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 Open Access
α\((\psi,\phi)\) Contractive mappings on quasipartial metric spaces
 Erdal Karapınar^{1, 2}Email author,
 Leila Gholizadeh^{3},
 Hamed H Alsulami^{2} and
 Maha Noorwali^{2}
https://doi.org/10.1186/s136630150352z
© Karapınar et al. 2015
 Received: 8 April 2015
 Accepted: 14 June 2015
 Published: 3 July 2015
Abstract
In this paper, we consider α\((\psi,\phi)\) contractive mappings in the setting of quasipartial metric spaces and verify the existence of a fixed point on such spaces. Moreover, we present some examples and applications in integral equations of our obtained results.
Keywords
 quasipartial metric space
 fixed point
 αadmissible
 contractive mapping
MSC
 46T99
 46N40
 47H10
 54H25
1 Introduction and preliminaries
One of the most interesting extensions of distance function was reported by Matthews [1] by introducing the notion of a partial metric in which selfdistance need not be zero. In this celebrated report, Matthews [1] successfully characterized the distinguished result, Banach contraction mapping, in the setting of partial metric spaces. Later, many authors have generalized some fixed point theorems on such a space, see e.g. [1–24] and the related references therein. Very recently, Karapınar et al. [13] presented quasipartial metric spaces and investigated the existence and uniqueness of certain operators in the context of quasipartial metric spaces.
Throughout this paper, we suppose that \(\mathbb{R}^{+}_{0} = [0, +\infty )\), \(\mathbb{N}_{0} = \mathbb{N} \cup\{0\}\), where \(\mathbb{N}\) denotes the set of all positive integers. First, we recall some basic concepts and notations. For more information, see [1, 13].
Definition 1.1
 (QM1)
\(d(x,y)=0\Leftrightarrow x=y\),
 (QM2)
\(d(x,y)\leq d(x,z)+ d(z,y)\).
Definition 1.2
 (PM1)
\(x =y\Leftrightarrow p(x,x)=p(x,y)= p(y,y)\),
 (PM2)
\(p(x,x) \leq p(x,y)\),
 (PM3)
\(p(x,y)= p(y,x)\),
 (PM4)
\(p(x,y)\leq p(x,z)+ p(z,y)p(z,z)\).
Definition 1.3
(See [13])
 (QPM1)
if \(q(x,x)=q(x,y)= q(y,y)\), then \(x =y\) (equality),
 (QPM2)
\(q(x,x) \leq q(x,y)\) (small selfdistances),
 (QPM3)
\(q(x,x)\leq q(y,x)\) (small selfdistances),
 (QPM4)
\(q(x,z)+ q(y,y)\leq q(x,y)+ p(y,z)\) (triangle inequality).
If \(q(x,y)=q(y,x)\) for all \(x,y \in X\), then \((X,q)\) becomes a partial metric space.
Lemma 1.1
(See [13])
Let \((X,q)\) be a quasipartial metric space. Then the following holds: If \(q(x,y)=0\) then \(x=y\).
Definition 1.4
(See [13])
 (i)a sequence \(\{x_{n}\} \subset X\) converges to \(x \in X\) if and only if$$q(x, x) = \lim_{n\rightarrow+\infty} q(x, x_{n}) = \lim _{n\rightarrow +\infty}q(x_{n},x); $$
 (ii)
a sequence \(\{x_{n}\} \subset X\) is called a Cauchy sequence if and only if \(\lim_{n,m\rightarrow+\infty}q(x_{n}, x_{m})\) and \(\lim_{n,m\rightarrow+\infty}q(x_{m}, x_{n})\) exist (and are finite);
 (iii)the quasipartial metric space is said to be complete if every Cauchy sequence \(\{x_{n}\} \subset X\) converges, with respect to \(\tau_{q}\), to a point \(x \in X\) such that$$q(x,x)=\lim_{n,m\rightarrow+\infty}q(x_{n}, x_{m})=\lim _{n,m\rightarrow +\infty}q(x_{m}, x_{n}); $$
 (iv)
a mapping \(f : X \rightarrow X\) is said to be continuous at \(x_{0} \in X\) if for every \(\varepsilon>0\), there exists \(\delta>0\) such that \(f(B(x_{0},\delta)) \subset B(f(x_{0}),\varepsilon)\).
Lemma 1.2
(See [13])
 (A)
The sequence \(\{x_{n}\}\) is Cauchy in \((X, q)\).
 (B)
The sequence \(\{x_{n}\} \) is Cauchy in \((X,p_{q})\).
 (C)
The sequence \(\{x_{n}\} \) is Cauchy in \((X,d_{p_{q}})\).
Lemma 1.3
(See [13])
 (A)
\((X, q)\) is complete.
 (B)
\((X,p_{q})\) is complete.
 (C)
\((X,d_{p_{q}})\) is complete.
In this paper, we shall handle Definition 1.5 in the following way.
Definition 1.5
(See [13])
 (ii)_{a} :

a sequence \(\{x_{n}\}\) in X is called a leftCauchy sequence if and only if for every \(\varepsilon>0\) there exists a positive integer \(N=N(\varepsilon)\) such that \(q(x_{n}, x_{m})<\varepsilon\) for all \(n>m>N\);
 (ii)_{b} :

a sequence \(\{x_{n}\}\) in X is called a rightCauchy sequence if and only if for every \(\varepsilon>0\) there exists a positive integer \(N=N(\varepsilon)\) such that \(q(x_{n}, x_{m})<\varepsilon\) for all \(m>n>N\);
 (iii)_{a} :

the quasipartial metric space is said to be leftcomplete if every leftCauchy sequence \(\{x_{n}\}\) in X is convergent;
 (iii)_{b} :

the quasipartial metric space is said to be rightcomplete if every leftCauchy sequence \(\{x_{n}\}\) in X is convergent.
Remark 1
It is clear that a sequence \(\{x_{n}\}\) in a quasipartial metric space is Cauchy if and only if it is leftCauchy and rightCauchy. Analogously, a quasipartial metric space \((X,q)\) is complete if and only if it is leftcomplete and rightcomplete.
Very recently, Samet et al. [14] introduced the concept αadmissible mappings and established various fixed point theorems for such mappings in complete metric spaces. Later, in 2013, Karapınar et al. [15] proved the existence and uniqueness of a fixed point for triangular αadmissible mappings. For more on αadmissible and triangular αadmissible mappings, see [14, 15].
Definition 1.6
[14]
Definition 1.7
[15]
Very recently, Popescu [16] improved the notion of αadmissible as follows.
Definition 1.8
[16]
Inspired by the notion of αadmissible defined by Popescu [16], we state the following definitions.
Definition 1.9
[16]
Note that a mapping T is αorbital admissible if it is both rightαorbital admissible and leftαorbital admissible.
Popescu [16] refined the notion of triangular αadmissible as follows.
Definition 1.10
[16]
Triangular αadmissible defined by Popescu [16] imposes the following definitions.
Definition 1.11
[16]
Notice that a mapping T is triangular αorbital admissible if it is both triangular rightαorbital admissible and triangular leftαorbital admissible.
It was noted in [16] that each αadmissible mapping is an αorbital admissible mapping and each triangular αadmissible mapping is a triangular αorbital admissible mapping. The converse is false, see e.g. [16], Example 7.
Definition 1.12
[16]
Let \((X,d)\) be a bmetric space, X is said αregular if for every sequence \(\{x_{n}\}\) in X such that \(\alpha(x_{n},x_{n+1})\geq1\) for all n and \(x_{n}\rightarrow x\in X\) as \(n\rightarrow\infty\), there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},x)\geq1\) for all k.
Lemma 1.4
[16]
Let \(T:X\to X\) be a triangular αorbital admissible mapping. Assume that there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\). Define a sequence \(\{x_{n}\}\) by \(x_{n+1}=Tx_{n}\) for each \(n\in\mathbb {N}_{0}\). Then we have \(\alpha(x_{n},x_{m})\geq1\) for all \(m,n \in\mathbb{N}\) with \(n< m\).
The following result can be easily derived from Lemma 1.4.
Lemma 1.5
Let \(T:X\to X\) be a triangular αorbital admissible mapping. Assume that there exists \(x_{0}\in X\) such that \(\alpha(Tx_{0},x_{0})\geq1\). Define a sequence \(\{x_{n}\}\) by \(x_{n+1}=Tx_{n}\) for each \(n\in\mathbb {N}_{0}\). Then we have \(\alpha(x_{m},x_{n})\geq1\) for all \(m,n \in\mathbb{N}\) with \(n< m\).
In this paper, we investigate and extend the existence of a fixed point of \((\psi,\phi)\)contractive mappings on quasipartial metric spaces via αadmissibility.
2 Main results
Definition 2.1
Let \((X,q)\) be a quasipartial metric space where X is a nonempty set. We say that X is said to be αleftregular if for every sequence \(\{x_{n}\}\) in X such that \(\alpha(x_{n+1},x_{n})\geq1\) for all n and \(x_{n}\rightarrow x\in X\) as \(n\rightarrow\infty\), there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha (x,x_{n(k)})\geq1\) for all k. Analogously, a quasipartial metric space X is said to be an αrightregular if for every sequence \(\{x_{n}\}\) in X such that \(\alpha(x_{n},x_{n+1})\geq1\) for all n and \(x_{n}\rightarrow x\in X\) as \(n\rightarrow\infty\), there exists a subsequence \(\{x_{n(k)}\}\) of \(\{ x_{n}\}\) such that \(\alpha(x_{n(k)},x)\geq1\) for all k. We say that X is regular if it is both αleftregular and αrightregular.
Our first result is the following.
Theorem 2.1
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous or X is αregular.
Proof
Now, we shall show that \(\{x_{n}\}\) is a Cauchy sequence in the quasipartial metric space \((X,q)\), that is, the sequence \(\{x_{n}\}\) is leftCauchy and rightCauchy.
As a consequence of Theorem 2.1, we may state the following corollaries.
First, taking \(L = 0\) in Theorem 2.1, we have the following.
Corollary 2.1
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous or X is αregular.
Corollary 2.2
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous or X is αregular.
Proof
It follows by taking \(\psi(t)=t\) and \(\phi(t)=(1k)t\) in Theorem 2.1. □
Corollary 2.3
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous or X is αregular.
Proof
It is sufficient to take \(L=0\) in Corollary 2.2. □
Corollary 2.4
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous or X is αregular.
Proof
By following the lines in the proof of Theorem 2.1, we derive the desired result. We skip the details to avoid repetition. □
 (1)
λ is a Lebesgueintegrable mapping on each compact subset of \([0,+\infty)\),
 (2)
for every \(\epsilon> 0\), we have \(\int_{0}^{\epsilon}\lambda(s)\, ds>0\).
We have the following result.
Corollary 2.5
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous or X is αregular.
Proof
Taking \(L = 0\) in Corollary 2.5, we obtain the following result.
Corollary 2.6
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous or X is αregular.
 (\(\varphi_{1}\)):

φ is nondecreasing,
 (\(\varphi_{2}\)):

\(\sum_{n=0}^{+\infty} \varphi^{n}(t)\) converges for all \(t>0\).
Note that if \(\varphi\in\mathcal{F}\), φ is said to be a \((C)\)comparison function. It is easily proved that if φ is a \((C)\)comparison function, then \(\varphi(t) < t\) for any \(t > 0\). Our second main result is as follows.
Theorem 2.2
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous or X is αregular.
Proof
The following corollary is a generalization of Theorem 17 in [22].
Corollary 2.7
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha(Tx_{0},x_{0})\geq1\);
 (iii)
T is continuous or X is αregular.
We give the following two examples making effective our obtained results.
Example 2.1
Moreover, T is triangular αorbital admissible, \(\alpha(0,T0)\geq1\) and \(\alpha(T0,0)\geq 1\). Thus, by applying Theorem 2.1, n has a fixed point, which is \(u=0\).
Example 2.2
Example 2.3
By induction, we have \(\varphi^{n}(t)\leq t (\frac{t}{1+t})^{n}\) for all \(n\geq1\), so it is clear that φ is a \((C)\)comparison function. Now, we show that (41) is satisfied for all \(x,y\in X\). It suffices to prove it for \(x\leq y\). Consider the following six cases.
3 Consequences and final remarks
Now, we will show that many existing results in the literature can be deduced easily from our Corollary 2.7.
3.1 Standard fixed point theorems
Taking in Corollary 2.7 \(\alpha(x,y)=1\) for all \(x,y\in X\), we derive immediately the following fixed point theorem.
Corollary 3.1
Corollary 3.2
Proof
It is sufficient to take \(\varphi(t)=kt\), where \(k \in [0,1)\), in the above corollary. □
Corollary 3.3
Declarations
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Authors’ Affiliations
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