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Advances on the fixed point results via simulation function involving rational terms
Advances in Difference Equations volume 2021, Article number: 409 (2021)
Abstract
In this paper, we propose two new contractions via simulation function that involves rational expression in the setting of partial bmetric space. The obtained results not only extend, but also generalize and unify the existing results in two senses: in the sense of contraction terms and in the sense of the abstract setting. We present an example to indicate the validity of the main theorem.
1 Introduction and preliminaries
The origin of the fixed point theory goes back a century, to the pioneer work of Banach. Since the first study of Banach, researchers have been extended, improved, and generalized this very simple stated but at the same time very powerful theorem. For this purpose, the terms of the contraction inequality and the abstract structure of Banach’s theorem have been investigated. In this paper, we shall combine these two trends and introduce two new type contraction via simulation functions involving rational terms in the more general setting, partialbmetric space.
For the sake of the completeness of the manuscript, we shall recall some basic results and concepts here.
Theorem 1
([1])
Let \((\mathcal {A},\delta )\) be a complete metric space and be a mapping. If there exist , with \(\kappa _{1}+ \kappa _{2}<1\) such that
for all , then has a unique fixed point \(\mathsf {u}\in \mathcal {A}\) and the sequence converges to the fixed point u for all \(x\in \mathcal {A}\).
Theorem 2
([2])
Let \((\mathcal {A}, \delta )\) be a complete metric space and be a continuous mapping. If there exist \(\kappa _{1}, \kappa _{2}\in [0,1 )\), with \(\kappa _{1}+ \kappa _{2}<1\) such that
for all distinct , then possesses a unique fixed point in \(\mathcal {A}\).
We mention that over the last few years many interesting and different generalizations for rational contractions have been provided; see, for example [3–8].
Let Γ be the set of all nondecreasing and continuous functions \(\psi :[0,+\infty )\rightarrow [0,+\infty )\). such that \(\psi (0)=0\).
Definition 1
([9])
A function \(\eta :\mathbb{R}^{+}_{0}\times \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}\) is a ψsimulation function if there exists \(\psi \in \Gamma \) such that the following conditions hold:
 \((\eta _{1})\):

\(\eta (\mathsf {r},\mathsf {t})<\psi (\mathsf {t})\psi (\mathsf {r})\) for all \(\mathsf {r},\mathsf {t}\in \mathbb{R}^{+}\);
 \((\eta _{2})\):

if \(\{\mathsf {r}_{n}\},\{\mathsf {t}_{n}\}\) are two sequences in \([0,+\infty )\) such that \(\lim_{n\rightarrow +\infty }\mathsf {r}_{n}= \lim_{n\rightarrow +\infty }\mathsf {t}_{n}>0\), then
$$\begin{aligned} \limsup_{n\rightarrow +\infty }\eta (\mathsf {r}_{n},\mathsf {t}_{n})< 0. \end{aligned}$$(1.3)
We will denote by \(\mathcal{Z}_{\psi }\) the family of all ψsimulation functions; see e.g. [10–22]. It is clear, due to the axiom \((\eta _{1})\), that
Definition 2
([23])
On a nonempty set \(\mathcal {A}\), a function \(\rho :\mathcal {A}\times \mathcal {A}\rightarrow \mathbb{R}^{+}_{0}\) is a partial metric if the following conditions:
 \((\rho _{1})\):

;
 \((\rho _{2})\):

;
 \((\rho _{3})\):

;
 \((\rho _{4})\):

;
hold for all .
The pair \((\mathcal {A}, \rho )\) is called a partialmetric space.
Every partial metric ρ on \(\mathcal {A}\) generates a \(\mathsf{T}_{0}\) topology on \(\mathcal {A}\), that has a base of the set of all open balls , where an open ball for a partial metric ρ on \(\mathcal {A}\) is defined [23] as
for each and \(\mathsf{e}>0\).
If \((\mathcal {A}, \rho )\) is a partialmetric space and a sequence in \(\mathcal {A}\), then:

is convergent to a limit \(\mathsf {u}\in \mathcal {A}\), if ;

is a Cauchy sequence if exists and is finite.
Moreover, we say that the partialmetric space \((\mathcal {A},\rho )\) is complete if every Cauchy sequence in \(\mathcal {A}\) converges to a point \(\mathsf {u}\in \mathcal {A}\), that is,
Remark 1
The limit in a partial metric space may not be unique. For a sequence on \((\mathcal {A},\rho )\), we denote by the set of limit points (if there exist any),
We recall some results in the context of partialmetric spaces, necessary in our following considerations.
Lemma 1
Let \((\mathcal {A}, \rho )\) be a partialmetric space and be a sequence in \(\mathcal {A}\) such that . If , then there exist \(\mathsf{e}>0\) and subsequences , of such that
Lemma 2
([24])
Let be a Cauchy sequence on a complete partialmetric space \((\mathcal {A}, \rho )\). If there exists with , then , for every subsequence of .
Lemma 3
([25])
If , \(\{ \omega _{m} \} \) are two sequences in a partialmetric space \((\mathcal {A}, \rho )\) such that
then . Moreover, , for each \(\mathsf {u}\in \mathcal {A}\).
On a partialmetric space \((\mathcal {A}, \rho )\), a mapping is continuous at if and only if for every \(\mathsf{e}>0\), there exists \(\delta >0\) such that
( is continuous if it is continuous at every point .)
Lemma 4
([24])
On a complete partialmetric space \((\mathcal {A}, \rho )\), let be a continuous mapping and be a Cauchy sequence in \(\mathcal {A}\). If there exists with , then .
Definition 3
([26])
Let \(\mathcal {A}\) be a nonempty set and \(\mathsf {s}\geq 1\). A function \(\rho _{\mathsf{b}}:\mathcal {A}\times \mathcal {A}\rightarrow \mathbb{R}^{+}_{0}\) is a partial bmetric with a coefficient s if the following conditions hold for all
 \((\rho _{b}1)\):

;
 \((\rho _{b}2)\):

;
 \((\rho _{b}3)\):

;
 \((\rho _{b}4)\):

.
In this case, we say that \((\mathcal {A},\rho _{\mathsf{b}},\mathsf {s})\) is a partial bmetric space.
Example 1
([26])
Let \(\mathcal {A}\) be a nonempty set and .

if ρ is a partial metric on \(\mathcal {A}\), then the function \(\rho _{\mathsf{b}}\) defined as
(1.6)is a partial bmetric on \(\mathcal {A}\), with \(\mathsf {s}=2^{\lambda 1}\), for \(\lambda >1\).

if b is a bmetric and ρ is a partial metric on \(\mathcal {A}\), then the function
(1.7)is a partial bmetric on \(\mathcal {A}\).
A sequence in a partial bmetric space \((\mathcal {A}, \rho _{\mathsf{b}}, \mathsf {s})\) is said to be \(\rho _{\mathsf{b}}\)convergent to a point \(\mathsf {u}\in \mathcal {A}\) if
If the limit exists and it is finite, the sequence is said to be \(\rho _{\mathsf{b}}\)Cauchy. Moreover, if every \(\rho _{\mathsf{b}}\)Cauchy sequence in \(\mathcal {A}\) is \(\rho _{\mathsf{b}}\)convergent to \(\mathsf {u}\in \mathcal {A}\), that is
we say that the partial bmetric space \((\mathcal {A}, \rho _{\mathsf{b}}, \mathsf {s})\) is \(\rho _{\mathsf{b}}\)complete.
Remark 2
In [27] it is proved that a partial bmetric induces a bmetric, say \(\delta _{\mathsf {b}}\), with
for all .
On the other hand, in [28], the notion of 0\(\rho _{\mathsf{b}}\)completeness was introduced and the relation between 0\(\rho _{\mathsf{b}}\)completeness and \(\rho _{\mathsf{b}}\)completeness of a partial bmetric was established.
Definition 4
([28])
A sequence on a partial bmetric space \((\mathcal {A}, \rho _{\mathsf{b}}, \mathsf {s})\) is 0\(\rho _{\mathsf{b}}\)Cauchy if . Moreover, the space \((\mathcal {A}, \rho _{\mathsf{b}}, \mathsf {s})\) is said to be 0\(\rho _{\mathsf{b}}\)complete if for each 0\(\rho _{\mathsf{b}}\)Cauchy sequence in \(\mathcal {A}\), there is \(\mathsf {u}\in \mathcal {A}\), such that
Lemma 5
([28])
If the partial bmetric space \((\mathcal {A}, \rho _{\mathsf{b}}, \mathsf {s})\) is \(\rho _{\mathsf{b}}\)complete, then it is 0\(\rho _{\mathsf{b}}\)complete.
Lemma 6
([29])
Let \((\mathcal {A}, \rho _{\mathsf{b}}, \mathsf {s})\) be a partial bmetric space. If then and for all .
The next result is important in our future considerations.
Lemma 7
([30])
Let \((\mathcal {A}, \rho _{\mathsf{b}}, \mathsf {s}\geq 1)\) be a partial bmetric space, a mapping and a number \(\kappa \in [0,1)\). If is a sequence in \(\mathcal {A}\), where and
for each \(m\in \mathbb{N}\), then the sequence is 0\(\rho _{\mathsf{b}}\)Cauchy.
2 Main results
We start with the definition of simulation function for partial bmetric spaces.
Definition 5
Let \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}\geq 1)\) be a partial bmetric space. A bψsimulation function is a function \(\eta _{\mathsf {b}}:[0,+\infty )\times [0,+\infty )\rightarrow \mathbb{R}\) satisfying:
 \((\eta _{b1})\):

\(\eta _{\mathsf {b}}(\mathsf {r},\mathsf {t})<\psi (\mathsf {t})\psi (\mathsf {r})\) for all \(\mathsf {r},\mathsf {t}\in \mathbb{R}^{+}\);
 \((\eta _{b2})\):

if \(\{\mathsf {r}_{n}\},\{\mathsf {t}_{n}\}\) are two sequences in \([0,+\infty )\), such that for \(p>0\)
$$\begin{aligned} \limsup_{n\rightarrow +\infty }\mathsf {t}_{n}= \mathsf {s}^{p} \lim_{n\rightarrow +\infty }\mathsf {r}_{n}>0, \end{aligned}$$(2.1)then
$$\begin{aligned} \limsup_{n\rightarrow +\infty }\eta _{\mathsf {b}}\bigl(\mathsf {s}^{p} \mathsf {r}_{n}, \mathsf {t}_{n}\bigr)< 0. \end{aligned}$$(2.2)
We shall denote by \(\mathcal{Z}_{\psi _{b}}\) the family of all bψsimulation functions.
Example 2
Let \(\psi \in \Gamma \) and \(\gamma :[0,+\infty )\rightarrow [0,+\infty )\) be a function such that \(\limsup_{\mathsf {t}\rightarrow \mathsf {t}_{0}}\gamma ( \mathsf {t})<1\) for every \(\mathsf {t}_{0}>0\) and \(\phi (\mathsf {t})=0\) if and only if \(\mathsf {t}=0\). Then \({\eta _{\mathsf {b}}}(\mathsf {r}, \mathsf {t})=\gamma (\mathsf {t})\psi ( \mathsf {t})\psi (\mathsf {r})\), for \(\mathsf {r},\mathsf {t}\geq 0\) is a bψsimulation function.
Example 3
Let \(\psi \in \Gamma \) and \(\phi :[0,+\infty )\rightarrow [0,+\infty )\) be a function such that \(\lim_{\mathsf {t}\rightarrow \mathsf {t}_{0}}\phi (\mathsf {t})>0\) for every \(\mathsf {t}_{0}>0\) and \(\phi (\mathsf {t})=0\) if and only if \(\mathsf {t}=0\). Then \({\eta _{\mathsf {b}}}(\mathsf {r}, \mathsf {t})=\psi (\mathsf {t})\phi (\mathsf {t}) \psi (\mathsf {r})\), for \(\mathsf {r},\mathsf {t}\geq 0\) is a bψsimulation function.
Obviously, \((\eta _{b1})\) holds. Now, considering two sequences \(\{ \mathsf {r}_{n} \} \) and \(\{ \mathsf {t}_{n} \} \) in \((0,+\infty )\) such that (2.1) holds, we have
Thus, also \((\eta _{b2})\) holds, that is \({\eta _{\mathsf {b}}}\in \mathcal{Z}_{\psi _{b}}\).
Definition 6
Let \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}\geq 1)\) be a partial bmetric space. A mapping is called \((\eta _{\mathsf {b}})\)rational contraction of type A if there exists a function \(\eta _{\mathsf {b}}\in \mathcal{Z}_{\psi _{b}}\) such that
for every , where \(\mathcal{D}_{A}\) is defined as
With the purpose to simplify the demonstrations, we prefer in the sequel, to discuss separately, the cases
Theorem 3
Let \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}>1)\) be a \(\rho _{\mathsf{b}}\)complete partial bmetric space and be a \((\eta _{\mathsf {b}})\)rational contraction of type A. Then admits exactly one fixed point.
Proof
Let be an arbitrary but fixed point and be the sequence in \(\mathcal {A}\) defined as follows:
Thus, we can assume that for every \(m\in \mathbb{N}\). Indeed, if we suppose that there exists \(m_{0}\in \mathbb{N}\) such that . Taking into account (2.5) we get , that is, is a fixed point of . Therefore, substituting and in (2.4), we have
Moreover, by (2.3) we get
which implies
Now, taking into account \((\eta _{b1})\), the above inequality yields
or, equivalently,
Consequently, due to the monotony of the function ψ, we obtain
If there exists \(m_{1}\in \mathbb{N}\) such that , (2.6) becomes , which is a contradiction (because \(\mathsf {s}>1\)). Therefore, for any \(m\in \mathbb{N}\) we have
or
Denoting \(\frac{1}{\mathsf {s}^{p}}\) by κ, we have , with \(0\leq \kappa <1\). Thus, by Lemma 7 we see that the sequence is a 0\(\rho _{\mathsf{b}}\)Cauchy sequence on the \(\rho _{\mathsf{b}}\)complete partial bmetric space. Since by Lemma 5, the space is also 0\(\rho _{\mathsf{b}}\)complete, it follows that there exists \(\mathsf {u}\in \mathcal {A}\) such that
Now, we claim that
Assuming the contrary, we can find \(m_{0}\in \mathbb{N}\) such that
which is a contradiction. Thus, there exists a subsequence of such that
which implies
where
Therefore, letting \(l\rightarrow +\infty \) and keeping (2.8) in mind we get
On one hand, without loss of generality, we assume that , for infinitely many \(m\in \mathbb{N}\). Thus,
which by \((\eta _{b1})\) leads us to
Taking into account the nondecreasing property of ψ
On the other hand,
Letting \(m\rightarrow +\infty \) in the above inequality and keeping in mind (2.8) and (2.9) we get
Therefore, . Thus, letting and , by \((\eta _{b2})\) it follows \(\limsup_{m\rightarrow +\infty }\eta _{\mathsf {b}}(\mathsf {s}^{p} \mathsf {r}_{m}, \mathsf {t}_{m})<0\), which is a contradiction. Then , that is, u is a fixed point of .
As a last step, we establish uniqueness of the fixed point. Indeed, if we can find another point, \(\mathsf {z}\in \mathcal {A}\), \(\mathsf {z}\neq \mathsf {u}\) such that ,
which implies
which is a contradiction. Thus, \(\mathsf {u}=\mathsf {z}\). □
Example 4
Let the set \(\mathcal {A}= \{ 10,11,12,13 \} \) and \(\rho _{\mathsf{b}}\) be the partial bmetric on \(\mathcal {A}\) (\(\mathsf {s}=2\)), where We define the mapping , and we choose \(\phi \in \Gamma \), \(\phi (\mathsf {t})=\frac{\mathsf {t}}{2}\) and \(\eta _{\mathsf {b}}(\mathsf {r},\mathsf {t})= \frac{\frac{15}{16}\mathsf {t}\mathsf {r}}{2}\). It is easy to see that \(\eta _{\mathsf {b}}\in \mathcal{Z}_{\psi _{b}}\) (by taking \(\gamma (\mathsf {t})=\frac{15}{16}\) in Example 2). We have




10  10  0 
11  10  1 
12  10  4 
13  11  4 
and shall consider the following cases:

1.
For , we have , and then
which implies

2.
For we have , , , and then
which implies

3.
For we have , , , and then
which implies

4.
For we have , , , and then
which implies
Thus, the hypothesis of Theorem 3 are satisfied and is the fixed point of the mapping .
Definition 7
Let \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}>1)\) be a partial bmetric space. The mapping is said to be a \((\eta _{\mathsf {b}})\)rational contraction of type B if there exists \(\eta _{\mathsf {b}}\in \mathcal{Z}_{\psi _{b}}\) such that
for all , , where
Theorem 4
On a \(\rho _{\mathsf{b}}\)complete partial bmetric space \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}>1)\) any continuous \((\eta _{\mathsf {b}})\)rational contraction of type B, admits exactly one fixed point.
Proof
Let the sequence be defined by (2.5). Since , for each \(m\in \mathbb{N}\) (by similar reasoning as in the proof of Theorem 3), we have
which implies
where
Therefore
and since the function ψ is nondecreasing, we get, for any \(m\in \mathbb{N}\),
Moreover, if we get a contradiction, and then it follows that
and by Lemma (7), we conclude that is a 0\(\rho _{\mathsf{b}}\)Cauchy on a \(\rho _{\mathsf{b}}\)complete bpartialmetric space, and there exists \(\mathsf {u}\in \mathcal {A}\) such that .
Taking into account the continuity of the mapping , we have
that is, u is a fixed point of the mapping .
We claim that the fixed point of is unique. Let \(\mathsf {u},\mathsf {z}\in \mathcal {A}\) be two different fixed point of . Then
which implies
which is a contradiction. Therefore, \(\rho _{\mathsf{b}}(\mathsf {u}, \mathsf {z})=0\), that is (by Lemma 6), \(\mathsf {u}=\mathsf {z}\). □
Example 5
Let the set \(\mathcal {A}=[0,1]\), and \(\rho _{\mathsf{b}}:\mathcal {A}\times \mathcal {A}\rightarrow [0,+\infty )\), be a partial bmetric on \(\mathcal {A}\). Let the continuous mapping be defined by and the functions \(\psi \in \Gamma \), \(\eta _{\mathsf {b}}\in \mathcal{Z}_{\psi _{b}}\), where \(\psi (\mathsf {t})=\frac{\mathsf {t}}{2}\) and \(\eta _{\mathsf {b}}(\mathsf {r},\mathsf {t})=\frac{8}{9}(\frac{\mathsf {t}}{2}) \frac{\mathsf {r}}{2}\).
We verify that is a \((\eta _{\mathsf {b}})\)ψrational contraction of type B.

1.
For , if , (the case is similar), we have
Therefore,
which implies

2.
For , if , (the case is similar), we have
Therefore,
which implies

3.
For , we have
Therefore,
which implies
Therefore, all the hypotheses of Theorem 2.10 are satisfied and is the unique fixed point of .
Removing the condition in Theorem 3, respectively, Theorem 4, we immediately obtain the next results.
Corollary 1
Let \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}>1)\) be a \(\rho _{\mathsf{b}}\)complete partial bmetric space and be a mapping such that there exists \(\eta _{\mathsf {b}}\in \mathcal{Z}_{\psi _{b}}\) such that
for all , where \(\mathcal{D_{A}}\) is defined by (2.4). Then has a unique fixed point.
Corollary 2
Let \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}>1)\) be a \(\rho _{\mathsf{b}}\)complete partial bmetric space and be a continuous mapping such that there exists \(\eta _{\mathsf {b}}\in \mathcal{Z}_{\psi _{b}}\) such that
for all distinct , where \(\mathcal{D}_{B}\) is defined by (2.11). Then has a unique fixed point.
Corollary 3
Let be a mapping on a \(\rho _{\mathsf{b}}\)complete partial bmetric space \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}>1)\). Suppose that \(\psi \in \Gamma \) and \(\phi :[0,+\infty )\rightarrow [0,+\infty )\) is a function such that \(\liminf_{\mathsf {t}\rightarrow \mathsf {t}_{0}}\phi ( \mathsf {t})>0\), for \(\mathsf {t}_{0}>0\) and \(\phi (\mathsf {t})=0 \Leftrightarrow \mathsf {t}=0\). If for every \(\mathsf {r}, \mathsf {t}\in \mathcal {A}\)
which implies
then admits a unique fixed point.
Proof
Let \(\eta _{\mathsf {b}}(\mathsf {r}, \mathsf {t})=\psi (\mathsf {t})\phi (\mathsf {t}) \psi (\mathsf {r})\) in Theorem 3 and take into account Example 2. □
Corollary 4
Let be a continuous mapping on a \(\rho _{\mathsf{b}}\)complete partial bmetric space \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}>1)\). Suppose that \(\psi \in \Gamma \) and \(\phi :[0,+\infty )\rightarrow [0,+\infty )\) is a function such that \(\liminf_{\mathsf {t}\rightarrow \mathsf {t}_{0}}\phi ( \mathsf {t})>0\), for \(\mathsf {t}_{0}>0\) and \(\phi (\mathsf {t})=0 \Leftrightarrow \mathsf {t}=0\). If for every distinct \(\mathsf {r}, \mathsf {t}\in \mathcal {A}\)
which implies
then admits a unique fixed point.
Proof
Let \(\eta _{\mathsf {b}}(\mathsf {r}, \mathsf {t})=\psi (\mathsf {t})\phi (\mathsf {t}) \psi (\mathsf {r})\) in Theorem 4 and take into account Example 3. □
Corollary 5
Let be a mapping on a \(\rho _{\mathsf{b}}\)complete partial bmetric space \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}>1)\). Suppose that \(\psi \in \Gamma \) and \(\gamma :[0,+\infty )\rightarrow [0,1)\) is a function such that \(\limsup_{\mathsf {t}\rightarrow \mathsf {t}_{0}}\gamma ( \mathsf {t})<1\), for \(\mathsf {t}_{0}>0\) and \(\gamma (\mathsf {t})=0 \Leftrightarrow \mathsf {t}=0\). If for every \(\mathsf {r}, \mathsf {t}\in \mathcal {A}\)
which implies
then admits a unique fixed point.
Proof
Let \(\eta _{\mathsf {b}}(\mathsf {r},\mathsf {t})=\gamma (\mathsf {t})\psi (\mathsf {t}) \psi (\mathsf {r})\) in Theorem 3 and take into account Example 2. □
Corollary 6
Let be a continuous mapping on a \(\rho _{\mathsf{b}}\)complete partial bmetric space \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}>1)\). Suppose that \(\psi \in \Gamma \) and \(\gamma :[0,+\infty )\rightarrow [0,1)\) is a function such that \(\limsup_{\mathsf {t}\rightarrow \mathsf {t}_{0}}\gamma ( \mathsf {t})<1\), for \(\mathsf {t}_{0}>0\) and \(\gamma (\mathsf {t})=0 \Leftrightarrow \mathsf {t}=0\). If for every \(\mathsf {r}, \mathsf {t}\in \mathcal {A}\), with ,
which implies
then admits a unique fixed point.
Proof
Let \(\eta _{\mathsf {b}}(\mathsf {r}, \mathsf {t})=\gamma (\mathsf {t})\psi (\mathsf {t}) \psi (\mathsf {r})\) in Theorem 4 and take into account Example 2. □
We will prove below results similar to those stated in Theorems 3, 4 that can be formulated for the case \(\mathsf {s}=1\).
Theorem 5
Let \((\mathcal {A},\rho )\) be a \(\rho _{\mathsf{b}}\)complete partialmetric space and be a mapping. If there exists a function \(\eta \in \mathcal{Z}_{\psi }\) such that
for every distinct , where \(\mathcal{D}^{1}_{A}\) is defined as
then admits exactly one fixed point.
Proof
For , let be the sequence defined by (2.5), , for any \(m\in \mathbb{N}\).
First of all, we claim that . From (2.13), we have
which implies
Consequently, we get
which, since ψ is nondecreasing, implies
Therefore, the sequence is decreasing, so, we can find \(\theta \geq 0\) such that . On the other hand, it is easy to see that , as well. Assuming that \(\theta >0\), from (\(\eta _{2}\)) and (2.13) it follows that
which is a contradiction. So, we found that
We claim that is a Cauchy sequence. If we suppose that , there exist two subsequences , of the sequence and a number \(\mathsf{e}>0\) such that .
Moreover, by Lemma 1, we have
Looking on the definition of the function \(\mathcal{D}_{A}^{1}\), we have
and keeping in mind (2.15) and (2.16) we get
Now, letting and , by \((\eta _{2})\) we get
On the other hand, by (2.15), we have
Thus, by the triangle inequality and taking into account (2.20), we get
and then . Therefore,
which implies
which contradicts (2.19). Thus,
and is a Cauchy sequence in the complete partialmetric space \((\mathcal {A},\rho )\). This implies that there exists \(\mathsf {u}\in \mathcal {A}\) such that
We shall prove that . By \((\rho _{b2})\), we get
which implies
Thus, by the nondecreasing property of ψ, we obtain
and using (2.21) we get . Thus, and u is a fixed point of .
In order to show the uniqueness of the fixed point, let \(\mathsf {u}, \mathsf {z}\in \mathcal {A}\) such that and . We have
which implies
which is a contradiction. Thus, we conclude that u is the unique fixed point of . □
Theorem 6
Let \((\mathcal {A},\rho )\) be a \(\rho _{\mathsf{b}}\)complete partialmetric space and be a continuous mapping. If there exists a function \(\eta \in \mathcal{Z}_{\psi }\) such that
holds for every , where \(\mathcal{D}^{1}_{A}\) is defined as
then admits exactly one fixed point.
Proof
Let and consider the sequence , with . We assume that for each \(m\in \mathbb{N}\) because we remark that, on the contrary, if there exits \(l_{0}\) such that , that is is a fixed point for the mapping , then by (2.23), for any terms and we have
On the other hand, by (2.22),
which implies
But \(\psi \in \Gamma \) and then
If for some m, then (2.24) becomes , which is a contradiction. Then, for each \(m\geq 0\), , the inequality (2.24) yields
Thus, the sequence is decreasing, so it is convergent (being bounded from below). In this case, we can find a real number such that . Assume that , let and . Since
from \((\eta _{2})\) we have
This is a contradiction, so that
As a next step, we claim that is a Cauchy sequence in \((\mathcal {A}, \rho )\). Reasoning by contradiction, we suppose that . Then, by Lemma 1, there exist the subsequences , of the sequence , with \(q_{l}>m_{l}>l\), and a number \(\mathsf{e}>0\) such that and
Now, according to (2.25), there exists \(n_{1}\in \mathbb{N}\), such that
and \(n_{2}\in \mathbb{N}\), such that
Therefore, for \(l>\max \{ n_{1},n_{2} \} \) we have
and we can conclude . Thus,
which implies
On the other hand,
and \((\eta _{2})\) implies
which contradicts (2.26). Therefore, is a Cauchy sequence in a ρcomplete partialmetric space \((\mathcal {A}, \rho )\) and there exists \(\mathsf {u}\in \mathcal {A}\) such that
On the other hand, due to the continuity of the mapping , we get
Consequently, from (2.27), (2.28), on account of Lemma 3, we see that u is a fixed point of . The uniqueness of the fixed point follows immediately as in the previous theorem. □
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Karapınar, E., Chen, CM., Alghamdi, M.A. et al. Advances on the fixed point results via simulation function involving rational terms. Adv Differ Equ 2021, 409 (2021). https://doi.org/10.1186/s1366202103564w
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DOI: https://doi.org/10.1186/s1366202103564w
MSC
 47H10
 54H25
Keywords
 Simulation functions
 Contraction
 Fixed point