \(\lim_{n,m\to\infty}M^{*}(x_{n},x_{m})\neq0\), there exist \(\varepsilon>0\) and \(\{l_{k}\}\subset\mathbb{N}\) such that
Suppose that k is the smallest integer which satisfies the above equation such that
Now by (m4) we have
Thus
which means
On the other hand,
so we have
Again by (m4) we have
and
and taking the limit as \(k\to+\infty\), together with (6) and (7), we have
Now by (4), (7), and (8) we have
Therefore we get
Letting \(k\to\infty\) in the above inequality, we get
so
Using the properties of F, ψ, and φ, we obtain \(\psi(\varepsilon)=0\) or \(\varphi(\varepsilon)=0\), and then \(\varepsilon=0\), which is a contradiction. Therefore \(\{x_{n}\}\) is an M-Cauchy sequence. Now, by the completeness of X, \(x_{n}\to x\) for some \(x\in X\) in the \(\tau_{m}\) topology, i.e.,
and
However, \(\lim_{n\to\infty} m_{x_{n},x} =0\), hence \(\lim_{n\to\infty}\mu (x_{n},x)=0\), and by Remark 1.10
Now suppose (a) holds. Then T is continuous and we have
i.e.,
and similar to the above, we have \(\lim_{n\to\infty} m_{x_{n+1},Tx} =0\). Hence \(\lim_{n\to \infty}\mu(x_{n+1},Tx)=0\) and by Remark 1.10, \(\mu(Tx,Tx)=0\). On the other hand, \(x_{n}\to x\) as \(n\to\infty\) so by Lemma 2.3, we get
but we have
Thus
therefore \(\mu(x,Tx)=\mu(Tx,Tx)=\mu(x,x)=0\) and by (m1) we get
Next suppose (b) holds. Then \(\alpha(x,Tx)\geq1\). Now by (4) we have
that is, \(\psi(\mu(Tx_{n},Tx))\leq F( \psi(\mu(x_{n},x)),\varphi(\mu (x_{n},x))) \leq\psi(\mu(x_{n},x))\), and so we get
On the other hand,
Thus \(Tx_{n} \to Tx \) in the \(\tau_{m}\) topology.
The proof of \(Tx=x\) follows as in (a). □
Theorem 4.2
Let \((X,\mu) \) be a complete M-metric space and \(T: X\to X\) be an α-admissible mapping. Suppose that the following condition is satisfied:
for all \(x,y \in X\), where \(\psi\in\Psi\), \(\varphi\in\Phi\), and \(F\in\mathcal{C}\). Suppose that either
-
(a)
T is continuous,
or
-
(b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\), \(\alpha(x_{n},x_{n+1})\geq1\) for all n, then \(\alpha (x,Tx)\geq1\).
If there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), then T has a fixed point.
Proof
Let \(x_{0} \in X \) be such that \(\alpha(x_{0},Tx_{0})\geq1\). Define a sequence \(\{x_{n}\}\) in X by \(x_{n}=T^{n} x_{0}=Tx_{n-1}\) for all \(n\in\mathbb{N}\). Since T is an α-admissible mapping and \(\alpha(x_{0},Tx_{0})\geq 1\), we deduce that \(\alpha(x_{1},x_{2})=\alpha(Tx_{0},T^{2}x_{0})\geq1\). Continuing this process, we get \(\alpha(x_{n},Tx_{n})\geq1\) for all \(n\in\mathbb{N}\cup\{ 0\}\). From inequality (9) we have
Then we have
Now similar to the proof in Theorem 4.1, we get
Now we prove that \(\{x_{n}\}\) is an M-Cauchy sequence in \((X,\mu)\). We have
and
On the other hand,
so
We show
Let
If \(\lim_{n,m\to\infty}M^{*}(x_{n},x_{m})\neq0\), there exist \(\varepsilon>0\) and \(\{l_{k}\}\subset\mathbb{N}\) such that
Suppose that k is the smallest integer which satisfies the above equation such that
Again as in the proof in Theorem 4.1, we obtain that
and
Now by (9), (12), and (13) we have
Therefore we get
Letting \(k\to\infty\) in the above inequality, we get
so
Using the properties of functions F, ψ, and φ, we obtain that \(\psi(\varepsilon)=0\), or \(\varphi(\varepsilon)=0\), and then \(\varepsilon=0\), which is a contradiction. Therefore \(\{x_{n}\}\) is an M-Cauchy sequence.
Now, by the completeness of X, \(x_{n}\to x\) for some \(x\in X\) in the \(\tau_{m}\) topology, i.e.,
and
However, \(\lim_{n\to\infty} m_{x_{n},x} =0\), hence \(\lim_{n\to\infty}\mu (x_{n},x)=0\) and by Remark 1.10
Now suppose (a) holds. Then, as in the proof in Theorem 4.1, we have \(Tx=x\). Next suppose (b) holds. Then \(\alpha(x,Tx)\geq1\). From (9) we have
that is, \(\psi(\mu(Tx_{n},Tx))\leq F( \psi(\mu(x_{n},x)),\varphi(\mu (x_{n},x))) \leq\psi(\mu(x_{n},x))\), and so we get
On the other hand,
Thus \(Tx_{n} \to Tx \) in the \(\tau_{m}\) topology.
The proof of \(Tx=x\) follows as in (a). □
Theorem 4.3
Let \((X,\mu) \) be a complete M-metric space and \(T: X\to X\) be an α-admissible mapping. Suppose that the following condition is satisfied:
for all \(x,y \in X\), where \(\psi\in\Psi\), \(\varphi\in\Phi\), and \(F\in\mathcal{C}\). Suppose that either
-
(a)
T is continuous,
or
-
(b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\), \(\alpha(x_{n},x_{n+1})\geq1\) for all n, then \(\alpha (x,Tx)\geq1\).
If there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), then T has a fixed point.
Proof
Let \(x_{0} \in X \) be such that \(\alpha(x_{0},Tx_{0})\geq1\). Define a sequence \(\{x_{n}\}\) in X by \(x_{n}=T^{n} x_{0}=Tx_{n-1}\) for all \(n\in\mathbb{N}\). Since T is an α-admissible mapping and \(\alpha(x_{0},Tx_{0})\geq 1\), we deduce that \(\alpha(x_{1},x_{2})=\alpha(Tx_{0},T^{2}x_{0})\geq1\). Continuing this process, we get \(\alpha(x_{n},Tx_{n})\geq1\) for all \(n\in\mathbb{N}\cup\{ 0\}\). From inequality (14) we have
Then we have
Now, similar to the proof in Theorem 4.1, we get
Now we prove that \(\{x_{n}\}\) is an M-Cauchy sequence in \((X,\mu)\). We have
and
On the other hand,
so
We show
Let
If \(\lim_{n,m\to\infty}M^{*}(x_{n},x_{m})\neq0\), there exist \(\varepsilon>0\) and \(\{l_{k}\}\subset\mathbb{N}\) such that
Suppose that k is the smallest integer which satisfies the above equation such that
Again as in the proof in Theorem 4.1, we obtain that
and
Now by (14), (17), and (18) we have
Therefore we get
Letting \(k\to\infty\) in the above inequality, we get
so
Using the properties of functions F, ψ, and φ, we obtain that \(\psi(\varepsilon)=0\), or \(\varphi(\varepsilon)=0\), then \(\varepsilon=0\), which is a contradiction. Therefore \(\{x_{n}\}\) is an M-Cauchy sequence.
Now, by the completeness of X, \(x_{n}\to x\) for some \(x\in X\) in the \(\tau_{m}\) topology, i.e.,
and
However, \(\lim_{n\to\infty} m_{x_{n},x} =0\), hence \(\lim_{n\to\infty}\mu (x_{n},x)=0\) and by Remark 1.10
Now suppose (a) holds. Then, as in the proof in Theorem 4.1, we have \(Tx=x\). Next suppose (b) holds. Then \(\alpha(x,Tx)\geq1\). From (14) we have
that is, \(\psi(\mu(Tx_{n},Tx))\leq F( \psi(\mu(x_{n},x)),\varphi(\mu (x_{n},x))) \leq\psi(\mu(x_{n},x))\), and so we get
On the other hand,
Thus \(Tx_{n} \to Tx \) in the \(\tau_{m}\) topology.
The proof of \(Tx=x\) follows as in (a). □
Theorem 4.4
Assume that all of the hypotheses of Theorems 4.1 or 4.2 or 4.3 hold. In addition, suppose the following condition is satisfied:
-
(c)
if \(Tx=x\) then \(\alpha(x,Tx)\geq1\).
Then the fixed point of T is unique.
Proof
Suppose that \(u,v\in X\) are two fixed points of T such that \(u\neq v\). Then \(\alpha(u,Tu)\geq1\) and \(\alpha(v,Tv)\geq1\).
For Theorem 4.1, we have
For Theorem 4.2, we have
For Theorem 4.3, we have
Therefore equations (19), (20), (21), (22), (23), and (24) imply that
and so from the properties of functions F, ψ, and φ, we have
Therefore by (m1)
□
5 Consequences
From Theorems 4.1, 4.2, and 4.3 we obtain the following corollaries as an extension of several known results in the literature.
If we let \(\varphi(t)=\psi(t)=t\), we get the following three corollaries.
Corollary 5.1
Let \((X,\mu) \) be a complete M-metric space and \(T: X\to X\) be an α-admissible mapping. Suppose that the following condition is satisfied:
for all \(x,y \in X\) and \(l\geq1\), where \(\psi\in\Psi\), \(\varphi\in \Phi\), and \(F\in\mathcal{C}\). Suppose that either
-
(a)
T is continuous,
or
-
(b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\), \(\alpha(x_{n},x_{n+1})\geq1\) for all n, then \(\alpha (x,Tx)\geq1\).
If there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), then T has a fixed point.
Corollary 5.2
Let \((X,\mu) \) be a complete M-metric space and \(T: X\to X\) be an α-admissible mapping. Suppose that the following condition is satisfied:
for all \(x,y \in X\), where \(\psi\in\Psi\), \(\varphi\in\Phi\), and \(F\in\mathcal{C}\). Suppose that either
-
(a)
T is continuous,
or
-
(b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\), \(\alpha(x_{n},x_{n+1})\geq1\) for all n, then \(\alpha (x,Tx)\geq1\).
If there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), then T has a fixed point.
Corollary 5.3
Let \((X,\mu) \) be a complete M-metric space and \(T: X\to X\) be an α-admissible mapping. Suppose that the following condition is satisfied:
for all \(x,y \in X\), where \(\psi\in\Psi\), \(\varphi\in\Phi\), and \(F\in\mathcal{C}\). Suppose that either
-
(a)
T is continuous,
or
-
(b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\), \(\alpha(x_{n},x_{n+1})\geq1\) for all n, then \(\alpha (x,Tx)\geq1\).
If there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), then T has a fixed point.
Lemma 5.4
([7])
Every p-metric and metric is an M-metric.
If we let \(\beta\in\mathcal{F}\), \(\varphi(t)=\psi(t)=t\) and \(F(s,t)=\beta(s)s\), by Lemma 5.4 we get three results of Hussein et al. [13] (they are the immediate consequences of our results).
Corollary 5.5
([13, Theorem 4])
Let \((X,d) \) be a complete metric space and \(T: X\to X\) be an α-admissible mapping. Assume that there exists a function \(\beta:\mathbb{R}^{+}\to[0,1]\) such that, for any bounded sequence \(\{t_{n}\}\) of positive reals, \(\beta(t_{n})\to1\) implies \(t_{n}\to0\) and
for all \(x,y \in X\) where \(l\geq1\). Suppose that either
-
(a)
T is continuous,
or
-
(b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\), \(\alpha(x_{n},x_{n+1})\geq1\) for all n, then \(\alpha (x,Tx)\geq1\).
If there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), then T has a fixed point.
Corollary 5.6
([13, Theorem 6])
Let \((X,d) \) be a complete metric space and \(T: X\to X\) be an α-admissible mapping. Assume that there exists a function \(\beta:\mathbb{R}^{+}\to[0,1]\) such that, for any bounded sequence \(\{t_{n}\}\) of positive reals, \(\beta(t_{n})\to1\) implies \(t_{n}\to0\) and
for all \(x,y \in X\). Suppose that either
-
(a)
T is continuous,
or
-
(b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\), \(\alpha(x_{n},x_{n+1})\geq1\) for all n, then \(\alpha (x,Tx)\geq1\).
If there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), then T has a fixed point.
Corollary 5.7
([13, Theorem 8])
Let \((X,d)\) be a complete metric space and \(T: X\to X\) be an α-admissible mapping. Assume that there exists a function \(\beta:\mathbb{R}^{+}\to[0,1]\) such that, for any bounded sequence \(\{t_{n}\}\) of positive reals, \(\beta(t_{n})\to1\) implies \(t_{n}\to0\) and
for all \(x,y \in X\). Suppose that either
-
(a)
T is continuous,
or
-
(b)
if \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\), \(\alpha(x_{n},x_{n+1})\geq1\) for all n, then \(\alpha (x,Tx)\geq1\).
If there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\), then T has a fixed point.
6 Conclusion
Recently, the authors in [17] introduced the class of α-ψ contractive type mappings and obtained a fixed point result for this new class of mappings in the set-up of metric spaces. Their result contains several well-known fixed point theorems including the Banach contraction principle. Matthews (1994) in [18] established fixed point theorems in partial metric spaces. The authors in [7] introduced M-metric spaces which extend p-metric spaces and the authors established some new fixed point theorems.
In this paper, we introduce the class of \(F(\psi,\varphi)\)-contractions and investigate the existence and uniqueness of fixed points for α-admissible mappings on M-metric spaces. We also show that the fixed point results in [13] and Geraghty’s theorem [10] (Theorem 1.1) are immediate consequences of our results. For further results, we refer the reader to [1–4, 6, 8–12, 19–21, 23].
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Monfared, H., Asadi, M., Azhini, M. et al. \(F(\psi,\varphi)\)-Contractions for α-admissible mappings on M-metric spaces. Fixed Point Theory Appl 2018, 22 (2018). https://doi.org/10.1186/s13663-018-0647-y
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DOI: https://doi.org/10.1186/s13663-018-0647-y