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, partial-b-metric 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

(1.1)

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

(1.2)

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 non-decreasing 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 ψ-simulation 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 ψ-simulation functions; see e.g. [10–22]. It is clear, due to the axiom \((\eta _{1})\), that

$$\begin{aligned} \sigma (\mathsf {r},\mathsf {r})< 0 \quad\text{{for all }}\mathsf {r}>0. \end{aligned}$$
(1.4)

Definition 2

([23])

On a non-empty 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 partial-metric 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 partial-metric 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 partial-metric 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 partial-metric spaces, necessary in our following considerations.

Lemma 1

Let \((\mathcal {A}, \rho )\) be a partial-metric space and be a sequence in \(\mathcal {A}\) such that . If , then there exist \(\mathsf{e}>0\) and subsequences , of such that

(1.5)

Lemma 2

([24])

Let be a Cauchy sequence on a complete partial-metric space \((\mathcal {A}, \rho )\). If there exists with , then , for every subsequence of .

Lemma 3

([25])

If , \(\{ \omega _{m} \} \) are two sequences in a partial-metric space \((\mathcal {A}, \rho )\) such that

then . Moreover, , for each \(\mathsf {u}\in \mathcal {A}\).

On a partial-metric 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 partial-metric 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 non-empty set and \(\mathsf {s}\geq 1\). A function \(\rho _{\mathsf{b}}:\mathcal {A}\times \mathcal {A}\rightarrow \mathbb{R}^{+}_{0}\) is a partial b-metric 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 b-metric space.

Example 1

([26])

Let \(\mathcal {A}\) be a non-empty set and .

  • if ρ is a partial metric on \(\mathcal {A}\), then the function \(\rho _{\mathsf{b}}\) defined as

    (1.6)

    is a partial b-metric on \(\mathcal {A}\), with \(\mathsf {s}=2^{\lambda -1}\), for \(\lambda >1\).

  • if b is a b-metric and ρ is a partial metric on \(\mathcal {A}\), then the function

    (1.7)

    is a partial b-metric on \(\mathcal {A}\).

A sequence in a partial b-metric 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

(1.8)

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

(1.9)

we say that the partial b-metric space \((\mathcal {A}, \rho _{\mathsf{b}}, \mathsf {s})\) is \(\rho _{\mathsf{b}}\)-complete.

Remark 2

In [27] it is proved that a partial b-metric induces a b-metric, say \(\delta _{\mathsf {b}}\), with

(1.10)

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 b-metric was established.

Definition 4

([28])

A sequence on a partial b-metric 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

(1.11)

Lemma 5

([28])

If the partial b-metric 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 b-metric 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 b-metric space, a mapping and a number \(\kappa \in [0,1)\). If is a sequence in \(\mathcal {A}\), where and

(1.12)

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 b-metric spaces.

Definition 5

Let \((\mathcal {A},\rho _{\mathsf{b}}, \mathsf {s}\geq 1)\) be a partial b-metric space. A b-ψ-simulation 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 b-ψ-simulation 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 b-ψ-simulation 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 b-ψ-simulation 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

$$\begin{aligned} \lim_{n\rightarrow +\infty }{\eta _{\mathsf {b}}}\bigl(\mathsf {s}^{p} \mathsf {r}_{n}, \mathsf {t}_{n}\bigr)= \lim_{n\rightarrow +\infty } \psi (\mathsf {t}_{n})-\phi (\mathsf {t}_{n})-\psi \bigl(\mathsf {s}^{p}\mathsf {r}_{n}\bigr) \leq -\phi (\mathsf {t}_{n})< 0. \end{aligned}$$

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 b-metric 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

(2.3)

for every , where \(\mathcal{D}_{A}\) is defined as

(2.4)

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 b-metric 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:

(2.5)

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

(2.6)

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

(2.7)

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 b-metric 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

(2.8)

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

(2.9)

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 non-decreasing 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 b-metric 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. 1.

    For , we have , and then

    which implies

  2. 2.

    For we have , , , and then

    which implies

  3. 3.

    For we have , , , and then

    which implies

  4. 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 b-metric 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

(2.10)

for all , , where

(2.11)

Theorem 4

On a \(\rho _{\mathsf{b}}\)-complete partial b-metric 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

(2.12)

where

Therefore

and since the function ψ is non-decreasing, 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 b-partial-metric 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 b-metric 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 \((\eta _{\mathsf {b}})\)-ψ-rational contraction of type B.

  1. 1.

    For , if , (the case is similar), we have

    Therefore,

    which implies

  2. 2.

    For , if , (the case is similar), we have

    Therefore,

    which implies

  3. 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 b-metric 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 b-metric 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 b-metric 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 b-metric 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 b-metric 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 b-metric 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 partial-metric space and be a mapping. If there exists a function \(\eta \in \mathcal{Z}_{\psi }\) such that

(2.13)

for every distinct , where \(\mathcal{D}^{1}_{A}\) is defined as

(2.14)

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 non-decreasing, 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

(2.15)

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

(2.16)

Looking on the definition of the function \(\mathcal{D}_{A}^{1}\), we have

(2.17)

and keeping in mind (2.15) and (2.16) we get

(2.18)

Now, letting and , by \((\eta _{2})\) we get

(2.19)

On the other hand, by (2.15), we have

(2.20)

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 partial-metric space \((\mathcal {A},\rho )\). This implies that there exists \(\mathsf {u}\in \mathcal {A}\) such that

(2.21)

We shall prove that . By \((\rho _{b2})\), we get

which implies

Thus, by the non-decreasing 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 partial-metric space and be a continuous mapping. If there exists a function \(\eta \in \mathcal{Z}_{\psi }\) such that

(2.22)

holds for every , where \(\mathcal{D}^{1}_{A}\) is defined as

(2.23)

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

(2.24)

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

$$\begin{aligned} 0\leq \limsup_{m\rightarrow +\infty }\eta (\mathsf {r}_{m}, \mathsf {t}_{m})< 0. \end{aligned}$$

This is a contradiction, so that

(2.25)

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

(2.26)

On the other hand,

and \((\eta _{2})\) implies

which contradicts (2.26). Therefore, is a Cauchy sequence in a ρ-complete partial-metric space \((\mathcal {A}, \rho )\) and there exists \(\mathsf {u}\in \mathcal {A}\) such that

(2.27)

On the other hand, due to the continuity of the mapping , we get

(2.28)

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. □