A unified view on common fixed point theorems for Ćirić quasicontraction maps
 Fei He^{1}Email author and
 XiaoYa Nan^{1}
https://doi.org/10.1186/s1366301503648
© He and Nan 2015
Received: 31 January 2015
Accepted: 23 June 2015
Published: 11 July 2015
Abstract
The main purpose of this paper is to unify and to generalize some common fixed point results in metric spaces with a Qfunction (or a wdistance) and the related results in dislocated metric spaces (also called metriclike spaces). First in the setting of dislocated quasimetric spaces we introduce the notion of weak 0σcompleteness which is weaker than 0σcompleteness. By using the new type of completeness, we establish some common fixed point theorems for two selfmappings satisfying a nonlinear contractive condition of Ćirić type. Our results unify and generalize many wellknown common fixed point theorems. Finally, we give some further applications of our main results.
Keywords
MSC
1 Introduction
In 1922, Banach [1] published his fixed point theorem, also known as Banach contraction principle. Over the years, various extensions and generalizations of this principle have appeared in the literature. Ćirić [2] introduced the notion of a quasicontraction as one of the most general contractive type mappings. A wellknown Ćirić result is that a quasicontraction possesses a unique fixed point. Das and Naik [3] proved some common fixed point theorems which generalized and unified the results of Ćirić [2] and Jungck [4]. Ume [5] obtained fixed point theorems in a complete metric space using the concept of a wdistance and improved fixed point theorems of Ćirić [2]. Ilić and Rakočević [6] established some common fixed point results for maps on metric spaces with wdistance and unified the results of Das and Naik [3] and Ume [5]. Recently, Di Bari and Vetro [7] obtained common fixed points for two selfmappings satisfying nonlinear quasicontractions of Ćirić type on metric spaces. Inspired by the ideas in [6, 7], He [8] established some common fixed point theorems for two selfmappings satisfying a nonlinear contractive condition of Ćirić type with a Qfunction which generalize and unify fixed point results in [6, 7].
In 1994, Matthews [9] introduced the notion of partial metric space as a part of the study of denotational semantics of dataflow networks. Further, Matthews showed that the Banach contraction principle is valid in partial metric spaces and can be applied in program verification. After that, many authors studied and generalized the results of Matthews (see, for example, [10–14]). In particular, Hitzler and Seda [15] introduced the concept of dislocated metric spaces as a generalization of metric spaces and partial metric spaces and presented variants of Banach contraction principle in such spaces. Zeyada et al. [16] introduced the notion of dislocated quasimetric space and generalized the results of Hitzler and Seda [15]. AminiHarandi [17] reintroduced the dislocated metric spaces under the name of metriclike spaces and proved some fixed point theorems in metriclike spaces. Shukla et al. [18] proved some common fixed point theorems in 0σcomplete metriclike spaces and generalized the results of AminiHarandi [17].
In this paper, we attempt to give a unified approach to the above mentioned fixed point results. First in the framework of dislocated quasimetric spaces we introduce the notion of weak 0σcompleteness, which is weaker than 0σcompleteness. In particular, we show that if p is a Qfunction (or a wdistance) on complete metric spaces X then \((X,p)\) is a weak 0σcomplete dislocated quasimetric space. By using the new type of completeness, we establish some common fixed point results on dislocated quasimetric spaces, which generalize and unify the results of Ume [5], Ilić and Rakočević [6], Di Bari and Vetro [7], He [8], AminiHarandi [17], Shukla et al. [18] and some others. As further applications of our results, we derive some common fixed point theorems in weak quasipartial metric spaces, in \(T_{0}\)quasipseudometric spaces and in uniform spaces.
2 Preliminaries
First we recall some definitions and results as regards the Qfunction and dislocated quasimetric spaces.
Definition 2.1
 (q1):

\(q(x,z)\leq q(x,y)+q(y,z)\), for any \(x,y,z\in X\),
 (q2):

if \(x\in X\) and \(\{y_{n}\}_{n\in\mathbf{N}}\) is a sequence in X which converges to a point y and \(q(x,y_{n})\leq M\) for some \(M=M(x)>0\), then \(q(x,y)\leq M\),
 (q3):

for any \(\varepsilon>0\), there exists \(\delta>0\) such that \(q(z,x)\leq\delta\) and \(q(z,y)\leq\delta\) imply \(d(x,y)\leq\varepsilon\).
 (q2′):

for any \(x\in X\), \(q(x,\cdot) : X\to[0,+\infty)\) is lower semicontinuous,
For some examples of Qfunctions and wdistances, the reader can refer to [19, 20]. The following lemma has been presented in [19, 20].
Lemma 2.2
 (i)
If \(q(x_{n},y)\leq\alpha_{n}\) and \(q(x_{n},z)\leq\beta_{n}\) for any \(n\in\mathbf{N}\), then \(y=z\). In particular, if \(q(x,y)=0\) and \(q(x,z)=0\), then \(y=z\).
 (ii)
If \(q(x_{n},y_{n})\leq\alpha_{n}\) and \(q(x_{n},z)\leq\beta_{n}\) for any \(n\in\mathbf{N}\), then \(\{y_{n}\}\) converges to z.
 (iii)
If \(q(x_{n},x_{m})\leq\alpha_{n}\) for any \(n,m\in\mathbf{N}\) with \(m>n\), then \(\{x_{n}\}\) is a Cauchy sequence.
 (iv)
If \(q(y,x_{n})\leq\alpha_{n}\) for any \(n\in\mathbf{N}\), then \(\{x_{n}\}\) is a Cauchy sequence.
Definition 2.3
 (σ1):

if \(\sigma(x,y)=\sigma(y,x)=0\) then \(x =y\),
 (σ2):

\(\sigma(x,y)= \sigma(y,x)\),
 (σ3):

\(\sigma(x,y)\leq\sigma(x,z)+ \sigma(z,y)\),
Remark 2.4
Let \((X,d)\) be a metric space and let q be a Qfunction on X. Then \((X,q)\) is a dislocated quasimetric space. Indeed, it is obvious that the function q satisfies (σ3). Let \(x,y\in X\) be such that \(q(x,y)=q(y,x)=0\). By (σ3), we have \(q(x,x)=q(y,y)=0\). From Lemma 2.2(i), \(q(x,x)=0\) and \(q(x,y)=0\) imply \(x=y\). Hence (σ1) is satisfied.
Definition 2.5
 (1)A sequence \(\{x_{n}\}\) in X converges to \(x\in X\) if and only if$$\lim_{n\to+\infty}\sigma(x,x_{n})=\lim_{n\to+\infty} \sigma(x_{n},x)= \sigma(x,x). $$
 (2)A sequence \(\{x_{n}\}\) in X is called a σCauchy sequence if and only ifexist (and are finite).$$\lim_{n,m\to+\infty}\sigma(x_{n},x_{m})\quad \mbox{and}\quad \lim_{n,m\to+\infty }\sigma(x_{m},x_{n}) $$
 (3)
A sequence \(\{x_{n}\}\) in X is called a 0σCauchy sequence if for any \(\varepsilon>0\), there exists \(n_{0}\in\mathbf{N}\) such that for all \(m,n>n_{0}\), \(\sigma(x_{n},x_{m})<\varepsilon\), that is, \(\sigma(x_{n},x_{m})\to0\) and \(\sigma(x_{m},x_{n})\to0\) as \(m,n\to+\infty\).
 (4)The space \((X,\sigma)\) is said to be complete if for each σCauchy sequence \(\{x_{n}\}\) in X, there exists \(x\in X\) such that$$\lim_{n\to+\infty}\sigma(x_{n},x)=\lim_{n\to+\infty} \sigma(x,x_{n})=\sigma (x,x)=\lim_{m,n\to\infty} \sigma(x_{n},x_{m}). $$
 (5)
The space \((X,\sigma)\) is said to be 0σcomplete if every 0σCauchy sequence \(\{x_{n}\}\) in X converges to a point \(x\in X\) such that \(\sigma(x,x)=0\).
It is obvious that every complete dislocated metric space is 0σcomplete, but the converse may not be true; see Example 3 in [18].
Now we introduce the following more extensive completeness for dislocated quasimetric spaces, which is weaker than the 0σcompleteness.
Definition 2.6
A dislocated quasimetric space \((X,\sigma)\) is called weak 0σcomplete, if for each 0σCauchy sequence \(\{x_{n}\}\), there exists \(x\in X\) such that \(\sigma(x_{n},x)\to0\) as \(n\to+\infty\).
It is not hard to see that every 0σcomplete dislocated quasimetric space is weak 0σcomplete. The following example shows that the converse assertions do not hold.
Example 2.7

\(\sigma(x,x)=0\) for all \(x\in X\),

\(\sigma(n,0)=0\) for all \(n\in\mathbf{N}\),

\(\sigma(0,n)=1\) for all \(n\in\mathbf{N}\), and

\(\sigma(n,m)=\frac{1}{n}\frac{1}{m}\) for all \(n,m\in\mathbf{N}\).
Remark 2.8
 (i)A sequence \(\{x_{n}\}\) in X rightconverges (leftconverges) to \(x\in X\) if and only if$$\lim_{n\to+\infty}\sigma(x_{n},x)= \sigma(x,x)\qquad \Bigl( \lim_{n\to+\infty }\sigma(x,x_{n})= \sigma(x,x)\Bigr). $$
 (ii)
A sequence \(\{x_{n}\}\) in X is called a 0σrightCauchy (0σleftCauchy) sequence if for any \(\varepsilon>0\), there exists \(n_{0}\in\mathbf{N}\) such that for all \(m>n>n_{0}\), \(\sigma(x_{m},x_{n})<\varepsilon\) (\(\sigma(x_{n},x_{m})<\varepsilon\)), that is, \(\sigma(x_{m},x_{n})\to0\) (\(\sigma(x_{n},x_{m})\to0\)) as \(m>n\to+\infty\).
 (iii)
The space \((X,\sigma)\) is said to be 0σrightcomplete (0σleftcomplete) if every 0σrightCauchy (0σleftCauchy) sequence \(\{x_{n}\}\) in X rightconverges (leftconverges) to a point \(x\in X\) such that \(\sigma(x,x)=0\).
Example 2.9

\(\sigma(n,n)=0\) for all \(n\in X\),

\(\sigma(m,n)=1\) for all \(m,n\in X\) with \(m< n\), and

\(\sigma(m,n)=\frac{1}{n}\frac{1}{m}\) for all \(m,n\in X\) with \(m>n\).
Proposition 2.10
Let \((X,d)\) be a complete metric space and let q be a Qfunction on X. Then \((X,q)\) is a weak 0qcomplete dislocated quasimetric space.
Proof
First \((X,q)\) is a dislocated quasimetric space; see Remark 2.4. Let \(\{x_{n}\}\) be a 0qCauchy sequence in \((X,q)\). Then for any \(\varepsilon>0\) there exists \(n_{0}\in\mathbf{N}\) such that \(q(x_{n},x_{m})<\varepsilon\) for all \(m,n>n_{0}\). In particular, we have \(q(x_{n},x_{m})<\varepsilon\) for all \(m>n>n_{0}\). By Lemma 2.2(iii), we see that \(\{x_{n}\}\) is a Cauchy sequence in \((X,d)\). Since \((X,d)\) is complete, there exists \(x\in X\) such that \(d(x_{n},x)\to0\) (\(n\to+\infty\)). Let \(n>n_{0}\) be given. Since \(x_{m}\to x\) (\(m\to+\infty\)) and \(q(x_{n},x_{m})<\varepsilon\) (\(m>n>n_{0}\)), by (q2) we get \(q(x_{n},x)\leq\varepsilon\) for all \(n>n_{0}\). This means that \(q(x_{n},x)\to0\) as \(n\to+\infty\). Hence \((X,q)\) is weak 0qcomplete. □
For the following definitions and notations we can refer to [7, 8].
Definition 2.11
Let f and g be selfmaps of a set X. If \(fx = gx\) for some \(x\in X\), then x is called a coincidence point of f and g. The pair f, g of selfmaps is weakly compatible if they commute at their coincidence points.
 (i)
ψ is nondecreasing,
 (ii)
\(\psi(0)=0\),
 (iii)
\(\lim_{x\to+\infty}(x\psi(x))=+\infty\), and
 (iv)
\(\lim_{t\to r^{+}}\psi(t)< r\) for all \(r>0\).
It is obvious that if \(\psi : [0,+\infty)\to[0,+\infty)\) is defined by \(\psi(t)=\lambda t\) for some \(\lambda\in[0,1)\), or \(\psi(t)=\ln(t+1)\), then \(\psi\in\Psi\).
Remark 2.12
(Remark 2.1 in [7])
If \(\psi\in\Psi\), then we have \(\psi(r)< r\) and \(\lim_{n\to+\infty}\psi^{n}(r)=0\) for all \(r>0\).
The following lemmas will be useful in the sequel.
Lemma 2.13
Proof
Lemma 2.14
Proof
Lemma 2.15
Let \((X,\sigma)\) be a dislocated quasimetric space and let \(f,g : X\to X\) be weakly compatible selfmappings such that \(f(X)\subset g(X)\). Suppose that f and g are a \((\psi,\sigma)\)quasicontraction with \(\psi\in\Psi\). If f and g have a coincidence point y, i.e., \(fy = gy\), then \(u=fy=gy\) is the unique common fixed point of f and g and \(\sigma(u,u)=0\).
Proof
Lemma 2.16
 (i)For each \(x_{0}\in X\) and \(n\in\mathbf{N}\), there exist \(k,l\in\mathbf{N}\) with \(k,l\leq n\) such that$$\delta_{\sigma}\bigl({\mathcal{O}}(x_{0},n)\bigr)=\max\bigl\{ \sigma(gx_{0},gx_{0}),\sigma (gx_{0},fx_{k}), \sigma(fx_{l},gx_{0})\bigr\} . $$
 (ii)For each \(x_{0}\in X\), there exists \(c>0\) such that$$\delta_{\sigma}\bigl({\mathcal{O}}(x_{0},\infty)\bigr)< c. $$
 (iii)
For each \(x_{0}\in X\), \(\{fx_{n}\}\) is a 0σCauchy sequence.
Proof
3 Main results
Now we begin to state our main result.
Theorem 3.1
 (D1)for every \(y\in X\) with \(fy\neq gy\),$$\inf\bigl\{ \sigma(gx,y)+\sigma(gx,fx)+\sigma(fx,gx) : x\in X\bigr\} >0. $$
Proof
Let \(x_{0}\in X\) be fixed. As \(f(X)\subset g(X)\), we may choose \(x_{1}\in X\) such that \(fx_{0}=gx_{1}\). If \(x_{n}\in X\) is given, we may choose \(x_{n+1}\in X\) such that \(fx_{n}=gx_{n+1}\). In this way we construct a fgsequence \(\{fx_{n}\}\) of initial point \(x_{0}\). Using Lemma 2.16(iii), we see that \(\{fx_{n}\}\) is a 0σCauchy sequence. Since X is weak 0σcomplete, there exists \(y\in X\) such that \(\sigma(fx_{n}, y)\to0\).
If f and g are weakly compatible, then by Lemma 2.15 we obtain \(u=fy\) is a unique common fixed point of f and g and \(\sigma(u,u)=0\). □
Theorem 3.2
 (D2)for every \(z\in X\) with \(fz\neq gz\),$$\inf\bigl\{ \sigma(gx,gz)+\sigma(gx,fx)+\sigma(fx,gx) : x\in X\bigr\} >0. $$
Proof
Let \(x_{0}\in X\) be fixed. By using the same way as in the proof Theorem 3.1, we may construct a fgsequence \(\{fx_{n}\}\) of initial point \(x_{0}\). Using Lemma 2.16(iii), we see that \(\{fx_{n}\}\) is a 0σCauchy sequence. Since \(f(X)\) or \(g(X)\) is a weak 0σcomplete subspace of X, there exists \(y\in g(X)\) such that \(\sigma(fx_{n}, y)\to0\). Let \(z\in X\) be such that \(gz=y\).
If f and g are weakly compatible, then by Lemma 2.15 we obtain \(u=fz\) is a unique common fixed point of f and g and \(\sigma(u,u)=0\). □
The following theorem shows that in Theorem 3.2 if the weak 0σcompleteness is replaced by the 0σcompleteness then the condition (D2) can be omitted.
Theorem 3.3
Let \((X,\sigma)\) be a dislocated quasimetric space and let \(f,g : X\to X\) be two selfmappings such that \(f(X)\subset g(X)\). Suppose that f and g are a \((\psi,\sigma)\)quasicontraction with \(\psi\in\Psi\). Let \(f(X)\) or \(g(X)\) is a 0σcomplete subspace of X. Then f and g have a coincidence point in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point u in X and \(\sigma(u,u)=0\).
Proof
Let \(x_{0}\in X\) be fixed. By using the same way as in the proof Theorem 3.1, we may construct a fgsequence \(\{fx_{n}\}\) of initial point \(x_{0}\). Using Lemma 2.16(iii), we see that \(\{fx_{n}\}\) is a 0σCauchy sequence. Since \(f(X)\) or \(g(X)\) is 0σcomplete, there exists \(y\in g(X)\) such that \(\sigma(fx_{n}, y)\to0\) and \(\sigma (y,fx_{n})\to0\).
If f and g are weakly compatible, then by Lemma 2.15 we obtain \(u=fy\) is a unique common fixed point of f and g and \(\sigma(u,u)=0\). □
Now, we present two examples to illustrate the obtained results.
Example 3.4
Example 3.5
4 Applications
4.1 Applications to metric spaces with Qfunction
From Theorem 3.1, Theorem 3.2 and Proposition 2.10, we can obtain the following results, which generalize Theorem 13 and Theorem 14 in [8].
Corollary 4.1
 (D1′):

for every \(y\in X\) with \(fy\neq gy\),$$\inf\bigl\{ q(gx,y)+q(gx,fx)+q(fx,gx) : x\in X\bigr\} >0. $$
Corollary 4.2
 (D2′):

for every \(z\in X\) with \(fz\neq gz\),$$\inf\bigl\{ q(gx,gz)+q(gx,fx)+q(fx,gx) : x\in X\bigr\} >0. $$
Remark 4.3
 (D1″):

for every \(y\in X\) with \(fy\neq gy\),$$\inf\bigl\{ q(gx,y)+q(gx,fx) : x\in X\bigr\} >0, $$
The next is an example where we can apply Theorem 3.1 but cannot apply Theorem 13 in [8].
Example 4.4

\(d(x,x)=0\) for all \(x\in X\),

\(d(n,0)=d(0,n)=1\) for all \(n\in\mathbf{N}\), and

\(d(n,m)=\frac{1}{n}\frac{1}{m}\) for all \(n,m\in\mathbf{N}\).
4.2 Application to dislocated metric spaces
 (i)For every \(z\in X\) with \(fz\neq gz\),$$\inf\bigl\{ \sigma(gx,gz)+\sigma(gx,fx) : x\in X\bigr\} >0. $$
 (ii)For every \(z\in X\) with \(fz\neq gz\),$$\inf\bigl\{ \sigma(gx,gz)+\sigma(gx,fx)+\sigma(fx,gx) : x\in X\bigr\} >0. $$
 (iii)For every \(z\in X\) with \(fz\neq gz\),$$\inf\bigl\{ \sigma(gx,gz)+\sigma(gz,gx)+\sigma(gx,fx)+\sigma(fx,gx) : x\in X\bigr\} >0. $$
Corollary 4.5
(Theorem 1 in [18])
Let \((X,\sigma)\) be a dislocated metric space (or metriclike space) and let \(f,g : X\to X\) be two selfmappings such that \(f(X)\subset g(X)\). Suppose that the mappings f and g are a \((\psi,\sigma)\)quasicontraction with \(\psi\in\Psi\). Let \(f(X)\) or \(g(X)\) is a 0σcomplete subspace of X. Then f and g have a coincidence point in X. If f and g are weakly compatible, then the mappings f and g have a unique common fixed point u in X and \(\sigma(u,u)=0\).
The following example shows that Theorem 3.3 is indeed a proper generalization of Theorem 1 in [18] (that is, Corollary 4.5).
Example 4.6
4.3 Application to weak quasipartial spaces
First, let us briefly recall some definitions and facts about partial metric spaces. For more details, we refer to [9, 12, 13].
Definition 4.7
 (pm1):

\(x=y\) if and only if \(p(x,x)=p(y,y)=p(x,y)\),
 (pm2):

\(p(x,x)\leq p(x,y)\),
 (pm3):

\(p(x,y)=p(y,x)\),
 (pm4):

\(p(x,y)\leq p(x,z)+p(z,y)p(z,z)\),
 (pm2′):

\(p(y,y)\leq p(x,y)\), for all \(x,y\in X\),
It is clear that the partial metric space is a weak partial metric space, as well as a quasipartial metric space. But the converse may not be true; see Example 12 in [13] and Example 2.2 in [12].
Now, we introduce a new type of partial metric space as follows.
Definition 4.8
Let \(p : X\times X\to[0,+\infty)\) be a function where X is a nonempty set. If the function p satisfies (pm1) and (pm4) in Definition 4.7, then p is called a weak quasipartial metric on X and the pair \((X,p)\) is called a weak quasipartial metric space.
It is clear that both weak partial metric space and quasipartial metric space are a weak quasipartial metric space, but the converse may not be true. A basic example of a weak quasipartial metric space but not a weak partial metric space or a quasipartial metric space is the pair \((X,p)\), where \(X=[0,1]\), \(p(x, y) = y\) for all \(x, y \in X\).
Definition 4.9
(Definition 6 in [13])
 (i)
a sequence \(\{x_{n}\}\) in X converges to \(x\in X\) if and only if \(p(x_{n},x)\to p(x,x)\) and \(p(x,x_{n})\to p(x,x)\) as \(n\to\infty\);
 (ii)
a sequence \(\{x_{n}\}\) in X is called a Cauchy sequence if and only if \(\lim_{m,n\to\infty}p(x_{n}, x_{m})\) and \(\lim_{m,n\to\infty }p(x_{m}, x_{n})\) exist (and are finite);
 (iii)
a sequence \(\{x_{n}\}\) in X is called a 0Cauchy sequence if and only if \(p(x_{n},x_{m})\to0\) and \(p(x_{m},x_{n})\to0\) as \(m,n\to\infty\);
 (iv)
the space \((X,p)\) is said to be complete if every Cauchy sequence \(\{x_{n}\}\) in X converges;
 (v)
the space \((X,p)\) is said to be 0complete if every 0Cauchy sequence \(\{x_{n}\}\) in X converges to a point \(x\in X\) such that \(p(x,x)=0\).
It is not hard to see that if \((X,p)\) is a 0complete weak quasipartial metric space, then \((X,p)\) is a 0pcomplete dislocated quasimetric space. Consequently, using Theorem 3.3 we obtain the following result which generalizes Theorem 3.3 in [10] and Theorem 1 in [14].
Corollary 4.10
Let \((X,p)\) be a weak quasipartial metric space and let \(f,g : X\to X\) be two selfmappings such that \(f(X)\subset g(X)\). Suppose that f and g are a \((\psi,p)\)quasicontraction with \(\psi\in\Psi\). Let \(f(X)\) or \(g(X)\) is a 0complete subspace of X. Then f and g have a coincidence point in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point u in X and \(p(u,u)=0\).
4.4 Applications to \(T_{0}\)quasipseudometric spaces
We recall some basic concept and facts about \(T_{0}\)quasipseudometric spaces. For more details, see, for example [22, 23].
 (i)
\(d(x,y)=d(y,x)=0\) if and only if \(x=y\);
 (ii)
\(d(x,y)\leq d(x,z)+d(z,y)\) for all \(x, y,z \in X\).
Given a \(T_{0}\)qpm d on X, the function \(d^{1}\) defined by \(d^{1}(x,y)=d(y,x)\) for all \(x,y\in X\), is also a \(T_{0}\)qpm, and the function \(d^{s}\) defined by \(d^{s}(x,y)=\max\{d(x,y), d(y,x)\}\) for all \(x,y\in X\), is a metric on X.
A \(T_{0}\)qpm space \((X,d)\) is called complete if every Cauchy sequence \(\{x_{n}\}\) in the metric space \((X,d^{s})\) converges with respect to the topology \(\tau_{d^{1}}\) (i.e., there exists \(z\in X\) such that \(d(x_{n},z)\to0\)). We see that the completeness for a \(T_{0}\)qpm space is very general.
It can be verified that if \((X,d)\) is a complete \(T_{0}\)quasipseudometric space, then \((X,d)\) is a weak 0dcomplete dislocated quasimetric spaces. Consequently, using Theorem 3.1 and Theorem 3.2 we obtain the following results.
Corollary 4.11
Corollary 4.12
4.5 Applications to uniform spaces
Now, we recall the notions of Edistance and Scompleteness in uniform spaces; see [24, 25].
Let \((X,\mathcal{U})\) be a uniform space and let p be an Adistance on X. Then the space \((X,\mathcal{U})\) is called Scomplete if for any sequence \(\{x_{n}\}\) in X with \(p(x_{n},x_{m})\to0\) as \(m,n\to+\infty\), there exists \(\bar{x}\in X\) such that \(p(x_{n},\bar{x})\to0\) as \(n\to+\infty\).
Lemma 4.13
 (a)
If \(p(x_{n},y)\leq\alpha_{n}\) and \(p(x_{n},z)\leq\beta_{n}\) for all \(n\in\mathbf{N}\), then \(y=z\). In particular, if \(p(x,y)=0\) and \(p(x,z)=0\), then \(y=z\).
 (b)
If \(p(x_{n},y_{n})\leq\alpha_{n}\) and \(p(x_{n},z)\leq\beta_{n}\) for all \(n\in\mathbf{N}\), then \(\{y_{n}\}\) converges to z.
 (c)
If \(p(x_{n},x_{m})\leq\alpha_{n}\) for all \(n,m\in\mathbf{N}\) with \(m>n\), then \(\{x_{n}\}\) is a Cauchy sequence.
From Lemma 4.13, we can easily show that if \((X,\mathcal{U})\) be a Scomplete separated uniform space such that p is an Edistance on X, then \((X,p)\) is a weak 0pcomplete dislocated quasimetric space. Consequently, using Theorem 3.1 and Theorem 3.2, we easily deduce the following corollaries.
Corollary 4.14
Corollary 4.15
Declarations
Acknowledgements
The authors are thankful to the referees and editor for their valuable comments and suggestions to improve this paper. The first author is supported by the National Natural Science Foundation of China (11471236) and by the Program of Higherlevel talents of Inner Mongolia University (30105125150, 30105135117). The second author is supported by the National Undergraduate Training Programs for Innovation (201410126027).
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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