On $\mathcal{L}$-simulation mappings in partial metric spaces

1.   Introduction

An abstraction of the classical distance goes back to ancient mathematician Euclid's postulate. The notion of a metric, an Euclidean distance, was axiomatically formulated by Fréchet and Haussdorff at the beginning of nineteen century. Since then this notion has been refined, generalized and improved in several directions. Among all, we focus on a partial metric, defined by Matthews ^[21] by relaxing the self-distance axiom. Indeed, a partial metric is a very important extension of the metric, which is very useful in the framework of theoretical computer science, in particular, domain theory and semantic.

Matthews ^[21] proved the analog of the Banach's contraction principle ^[10] in the setting of partial metric spaces. This pioneer work ^[21] initiates an attractive trend, see e.g. ^{[1,2,3,4,5,6,7,8,9,11,13,15,16,17,18,19,22,23,24,25,27,28,29,30,31]}.

For the readability of the paper, we recollect fundamental definitions and results.

Definition 1.1. For a non-empty set $X$, if a function $p:X\times X\rightarrow\mathbb [0, \infty)$ that satisfies:

(PM1) $p(\theta, \theta) = p(\theta, \vartheta) = p(\vartheta, \vartheta)\Longleftrightarrow \theta = \vartheta$;

(PM2) $p(\theta, \theta) \leq p(\theta, \vartheta)$;

(PM3) $p(\theta, \vartheta) = p(\vartheta, \theta)$;

(PM4) $p(\theta, \eta) + p(\vartheta, \vartheta) \leq p(\theta, \vartheta) + p(\vartheta, \eta)$.

for all $\theta, \vartheta, \eta \in X$, then it is called a "partial metric" and $(X, p)$ denotes a "partial metric space", in short, PMS.

Let $\{\xi_n\}$ be a sequence in a PMS $(X, p)$. We say that $\{\xi_n\}$ converges to $\xi\in X$ if the following inequality holds:

A sequence $\{\xi_n\}$ is called "fundamental" (Cauchy), if $\lim\limits_{n, m\to\infty}p(\xi_n, \xi_m)$ exists and is finite. A PMS $(X, p)$ is called complete if each fundamental sequence in $X$ converges to a point $\xi\in X$ such that

We need to mention that the strong correlation between metric and partial metric. A function $d_p : X \times X\to [0, \infty)$, defined by

for all $\xi, \eta\in X,$ forms a metric on $X.$

A sequence in the PMS $(X, p)$ is Cauchy if and only if it is a Cauchy sequence in the metric space $(X, d_p).$ Also, $(X, p)$ is complete if and only if the metric space $(X, d_p)$ is complete. Moreover,

The following lemmas are useful.

Lemma 1.1. ^[7,8,16] Let $(X, p)$ be a PMS. We have

$(1)$ if $p(\xi, \eta) = 0$, then $\xi = \eta$;

$(2)$ if $\xi\neq\eta$, then $p(\xi, \eta) > 0.$

Lemma 1.2. ^[7,8,16] Let $\xi_n \to \xi$ as $n \to\infty$ in the PMS $(X, p)$ such that $p(\xi, \xi) = 0.$ Then $p(\xi_n, \eta)\to p(\xi, \eta)$ as $n \to\infty$ for each $\eta\in X$.

In ^[14], for a mapping $\theta:(0, \infty)\rightarrow (1, \infty)$, the concept of $\theta$-contractions was defined as

$(\theta_1)$ $\theta$ is non-decreasing;

$(\theta_2)$ For any positive sequence $\{u_n\}$

$(\theta_3)$ there exists $(s, t)\in (0, 1)\times (0, \infty)$ such that

Let $\digamma$ be the class of functions $\theta$ verifying $(\theta_1)$-$(\theta_3)$. Very recently, Cho ^[12] introduced the concept of $\mathcal{L}$-simulation mappings. Let $\mathcal{L}$ be the family of all mappings $\xi: [1, \infty)\times [1, \infty)\rightarrow \mathbb{R}$ such that

$(\xi_1)$ $\xi(1, 1) = 1$;

$(\xi_2)$ $\xi(u, v) < \frac{v}{u}$ for all $u, v > 1$;

$(\xi_3)$ For any sequences $\{u_n\}$, $\{v_n\}$ in $(1, \infty)$ with $u_n < v_n$ for $n = 1, 2, 3, \ldots$

Any $\xi\in\mathcal{L}$ is said an $\mathcal{L}$-simulation function. Note that $\xi(u, u) < 1$ for each $u > 1$.

Let $\mathcal{Z}$ be the set of simulation functions in the sense of Khojasteh, Shukla and Radenović ^[20].

Definition 1.2. ^[20] A simulation function is a mapping $\zeta:[0, \infty)\times [0, \infty)\to\mathbb{R}$ satisfying the following conditions:

($\zeta_{1}$) $\zeta(0, 0) = 0$;

($\zeta_{2}$) $\zeta(u, v) < v-u$ for all $u, v > 0;$

($\zeta_{3}$) if $\{u_n\}$ and $\{v_n\}$ are sequences in $(0, \infty)$ such that $\lim\limits_{n\to\infty}u_n = \lim\limits_{n\to\infty}v_n = \ell\in(0, \infty),$ then

We mention that any $\mathcal{L}$-simulation function can be deduced from simulation functions of Khojasteh, Shukla and Radenović ^[20]. Indeed, for $u, v\geq 0$, let $\xi(e^u, e^v) = e^{\zeta(u, v)}$. If $\zeta\in\mathcal{Z}$, then $\xi\in\mathcal{L}$.

In this paper, we obtain some fixed point results for $\mathcal{L}$-simulation mappings using $\theta$-functions in the class of partial metric spaces. Some consequences are also derived. Moreover, we present some examples in support of the given results.

2.   Main results

First, let $\Theta$ be the set of continuous functions $\theta:(0, \infty)\rightarrow (1, \infty)$ satisfying $(\theta_1)$ and $(\theta_2)$. Since the range of $\theta$ is in $(1, \infty)$, we will consider $\mathcal{L}$-simulation functions instead of simulation functions ^[20] and we will get many consequences in this setting. Our first main result is the following theorem.

Theorem 2.1. Let $(X, p)$ be a complete PMS and $T:X\rightarrow X$ be a given mapping. Suppose that there exist $\xi\in\mathcal{L}$ and $\theta\in\Theta$ such that for all $x, y\in X$ with $p(Tx, Ty)\neq 0$ and $p(x, y)\neq 0$,

Then $T$ has a unique fixed point.

Proof. Define a sequence $\{\xi_n\}$ by $\xi_n = T^n\xi_0$ for all $n\geq0.$ If $p(\xi_{n}, \xi_{n+1}) = 0$ for some $n,$ then $\xi_{n} = \xi_{n+1} = T\xi_{n},$ that is, $\xi_{n}$ is a fixed point of $T$ and so the proof is completed. Suppose from now on that $p(\xi_{n}, \xi_{n+1}) > 0\quad\mbox{ for all}\, \, n = 0, 1, \ldots$

Step 1: We shall prove that

First, from the condition (2.1), we have

Consequently, we obtain that

which implies for all $n = 1, 2, 3, \ldots$,

Hence $\{p(\xi_{n}, \xi_{n+1})\}$ is a decreasing sequence, so there exists $r\geq 0$ such that

Assume that $r\neq 0.$ It follows from $(\theta_2)$ that

and so

From $(\xi_3)$,

which is contradiction. This implies that

Step 2: Now, we shall show that $\{\xi_n\}$ is a bounded sequence in $(X, p)$. We argue by contradiction. If $\{\xi_n\}$ is not bounded, then there exists a subsequence $\{\xi_{n(k)}\}$ of $\{\xi_n\}$ such that $n(1) = 1$ and for all $k = 1, 2, 3, \dots, n(k + 1)$ is the minimum integer greater than $n(k)$ with

for all $n(k)\leq l\leq n(k+1)-1.$ We have

Taking the limit as $k\to \infty$ and using (2.2), we get

Again, using (2.3), (2.4) and $(PM4)$, we have

and

By (2.5) and (2.6), we have

It follows from (2.7) and $(\theta_2)$ that

Having in mind from (2.3) that $p(\xi_{n(k+1)-1}, \xi_{n(k)}) > 0$ and $p(\xi_{n(k+1)}, \xi_{n(k)}) > 0$, so by using condition (2.1), we have

This implies that

From (2.8), (2.9) and $(\xi_3)$, we have

which is contradiction. Thus, $\{\xi_n\}$ is bounded.

Step 3: Here, we shall show that $\{\xi_n\}$ is a Cauchy sequence in $(X, p)$. It suffices to prove that $\{\xi_n\}$ is Cauchy in the metric space $(X, d_p)$. Consider,

It is clear that

Hence, there exists $R\geq 0$ such that

Assume that $R\neq 0.$ For each positive integer $k$, there exist $n(k)\geq m(k)\geq k$ such that

Taking $k\to \infty$ and using (2.10), we get

In view of (2.2), we have

By definition of $d_p$ and using (2.12), we get

From condition (2.1),

This implies that

Since $\theta$ is non-decreasing, we get that

By $(PM4)$, we have

From (2.2) and (2.13), we get

Applying (2.15) in $(\xi_3)$, we have

which is a contradiction. This proves that $R = 0$. We deduce that

Then $\{\xi_n\}$ is Cauchy in $(X, p)$.

Step 4: Existence and uniqueness of a fixed point of $T$.

The sequence $\{\xi_n\}$ is Cauchy in the complete PMS $(X, p)$, so there exists $u\in X$ such that

By (2.16),

We shall prove that $u$ is the unique fixed point of $T$. Without loss of generality, we may assume that for infinitely many $n$,

Let $x, y\in X$ such that $p(Tx, Ty)\neq 0$ and $p(x, y)\neq 0$. Applying (2.1), we have

In this case, we have

By (2.18) and (2.19), $p(\xi_n, Tu) < p(\xi_{n-1}, u)$ for infinitely many $n$. Using (2.17) and Lemma 1.2, we have

Thus, $p(Tu, u) = 0$, that is, $u = Tu.$ Finally, we show the uniqueness of the fixed point. Let $\xi_1, \xi_2\in X$ be two distinct fixed points of $T$. Then $\xi_1\neq \xi_2$, so $p(T\xi_1, T\xi_2) = p(\xi_1, \xi_2) > 0$. From condition (2.1),

It is a contradiction, so $\xi_1 = \xi_2.$

Since each metric space is a PMS, the following is the analog of Theorem 2.1 in the setting of metric spaces.

Corollary 2.1. Let $(X, d)$ be a complete metric metric space and $T:X\rightarrow X$ be a given mapping. Suppose that there exist $\xi\in\mathcal{L}$ and $\theta\in\Theta$ such that for all $x, y\in X$ with $d(Tx, Ty)\neq 0$,

Then $T$ has a unique fixed point.

Remark 2.1. Corollary 2.1 is a proper generalization of Banach contraction mapping principle, by taking $\zeta(t, s) = \frac{t}{s^k}$ where $k \in (0, 1)$ and $\theta(t) = e^t$.

The following corollary is a key result which guides us to derive several existing results.

Corollary 2.2. Let $(X, p)$ be a complete PMS and $T:X\rightarrow X$ be a given mapping such that for all $x, y\in X$ with $p(Tx, Ty)\neq 0$,

where $\varphi: [0, \infty) \rightarrow [0, \infty)$ is nondecreasing and lower semi-continuous such that $\varphi^{-1}(\{0\}) = \{0\}$. Then $T$ has a unique fixed point.

Proof. From condition (2.21), we have

Putting $\theta(t) = e^t$, we get

Also, define $\varphi(t) = \ln(\psi(\theta(t)))$, where $\psi: [1, \infty) \rightarrow [1, \infty)$ is nondecreasing and lower semi-continuous such that $\psi^{-1}(\{1\}) = \{1\}$, we get

By putting $\xi(t, s) = \frac{s}{t\psi(s)},$ we get

By Theorem 2.1, $T$ has a unique fixed point.

Remark 2.2. In Corollary 2.2, the condition $p(x, y)\neq 0$ is not required. Because, if $p(Tx, Ty)\neq 0$ in (2.21), necessarily we have $p(x, y)\neq 0$.

To illustrate Theorem 2.1, we present the following examples.

Example 2.1. Let $X = [0, 1]\cup [3,4]$ and $p:X\times X\to [0, \infty)$ be defined by $p(x, y) = \max\{x, y\}.$ It is clear that $(X, p)$ is a complete PMS. Consider the mapping $T:X\to X$ defined by

Choose $\theta(t) = e^t$ for all $t > 0$, and $\xi(t, s) = \frac{\sqrt{s}}{t}$ for all $t, s\ge 1.$

To prove that $T$ is a $\mathcal{L}$-simulation with respect to $\xi$, let $x, y\in X$ be such that $p(x, y)\neq 0$ and $p(Tx, Ty)\neq 0$. Then the case $x = y = 0$ is excluded. Here, we have the following possible cases:

Case 1. $x, y\in [0, 1]$ and $(x, y)\neq(0, 0)$. We have $\theta(p(Tx, Ty)) = e^{\frac{1}{3}\max\{x, y\}}$ and $\theta(p(x, y)) = e^{\max\{x, y\}}$. Hence

Case 2. $x, y \in [3,4].$ Then $\theta(p(Tx, Ty)) = e^{\frac{3}{2}},$ $\theta(p(x, y)) = e^{\max\{x, y\}}$ and

Case 3. $y \in [3,4]$ and $x\in [0, 1].$ Here, $\theta(p(Tx, Ty)) = e^{\frac{3}{2}}$ and $\theta(p(x, y)) = e^{y}$. Then

for each $y \in [3,4].$

Case 4. $x \in [3,4]$ and $y\in [0, 1].$ Then $\theta(p(Tx, Ty)) = e^{\frac{3}{2}}$ and $\theta(p(x, y)) = e^{x}$. Then

for all $x \in [3,4].$ Then $T$ is a $\mathcal{L}$-simulation with respect to $\xi$. Hence all conditions of Theorem 2.1 are satisfied, and then $T$ has a unique fixed point, which is 0.

On the other hand, if we replace the partial metric $p$ by the usual metric $d$ and choose $x = 1$ and $y = 3$, then $\theta(d(Tx, Ty)) = e^{\frac{7}{6}}$ and $\theta(d(x, y)) = e^{2}$. Hence

Thus, $T$ is not a $\mathcal{L}$-simulation with respect to $\xi$ in the usual metric space.

Example 2.2. Let $X = [0, \infty)$ and $p:X\times X\to [0, \infty)$ be defined by $p(x, y) = \max\{x, y\}.$ It is clear that $(X, p)$ is a complete PMS. Consider the mapping $T:X\to X$ defined by

Choose $\theta(t) = e^t$ for all $t > 0$, and $\xi(t, s) = \frac{{s^{\frac{1}{3}}}}{t}$ for all $t, s\ge 1.$

To prove that $T$ is a $\mathcal{L}$-simulation with respect to $\xi$, let $x, y\in X$ be such that $p(x, y)\neq 0$ and $p(Tx, Ty)\neq 0$. The case $x = y = 0$ is excluded. Here, we have the following possible cases:

Case 1. $x, y\in [0, 3]$ and $(x, y)\neq(0, 0)$. We have $\theta(p(Tx, Ty)) = e^{\frac{1}{4}\max\{x, y\}}$ and $\theta(p(x, y)) = e^{\max\{x, y\}}$. Then

Case 2. $x, y \in (3, \infty).$ Then $\theta(p(Tx, Ty)) = e^{\frac{1}{3}}$ and $\theta(p(x, y)) = e^{\max\{x, y\}}$. Then

Case 3. $x \in [0, 3]$ and $y\in (3, \infty).$ Here, $\theta(p(Tx, Ty)) = e^{\max\{\frac{x}{4}, \frac{1}{3}\}}$ and $\theta(p(x, y)) = e^{y}$. Now,

We have

Case 4. $y \in [0, 3]$ and $x\in (3, \infty).$ This case follows from case 3 by replacing $x$ and $y$ as $p(x, y) = p(y, x)$. Then $T$ is a $\mathcal{L}$-simulation with respect to $\xi$. Hence all conditions of Theorem 2.1 are satisfied, and then $T$ has a unique fixed point, which is 0.

To illustrate Corollary 2.2, we present the following example.

Example 2.3. Consider the PMS $(X, p)$ where $X = [0, \infty)$ and $p(x, y) = \max\{x, y\}.$ Consider the mapping $T:X\to X$ defined by

Define $\varphi: [0, \infty) \rightarrow [0, \infty)$ by

Now for $x, y \in X$ with $p(Tx, Ty) \ne 0$, we get $(x, y)\neq(0, 0)$. We have the following 3 cases:

Case 1. If $x, y \leq 2$ and $(x, y)\neq(0, 0)$, then $p(Tx, Ty) = \frac{1}{3}\max\{x, y\}$ and $\varphi(p(x, y)) = \frac{1}{5}\max\{x, y\}$. Thus we have

Case 2. If $x, y > 2$, then $p(Tx, Ty) = 1$ and $p(x, y) = \max\{x, y\} > 2$. Also, $\varphi(p(x, y)) = \frac{1}{2}$. Hence

Case 3. If $x \leq 2$ and $y > 2$ or $y \leq 2$ and $x > 2$. Without loss of generality, let us assume $x \leq 2$ and $y > 2$. Then $p(Tx, Ty) = 1$ and $p(x, y) = y > 2$. Also, $\varphi(p(x, y)) = \frac{1}{2}$. We obtain that

Thus all conditions of Corollary 2.2 are satisfied, and then $T$ has a unique fixed point, which is 0.

Open Problem

Romaguera ^[26] defined the concepts of $0$-Cauchyness and $0$-completeness of a partial metric space as follows:

Definition 2.1. ^[26] Let $(X, p)$ be a partial metric space and $\{x_n\}$ be any sequence in $X$ and $x \in X$. Then:

$(i)$ The sequence $\{x_n\}$ is called $0$-Cauchy if

$(ii)$ $(X, p)$ is called $0$-complete if for every $0$-Cauchy sequence $\{x_n\}$ in $X$, there exists $x \in X$ such that

It is straightforward that if the partial metric type $(X, p)$ is complete, then it is $0$-complete. Then since $0$-completeness is more general than completeness, it would be better to prove or disapprove Theorem 2.1 in the class of $0$-complete partial metric spaces.

Acknowledgments

The authors thanks anonymous referees for their remarkable comments, suggestion, and ideas that help to improve this paper.

Conflict of interest

The authors declare that they have no competing interests.