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α-\((\psi,\phi)\) Contractive mappings on quasi-partial metric spaces
Fixed Point Theory and Applications volume 2015, Article number: 105 (2015)
Abstract
In this paper, we consider α-\((\psi,\phi)\) contractive mappings in the setting of quasi-partial 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.
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 self-distance 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 quasi-partial metric spaces and investigated the existence and uniqueness of certain operators in the context of quasi-partial 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
A quasi-metric on a nonempty set X is a function \(d : X\times X\rightarrow[0, +\infty)\) such that for all \(x, y, z \in X\):
-
(QM1)
\(d(x,y)=0\Leftrightarrow x=y\),
-
(QM2)
\(d(x,y)\leq d(x,z)+ d(z,y)\).
A quasi-metric space is a pair \((X,d)\) such that X is a nonempty set and d is a quasi-metric on X.
Definition 1.2
A partial metric on a nonempty set X is a function \(p : X\times X\rightarrow[0, +\infty)\) such that for all \(x, y, z \in X\):
-
(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)
\(p(x,y)\leq p(x,z)+ p(z,y)-p(z,z)\).
A partial metric space is a pair \((X,p)\) such that X is a nonempty set and p is a partial metric on X.
Definition 1.3
(See [13])
A quasi-partial metric space on a nonempty set X is a function \(q: X\times X \rightarrow[0,+\infty)\) such that for all \(x, y, z \in X\):
-
(QPM1)
if \(q(x,x)=q(x,y)= q(y,y)\), then \(x =y\) (equality),
-
(QPM2)
\(q(x,x) \leq q(x,y)\) (small self-distances),
-
(QPM3)
\(q(x,x)\leq q(y,x)\) (small self-distances),
-
(QPM4)
\(q(x,z)+ q(y,y)\leq q(x,y)+ p(y,z)\) (triangle inequality).
A quasi-partial metric space is a pair \((X,q)\) such that X is a nonempty set and q is a partial metric on X.
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 quasi-partial metric space. Then the following holds: If \(q(x,y)=0\) then \(x=y\).
For a partial metric p on X, the function \(d_{p} : X \times X\rightarrow[0,+\infty)\) defined by
is a metric on X. For a quasi-partial metric q on X, the function \(d_{q} : X \times X\rightarrow[0,+\infty)\) defined by
is a metric on X and
is a partial metric on X. Notice also that for a quasi-partial metric q on X, the function \(d_{m}^{q} : X \times X\rightarrow[0,+\infty)\) defined by
is a metric on X.
Definition 1.4
(See [13])
Let \((X, q)\) be a quasi-partial metric space. Then:
-
(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 quasi-partial 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])
Let \((X, q)\) be a quasi-partial metric space. Let \((X,p_{q})\) be the corresponding partial metric space, and let \((X,d_{p_{q}})\) be the corresponding metric space. The following statements are equivalent:
-
(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])
Let \((X, q)\) be a quasi-partial metric space. Let \((X,p_{q})\) be the corresponding partial metric space, and let \((X,d_{p_{q}})\) be the corresponding metric space. The following statements are equivalent:
-
(A)
\((X, q)\) is complete.
-
(B)
\((X,p_{q})\) is complete.
-
(C)
\((X,d_{p_{q}})\) is complete.
Moreover,
In this paper, we shall handle Definition 1.5 in the following way.
Definition 1.5
(See [13])
Let \((X, q)\) be a quasi-partial metric space. Then:
- (ii)a :
-
a sequence \(\{x_{n}\}\) in X is called a left-Cauchy 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 right-Cauchy 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 quasi-partial metric space is said to be left-complete if every left-Cauchy sequence \(\{x_{n}\}\) in X is convergent;
- (iii)b :
-
the quasi-partial metric space is said to be right-complete if every left-Cauchy sequence \(\{x_{n}\}\) in X is convergent.
Remark 1
It is clear that a sequence \(\{x_{n}\}\) in a quasi-partial metric space is Cauchy if and only if it is left-Cauchy and right-Cauchy. Analogously, a quasi-partial metric space \((X,q)\) is complete if and only if it is left-complete and right-complete.
Very recently, Samet et al. [14] introduced the concept α-admissible 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 α-admissible mappings. For more on α-admissible and triangular α-admissible mappings, see [14, 15].
Definition 1.6
[14]
Let T be a self-mapping on X and \(\alpha: X \times X \rightarrow [0, +\infty)\) be a function. We say that T is an α-admissible mapping if
Definition 1.7
[15]
Let T be a self-mapping on X and \(\alpha: X \times X \rightarrow [0, +\infty)\) be a function. We say that T is a triangular α-admissible mapping if T is α-admissible and
Very recently, Popescu [16] improved the notion of α-admissible as follows.
Definition 1.8
[16]
Let \(T:X\rightarrow X\) be a self-mapping and \(\alpha:X\times X\rightarrow[0,\infty)\) be a function. Then T is said to be α-orbital admissible if
Inspired by the notion of α-admissible defined by Popescu [16], we state the following definitions.
Definition 1.9
[16]
Let \(T:X\rightarrow X\) be a self-mapping and \(\alpha:X\times X\rightarrow[0,\infty)\) be a function. Then T is said to be right-α-orbital admissible if
and be left-α-orbital admissible if
Note that a mapping T is α-orbital admissible if it is both right-α-orbital admissible and left-α-orbital admissible.
Popescu [16] refined the notion of triangular α-admissible as follows.
Definition 1.10
[16]
Let \(T:X\rightarrow X\) be a self-mapping and \(\alpha:X\times X\rightarrow[0,\infty)\) be a function. Then T is said to be triangular α-orbital admissible if T is α-orbital admissible and
Triangular α-admissible defined by Popescu [16] imposes the following definitions.
Definition 1.11
[16]
Let \(T:X\rightarrow X\) be a self-mapping and \(\alpha:X\times X\rightarrow[0,\infty)\) be a function. Then T is said to be triangular α-orbital admissible if T is right-α-orbital admissible and
and be triangular left-α-orbital admissible if T is α-orbital admissible and
Notice that a mapping T is triangular α-orbital admissible if it is both triangular right-α-orbital admissible and triangular left-α-orbital admissible.
It was noted in [16] that each α-admissible mapping is an α-orbital admissible mapping and each triangular α-admissible mapping is a triangular α-orbital admissible mapping. The converse is false, see e.g. [16], Example 7.
Definition 1.12
[16]
Let \((X,d)\) be a b-metric space, X is said α-regular 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 α-orbital 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 α-orbital 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 quasi-partial metric spaces via α-admissibility.
2 Main results
In this section, we shall present the main theorem of the paper. For our aim, we need to define the following class of auxiliary mappings: Let Λ be set of functions \(\varphi: [0,+\infty)\rightarrow [0,+\infty)\) such that \(\varphi^{-1}(\{0\})=\{0\}\):
Let \((X,q)\) be a quasi-partial metric space. We consider the following expressions:
for all \(x,y\in X\).
Definition 2.1
Let \((X,q)\) be a quasi-partial metric space where X is a nonempty set. We say that X is said to be α-left-regular 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 quasi-partial metric space X is said to be an α-right-regular 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 α-left-regular and α-right-regular.
Our first result is the following.
Theorem 2.1
Let \((X,q)\) be a complete quasi-partial metric space. Let \(T:X\rightarrow X\) be a self-mapping. Assume that there exist \(\psi \in\Psi\), \(\phi\in\Phi\), \(L\geq0\) and a function \(\alpha: X \times X \rightarrow[0, +\infty)\) such that for all \(x,y\in X\),
Also, suppose that the following assertions hold:
-
(i)
T is triangular α-orbital 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 α-regular.
Then T has a fixed point \(u\in X\) and \(q(u,u)=0\).
Proof
We construct a sequence \(\{x_{n}\}\) in X in the following way:
If \(q(x_{n_{0}}, x_{n_{0}+1})=0\) for some \(n_{0}\geq0\), then we have \(x_{n_{0}}=x_{n_{0}+1}=Tx_{n_{0}}\), that is, \(x_{n_{0}}\) is the fixed point of T. Consequently, we suppose that \(q(x_{n}, x_{n+1})> 0\) for all \(n\in\mathbb{N}_{0}\).
By (ii), we have \(\alpha(x_{0},Tx_{0})\geq1\) and \(\alpha (Tx_{0},x_{0})\geq1\). On account of (i), we derive that
Recursively, we obtain that
Regarding (6) and (7), we find that
where
and
Thus, we conclude from (8) that
by taking (10) and (9) into account.
If for some n we have \(\max\{q(x_{n-1},x_{n}), q(x_{n},x_{n+1})\}=q(x_{n},x_{n+1})\), then (11) yields that
Consequently, we conclude that \(\phi(q(x_{n}, x_{n+1}))=0\). Since \(\phi^{-1}(\{0\})=\{0\}\), we get \(q(x_{n}, x_{n+1})=0\), which contradicts the assumption that \(q(x_{n}, x_{n+1})>0\) for all \(n\in \mathbb{N}_{0}\). Thus, we have
Hence, (8) turns into
Due to the property of the auxiliary function ψ, we have
Eventually, we observe that the sequence \(\{q(x_{n}, x_{n+1})\}\) is non-increasing. So, there exists \(\delta\geq0\) such that
If \(\delta> 0\), taking \(\limsup_{n\rightarrow+\infty }\) in inequality (11), by keeping (10) and (12) in the mind, we obtain that
By continuity of ψ and lower semi-continuity of ϕ, we obtain \(\psi(\delta)\leq\psi(\delta)-\phi(\delta)\), which is a contradiction. So,
Analogously, we derive that
Now, we shall show that \(\{x_{n}\}\) is a Cauchy sequence in the quasi-partial metric space \((X,q)\), that is, the sequence \(\{x_{n}\}\) is left-Cauchy and right-Cauchy.
Suppose that \(\{x_{n}\}\) is not a left-Cauchy sequence in \((X,q)\). Then there is \(\varepsilon> 0\) such that for each integer k there exist integers \(n(k) > m(k) > k\) such that
Further, corresponding to \(m(k)\), we can choose \(n(k)\) so that it is the smallest integer with \(n(k)>m(k)\) satisfying (16). Consequently, we have
Due to the triangle inequality, we have
Letting \(k \to\infty\) and taking (14) into account, we get that
On the other hand, again by the triangle inequality, we find that
and
Letting \(k \to\infty\) and taking (14), (15), (19), (20), (21) into account, we derive that
Letting \(k \to\infty\) and taking (14), (18), (20), (23) into account, we derive that
Since T is triangular α-orbital admissible, from Lemma 1.4 and Lemma 1.5 we derive that
Regarding (6) and (25), we find that
where
and
Regarding (14) and (15), we note that
On the other hand, we get that
due to the limits (14), (15), (19).
From the above observation, letting \(k \to\infty\) in (26), we obtain
So, \(\phi(\varepsilon)=0\), which is a contradiction with respect to the fact that \(\varepsilon> 0\). Thus \(\{x_{n}\}\) is a left-Cauchy sequence in the metric space \((X,q)\). Analogously, we derive that \(\{x_{n}\}\) is a right-Cauchy sequence in the metric space \((X,q)\). Since \((X,q)\) is complete, then from Lemma 1.3 \((X,d_{p_{q}})\) is a complete metric space. Therefore, the sequence \(\{x_{n}\}\) converges to a point \(u\in X\) in \((X,d_{p_{q}})\), that is,
Again, from Lemma 1.3,
On the other hand, by (14) and the condition (QPM2) from Definition 1.3,
So, it follows that
Now, for proving fixed point of T, first we suppose that T is continuous, then we have
So, u is a fixed point of T.
As the last step, suppose that X is α-regular. Hence it is α-right-regular, then there exists a subsequence \(\{x_{n(k)}\} \) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},u)\geq1\) for all k. Now, we show that \(q(u,T u)=0\). Assume this is not true, then from (6) we obtain
where
It is obvious that \(\lim_{k\rightarrow+\infty }q(x_{n(k)},Tu)=q(u,T u)\). Therefore, using (14) and (32), we deduce that
Also,
because (14), (15) and (31) give \(\lim_{n\rightarrow+\infty }d_{m}^{q}(x_{n(k)},Tx_{n(k)})=0\). Now, by using the properties of ψ and ϕ and taking the upper limit as \(n\rightarrow +\infty\), we obtain
Then \(\phi(q(u,T u))=0\), i.e., \(q(u,T u)=0\), and so \(Tu=u\). Now, we conclude that T has a fixed point \(u\in X\) and \(q(u,u)=0\). □
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
Let \((X,q)\) be a complete quasi-partial metric space. Let \(T:X\rightarrow X\) be a self-mapping. Suppose that there exist \(\psi\in\Psi\), \(\phi\in\Phi\) and a function \(\alpha: X \times X \rightarrow[0, +\infty)\) such that for all \(x,y\in X\),
Also, suppose that the following assertions hold:
-
(i)
T is triangular α-orbital 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 α-regular.
Then T has a fixed point \(u\in X\) and \(q(u,u)=0\).
Corollary 2.2
Let \((X,q)\) be a complete quasi-partial metric space. Let \(T:X\rightarrow X\) be a self-mapping. Suppose that there exist \(k\in [0,1)\), \(L\geq0\) and a function \(\alpha: X \times X \rightarrow[0, +\infty)\) such that for all \(x,y\in X\),
Also, suppose that the following assertions hold:
-
(i)
T is triangular α-orbital 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 α-regular.
Then T has a fixed point \(u\in X\) and \(q(u,u)=0\).
Proof
It follows by taking \(\psi(t)=t\) and \(\phi(t)=(1-k)t\) in Theorem 2.1. □
Corollary 2.3
Let \((X,q)\) be a complete quasi-partial metric space. Let \(T:X\rightarrow X\) be a self-mapping. Suppose that there exist \(k\in [0,1)\), \(L\geq0\) and a function \(\alpha: X \times X \rightarrow[0, +\infty)\) such that for all \(x,y\in X\),
Also, suppose that the following assertions hold:
-
(i)
T is triangular α-orbital 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 α-regular.
Then T has a fixed point \(u\in X\) and \(q(u,u)=0\).
Proof
It is sufficient to take \(L=0\) in Corollary 2.2. □
Corollary 2.4
Let \((X,q)\) be a complete quasi-partial metric space. Let \(T:X\rightarrow X\) be a self-mapping. Suppose that there exist \(k\in [0,1)\), \(L\geq0\) and a function \(\alpha: X \times X \rightarrow[0, +\infty)\) such that for all \(x,y\in X\),
Also, suppose that the following assertions hold:
-
(i)
T is triangular α-orbital 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 α-regular.
Then T has a fixed point \(u\in X\) and \(q(u,u)=0\).
Proof
By following the lines in the proof of Theorem 2.1, we derive the desired result. We skip the details to avoid repetition. □
Denote by \(\Lambda'\) the set of functions \(\lambda: [0,+\infty )\rightarrow[0,+\infty)\) satisfying the following hypotheses:
-
(1)
λ is a Lebesgue-integrable 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
Let \((X,q)\) be a complete quasi-partial metric space. Let \(T:X\rightarrow X\) be a self-mapping. Suppose that there exist \(\lambda,\beta\in\Lambda'\), \(L\geq0\) and a function \(\alpha: X \times X \rightarrow[0, +\infty)\) such that for all \(x,y\in X\),
Also, suppose that the following assertions hold:
-
(i)
T is triangular α-orbital 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 α-regular.
Then T has a fixed point \(u\in X\) and \(q(u,u)=0\).
Proof
It follows from Theorem 2.1 by taking
and
□
Taking \(L = 0\) in Corollary 2.5, we obtain the following result.
Corollary 2.6
Let \((X,q)\) be a complete quasi-partial metric space. Let \(T:X\rightarrow X\) be a self-mapping. Suppose that there exist \(\lambda,\beta\in\Lambda'\) and a function \(\alpha: X \times X \rightarrow[0, +\infty)\) such that for all \(x,y\in X\),
Also, suppose that the following assertions hold:
-
(i)
T is triangular α-orbital 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 α-regular.
Then T has a fixed point \(u\in X\) and \(q(u,u)=0\).
Now, let \(\mathcal{F}\) be the set of functions \(\varphi:[0,+\infty )\rightarrow[0,+\infty)\) satisfying the following hypotheses:
- (\(\varphi_{1}\)):
-
φ is non-decreasing,
- (\(\varphi_{2}\)):
-
\(\sum_{n=0}^{+\infty} \varphi^{n}(t)\) 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
Let \((X,q)\) be a complete quasi-partial metric space. Let \(T:X\rightarrow X\) be a mapping such that there exist \(\varphi\in \mathcal{F}\), \(L\geq0\) and a function \(\alpha: X \times X \rightarrow[0, +\infty)\) such that for all \(x,y\in X\),
Also, suppose that the following assertions hold:
-
(i)
T is triangular α-orbital 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 α-regular.
Then T has a fixed point \(u\in X\) and \(q(u,u)=0\).
Proof
By following the lines in the proof of Theorem 2.1, we derive that
By combining (41) and (42), we find that
So,
Now, similar to proving Theorem 2.1, we obtain
and
Therefore
If, for some \(n\geq1\), we have \(q(x_{n-1},x_{n})\leq q(x_{n},x_{n+1})\). So, from (43), we obtain that
which is a contradiction. Thus, for all \(n\geq1\), we have
Using (43) and (44), we get that
Iteratively, we find that
for all \(n\geq0\). By the triangle inequality, we get, for \(m>n\),
Since \(\sum_{k=0}^{+\infty} \varphi^{k}(q(x_{0},x_{1}))<\infty\), then \(\lim_{n,m\rightarrow+\infty}q(x_{n}, x_{m})=0\). So, \(\{ x_{n}\}\) is a right-Cauchy sequence in \((X,q)\). Similarly, since \(\alpha(Tx_{0},x_{0})\geq1\), we get
So, \(\{x_{n}\}\) is a left-Cauchy sequence in \((X,q)\). By part (ii) of Definition 1.5, \(\{x_{n}\}\) is a Cauchy sequence in a complete quasi-partial metric space \((X,q)\), then \(\{x_{n}\} \) converges, with respect to \(\tau_{q}\), to a point \(u\in X\) such that
Suppose that T is continuous, then we have
So, u is a fixed point of T.
Now, suppose that X is α-regular. Hence it is both α-left-regular and α-right-regular. Thus, there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha (x_{n(k)},u)\geq1\) for all k. Now, we claim that \(q(u, T u) = 0\). Suppose the contrary, then \(q(u, T u) > 0\). By (41), we have
where
By (51), we have
and
Therefore
which is a contradiction. That is, \(q(u, Tu)=0\). Thus, we obtained that u is a fixed point for T and \(q(u,u)=0\). □
The following corollary is a generalization of Theorem 17 in [22].
Corollary 2.7
Let \((X,q)\) be a complete quasi-partial metric space. Let \(n:X\rightarrow X\) be a mapping such that there exist \(\varphi\in \mathcal{F}\) and a function \(\alpha: X \times X \rightarrow[0, +\infty)\) such that for all \(x,y\in X\),
Also, suppose that the following assertions hold:
-
(i)
T is triangular α-orbital 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 α-regular.
Then T has a fixed point \(u\in X\) and \(q(u,u)=0\).
We give the following two examples making effective our obtained results.
Example 2.1
Let \(X= [0,+\infty)\) and \(q(x, y) = | x-y|+ x\) for all \(x, y \in X\). Then \((X,q)\) is a complete quasi-partial metric space. Consider \(T:X\rightarrow X\) defined by
Define \(\alpha:X\times X\rightarrow[0,+\infty)\) by
Take \(\psi(t)=\frac{2t}{3}\) and \(\phi(t)=\frac{t}{3}\) for all \(t\geq0\). Note that \(\psi\in\Psi\) and \(\phi\in\Phi\). Take \(x\leq y\), then
for all \(L\geq0\). Now, let \(y< x\), then
We have two possibilities for \(M(x,y)\).
Case 1: if \(M(x,y)=2x-y\) then
for all \(L\geq0\).
Case 2: if \(M(x,y)=\frac{3 x}{2}\) then
for all \(L\geq0\).
Moreover, T is triangular α-orbital 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
Let \(X= [0,+\infty)\) and \(q(x, y) = \max\{x,y\}\) for all \(x, y \in X\). Then \((X,q)\) is a complete quasi-partial metric space. Consider \(n:X\rightarrow X\) defined by
Define \(\alpha:X\times X\rightarrow[0,+\infty)\) by
Take \(\psi(t)=1\) and \(\phi(t)=\frac{1}{1+t}\) for all \(t\geq0\). Note that \(\psi\in\Psi\) and \(\phi\in\Phi\). Take \(x\leq y\), then
for all \(L\geq0\). Moreover, T is triangular α-orbital 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 2.3
Let \(X = [0, 2]\) and \(q:X\times X\rightarrow\mathbb{R}^{+}\) be defined by \(q(x, y) =\max\{x, y\}\). Then \((X,q)\) is a complete quasi-partial metric space. Define \(T : X \rightarrow X\) by
and \(\alpha:X\times X\rightarrow[0,+\infty)\) by
Take \(\varphi:[0,+\infty)\rightarrow[0,+\infty)\) defined by
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.
Case 1. Let \(x,y\in[0, 1)\), then
Case 2. Let \(x, y\in[1,2)\), then
Case 3. Let \(x= y=2\), then
Case 4. Let \(x\in[0, 1)\) and \(y\in[1,2)\), then
Case 5. Let \(x\in[0, 1)\) and \(y=2\), then
Case 6. Let \(x\in[1, 2)\) and \(y=2\), then
Since, for all \(x, y\in X\), \(LN(x,y)\geq0\), so we have
Moreover, T is triangular α-orbital admissible, \(\alpha (0,T0)\geq1\) and \(\alpha(n0,0)\geq1\). Then all the required hypotheses of Theorem 2.2 are satisfied. So, T has a fixed point, which is \(u=0\).
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
Let \((X,q)\) be a complete quasi-partial metric space and \(T: X\to X\) be a given mapping. Suppose that there exists a function \(\varphi\in \mathcal{F}\) such that
for all \(x,y\in X\). Then T has a unique fixed point.
Corollary 3.2
Let \((X,q)\) be a complete quasi-partial metric space and \(T: X\to X\) be a given mapping. Suppose that there exists \(k \in[0,1)\) such that
for all \(x,y\in X\). Then T has a unique fixed point.
Proof
It is sufficient to take \(\varphi(t)=kt\), where \(k \in [0,1)\), in the above corollary. □
Corollary 3.3
Let \((X,q)\) be a complete quasi-partial metric space and \(T: X\to X\) be a given mapping. Suppose that there exists \(k \in[0,1)\) such that
for all \(x,y\in X\). Then T has a unique fixed point.
Remark 2
As we show in Corollaries 3.1-3.3, we can list some more results as a consequence of our main theorems by choosing the auxiliary function \(\alpha: X\times X \to[0,\infty)\) as it was done in e.g. [14, 17].
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Karapınar, E., Gholizadeh, L., Alsulami, H.H. et al. α-\((\psi,\phi)\) Contractive mappings on quasi-partial metric spaces. Fixed Point Theory Appl 2015, 105 (2015). https://doi.org/10.1186/s13663-015-0352-z
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DOI: https://doi.org/10.1186/s13663-015-0352-z