Some fixed point results on quasibmetriclike spaces
 Marija Cvetković^{1},
 Erdal Karapınar^{2, 3}Email author and
 Vladimir Rakocević^{1}
https://doi.org/10.1186/s1366001508978
© Cvetković et al. 2015
Received: 27 August 2015
Accepted: 17 November 2015
Published: 2 December 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.
Keywords
MSC
1 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]
 (b_{1}):

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

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

\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
Definition 1.2
[9]
 (bm_{1}):

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

\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
Definition 1.3
[10]
 (bM_{1}):

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

\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
Example 1.4
Definition 1.5
(see e.g. [10])
 (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$$\lim_{n \to\infty}d(u, x_{n}) = d(u,u) =\lim _{n,m\to\infty}d(x_{n}, x_{m})=0,\quad \text{where } m\geq n. $$
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
Very recently, Popescu [13] improved these notions as follows.
Definition 1.7
[13]
Triangular αadmissible mappings defined by Popescu [13] impose the following definitions.
Definition 1.8
[13]
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
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.
2 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]
 (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
 (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 byis increasing and continuous at 0.$$p_{s}(t)= \sum_{k=0}^{\infty}s^{k}\psi^{k}(t)\quad \textit{for all } t\in[0,\infty) $$
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
Theorem 2.5
 (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.
Proof
Example 2.6
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
Theorem 2.8
 (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.
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.
 (U)
For all \(x,y\in\operatorname{Fix}(f)\), either \(\alpha(x,y)\geq1\) or \(\alpha (y,x)\geq1\).
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 \(d(x^{*},y^{*})>0\).
Definition 2.10
Theorem 2.11
 (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.
Proof
Theorem 2.12
Adding condition (U) to hypotheses of Theorem 2.11, we obtain the uniqueness of a fixed point of T.
Proof
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
Definition 2.14
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
 (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.
Definition 2.16
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
 (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.
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
 (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.
Proof
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
Definition 2.21
Theorem 2.22
 (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.
Proof
3 Consequences
In this section, we will list some consequences of our main results.
3.1 For standard quasibmetriclike
Corollary 3.1
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
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
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
3.2 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
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
 (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.
Proof
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
 (i)
there exists \(x_{0}\in X\) such that \(x_{0}\preceq fx_{0}\) and \(fx_{0}\preceq x_{0}\);
 (ii)
f is continuous.
Declarations
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.
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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