Some extensions for Geragthy type contractive mappings
 Erdal Karapınar^{1, 2}Email author,
 Hamed H Alsulami^{2} and
 Maha Noorwali^{2}
https://doi.org/10.1186/s1366001508301
© Karapınar et al. 2015
Received: 16 July 2015
Accepted: 15 September 2015
Published: 26 September 2015
Abstract
In this paper, we establish some fixed point theorems on some extensions of Geragthy contractive type mappings in the context of bmetriclike spaces.
Keywords
MSC
1 Introduction and preliminaries
One of the interesting extensions of the notion of a metric space is the dislocated space, introduced by Hitzler [1]. This notion was rediscovered by AminiHarandi [2] and given the name of a metriclike space.
Definition 1.1
 (σ1):

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

\(\sigma(x,y)=\sigma(y,x)\);
 (σ3):

\(\sigma(x,y)\leq\sigma(x,z)+\sigma(z,y)\);
Throughout this paper, we suppose that \(\mathbb{N}_{0} = \mathbb{N} \cup\{0\}\) where \(\mathbb{N}\) denotes the set of all positive integers. Further, the symbols \(\mathbb{R^{+}}\) and \(\mathbb{R}^{+}_{0}\) denotes the set of positive reals and the set of nonnegative reals. First, we recall some basic concepts and notations.
The concept of a bmetric was introduced by Czerwik [3] as a generalization of the metric (see also Bakhtin [4] and Bourbaki [5]) to extend the celebrated Banach contraction mapping principle. Following this initial paper of Czerwik [3], 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 e.g. [6–10] and references therein).
Definition 1.2
[3]
 (\(bM_{1}\)):

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

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

\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
In what follows, we recall the notion of bmetriclike space which is an interesting generalization of both bmetric space and metriclike space.
Definition 1.3
[11]
 (\(bML_{1}\)):

if \(d(x, y) =0\) then \(x = y\);
 (\(bML_{2}\)):

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

\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
Example 1.4
Remark 1.5
 (S1)
\(d^{s}(x,x)=0\) for all \(x \in X\).
Definition 1.6
[11]
 (1)
a sequence \(\{x_{n}\} \) in X is called convergent to \(x\in X\) if and only if \(\lim_{n\to\infty} d(x_{n},x)=d(x,x)\);
 (2)
a sequence \(\{x_{n}\} \) in X is called Cauchy sequence if and only if \(\lim_{n,m\to\infty} d(x_{n},x_{m})\) exists and finite;
 (3)\((X,d)\) is complete if and only if every Cauchy sequence \(\{x_{n}\} \) in X converges to \(x\in X \) so that$$\lim_{n\rightarrow\infty} d(x_{n},x)=d(x,x)=\lim _{m,n\rightarrow \infty} d(x_{n},x_{m}). $$
Proposition 1.7
[11]
 (1)
x is unique.
 (2)
\(\frac{1}{s} d(x,y) \leq\lim_{n\to\infty} d(x_{n},y) \leq s d(x,y) \) for all \(y\in X\).
Lemma 1.8
[11]
Lemma 1.9
[12]
Notice that, in general, a bmetriclike mapping does not need to be continuous.
The notion of αadmissible and triangular αadmissible mappings were introduced by Samet et al. [13] and Karapınar et al. [14], respectively.
Definition 1.10
For more details on αadmissible and triangular αadmissible mappings, see e.g. [13–17].
Very recently, Popescu [18] refined the notion of triangular αorbital admissible as follows.
Definition 1.11
[18]
As mentioned in [18] each αadmissible (respectively, triangular αadmissible) mapping is an αorbital admissible (respectively, triangular αorbital admissible) mapping. In the following example we shall show that the converse is not true.
Example 1.12
Lemma 1.13
[18]
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 \(n< m\).
Lemma 1.14
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 \(n< m\).
We characterize the notion of αregular in the setting of a bmetriclike space.
Definition 1.15
(cf. [18])
Let \((X,d)\) be a bmetriclike space, X is said to be αregular, if for every sequence \(\{x_{n}\}\) in X such that \(\alpha(x_{n},x_{n+1})\geq1\) (respectively, \(\alpha(x_{n+1},x_{n})\geq1\)) 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 \(\alpha(x_{n_{k}},x)\geq1\) (respectively, \(\alpha(x,x_{n_{{k}}})\geq1\)) for all k.
In this paper, we shall prove the existence and uniqueness of a fixed point for certain operators in the setting of bmetriclike spaces. The presented results improve, extend, and unify a number of existing results in the literature.
2 Main result for bmetriclike spaces
Definition 2.1
Remark 2.2
Theorem 2.3
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
T is continuous.
Proof
By (ii) there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\). Define a sequence \(\{x_{n}\}\subset X\) by \(x_{n+1}=Tx_{n}\) for all \(n\in \mathbb{N}_{0}\). As T is triangular αorbital admissible, by Lemma 1.13 we have \(\alpha(x_{n},x_{n+1})\geq1\) for all \(n\in\mathbb{N}_{0}\). Throughout the proof, we suppose that \(x_{n}\neq x_{n+1}\) for all \(n\in \mathbb{N}_{0}\). Indeed, if there exists \(n_{0}\) such that \(x_{n_{0}}=x_{n_{0}+1}\), then \(x_{n_{0}}\) becomes the fixed point of T, which completes the proof.
In what follows, we replace the condition of continuity of the operator by the condition of αregularity of the space.
Theorem 2.4
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
X is αregular and d is continuous.
Proof
Since X is αregular, \(\alpha(x_{n}, x_{n+1})\geq1\) for all n. Due to the fact that \(\lim_{n\to\infty}x_{n}=u\), there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n_{k}},u)\geq 1\) for all k. To prove that u is a fixed point for T, suppose on the contrary that \(d(u,Tu)>0\).
 (H)
For all \(x,y\in\operatorname{Fix}(T)\), either \(\alpha(x,y)\geq1\) or \(\alpha (y,x)\geq1\).
Theorem 2.5
Adding condition (H) to the hypotheses of Theorem 2.3 (or Theorem 2.4), we obtain the uniqueness of the fixed point of T.
Proof
Definition 2.6
Definition 2.7
Theorem 2.8
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
T is continuous.
Proof
We shall use the same techniques as in the proof of Theorem 2.3. First of all, we shall construct a sequence \(\{x_{n}\}\subset X\) where \(x_{n+1}=Tx_{n}\) for which \(\alpha(x_{n},x_{n+1})\geq1\) and \(x_{n}\neq x_{n+1}\) for all \(n\in\mathbb{N}_{0}\).
Consequently, we get \(K(x_{n},x_{n+1})\leq\max\{ d(x_{n},x_{n+1}),d(x_{n+1},x_{n+2})\}\).
Theorem 2.9
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
X is αregular and d is continuous.
Proof
Hence \(\lim_{k\to\infty} K(x_{n_{k}},u)=0\) and as in the proof of Theorem 2.4 \(\lim_{k\to\infty}N(x_{n_{k}},u)=0\).
Thus taking the limit as \(k \rightarrow\infty\) on both sides of (38) and keeping in mind that ψ and d are continuous we have \(\psi(d(u,Tu))\leq0\). Hence \(d(u,Tu)=0\); therefore \(Tu=u\). □
Theorem 2.10
Adding condition (H) to the hypotheses of Theorem 2.8 (or Theorem 2.9), we obtain uniqueness of the fixed point of T.
Proof
Theorem 2.11
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
T is continuous.
Proof
Hence, we get \(Q(x_{n},x_{n+1})=d(x_{n},x_{n+1})\).
By using (40) we get \(\psi(s^{2}d(x_{n+1},x_{n+2}))\leq\psi (d(x_{n},x_{n+1}))\). Since ψ is increasing we have \(d(x_{n+1},x_{n+2})<\frac {1}{s^{2}}d(x_{n},x_{n+1})\). If \(s>1\) then, as in the proof of Theorem 2.3, by using Lemma 1.8, we conclude that \(\{x_{n}\}\) is a Cauchy sequence and \(\lim_{n,m\to\infty }d(x_{n},x_{m})=0\). If \(s=1\), by verbatim of the proof of Theorem 2.3, we deduce that \(\{x_{n}\}\) is a Cauchy sequence.
Since \((X,d)\) is complete, there exists \(u\in X\) such that \(0=\lim_{n,m\to\infty}d(x_{n},x_{m})= \lim_{n\to\infty}d(x_{n},u)=d(u,u)\). Now, since T is continuous, \(Tu=T(\lim_{n\to\infty}x_{n})=\lim_{n\to\infty}Tx_{n}=\lim_{n\to\infty}x_{n+1}=u\) and u is a fixed point for T. □
Theorem 2.12
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
X is αregular and d is continuous.
Proof
Theorem 2.13
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
T is continuous.
Proof
By (ii) there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\). Define a sequence \(\{x_{n}\}\subset X\) by \(x_{n+1}=Tx_{n}\) for all \(n\in \mathbb{N}_{0}\). As T is triangular αorbital admissible, by Lemma 1.13 we have \(\alpha(x_{n},x_{n+1})\geq1\) for all \(n\in\mathbb{N}_{0}\). Notice that if there exists a natural number \(n_{0}\) such that \(x_{n_{0}}= x_{n_{0}+1}\), then the proof is complete. To avoid this trivial case, from now on, we assume that \(x_{n}\neq x_{n+1}\) for all \(n\in\mathbb{N}_{0}\).
Case (i): If \(s>1\), then, since \(\frac{1}{s^{2}}>0\) and \(s\frac {1}{s^{2}}=\frac{1}{s}<1\), by Lemma 1.8, \(\{x_{n}\}\) is a Cauchy sequence and \(\lim_{n,m\to\infty}d(x_{n},x_{m})=0\).
Since T is continuous, \(Tu=T(\lim_{n\to\infty}x_{n})=\lim_{n\to\infty}Tx_{n}=\lim_{n\to\infty}x_{n+1}=u\) and u is a fixed point for T. □
Theorem 2.14
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
X is αregular and d is continuous.
Theorem 2.15
Adding condition (H) to the hypotheses of Theorem 2.13 (or Theorem 2.14), we obtain the uniqueness of the fixed point of T.
Remark 2.16
Notice that we get several corollaries by replacing the auxiliary functions ψ and β in a proper way. In particular, by taking \(\psi(t)=t\) we find the extended version of several existing results.
3 Expected consequences
In this section, we shall consider some immediate consequences of our main results.
The following result is obtained by letting \(L=0\) in Theorem 2.3 or 2.4.
Corollary 3.1
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
T is continuous or (iii)′ X is αregular and d is continuous.
Adding condition (H) to the hypothesis of Corollary 3.1, we guarantee the uniqueness of the fixed point.
Again by letting \(L=0\) in Theorem 2.5 and Theorem 2.10 we get two more corollaries as Corollary 3.1. We skip the details regarding the volume of the paper.
Corollary 3.2
 (a)
\(\alpha(x,y) d(Tx,Ty) \leq\frac{1}{2 s^{3}} M(x,y)\);
 (b)
\(\alpha(x,y) d(Tx,Ty) \leq\frac{1}{2 s^{3}} K(x,y)\);
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
T is continuous or (iii)′ X is αregular and d is continuous.
Proof
It is sufficient to take \(L=0\), \(\psi(t)=t\), and \(\beta(t)= \frac{1}{2 s}\) in Theorem 2.5 and Theorem 2.10 (and thus, Theorem 2.3 or Theorem 2.4, Theorem 2.8 or Theorem 2.9, respectively). □
Adding condition (H) to the hypothesis of Corollary 3.2, we guarantee the uniqueness of the fixed point.
Corollary 3.3
 (c)
\(\alpha(x,y) d(Tx,Ty) \leq\frac{1}{2 s^{3}} Q(x,y)\),
 (i)
T is triangular αorbital admissible;
 (ii)
there exists \(x_{0}\in X\) such that \(\alpha(x_{0},Tx_{0})\geq1\);
 (iii)
T is continuous or (iii)′ X is αregular and d is continuous.
Proof
It is sufficient to take \(L=0\), \(\psi(t)=t\), and \(\beta(t)= \frac{1}{2 s}\) in Theorem 2.11 or Theorem 2.12, respectively. □
3.1 For standard bmetriclike spaces
If we set \(\alpha(x,y)=1\) for all \(x,y \in X\) in Theorem 2.5, then we derive the following results.
Corollary 3.4
If we set \(\alpha(x,y)=1\) for all \(x,y \in X\) in Theorem 2.10, then we derive the following results.
Corollary 3.5
If we set \(\alpha(x,y)=1\) for all \(x,y \in X\) in Theorem 2.11, then we derive the following results.
Corollary 3.6
If take \(L=0\) in Corollaries 3.43.6, we get three more consequences. Regarding the volume of the paper, we skip the details.
Corollary 3.7
Proof
It follows from Theorem 2.15 by \(\alpha(x,y)=1\) for all \(x,y \in X\). □
3.2 For bmetriclike spaces endowed with a partial order
In this section, from our main results, we shall derive easily various fixed point results on a bmetriclike space endowed with a partial order. We, first, recall some notions.
Definition 3.8
Definition 3.9
Let \((X,\preceq)\) be a partially ordered set. A sequence \(\{x_{n}\} \subset X\) is said to be nondecreasing (respectively, nonincreasing) with respect to ⪯ if \(x_{n}\preceq x_{n+1}\) (respectively, \(x_{n+1}\preceq x_{n} \) for all n).
Definition 3.10
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 \(\{x_{n}\}\subset X\) 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, \(x_{n_{k}}\succeq x\)) for all k.
We have the following result.
Corollary 3.11
 (i)
there exists \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\);
 (ii)
T is continuous or (ii)′ \((X,\preceq,d)\) is regular and d is continuous.
Proof
In an analogous way, we derive the following results from Theorem 2.8 and Theorem 2.11, respectively.
Corollary 3.12
 (i)
there exists \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\);
 (ii)
T is continuous or (ii)′ \((X,\preceq,d)\) is regular and d is continuous.
Corollary 3.13
 (i)
there exists \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\);
 (ii)
T is continuous or (ii)′ \((X,\preceq,d)\) is regular and d is continuous.
Corollary 3.14
 (i)
there exists \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\);
 (ii)
T is continuous or (ii)′ \((X,\preceq,d)\) is regular and d is continuous.
Example 3.15
Example 3.16
Let \(X=\{ 0,1,2\}\) define \(d:X \times X \rightarrow[0,\infty) \) by \(d(x,y)=(\max\{x,y\})^{\frac{3}{2}}\). Then \((X,d)\) is complete bmetriclike space with constant \(s=2^{\frac{3}{2}1}=2^{\frac {1}{2}}\) such that d is continuous. Define \(T:X \rightarrow X\) by \(T=\{(0,0), (1,0), (2,0)\}\).
Let \(\psi(t)=t\), \(\beta(t)=\frac{1}{2^{\frac{1}{2}}} e^{t}\) or \(\beta(t)=\frac{1}{2^{\frac{1}{2}}+t}\), then clearly \(\psi\in\Psi\) and \(\beta\in\mathcal{F}_{2^{\frac{1}{2}}}\). Note that \(K(0,1)=1\), \(K(0,2)=K(1,2)=2^{\frac{3}{2}}\), and clearly T satisfies (56) with \(L=0\) Therefore, by Corollary 3.5, T has a fixed point \(x=0\).
4 Conclusion
It is clear that we can list several more results by replacing the bmetriclike space, with some other abstract space, such as a bmetric space, a metric space, a metriclike space, a partial metric space, and so on.
Declarations
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Authors’ Affiliations
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