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Some fixed point results on quasibmetriclike spaces
Journal of Inequalities and Applications volume 2015, Article number: 374 (2015)
Abstract
In this paper, we investigate the existence and uniqueness of a fixed point of certain operators in the setting of complete quasibmetriclike spaces via admissible mappings. Our results improve, extend, and unify several wellknown existence results.
Introduction and preliminaries
Throughout this paper, we denote \(\mathbb{R}^{+}_{0} = [0, +\infty)\) and \(\mathbb{N}_{0} = \mathbb{N} \cup\{0\}\), where \(\mathbb{N}\) is the set of all positive integers. First, we recall some basic concepts and notation.
The concept of bmetric was introduced by Czerwik [1] as a generalization of metric (see also Bakhtin [2, 3]) to extend the celebrated Banach contraction mapping principle. Following the initial paper of Czerwik [1], a number of researchers in nonlinear analysis investigated the topology of the paper and proved several fixed point theorems in the context of complete bmetric spaces (see [4–8] and references therein).
Definition 1.1
[1]
Let X be a nonempty set, and \(s\geq1\) be a given real number. A mapping \(d : X \times X\to[0, +\infty)\) is said to be a bmetric if for all \(x, y, z \in X\), the following conditions are satisfied:
 (b_{1}):

\(d(x, y) =0\) if and only if \(x = y\);
 (b_{2}):

\(d(x, y) = d(y,x)\);
 (b_{3}):

\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
In this case, the pair \((X, d)\) is called a bmetric space (with constant s).
Definition 1.2
[9]
Let X be a nonempty set, and \(s\geq1\) be a given real number. A mapping \(d : X \times X\to[0, +\infty)\) is said to be a quasibmetric if for all \(x, y, z \in X\), the following conditions are satisfied:
 (bm_{1}):

\(d(x, y) =0\) if and only if \(x = y\);
 (bm_{2}):

\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
In this case, the pair \((X,d)\) is called a quasibmetric space (with constant s).
Definition 1.3
[10]
Let X be a nonempty set, and \(s\geq1\) be a given real number. A mapping \(d : X \times X\to[0, +\infty)\) is said to be a quasibmetriclike if for all \(x, y, z \in X\), the following conditions are satisfied:
 (bM_{1}):

\(d(x, y) =0\) implies \(x = y\);
 (bM_{2}):

\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
In this case, the pair \((X, d)\) is called a quasibmetriclike space (with constant s).
Example 1.4
Let \(X=\{0,\frac{1}{2},\frac{1}{3}\} \cup [1, \infty)\), and let \(d : X \times X\to[0, +\infty)\) be defined as
It is clear that \((X,d)\) is a quasibmetriclike space with constant \(s=9\).
Definition 1.5
(see e.g. [10])
Let \((X, d)\) be a quasibmetriclike space. Then:
 (i)_{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 \(d(x_{n}, x_{m})<\varepsilon\) 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 \(d(x_{n}, x_{m})<\varepsilon\) for all \(m>n>N\);
 (iii)_{a} :

a quasipartial metric space is said to be leftcomplete if every leftCauchy sequence \(\{x_{n}\}\) in X converges with respect to d to a point \(u \in X\) such that
$$\lim_{n \to\infty}d( x_{n},u) = d(u,u) =\lim _{n,m\to\infty}d(x_{m}, x_{n})=0,\quad \text{where } m\geq n; $$  (iii)_{b} :

a quasipartial metric space is said to be rightcomplete if every leftCauchy sequence \(\{x_{n}\}\) in X converges with respect to d to a point \(u \in X\) such that
$$\lim_{n \to\infty}d(u, x_{n}) = d(u,u) =\lim _{n,m\to\infty}d(x_{n}, x_{m})=0,\quad \text{where } m\geq n. $$
Let \((X, d)\) and \((Y, \alpha)\) be quasibmetriclike spaces, and let \(f : X \to Y\) be a continuous mapping. Then
In 2012, Samet et al. [11] introduced the concept of αadmissible mappings, and in 2013, Karapınar et al. [12] improved this notion as triangular αadmissible mappings.
Definition 1.6
Let \(\alpha: X \times X \rightarrow[0, +\infty)\) be a function. A selfmapping f is called an αadmissible mapping if
for all \(x,y \in X\). If, further, f satisfies the condition
for all \(x,y,z \in X\), then it is called triangular αadmissible mapping.
Very recently, Popescu [13] improved these notions as follows.
Definition 1.7
[13]
Let \(\alpha:X\times X\rightarrow[0,\infty)\) be a function. If \(f: X\to X\) satisfies the condition
for all \(x\in X\), then it is called a rightαorbital admissible mapping. If f satisfies the condition
for all \(x\in X\), then it is called a leftαorbital admissible mapping. Furthermore, if f is both rightαorbital admissible and leftαorbital admissible, then f is called an αorbital admissible mapping.
Triangular αadmissible mappings defined by Popescu [13] impose the following definitions.
Definition 1.8
[13]
Let \(f:X\rightarrow X\) be a selfmapping, and \(\alpha:X\times X\rightarrow[0,\infty)\) be a function. Then f is said to be triangular rightαorbital admissible if f is rightαorbital admissible and
and is said to be triangular leftαorbital admissible if f is αorbital admissible and
If T satisfies both (T2)′ and (T2)″, then it is called triangular αorbital admissible.
It is easy to conclude that each αadmissible mapping is an αorbital admissible mapping and each triangular αadmissible mapping is a triangular αorbital admissible mapping. However, the converses of the statements are false. In the following example, we see that a mapping that is triangular αorbital admissible need not be triangular αadmissible.
Example 1.9
Let \(X = \{x_{i}: i=1,\ldots, n\}\) for some \(n\geq4\), and \(d : X \times X \to\mathbb{R}_{0}^{+}\) with \(d(x, y) = xy\). We define a selfmapping \(f : X \to X \) such that \(fx_{i}=x_{i}\) for \(i=1,2\), \(fx_{i}=x_{j}\) for \(i,j \in\{ 3,4\}\), \(i\neq j\), \(fx_{i}=x_{i+1}\) for \(i \in\{5,\ldots, n1\}\), and \(fx_{n}=fx_{5}\). Moreover, let \(\alpha: X \times X \to\mathbb{R}_{0}^{+}\) be such that
Note that f is αorbital admissible since \(\alpha(x_{3},fx_{3}) = \alpha(x_{3},x_{4}) = 1 \) and \(\alpha(x_{4},fx_{4}) = \alpha(x_{4},x_{3}) =1 \). On the other hand, we have \(\alpha(x_{1},x_{3}) = \alpha(x_{3},x_{2}) = 1 \), but \(\alpha(x_{1},x_{2})=0\). Hence, T is not triangular αadmissible.
Definition 1.10
[13]
Let \((X,d)\) be a quasibmetriclike space. Then X is said to be 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.
Main result
The notion of \((b)\)comparison was introduced by Berinde [14] in order to extend the notion of \((c)\)comparison.
Definition 2.1
[14]
Let \(s\geq1\) be a real number. A mapping \(\psi:\mathbb{R}_{0}^{+}\to\mathbb{R}_{0}^{+}\) is called a \((b)\)comparison function if the following conditions are fulfilled:

(1)
ψ is monotone increasing;

(2)
there exist \(k_{0}\in\mathbb{N}\), \(a\in(0,1)\), and a convergent series of nonnegative terms \(\sum_{k=1}^{\infty}v_{k}\) such that \(s^{k+1}\psi ^{k+1}(t)\leq a s^{k}\psi^{k}(t)+v_{k}\) for all \(k\geq k_{0}\) and \(t\in[0,\infty)\).
The class of \((b)\)comparison functions will be denoted by \(\Psi_{b}\). Notice that the notion of a \((b)\)comparison function reduces to the concept of a \((c)\)comparison function if \(s=1\).
The following lemma will be used in the proof of our main result.
Lemma 2.2
Let \(s\geq1\) be a real number. If \(\psi :\mathbb{R}_{0}^{+}\to\mathbb{R}_{0}^{+}\) is a \((b)\)comparison function, then:

(1)
the series \(\sum_{k=0}^{\infty}s^{k}\psi^{k}(t)\) converges for any \(t\in\mathbb{R}_{0}^{+}\);

(2)
the function \(p_{s}:[0,\infty)\to[0,\infty)\) defined by
$$p_{s}(t)= \sum_{k=0}^{\infty}s^{k}\psi^{k}(t)\quad \textit{for all } t\in[0,\infty) $$is increasing and continuous at 0.
Remark 2.3
It is easy to see that if \(\psi(t) \in\Psi _{b}\), then \(\psi(t)< t\) for all \(t>0\). In fact, if there is a \(t^{*}>0\) such that \(\psi(t^{*})\geq t^{*}\), then we have \(\psi^{2}(t^{*})\geq\psi(t^{*})\geq t^{*}\) (since ψ is increasing). Continuing in the same manner, we get \(\psi^{n}(t^{*})\geq t^{*}>0\), \(n\in \mathbb{N}\). This contradicts Lemma 2.2.
Definition 2.4
Let \((X,d)\) be a complete quasibmetriclike space with a constant \(s \geq1\). A selfmapping \(f: X \to X\) is called \((\alpha ,\psi)\)contractive mapping if there exist two functions \(\psi\in \Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition:
for all \(x,y \in X\).
Theorem 2.5
Let \((X,d) \) be a complete quasibmetriclike space, and let \(f: X \to X\) be an \((\alpha,\psi)\)contractive mapping. Suppose also that

(i)
f is αorbital admissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},fx_{0}) \geq1\) and \(\alpha(fx_{0},x_{0})\geq1\);

(iii)
f is continuous.
Then f has a fixed point u in X, and \(d(u,u)=0\).
Proof
By (ii) there exists \(x_{0}\in X\) such that \(\alpha(x_{0},fx_{0})\geq1\) and \(\alpha(fx_{0},x_{0})\geq1\). Define the iterative sequence \(\{x_{n}\}\) in X by \(x_{n+1}=fx_{n}\) for all \(n\in\mathbb{N}_{0}\). Note that if there exists \(n_{0} \in\mathbb{N}_{0}\) such that \(x_{n_{0}}= x_{n_{0}+1}\), then \(x_{n_{0}}\) becomes a fixed point, which completes the proof. Hence, throughout the proof, we suppose that \(x_{n}\neq x_{n+1}\) for all \(n\in\mathbb{N}_{0}\). Regarding the fact that f is αorbital admissible, from (ii) we derive that
Inductively, we get that
Analogously, again by (ii) and the fact that f is αorbital admissible we find that
Consequently, we observe that
From (2.1), by taking \(x=x_{n}\) and \(y=x_{n1}\), we find that
In view of Remark 2.3, we get that
By analogy, again by (2.1) and by substituting \(x=x_{n1}\) and \(y=x_{n}\), we have
Consequently,
From (2.4) and (2.5) we derive that
By Lemma 2.2(1) and letting \(n \to\infty\) in (2.6), we have \(\lim_{n\rightarrow\infty}d(x_{n},x_{n+1})=\lim_{n\rightarrow \infty}d(x_{n+1}, x_{n})=0\).
We further prove that the sequence \(\{x_{n}\}\) is rightCauchy and leftCauchy. For all \(n,p\in\mathbb{N}\), we have
By letting \(n,p \to\infty\) we get that
that is, the sequence \(\{x_{n}\}\) is rightCauchy.
Analogously,
that is, the sequence \(\{x_{n}\}\) is leftCauchy. As a result, the sequence \(\{x_{n}\}\) is a Cauchy sequence. Since \((X,d)\) is complete, there exists a point \(u \in X\) such that
Since f is continuous, we have
□
Example 2.6
Let \((X,d)\) be a quasi bmetric like space defined in Example 1.4, and let the mapping \(f:X\mapsto X\) be defined as
Let \(\psi(t)=\frac{t}{2}\), \(t\geq0\), and let \(\alpha:X\times X\to [0,\infty)\) be defined as
Then \(\psi\in\Psi_{b}\), and f is an \((\alpha,\psi)\)contractive mapping. Since the conditions of Theorem 2.5 are satisfied, it follows that f has a fixed point in X.
It is possible to remove the heavy condition of continuity of the selfmapping f in Theorem 2.5. For this purpose, we need the following result, which is inspired from the results in [17].
Lemma 2.7
Let \((X,d)\) be a quasibmetriclike space with constant s and assume that \(\{x_{n}\}\) and \(\{y_{n}\}\) are sequences in X converging to x and y, respectively. Then
In particular, if \(d(x,y)=0 \), then \(\lim_{n\rightarrow \infty} d(x_{n},y_{n})=0 \).
Moreover, for each \(z\in X \), we have
If \(d(x,x)=0\), then
Theorem 2.8
Let \((X,d) \) be a complete quasibmetriclike space, and let \(f: X \to X\) be an \((\alpha,\psi)\)contractive mapping. Suppose also that

(i)
f is αorbital admissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},fx_{0}) \geq1\) and \(\alpha(fx_{0},x_{0})\geq1\);

(iii)
X is αregular.
Then f has a fixed point u in X, and \(d(u,u)=0\).
Proof
By verbatim of the proof of Theorem 2.5 we find an iterative sequence \(\{x_{n}\}\) that converges to a point \(u \in X\) such that (2.7) holds.
Since \(d(u,u)=0\), by Lemma 2.7 we have
By letting \(n \to\infty\) in these inequalities we derive that \(\frac {1}{s} d(u,fu) =0\) and hence \(fu=u\). □
It is natural to consider the uniqueness of a fixed point of an \((\alpha,\psi)\)contractive mapping. We notice that we need to add an additional condition to guarantee the uniqueness.

(U)
For all \(x,y\in\operatorname{Fix}(f)\), either \(\alpha(x,y)\geq1\) or \(\alpha (y,x)\geq1\).
Here, \(\operatorname{Fix}(f)\) denotes the set of all fixed points of f.
Theorem 2.9
Adding condition (U) to hypotheses of Theorem 2.5 (or Theorem 2.8), we obtain the uniqueness of a fixed point of f.
Proof
Suppose that \(x^{*}\) and \(y^{*}\) are two distinct fixed points of f, so that \(d(x^{*},y^{*})>0\).
If, for example, \(\alpha(x^{*},y^{*})\geq1\), then
which is a contradiction. □
Definition 2.10
Let \((X,d)\) be a complete quasibmetriclike space with a constant \(s \geq1\). A selfmapping \(f: X \to X\) is called a generalized \((\alpha,\psi)\)contractive mapping of type \((A)\) if there exist two functions \(\psi\in\Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition:
for all \(x,y \in X\), where
Theorem 2.11
Let \((X,d) \) be a complete quasibmetriclike space, and let \(f: X \to X\) be a generalized \((\alpha,\psi)\)contractive mapping of type \((A)\). Assume that

(i)
f is αorbital admissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},fx_{0}) \geq1\) and \(\alpha(fx_{0},x_{0})\geq1\);

(iii)
f is continuous.
Then f has a fixed point u in X, and \(d(u,u)=0\).
Proof
As in the proof of Theorem 2.5, we construct an iterative sequence \(x_{n+1}=fx_{n}\), \(n\in\mathbb{N}_{0}\), where the existence of \(x_{0}\in X\) is guaranteed by (ii). By the same reason as in the proof of Theorem 2.5, we may assume that \(x_{n}\neq x_{n+1}\) for all \(n\in\mathbb{N}_{0}\), and we can conclude that
From (2.9) we have
for all \(n \in\mathbb{N} \), where
If \(M(x_{n1},x_{n})=d(x_{n},x_{n+1})\), then since we assumed that \(x_{n}\neq x_{n+1}\),
which is a contradiction. It allows us to conclude that \(M(x_{n1},x_{n})=d(x_{n1},x_{n})\), \(n\in\mathbb{N}\).
Thus,
and
Analogously, letting \(x=x_{n}\) and \(y=x_{n1}\) in (2.9), we get
for all \(n \in\mathbb{N} \), where
For the estimation of \(d(x_{n+1},x_{n})\), we will consider three different cases.
Case 1. If \(M(x_{n},x_{n1})=d(x_{n1},x_{n})\), then, by (2.13),
Case 2. If \(M(x_{n},x_{n1})=d(x_{n},x_{n+1})\), then
By Remark 2.3 we find that
Case 3. Otherwise, \(M(x_{n},x_{n1})=d(x_{n},x_{n1})\) and
Observing (2.14) and (2.15), it follows that, for any \(n\in \mathbb{N}\),
Obviously, in all considered cases, we deduce that
Since ψ is an increasing function, let
Consequently, we have that \(d(x_{n+1},x_{n})\leq\psi^{n}(v)\) and \(d(x_{n},x_{n+1})\leq\psi^{n}(v)\). By applying (bM_{2}) for any \(n,p\in\mathbb{N}\) it follows that
Therefore, \(\lim_{n,p\to\infty} d(x_{n},x_{n+p})=0\) and, likewise, \(\lim_{n,p\to\infty} d(x_{n+p},x_{n})=0\). Since, X is complete, there exists \(u\in X\) such that \(\lim_{n\rightarrow \infty}x_{n}=u\) and
Furthermore, f is a continuous mapping, and hence \(u=\lim_{n\rightarrow\infty} x_{n}=\lim_{n\rightarrow\infty}fx_{n1}=fu\). □
Theorem 2.12
Adding condition (U) to hypotheses of Theorem 2.11, we obtain the uniqueness of a fixed point of T.
Proof
Suppose that \(fx^{*}=x^{*}\) and \(fy^{*}=y^{*}\). Then
so that \(d(x^{*},y^{*})=0 \Rightarrow x^{*}=y^{*}\). □
In the following example, we show the existence of a function satisfying conditions of Theorem 2.11 but not satisfying conditions of Theorem 2.5.
Example 2.13
Let \((X,d)\) be a quasibmetriclike space described in Example 1.4, and \(f:X\mapsto X\) the mapping
Let \(\psi(t)=\frac{t}{2}\), \(t\geq0\), and let \(\alpha:X\times X\to [0,\infty)\) be defined as
Then (2.1) does not hold, for example, for \(x=0\) and \(y=\frac {1}{3}\), but (2.9) holds, f has a unique fixed point \(u=\frac {1}{2}\), and \(d(u,u)=0\).
Definition 2.14
Let \((X,d)\) be a complete quasibmetriclike space with a constant \(s \geq1\). A selfmapping \(f: X \to X\) is called a generalized \((\alpha,\psi)\)contractive mapping of type \((B)\) if there exist two functions \(\psi\in\Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition:
for all \(x,y \in X\), where
The following theorem can be deduced from the inequality \(N(x,y) \leq M(x,y)\) for all x, y, together with the monotonicity of ψ.
Theorem 2.15
Let \((X,d) \) be a complete quasibmetriclike space, and let \(f: X \to X\) be a generalized \((\alpha,\psi)\)contractive mapping of type \((B)\). Assume that

(i)
f is αorbital admissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},fx_{0}) \geq1\) and \(\alpha(fx_{0},x_{0})\geq1\);

(iii)
f is continuous.
Then f has a fixed point u in X, and \(d(u,u)=0\).
Definition 2.16
Let \((X,d)\) be a complete quasibmetriclike space with a constant \(s \geq1\). A selfmapping \(f: X \to X\) is called a generalized \((\alpha,\psi)\)contractive mapping of type \((C)\) if there exist two functions \(\psi\in\Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition:
for all \(x,y \in X\), where
The following theorem is easily observed from Theorem 2.11 since inequality (2.9) can be easily derived from inequality (2.20).
Theorem 2.17
Let \((X,d) \) be a complete quasibmetriclike space, and let \(f: X \to X\) be a generalized \((\alpha,\psi)\)contractive mapping of type \((C)\). Assume that

(i)
f is αorbital admissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},fx_{0}) \geq1\) and \(\alpha(fx_{0},x_{0})\geq1\);

(iii)
f is continuous.
Then f has a fixed point u in X, and \(d(u,u)=0\).
In the next theorems, we establish a fixed point result for a generalized \((\alpha,\psi)\)contractive mapping of type \((C)\) without any continuity assumption on the mapping f.
Theorem 2.18
Let \((X,d) \) be a complete quasibmetriclike space, and let \(f: X \to X\) be a generalized \((\alpha,\psi)\)contractive mapping of type \((C)\). Suppose that

(i)
f is αorbital admissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},fx_{0}) \geq1\) and \(\alpha(fx_{0},x_{0})\geq1\);

(iii)
X is αregular.
Then f has a fixed point u in X, and \(d(u,u)=0\).
Proof
As in the proof of Theorem 2.5, we consider an iterative sequence \(\{x_{n}\}\), and we obtain the existence of \(u\in X\) such that (2.17) holds. By Lemma 2.7 we get
where
According to (2.17) and the fact that \(\lim_{n\rightarrow\infty }d(x_{n1},x_{n})=0\), it remains to discuss only the case \(M(x_{n},u)=d(u,fu)\) because otherwise it follows \(d(u,fu)=0 \Rightarrow u=fu\).
Notice that, under this assumption, \(d(u,fu)\leq\psi(d(u,fu))\) also implies \(d(u,fu)=0\) since \(\psi(t)< t\) for any \(t>0\). Hence, u is a fixed point of the mapping f. □
Theorem 2.19
Adding condition (U) to hypotheses of Theorem 2.17 (or Theorem 2.18), we obtain the uniqueness of a fixed point of T.
Example 2.20
Let \((X,d)\) be a quasi bmetric like space defined in Example 1.4, and let the mapping \(f:X\mapsto X\) be defined as
Let \(\psi(t)=\frac{t}{2}\), \(t\geq0\), and \(\alpha:X\times X\to [0,\infty)\) be defined as
Then \(\psi\in\Psi_{b}\), and f is a generalized \((\alpha,\psi )\)contractive mapping of type \((C)\). Since the conditions of Theorem 2.18 are satisfied, it follows that f has a fixed point in X.
Definition 2.21
Let \((X,d)\) be a complete quasibmetriclike space with a constant \(s \geq1\). A selfmapping \(f: X \to X\) is called a generalized \((\alpha,\psi)\)contractive mapping of type \((D)\) if there exist two functions \(\psi\in\Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition:
for all \(x,y \in X\), where
Theorem 2.22
Let \((X,d)\) be a complete quasibmetriclike space, and let \(f: X \to X\) be a generalized \((\alpha,\psi)\)contractive mapping of type \((D)\). Suppose that

(i)
f is triangular αorbital admissible;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},fx_{0}) \geq1\) and \(\alpha(fx_{0},x_{0})\geq1\);

(iii)
X is αregular.
Then f has a fixed point in X, that is, there exists \(u \in X\) such that \(fu=u\) and \(d(u,u)=0\).
Proof
As in the proof of Theorem 2.11, we consider an iterative sequence \(\{x_{n}\}\) and obtain the existence of \(u\in X\) such that (2.17) holds. By Lemma 2.7 we get
where
If \(N(x_{n},u)=d(x_{n1},u)\), then we conclude the result due to (2.17). Taking \(\lim_{n\rightarrow\infty}d(x_{n1}, x_{n})=0\) into account, we deduce that \(\lim_{n\rightarrow\infty} N(x_{n},u)=\frac{ d(u,fu)}{2s}\). Notice that, under this assumption, \(\frac{ 1}{s}d(u,fu)\leq\psi (\frac{ d(u,fu)}{2s})\) also implies \(d(u,fu)=0\) since \(\psi(t)< t\) for any \(t>0\). Hence, u is a fixed point of the mapping f. □
Theorem 2.23
Adding condition (U) to hypotheses of Theorem 2.15 (and respectively, Theorem 2.22), we obtain the uniqueness of a fixed point of T.
Consequences
In this section, we will list some consequences of our main results.
For standard quasibmetriclike
Corollary 3.1
Let \((X,d)\) be a complete quasibmetriclike space, and let \(f: X \to X\) be a mapping such that
for all \(x,y \in X\), where \(\psi\in\Psi_{b}\). If f is continuous, then f has a fixed point u in X, and \(d(u,u)=0\).
Proof
The proof of Corollary 3.1 follows from Theorem 2.12 by taking \(\alpha(x,y)=1\) for all \(x,y \in X\), so (ii) is satisfied for any \(x_{0}\in X\), f is obviously an αorbital admissible, and (U) holds. Inequality (3.1) allows us to conclude that f is a generalized \((\alpha,\psi)\)contractive mapping of type \((A)\). □
Corollary 3.2
Let \((X,d)\) be a complete quasibmetriclike space, and let \(f: X \to X\) be a continuous mapping such that
for all \(x,y \in X\), where \(\psi\in\Psi_{b}\). Then f has a fixed point u in X, and \(d(u,u)=0\).
Proof
The proof of Corollary 3.2 follows from Theorem 2.15 by taking \(\alpha(x,y)=1\) for all \(x,y \in X\) since then (2.18) follows from (3.2). □
Notice that the continuity condition of f in Corollary 3.1 can be removed by adding an extra term s.
Corollary 3.3
Let \((X,d) \) be a complete quasibmetriclike space, and let \(f: X \to X\) be a mapping such that
for all \(x,y \in X\), where \(\psi\in\Psi_{b}\). Then f has a fixed point u in X such that \(d(u,u)=0\).
Proof
The proof of Corollary 3.3 follows from Theorem 2.18 by taking \(\alpha(x,y)=1\) for all \(x,y \in X\). Then f is an αorbital admissible mapping, and both inequalities in (ii) hold for any \(x_{0}\in X\). Notice that since \(\alpha(x,y)=1\), any constructive sequence turns to be regular, and thus X is αregular. □
Corollary 3.4
Let \((X,d) \) be a complete quasibmetriclike space, and let \(f: X \to X\) be a mapping such that
for all \(x,y \in X\), where \(\psi\in\Psi_{b}\). Then f has a fixed point u in X such that \(d(u,u)=0\).
Proof
The proof of Corollary 3.4 follows from Theorem 2.8 by taking \(\alpha(x,y)=1\) for all \(x,y \in X\) and observing that X is αregular and that (i) and (ii) hold. □
For standard quasibmetriclike spaces with a partial order
In this section, we deduce various fixed point results on a quasibmetriclike space endowed with a partial order. We, first, recollect some basic notions and notation.
Definition 3.5
Let \((X,\preceq)\) be a partially ordered set, and \(f: X\to X\) be a given mapping. We say that f is nondecreasing with respect to ⪯ if for all \(x,y\in X\),
Definition 3.6
Let \((X,\preceq)\) be a partially ordered set. A sequence \(\{x_{n}\} \subseteq X\) is said to be nondecreasing (respectively, nonincreasing) with respect to ⪯ if \(x_{n}\preceq x_{n+1}\), \(n\in\mathbb{N}\) (respectively, \(x_{n+1}\preceq x_{n}\), \(n\in\mathbb{N}\)).
Definition 3.7
Let \((X,\preceq)\) be a partially ordered set, and d be a bmetriclike on X. We say that \((X,\preceq,d)\) is regular if for every nondecreasing (respectively, nonincreasing) sequence \(\{x_{n}\} \subseteq X\) such that \(x_{n}\to x\in X\) as \(n\to\infty\), there exists a subsequence \(\{ x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{k}}\preceq x\) (respectively, \(x\preceq x_{n_{k}}\)) for all k.
We have the following result.
Corollary 3.8
Let \((X,\preceq)\) be a partially ordered set (which does not contain an infinite totally unordered subset), and d be a bmetriclike on X with constant \(s\geq1\) such that \((X,d)\) is complete. Let \(f: X\to X\) be a nondecreasing mapping with respect to ⪯. Suppose that there exists \(\psi\in\Psi_{b}\) such that
for all \(x,y \in X\) with \(x\preceq y\) or \(y\preceq x\), where \(M(x,y)\) is defined as in (2.10). Suppose also that the following conditions hold:
 (i):

there exists \(x_{0}\in X\) such that \(x_{0}\preceq fx_{0}\) and \(fx_{0}\preceq x_{0}\);
 (ii):

f is continuous or
 (ii)′:

\((X,\preceq,d)\) is regular, and d is continuous.
Then f has a fixed point \(u \in X\) with \(d(u,u)=0\). Moreover, if for all \(x,y\in X\), there exists \(z\in X\) such that \(x\preceq z\) and \(y\preceq z\), then f has a unique fixed point.
Proof
Define the mapping \(\alpha: X\times X\to[0,\infty)\) by
Clearly, f satisfies (2.20), that is,
for all \(x,y\in X\). From condition (i) we have \(x_{0}\preceq fx_{0}\) and \(fx_{0}\preceq x_{0}\). Moreover, for all \(x,y\in X\), from the monotone property of f we have
Hence, the selfmapping f is αadmissible. Similarly, we can prove that f is triangular αadmissible and so triangular αorbital admissible. Now, if f is continuous, then the existence of a fixed point follows from Theorem 2.17.
Suppose that \((X,\preceq,d)\) is regular. Let \(\{x_{n}\}\) be a sequence in X such that \(\alpha(x_{n},x_{n+1})\geq1\) for all n and \(x_{n} \rightarrow x\in X\) as \(n\rightarrow\infty\). By the regularity hypothesis, since X does not contain an infinite totally unordered subset, there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{k}}\preceq x\) or \(x\preceq x_{n_{k}}\) for all k.
This implies from the definition of α that \(\alpha (x_{n_{k}},x)\geq1\) for all k. In this case, the existence of a fixed point follows again from Theorem 2.18. □
Corollary 3.9
Let \((X,\preceq)\) be a partially ordered set (which does not contain an infinite totally unordered subset), and d be a bmetriclike on X with constant \(s\geq1\) such that \((X,d)\) is complete. Let \(f: X\to X\) be a nondecreasing mapping with respect to ⪯. Suppose that there exists \(\psi\in\Psi_{b}\) such that
for all \(x,y \in X\) with \(x\succeq y\) or \(y\succeq x\). Suppose also that the following conditions hold:

(i)
there exists \(x_{0}\in X\) such that \(x_{0}\preceq fx_{0}\) and \(fx_{0}\preceq x_{0}\);

(ii)
f is continuous.
Then T has a fixed point \(u \in X\) with \(d(u,u)=0\). Moreover, if for all \(x,y\in X\), there exists \(z\in X\) such that \(x\preceq z\) and \(y\preceq z\), we have the uniqueness of a fixed point.
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Acknowledgements
The authors are grateful to the reviewers for their careful reviews and useful comments. The first and third authors were supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.
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Cvetković, M., Karapınar, E. & Rakocević, V. Some fixed point results on quasibmetriclike spaces. J Inequal Appl 2015, 374 (2015). https://doi.org/10.1186/s1366001508978
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DOI: https://doi.org/10.1186/s1366001508978
MSC
 46T99
 46N40
 47H10
 54H25
Keywords
 bmetric like space
 fixed point
 αadmissible
 contractive mapping