On ({mathcal{F}})-Contractions for Weak α-Admissible Mappings in Metric-Like Spaces

Abstract

In the paper, we consider some fixed point results of $\mathcal{F}$-contractions for triangular $\alpha$-admissible and triangular weak $\alpha$-admissible mappings in metric-like spaces. The results on $\mathcal{F}$-contraction type mappings in the context of metric-like spaces are generalized, improved, unified, and enriched. We prove the main result but using only the property ($\mathcal{F}1$) of the strictly increasing mapping $\mathcal{F}:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$. Our approach gives a proper generalization of several results given in current literature.

1. Introduction and Preliminaries

First, we recall some notions introduced recently in several papers.

In 2012, Samet et al. [1] introduced the concept of $\alpha$-admissible mappings as follows.

Definition 1.

Let $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ and $\alpha :{\mathcal{X}}^2\to [0,+\infty )$. Then, $\mathcal{T}$ is called α-admissible if for all $\xi ,\zeta \in \mathcal{X}$ with $\alpha \left(\xi ,\zeta \right)\ge 1$ implies $\alpha \left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\ge 1$.

Furthermore, one says that $\mathcal{T}$ is a triangular $\alpha$-admissible mapping if it is $\alpha$-admissible and if

$$\alpha \left(\xi ,\eta \right)\ge 1\mathrm{and}\alpha \left(\eta ,\zeta \right)\ge 1\mathrm{implies}\alpha \left(\xi ,\zeta \right)\ge 1,\xi ,\zeta ,\eta \in \mathcal{X}.$$

For triangular $\alpha$-admissible mapping, the following result is known ([2], Lemma 7):

Lemma 1.

Let $\mathcal{T}$ be a triangular α-admissible mapping. Assume that there exists ${\xi }_0\in \mathcal{X}$ such that $\alpha \left({\xi }_0,\mathcal{T}{\xi }_0\right)\ge 1.$ Define sequence $\left\{{\xi }_n\right\}$ by ${\xi }_n={\mathcal{T}}^n{\xi }_0.$ Then,

$$\alpha \left({\xi }_m,{\xi }_n\right)\ge 1\mathit{for}\mathit{all}m,n\in \mathbb{N}\cup \left\{0\right\}\mathit{with}m<n.$$

In [3], the author presented the notion of weak $\alpha$-admissible mappings as follows:

Definition 2.

Let $\mathcal{X}$ be a nonempty set and let $\alpha :{\mathcal{X}}^2\to [0,+\infty )$ be a given mapping. A mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is said to be a weak α-admissible one if the following condition holds:

$$\mathrm{for}\xi \in \mathcal{X}\mathrm{with}\alpha \left(\xi ,\mathcal{T}\xi \right)\ge 1implies\alpha \left(\mathcal{T}\xi ,{\mathcal{T}}^2\xi \right)\ge 1.$$

(1)

Remark 1.

It is customary to write $A\left(\mathcal{X},\alpha \right)$ and $WA\left(\mathcal{X},\alpha \right)$ as the collection of all (triangular) α-admissible mappings on $\mathcal{X}$ and the collection of all (triangular) weak α-admissible mappings on $\mathcal{X}$ (see[3]). One can verify that $A\left(\mathcal{X},\alpha \right)\subseteq WA\left(\mathcal{X},\alpha \right).$

Now, we recall some basic concepts, notations, and known results from partial metric and metric-like spaces. In 1994 Matthews ([4]) introduced notion of partial metric space as follows.

Definition 3.

Let $\mathcal{X}$ be a nonempty set. A mapping $d_{pm}:{\mathcal{X}}^2\to [0,+\infty )$ is said to be a partial metric on $\mathcal{X}$ if for all $\xi ,\zeta ,\eta \in \mathcal{X}$ the following four conditions hold:

(1) 

$\xi =\zeta$ if and only if $d_{pm}\left(\xi ,\xi \right)=d_{pm}\left(\xi ,\zeta \right)=d_{pm}\left(\zeta ,\zeta \right)$;

(2) 

$d_{pm}\left(\xi ,\xi \right)\le d_{pm}\left(\xi ,\zeta \right)$;

(3) 

$d_{pm}\left(\xi ,\zeta \right)=d_{pm}\left(\zeta ,\xi \right)$;

(4) 

$d_{pm}\left(\xi ,\eta \right)\le d_{pm}\left(\xi ,\zeta \right)+d_{pm}\left(\zeta ,\eta \right)-d_{pm}\left(\zeta ,\zeta \right).$

In this case, the pair $\left(\mathcal{X},d_{pm}\right)$ is called a partial metric space. Obviously, every metric space is a partial metric space. The inverse is not true. Indeed, let $\mathcal{X}=[0,+\infty )$ and $d_{pm}\left(\xi ,\zeta \right)=max\left\{\xi ,\zeta \right\}$. Under these conditions $\left(\mathcal{X},d_{pm}\right)$ is a partial metric space but is not a metric space because $d_{pm}\left(1,1\right)=1>0$. For more details, see ([5,6,7,8,9,10,11]).

For the following notion see [12].

Definition 4.

Let $\mathcal{X}$ be a nonempty set. A mapping $d_{ml}:{\mathcal{X}}^2\to [0,+\infty )$ is said to be a metric-like on $\mathcal{X}$ if for all $\xi ,\zeta ,\eta \in \mathcal{X}$ the following three conditions hold:

(1) 

$d_{ml}\left(\xi ,\zeta \right)=0$ implies $\xi =\zeta$;

(2) 

$d_{ml}\left(\xi ,\zeta \right)=d_{ml}\left(\zeta ,\xi \right)$;

(3) 

$d_{ml}\left(\xi ,\eta \right)\le d_{ml}\left(\xi ,\zeta \right)+d_{ml}\left(\zeta ,\eta \right)$.

The pair $\left(\mathcal{X},d_{ml}\right)$ is called a metric-like space or dislocated metric space by some authors. A metric-like mapping $d_{ml}$ on $\mathcal{X}$ satisfies all the conditions of a metric except that $d_{ml}\left(\xi ,\xi \right)$ may be positive for some $\xi \in \mathcal{X}$. The following is a list of some metric-like spaces:

1. $\left(ℝ,d_{ml}\right),$ where $d_{ml}\left(\xi ,\zeta \right)=max\left\{\left|\xi \right|,\left|\zeta \right|\right\}$ for all $\xi ,\zeta \in ℝ.$

One can see that $\left(ℝ,d_{ml}\right)$ is a metric-like space, but it is not a metric space, due to the fact that $d_{ml}\left(\left|-2\right|,\left|-2\right|\right)=2>0.$ On the other hand, $\left(ℝ,d_{ml}\right)$ is a partial metric space.

2. $\left([0,+\infty ),d_{ml}\right),$ where $d_{ml}\left(\xi ,\zeta \right)=\xi +\zeta$ for all $\xi ,\zeta \in [0,+\infty ).$

It is clear that $\left([0,+\infty ),d_{ml}\right)$ is a metric-like space where $d_{ml}\left(\xi ,\xi \right)>0$ for each $\xi >0.$ Since $d_{ml}\left(2,2\right)=2+2=4>3=2+1=d_{ml}\left(2,1\right)$, it follows that $d_{ml}\left(\xi ,\xi \right)\le d_{ml}\left(\xi ,\zeta \right)$ does not hold. Hence, $\left([0,+\infty ),d_{ml}\right)$ is not a partial metric space.

3. $(\mathcal{X},d_{ml}),$ where $\mathcal{X}=\{0,1,2\}$ and $d_{ml}(0,0)=d_{ml}(1,1)=0,d_{ml}(2,2)=\frac{5}{2}$, $d_{ml}(0,2)=d_{ml}(2,0)=2,$ $d_{ml}(1,2)=d_{ml}(2,1)=3,d_{ml}(0,1)=d_{ml}(1,0)=\frac{3}{2}.$

It is clear that $\left(\mathcal{X},d_{ml}\right)$ is a metric-like (that is a dislocated metric) space with $d_{ml}\left(2,2\right)>0$. This means that $\left(\mathcal{X},d_{ml}\right)$ is not a standard metric space. However, $\left(\mathcal{X},d_{ml}\right)$ is also not a partial metric space because $d_{ml}\left(2,2\right)\cancel{\leqq }{d}_{ml}\left(2,0\right).$

4. $\left(\mathcal{X},d_{ml}\right),$ where $\mathcal{X}=C\left(\left[0,1\right],ℝ\right)$ is the set of real continuous functions on $\left[0,1\right]$ and $d_{ml}\left(f,g\right)={sup}_{t\in \left[0,1\right]}\left(\left|f\left(t\right)\right|+\left|g\left(t\right)\right|\right)$ for all $f,g\in C\left(\left[0,1\right],ℝ\right).$

This is an example of metric-like space that is not a partial metric space. Indeed, for $f\left(t\right)=2t,$ we obtain $d_{ml}\left(f,f\right)={sup}_{t\in \left[0,1\right]}\left(2t+2t\right)=4>0.$ Putting $g\left(t\right)\equiv 0$ for all $t\in \left[0,1\right],$ we obtain that $d_{ml}\left(f,f\right)=4\cancel{\leqq }{d}_{ml}\left(f,g\right)=d_{ml}\left(f,0\right)=2.$

Note that some of the metric-like spaces given in the list are not partial metric spaces. It is clear that a partial metric space is a metric-like space and the inverse is not true. Now, we give the definitions of convergence and Cauchyiness of the sequences in metric-like space (see [12]).

Definition 5.

Let $\left\{{\xi }_n\right\}$ be a sequence in a metric-like space $\left(\mathcal{X},d_{ml}\right)$.

(i) 

The sequence $\left\{{\xi }_n\right\}$ is said to be convergent to $\xi \in \mathcal{X}$ if ${lim}_{n\to +\infty }{d}_{ml}\left({\xi }_n,\xi \right)=d_{ml}\left(\xi ,\xi \right);$

(ii) 

The sequence $\left\{{\xi }_n\right\}$ is said to be $d_{ml}$—Cauchy in $\left(\mathcal{X},d_{ml}\right)$ if ${lim}_{n,m\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_m\right)$ exists and is finite;

(iii) 

A metric-like space $\left(\mathcal{X},d_{ml}\right)$ is $d_{ml}$—complete if for every $d_{ml}-$ Cauchy sequence $\left\{{\xi }_n\right\}$ in $\mathcal{X}$ there exists an $\xi \in \mathcal{X}$ such that ${lim}_{n,m\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_m\right)=d_{ml}\left(\xi ,\xi \right)={lim}_{n\to +\infty }{d}_{ml}\left({\xi }_n,\xi \right).$

More details on partial metric and metric-like spaces can be found in ([5,6,7,11,13,14,15,16,17,18]), and information on other classes of generalized metric spaces and contractive mappings can be found in: ([1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]).

Remark 2.

In metric-like space (as in the partial metric space), the limit of a sequence need not be unique and a convergent sequence need not be a $d_{ml}$—Cauchy sequence (see examples in Remark 1.4 (1) and (2) in [10]). However, if the sequence $\left\{{\xi }_n\right\}$ is $d_{ml}-$ Cauchy such that ${lim}_{n,m\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_m\right)=0$ in $d_{ml}-$ complete metric-like space $\left(\mathcal{X},d_{ml}\right)$, then the limit of such sequence is unique. Indeed, in such a case if ${\xi }_n\to \xi$ as $n\to +\infty$, we get that $d_{ml}\left(\xi ,\xi \right)=0$ (by (iii) of Definition 5). Now, if ${\xi }_n\to \xi ,{\xi }_n\to \zeta$ and $\xi \ne \zeta ,$ we obtain

$$d_{ml}\left(\xi ,\zeta \right)\le d_{ml}\left(\xi ,{\xi }_n\right)+d_{ml}\left({\xi }_n,\zeta \right)\to d_{ml}\left(\xi ,\xi \right)+d_{ml}\left(\zeta ,\zeta \right)=0+0=0.$$

(2)

By (1) from Definition 4, it follows that $\xi =\zeta ,$ which is a contradiction.

Now, we give the definition of the continuity for self-mapping $\mathcal{T}$ defined on a metric-like space $\left(\mathcal{X},d_{ml}\right)$ as follows (see for example [10,11,34]):

Definition 6.

Let $\left(\mathcal{X},d_{ml}\right)$ be a metric-like space and $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ be a self-mapping. We say that $\mathcal{T}$ is $d_{ml}-$ continuous in point $\xi \in \mathcal{X}$ if ${lim}_{n\to +\infty }{d}_{ml}\left(\mathcal{T}{\xi }_n,\mathcal{T}\xi \right)=d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\xi \right)$, for each sequence $\left\{{\xi }_n\right\}\subseteq \mathcal{X}$ such that ${lim}_{n\to +\infty }{d}_{ml}\left({\xi }_n,\xi \right)=d_{ml}\left(\xi ,\xi \right)$. In other words, the mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is $d_{ml}-$ continuous if the following holds true:

$${\xi }_n\stackrel{d_{ml}}{\to }\xi implies\mathcal{T}{\xi }_n\stackrel{d_{ml}}{\to }\mathcal{T}\xi .$$

(3)

Definition 7.

Let $\left(\mathcal{X},d_{ml}\right)$ be a metric-like space. A sequence $\left\{{\xi }_n\right\}$ in it is called $0-d_{ml}-$ Cauchy sequence if ${lim}_{n,m\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_m\right)=0$. The space $\left(\mathcal{X},d_{ml}\right)$ is said to be $0-d_{ml}-$ complete if every $0-d_{ml}-$ Cauchy sequence in $\mathcal{X}$ converges to a point $\xi \in \mathcal{X}$ such that $d_{ml}\left(\xi ,\xi \right)=0.$

It is obvious that every $0-d_{ml}-$ Cauchy sequence is a $d_{ml}-$ Cauchy sequence in $\left(\mathcal{X},d_{ml}\right)$ and every $d_{ml}-$ complete metric-like space is a $0-d_{ml}-$ complete metric-like space. In addition, every $0-$ complete partial metric space $\left(\mathcal{X},d_{ml}\right)$ is a $0-d_{ml}-$ complete metric-like space. In the sequel, some results on metric-like spaces are given. Proofs to most of the results are self-evident.

Proposition 1.

Let $\left(\mathcal{X},d_{ml}\right)$ be a metric-like space. Then, we have the following:

(i) 

If the sequence $\left\{{\xi }_n\right\}$ converges to $\xi \in \mathcal{X}$ as $n\to +\infty$ and if $d_{ml}\left(\xi ,\xi \right)=0,$ then, for all $\zeta \in \mathcal{X}$, it follows that $d_{ml}\left({\xi }_n,\zeta \right)\to d_{ml}\left(\xi ,\zeta \right);$

(ii) 

If $d_{ml}\left(\xi ,\zeta \right)=0$, then $d_{ml}\left(\xi ,\xi \right)=d_{ml}\left(\zeta ,\zeta \right)=0;$

(iii) 

If $\left\{{\xi }_n\right\}$ is a sequence such that ${lim}_{n\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_{n+1}\right)=0$, then

${lim}_{n\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_n\right)={lim}_{n\to +\infty }{d}_{ml}\left({\xi }_{n+1},{\xi }_{n+1}\right)=0;$

(iv) 

If $\xi \ne \zeta$, then $d_{ml}\left(\xi ,\zeta \right)>0;$

(v) 

$d_{ml}\left(\xi ,\xi \right)\le \frac{2}{n}{\displaystyle \sum _{i=1}^n}{d}_{ml}\left(\xi ,{\xi }_i\right)$ holds for all $\xi ,{\xi }_i\in \mathcal{X},$ where $1\le i\le n;$

(vi) 

Let $\left\{{\xi }_n\right\}$ be a sequence such that ${lim}_{n\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_{n+1}\right)=0.$ If ${lim}_{n,m\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_m\right)\ne 0,$ then there exists $\epsilon >0$ and sequences $\left\{m\left(k\right)\right\}$ and $\left\{n\left(k\right)\right\}$ such that $n\left(k\right)>m\left(k\right)>k,$ and the following sequences tend to ε when $k\to +\infty :$

$$\begin{array}{c}\left\{d_{ml}\left({\xi }_{n\left(k\right)},{\xi }_{m\left(k\right)}\right)\right\},\left\{d_{ml}\left({\xi }_{n\left(k\right)+1},{\xi }_{m\left(k\right)}\right)\right\},\left\{d_{ml}\left({\xi }_{n\left(k\right)},{\xi }_{m\left(k\right)-1}\right)\right\},\\ \left\{d_{ml}\left({\xi }_{n\left(k\right)+1},{\xi }_{m\left(k\right)-1}\right)\right\},\left\{d_{ml}\left({\xi }_{n\left(k\right)+1},{\xi }_{m\left(k\right)+1}\right)\right\}.\end{array}$$

(4)

Notice that, if the condition (vi) is satisfied then the sequences $d_{ml}\left({\xi }_{n\left(k\right)+q},{\xi }_{m\left(k\right)}\right)$ and $d_{ml}\left({\xi }_{n\left(k\right)+q},{\xi }_{m\left(k\right)+1}\right)$ also converge to $\epsilon$ when $k\to +\infty ,$ where $q\in \mathbb{N}.$ For more details on (i)–(vi), the reader can see in ([26,27,36]). The concept of $\mathcal{F}$-contraction was introduced by Wardowski in [16] (for more details, see also: [5,9,14,15,16,17,18,24,28,31,32,33]).

Definition 8.

Let $\mathcal{F}:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ be a mapping satisfying the following:

($\mathcal{F}1$)

$\mathcal{F}$ is a strictly increasing, that is, for $\alpha ,\beta \in \left(0,+\infty \right)$, $\alpha <\beta$ implies $\mathcal{F}\left(\alpha \right)<\mathcal{F}\left(\beta \right),$

($\mathcal{F}2$)

For each sequence $\left\{{\alpha }_n\right\}\subset \left(0,+\infty \right)$, ${lim}_{n\to +\infty }{\alpha }_n=0$ if and only if ${lim}_{n\to +\infty }\mathcal{F}\left({\alpha }_n\right)=-\infty ,$

($\mathcal{F}3$)

There exists $k\in \left(0,1\right)$ such that ${lim}_{\alpha \to 0^{+}}{\alpha }^k\mathcal{F}\left(\alpha \right)=0.$

Definition 9.

Let $\left(\mathcal{X},d\right)$ be a metric space. A mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is said to be an $\mathcal{F}$-contraction if there exist $\mathcal{F}:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ satisfying ($\mathcal{F}1$), ($\mathcal{F}2$) and ($\mathcal{F}3$) and $\tau >0$ such that

$$d\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)>0implies\tau +\mathcal{F}\left(d\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)\le \mathcal{F}\left(d\left(\xi ,\zeta \right)\right),$$

(5)

for all $\xi ,\zeta \in \mathcal{X}.$

In 2014, Piri and Kumam [32] investigated some fixed point results concerning $\mathcal{F}$ contraction in complete metric spaces by replacing the condition ($\mathcal{F}3$) with the condition:

($\mathcal{F}{3}^{\prime }$)

$\mathcal{F}$ is continuous on $\left(0,+\infty \right).$

Recently, in 2018, Qawaqueh et al. ([9]) defined and proved the following:

Definition 10.

Let $\left(\mathcal{X},d_{ml}\right)$ be a metric-like space and $\alpha :{\mathcal{X}}^2\to [0,+\infty )$. A mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is said to be an $\left(\alpha ,\beta ,\mathcal{F}\right)$-Geraghty contraction mapping if there exist $\beta \in \mathcal{G}$ and $\tau >0$ such that, for all $\xi ,\zeta \in \mathcal{X}$ with $d\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)>0$ and $\alpha \left(\xi ,\zeta \right)\ge 1,$

$$\alpha \left(\xi ,\zeta \right)\left(\tau +\mathcal{F}\left(d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)\right)\le \beta \left(\mathcal{M}\left(\xi ,\zeta \right)\right)\mathcal{F}\left(\mathcal{M}\left(\xi ,\zeta \right)\right),$$

(6)

where

$$\begin{array}{c}\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right),\frac{d_{ml}\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\mathcal{T}\xi ,\zeta \right)}{4},\right.\\ \left.\frac{\left[1+d_{ml}\left(\xi ,\mathcal{T}\xi \right)\right]d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)}{1+d_{ml}\left(\xi ,\zeta \right)}\right\},\end{array}$$

(7)

$\mathcal{F}:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ is strictly increasing function satisfying ($\mathcal{F}1$), ($\mathcal{F}2$) and ($\mathcal{F}3$) and $\mathcal{G}$ is a family of all functions $\beta :[0,+\infty )\to [0,1)$ which satisfy the condition: $\beta \left(t_n\right)\to 1$ implies $t_n\to 0$ as $\to +\infty .$

It is worth noticing that authors in [9] denote with $E\left(\mathcal{X},\alpha ,\beta ,\mathcal{F}\right)$ the collection of all almost generalized $\left(\alpha ,\beta ,\mathcal{F}\right)$-contractive mappings. However, it is not clear what “almost generalized $\left(\alpha ,\beta ,\mathcal{F}\right)$-contractive mappings” mean.

Theorem 1.

Let $\left(\mathcal{X},d_{ml}\right)$ be a metric-like space and $\alpha :{\mathcal{X}}^2\to [0,+\infty )$. A mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ be an $\left(\alpha ,\beta ,\mathcal{F}\right)$-Geraghty contraction mapping. Assume that the following conditions are satisfied:

(i) 

$\mathcal{T}\in E\left(\mathcal{X},\alpha ,\beta ,\mathcal{F}\right)\cap WA\left(\mathcal{X},\alpha \right)$.

(ii) 

There exists ${\xi }_0\in \mathcal{X}$ such that $d_{ml}\left({\xi }_0,\mathcal{T}{\xi }_0\right)\ge 1$.

(iii) 

$\mathcal{T}$ is $d_{ml}-$ continuous.

Then, $\mathcal{T}$ has a unique fixed point $\eta \in \mathcal{X}$ with $d_{ml}\left(\eta ,\eta \right)=0.$

2. Main Result

In this section, we improve the whole concept by introducing a new definition and new approaches. Firstly, we introduce the following:

Definition 11.

Let $\left(\mathcal{X},d_{ml}\right)$ be a metric-like space and $\alpha :{\mathcal{X}}^2\to [0,+\infty )$. A mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is said to be a triangular $\left(\alpha ,\mathcal{F}\right)$-contraction one if there exists $\tau >0$ such that, for all $\xi ,\zeta \in \mathcal{X}$ with $d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)>0$ and $\alpha \left(\xi ,\zeta \right)\ge 1$ holds true,

$$\alpha \left(\xi ,\zeta \right)\left(\tau +\mathcal{F}\left(d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)\right)\le \mathcal{F}\left(\mathcal{M}\left(\xi ,\zeta \right)\right),$$

(8)

where

$$\begin{array}{c}\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right),\frac{d_{ml}\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\zeta ,\mathcal{T}\xi \right)}{2},\right.\\ \left.\frac{\left[1+d_{ml}\left(\xi ,\mathcal{T}\xi \right)\right]d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)}{1+d_{ml}\left(\xi ,\zeta \right)}\right\},\end{array}$$

(9)

$\mathcal{F}:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ is strictly increasing function.

Example 3 from [9], for instance, illustrates the validity of this definition but without the function $\beta :\left[0,+\infty \right)\to \left[0,1\right)$. Definition 11 is an improvement of the definition given in [9] in several directions. Now, we prove the main result of our paper:

Theorem 2.

Let $\left(\mathcal{X},d_{ml}\right)$ be a $0-d_{ml}-$ complete metric-like space and $\alpha :{\mathcal{X}}^2\to [0,+\infty )$. Assume that a mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is a triangular $\left(\alpha ,\mathcal{F}\right)$-contraction one. Suppose further that the following conditions are satisfied:

(i) 

$\mathcal{T}\in WA\left(\mathcal{X},\alpha \right);$

(ii) 

There exists ${\xi }_0\in \mathcal{X}$ such that $\alpha \left({\xi }_0,\mathcal{T}{\xi }_0\right)\ge 1;$

(iii) 

$\mathcal{T}$ is $d_{ml}-$ continuous.

Then, $\mathcal{T}$ has a unique fixed point $\widehat{\xi }\in \mathcal{X}$ with $d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)=0.$

Proof. 

First of all, we show the following two claims:

I.

If $\widehat{\xi }$ is a fixed point of $\mathcal{T}$ then $d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)=0.$

II.

The uniqueness of a possible fixed point.

Firstly, we prove I. Indeed, if $\widehat{\xi }$ is a fixed point of $\mathcal{T}$ and if $d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)>0,$ then, putting $\xi =\zeta =\widehat{\xi }$ in (8), we get

$$\tau +\mathcal{F}\left(d_{ml}\left(\mathcal{T}\widehat{\xi },\mathcal{T}\widehat{\xi }\right)\right)\le \alpha \left(\widehat{\xi },\widehat{\xi }\right)\left(\tau +\mathcal{F}\left(d_{ml}\left(\mathcal{T}\widehat{\xi },\mathcal{T}\widehat{\xi }\right)\right)\right)\le \mathcal{F}\left(\mathcal{M}\left(\widehat{\xi },\widehat{\xi }\right)\right),$$

(10)

where

$$\begin{array}{ccc}\hfill \mathcal{M}\left(\widehat{\xi },\widehat{\xi }\right)& =& max\left\{d_{ml}\left(\widehat{\xi },\widehat{\xi }\right),d_{ml}\left(\widehat{\xi },\mathcal{T}\widehat{\xi }\right),d_{ml}\left(\widehat{\xi },\mathcal{T}\widehat{\xi }\right),\frac{d_{ml}\left(\widehat{\xi },\mathcal{T}\widehat{\xi }\right)+d_{ml}\left(\widehat{\xi },\mathcal{T}\widehat{\xi }\right)}{2},\right.\hfill \\ & & \left.\frac{\left[1+d_{ml}\left(\widehat{\xi },\mathcal{T}\widehat{\xi }\right)\right]d_{ml}\left(\widehat{\xi },\mathcal{T}\widehat{\xi }\right)}{1+d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)}\right\}\hfill \end{array}$$

$$=max\left\{d_{ml}\left(\widehat{\xi },\widehat{\xi }\right),d_{ml}\left(\widehat{\xi },\widehat{\xi }\right),d_{ml}\left(\widehat{\xi },\widehat{\xi }\right),d_{ml}\left(\widehat{\xi },\widehat{\xi }\right),\frac{\left[1+d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)\right]d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)}{1+d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)}\right\}=d_{ml}\left(\widehat{\xi },\widehat{\xi }\right).$$

Then, from (10), it follows

$$\tau +\mathcal{F}\left(d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)\right)\le \mathcal{F}\left(d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)\right),$$

which is a contradiction. Hence, the assumption that $d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)>0$ is wrong. We proved claim I.

Now, we shall prove II. Suppose that $\mathcal{T}$ has two distinct fixed point $\widehat{\xi }$ and $\widehat{\zeta }$ in $\mathcal{X}.$ By (I), we get $d\left(\widehat{\xi },\widehat{\xi }\right)=d_{ml}\left(\widehat{\zeta },\widehat{\zeta }\right)=0.$ Since $d_{ml}\left(\widehat{\xi },\widehat{\zeta }\right)=d_{ml}\left(\mathcal{T}\widehat{\xi },\mathcal{T}\widehat{\zeta }\right)>0$ and $\alpha \left(\widehat{\xi },\widehat{\zeta }\right)\ge 1$, according to (8), we get:

$$\tau +\mathcal{F}\left(d_{ml}\left(\mathcal{T}\widehat{\xi },\mathcal{T}\widehat{\zeta }\right)\right)\le \alpha \left(\widehat{\xi },\widehat{\zeta }\right)\left(\tau +\mathcal{F}\left(d_{ml}\left(\mathcal{T}\widehat{\xi },\mathcal{T}\widehat{\zeta }\right)\right)\right)\le \mathcal{F}\left(\mathcal{M}\left(\widehat{\xi },\widehat{\zeta }\right)\right),$$

(11)

where

$$\begin{array}{c}\mathcal{M}\left(\widehat{\xi },\widehat{\zeta }\right)=max\left\{d_{ml}\left(\widehat{\xi },\widehat{\zeta }\right),d_{ml}\left(\widehat{\xi },\widehat{\xi }\right),d_{ml}\left(\widehat{\zeta },\widehat{\zeta }\right),\frac{d_{ml}\left(\widehat{\xi },\widehat{\zeta }\right)+d_{ml}\left(\widehat{\zeta },\widehat{\xi }\right)}{2},\frac{\left[1+d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)\right]d_{ml}\left(\widehat{\zeta },\widehat{\zeta }\right)}{1+d_{ml}\left(\widehat{\xi },\widehat{\zeta }\right)}\right\}\\ =max\left\{d_{ml}\left(\widehat{\xi },\widehat{\zeta }\right),0,0,d_{ml}\left(\widehat{\xi },\widehat{\zeta }\right),\frac{\left[1+0\right]·0}{1+0}\right\}=d_{ml}\left(\widehat{\xi },\widehat{\zeta }\right).\end{array}$$

In other words, taking $\alpha \left(\widehat{\xi },\widehat{\zeta }\right)\ge 1$ into consideration,

$$\tau +\mathcal{F}\left(d_{ml}\left(\widehat{\xi },\widehat{\zeta }\right)\right)\le \mathcal{F}\left(d_{ml}\left(\widehat{\xi },\widehat{\zeta }\right)\right)$$

(12)

is a contradiction. Hence, the uniqueness of fixed point is proved.

In the sequel, we prove the existence of the fixed point of $\mathcal{T}$.

Let ${\xi }_0\in \mathcal{X}$ be such that $\alpha \left({\xi }_0,\mathcal{T}{\xi }_0\right)\ge 1.$ Furthermore, we define the sequence $\left\{{\xi }_n\right\}$ in $\mathcal{X}$ with ${\xi }_{n+1}=\mathcal{T}{\xi }_n$ for all $n\in \mathbb{N}\cup \left\{0\right\}$. If ${\xi }_k={\xi }_{k+1}$ for some $k\in \mathbb{N}\cup \left\{0\right\}$, then by the previous, ${\xi }_k$ is a unique fixed point of $\mathcal{T}$ and the proof of the theorem is finished. Now, let us suppose that ${\xi }_n\ne {\xi }_{n+1}$ for all $n\in \mathbb{N}\cup \left\{0\right\}$. Since $\mathcal{T}\in WA\left(\mathcal{X},\alpha \right)$ and $\alpha \left({\xi }_0,\mathcal{T}{\xi }_0\right)\ge 1$, we have

$$\alpha \left({\xi }_1,{\xi }_2\right)=\alpha \left(\mathcal{T}{\xi }_0,\mathcal{T}\mathcal{T}{\xi }_0\right)\ge 1,\alpha \left({\xi }_2,{\xi }_3\right)=\alpha \left(\mathcal{T}{\xi }_1,\mathcal{T}\mathcal{T}{\xi }_1\right)\ge 1.$$

Using this process again, we get $\alpha \left({\xi }_n,{\xi }_{n+1}\right)\ge 1$.

Because $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is a triangular $\left(\alpha ,\mathcal{F}\right)$-contraction mapping with $\alpha \left(\mathcal{T}{\xi }_{n-1},\mathcal{T}\mathcal{T}{\xi }_{n-1}\right)=\alpha \left({\xi }_n,{\xi }_{n+1}\right)\ge 1$, we have according to Lemma 1:

$$0<\tau +\mathcal{F}\left(d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)\right)$$

$$\le \alpha \left({\xi }_n,{\xi }_{n+1}\right)\left(\tau +\mathcal{F}\left(d_{ml}\left(\mathcal{T}{\xi }_{n-1},\mathcal{T}{\xi }_n\right)\right)\right)\le \mathcal{F}\left(\mathcal{M}\left({\xi }_{n-1},{\xi }_n\right)\right),$$

(13)

where

$$\begin{array}{ccc}& & \mathcal{M}\left({\xi }_{n-1},{\xi }_n\right)\hfill \\ & =& max\left\{d_{ml}\left({\xi }_{n-1},{\xi }_n\right),d_{ml}\left({\xi }_{n-1},\mathcal{T}{\xi }_{n-1}\right),d_{ml}\left({\xi }_n,\mathcal{T}{\xi }_n\right),\frac{d_{ml}\left({\xi }_{n-1},\mathcal{T}{\xi }_n\right)+d_{ml}\left(\mathcal{T}{\xi }_{n-1},{\xi }_n\right)}{2},\right.\hfill \\ & & \left.\frac{\left[1+d_{ml}\left({\xi }_{n-1},\mathcal{T}{\xi }_{n-1}\right)\right]d_{ml}\left({\xi }_n,\mathcal{T}{\xi }_n\right)}{1+d_{ml}\left({\xi }_{n-1},{\xi }_n\right)}\right\}\hfill \end{array}$$

$$=max\left\{d_{ml}\left({\xi }_{n-1},{\xi }_n\right),d_{ml}\left({\xi }_n,{\xi }_{n+1}\right),\frac{d_{ml}\left({\xi }_{n-1},{\xi }_{n+1}\right)+d_{ml}\left({\xi }_n,{\xi }_n\right)}{2},d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)\right\}$$

$$\le max\left\{d_{ml}\left({\xi }_{n-1},{\xi }_n\right),d_{ml}\left({\xi }_n,{\xi }_{n+1}\right),\frac{d_{ml}\left({\xi }_{n-1},{\xi }_n\right)+d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)-d_{ml}\left({\xi }_n,{\xi }_n\right)+d_{ml}\left({\xi }_n,{\xi }_n\right)}{2}\right\}$$

$$\begin{array}{ccc}& =& max\left\{d_{ml}\left({\xi }_{n-1},{\xi }_n\right),d_{ml}\left({\xi }_n,{\xi }_{n+1}\right),\frac{d_{ml}\left({\xi }_{n-1},{\xi }_n\right)+d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)}{2}\right\}\hfill \\ & \le & max\left\{d_{ml}\left({\xi }_{n-1},{\xi }_n\right),d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)\right\}.\hfill \end{array}$$

If $max\left\{d_{ml}\left({\xi }_{n-1},{\xi }_n\right),d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)\right\}=d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)$, then a contradiction follows from

$$0<\tau +\mathcal{F}\left(d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)\right)\le \mathcal{F}\left(d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)\right).$$

(14)

Thus, we conclude that $max\left\{d_{ml}\left({\xi }_{n-1},{\xi }_n\right),d_{ml}\left({\xi }_n,x_{n+1}\right)\right\}=$$d\left({\xi }_{n-1},{\xi }_n\right)$ for all $n\in \mathbb{N}$. Therefore, since $\alpha \left({\xi }_n,{\xi }_{n+1}\right)\ge 1$, we have

$$\tau +\mathcal{F}\left(d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)\right)<\mathcal{F}\left(d_{ml}\left({\xi }_{n-1},{\xi }_n\right)\right),$$

where from one can conclude that $d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)<d_{ml}\left({\xi }_{n-1},{\xi }_n\right)$ for all $n\in \mathbb{N}$. This further means that there exists ${lim}_{n\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_{n+1}\right)=\overline{d_{ml}}\ge 0$. If $\overline{d_{ml}}>0$, we obtain a contradiction since by ($\mathcal{F}1$), it follows:

$$\tau +\mathcal{F}\left(\overline{d_{ml}}+0\right)\le \mathcal{F}\left(\overline{d_{ml}}+0\right),$$

where $\mathcal{F}\left(\overline{d_{ml}}+0\right)={lim}_{n\to +\infty }\mathcal{F}\left(d_{ml}\left({\xi }_n,{\xi }_{n+1}\right)\right)$. We use the fact that strictly increasing function $\mathcal{F}:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ has a left and right limit in every point from $\left(0,+\infty \right)$. Hence, we obtain that ${lim}_{n\to +\infty }{d}_{ml}\left({\xi }_n,{\xi }_{n+1}\right)=0$. Now, we prove that the sequence ${\left\{{\xi }_n\right\}}_{n\in \mathbb{N}\cup \left\{0\right\}}$ is a $d_{ml}-$ Cauchy sequence by supposing the contrary. When we put $\xi ={\xi }_{m\left(k\right)},\zeta ={\xi }_{n\left(k\right)}$ in (8), we get

$$\alpha \left({\xi }_{m\left(k\right)},{\xi }_{n\left(k\right)}\right)\left(\tau +\mathcal{F}\left(d_{ml}\left({\xi }_{m\left(k\right)+1},{\xi }_{n\left(k\right)+1}\right)\right)\right)\le \mathcal{F}\left(\mathcal{M}\left({\xi }_{m\left(k\right)},{\xi }_{n\left(k\right)}\right)\right),$$

(15)

where

$$\begin{array}{ccc}\hfill \mathcal{M}\left({\xi }_{m\left(k\right)},{\xi }_{n\left(k\right)}\right)& =& max\left\{d_{ml}\left({\xi }_{m\left(k\right)},{\xi }_{n\left(k\right)}\right),d_{ml}\left({\xi }_{m\left(k\right)},{\xi }_{m\left(k\right)+1}\right),d_{ml}\left({\xi }_{n\left(k\right)},{\xi }_{n\left(k\right)+1}\right),\right.\hfill \\ & & \frac{d_{ml}\left({\xi }_{m\left(k\right)},{\xi }_{n\left(k\right)+1}\right)+d_{ml}\left({\xi }_{n\left(k\right)},{\xi }_{m\left(k\right)+1}\right)}{2},\hfill \end{array}$$

$$\left.\frac{\left[1+d_{ml}\left({\xi }_{m\left(k\right)},{\xi }_{m\left(k\right)+1}\right)\right]d_{ml}\left({\xi }_{n\left(k\right)},{\xi }_{n\left(k\right)+1}\right)}{1+d_{ml}\left({\xi }_{m\left(k\right)},{\xi }_{n\left(k\right)}\right)}\right\}\to max\left\{\epsilon ,0,0,\frac{\epsilon +\epsilon }{2},\frac{\left[1+0\right]·0}{1+\epsilon }\right\}=\epsilon .$$

Since $\alpha \left({\xi }_{m\left(k\right)},{\xi }_{n\left(k\right)}\right)\ge 1$ from the previous inequality, we get

$$\tau +\mathcal{F}\left(d_{ml}\left({\xi }_{m\left(k\right)+1},{\xi }_{n\left(k\right)+1}\right)\right)<\mathcal{F}\left(\mathcal{M}\left({\xi }_{m\left(k\right)},{\xi }_{n\left(k\right)}\right)\right),$$

(16)

that is,

$$\tau +\mathcal{F}\left(\epsilon +0\right)\le \mathcal{F}\left(\epsilon +0\right).$$

(17)

We obtain the contradiction, which means that the sequence ${\left\{{\xi }_n\right\}}_{n\in \mathbb{N}\cup \left\{0\right\}}$ is a $0-d_{ml}-$ Cauchy. This means that there exists a unique (by Remark 2) point $\widehat{\xi }\in \mathcal{X}$ such that

$$d_{ml}\left(\widehat{\xi },\widehat{\xi }\right)=\underset{n\to +\infty }{lim}{d}_{ml}\left({\xi }_n,\widehat{\xi }\right)=\underset{n,m\to +\infty }{lim}{d}_{ml}\left({\xi }_n,{\xi }_m\right)=0.$$

(18)

Since the mapping $\mathcal{T}$ is $d_{ml}-$ continuous, we get that ${lim}_{n\to +\infty }{d}_{ml}\left(\mathcal{T}{\xi }_n,\mathcal{T}\widehat{\xi }\right)=d_{ml}\left(\mathcal{T}\widehat{\xi },\mathcal{T}\widehat{\xi }\right)$, i.e., ${lim}_{n\to +\infty }{d}_{ml}\left({\xi }_{n+1},\mathcal{T}\widehat{\xi }\right)=d_{ml}\left(\mathcal{T}\widehat{\xi },\mathcal{T}\widehat{\xi }\right)$. According to Remark 2, it follows that $\mathcal{T}\widehat{\xi }=\widehat{\xi }$, that is, $\widehat{\xi }$ is a fixed point of $\mathcal{T}.$  □

Remark 3.

The following results are immediate corollaries of Theorem 2. Indeed, replacing $\mathcal{M}\left(\xi ,\zeta \right)$ in (8) with one of the following sets:

$$max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)\right\},$$

$$max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right),\frac{d_{ml}\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\zeta ,\mathcal{T}\xi \right)}{2}\right\},$$

$$\mathrm{and}max\left\{d_{ml}\left(\xi ,\zeta \right),\frac{d_{ml}\left(\xi ,\mathcal{T}\xi \right)+d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)}{2},\frac{d_{ml}\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\zeta ,\mathcal{T}\xi \right)}{2}\right\},$$

we get that Theorem 2 also holds true.

Immediate consequences of Theorem 2 are the following new contractive conditions that compliment the ones given in [23,35].

Corollary 1.

Let $\left(\mathcal{X},d_{ml}\right)$ be a $0-d_{ml}-$ complete $0-d_{ml}-$ metric-like space and ${\alpha }_i:{\mathcal{X}}^2\to [0,+\infty )$. Assume that a mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is a triangular $\left({\alpha }_i,\mathcal{F}\right)$- contraction where $\mathcal{F}:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ is the strictly increasing mapping. Suppose further that the following conditions are satisfied:

(i) 

$\mathcal{T}\in WA\left(\mathcal{X},{\alpha }_i\right)$;

(ii) 

There exists ${\xi }_0\in \mathcal{X}$ such that ${\alpha }_i\left({\xi }_0,\mathcal{T}{\xi }_0\right)\ge 1,i=\overline{1,6};$

(iii) 

$\mathcal{T}$ is $d_{ml}-$ continuous.

In addition, suppose that there exist ${\tau }_i>0,$$i=\overline{1,6}$ and, for all $\xi ,\zeta \in \mathcal{X}$ with $d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)>0$ and ${\alpha }_i\left(\xi ,\zeta \right)\ge 1,i=\overline{1,6}$, the following inequalities hold true:

$${\alpha }_1\left(\xi ,\zeta \right)\left({\tau }_1+d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)\le \mathcal{M}\left(\xi ,\zeta \right)$$

$${\alpha }_2\left(\xi ,\zeta \right)\left({\tau }_2+exp\left(d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)\right)\le exp\left(\mathcal{M}\left(\xi ,\zeta \right)\right)$$

$${\alpha }_3\left(\xi ,\zeta \right)\left({\tau }_3-\frac{1}{d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)}\right)\le -\frac{1}{\mathcal{M}\left(\xi ,\zeta \right)}$$

$${\alpha }_4\left(\xi ,\zeta \right)\left({\tau }_4-\frac{1}{d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)}+d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)\le -\frac{1}{\mathcal{M}\left(\xi ,\zeta \right)}+\mathcal{M}\left(\xi ,\zeta \right)$$

$${\alpha }_5\left(\xi ,\zeta \right)\left({\tau }_5+\frac{1}{1-exp\left(d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)}\right)\le \frac{1}{1-exp\left(\mathcal{M}\left(\xi ,\zeta \right)\right)}$$

$${\alpha }_6\left(\xi ,\zeta \right)\left({\tau }_6+\frac{1}{exp\left(-d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)-exp\left(d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)}\right)\le \frac{1}{exp\left(-\mathcal{M}\left(\xi ,\zeta \right)\right)-exp\left(\mathcal{M}\left(\xi ,\zeta \right)\right)}$$

where $\mathcal{M}\left(\xi ,\zeta \right)$ is one of the following sets:

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right),\frac{d_{ml}\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\zeta ,\mathcal{T}\xi \right)}{2},\frac{\left[1+d_{ml}\left(\xi ,\mathcal{T}\xi \right)\right]d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)}{1+d_{ml}\left(\xi ,\zeta \right)}\right\}$$

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right),\frac{d_{ml}\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\zeta ,\mathcal{T}\xi \right)}{2}\right\}$$

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),\frac{d_{ml}\left(\xi ,\mathcal{T}\xi \right)+d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)}{2},\frac{d_{ml}\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\zeta ,\mathcal{T}\xi \right)}{2}\right\}$$

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)\right\}$$

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right)\right\}=d_{ml}\left(\xi ,\zeta \right).$$

Then, in each of these cases, $\mathcal{T}$ has a unique fixed point in $\mathcal{X}$.

Proof. 

If we put ${\alpha }_i\left(\xi ,\zeta \right)=\alpha \left(\xi ,\zeta \right)$, $i=\overline{1,6}$ and $\mathcal{F}\left(\iota \right)=\iota$, $\mathcal{F}\left(\iota \right)=exp\left(\iota \right)$, $\mathcal{F}\left(\iota \right)=-\frac{1}{\iota },\mathcal{F}\left(\iota \right)=-\frac{1}{\iota }+\iota$, $\mathcal{F}\left(\iota \right)=\frac{1}{1-exp\left(\iota \right)}$, $\mathcal{F}\left(\iota \right)=\frac{1}{exp\left(-\iota \right)-exp\left(\iota \right)}$ in Theorem 2, respectively, then every of the functions $\iota \mapsto \mathcal{F}\left(\iota \right)$ is strictly increasing on $\left(0,+\infty \right)$, and the result follows according to Theorem 2.  □

Remark 4.

Putting ${\alpha }_i\left(\xi ,\zeta \right)=1$ for all $\xi ,\zeta \in \mathcal{X},i=\overline{1,6}$ in the previous corollary, we get the following six new contractive conditions:

$${\tau }_1+d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\le \mathcal{M}\left(\xi ,\zeta \right)$$

$${\tau }_2+exp\left(d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)\le exp\left(\mathcal{M}\left(\xi ,\zeta \right)\right)$$

$${\tau }_3-\frac{1}{d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)}\le -\frac{1}{\mathcal{M}\left(\xi ,\zeta \right)}$$

$${\tau }_4-\frac{1}{d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)}+d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\le -\frac{1}{\mathcal{M}\left(\xi ,\zeta \right)}+\mathcal{M}\left(\xi ,\zeta \right)$$

$${\tau }_5+\frac{1}{1-exp\left(d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)}\le \frac{1}{1-exp\left(\mathcal{M}\left(\xi ,\zeta \right)\right)}$$

$${\tau }_6+\frac{1}{exp\left(-d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)-exp\left(d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)}\le \frac{1}{exp\left(-\mathcal{M}\left(\xi ,\zeta \right)\right)-exp\left(\mathcal{M}\left(\xi ,\zeta \right)\right)}$$

where $\mathcal{M}\left(\xi ,\zeta \right)$ is one of the following sets:

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right),\frac{d\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\zeta ,\mathcal{T}\xi \right)}{2},\frac{\left[1+d_{ml}\left(\xi ,\mathcal{T}\xi \right)\right]d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)}{1+d_{ml}\left(\xi ,\zeta \right)}\right\}$$

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right),\frac{d_{ml}\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\zeta ,\mathcal{T}\xi \right)}{2}\right\}$$

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),\frac{d_{ml}\left(\xi ,\mathcal{T}\xi \right)+d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)}{2},\frac{d_{ml}\left(\xi ,\mathcal{T}\zeta \right)+d_{ml}\left(\zeta ,\mathcal{T}\xi \right)}{2}\right\}$$

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right)\right\}$$

$$\mathcal{M}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right)\right\}=d_{ml}\left(\xi ,\zeta \right).$$

In every one of these cases, $\mathcal{T}$ has a unique fixed point in $\mathcal{X}.$ The result can simply be obtained by putting ${\alpha }_i\left(\xi ,\zeta \right)=1,i=\overline{1,6}$ and $\mathcal{F}\left(\iota \right)=\iota ,\mathcal{F}\left(\iota \right)=exp\left(\iota \right),\mathcal{F}\left(\iota \right)=-\frac{1}{\iota },\mathcal{F}\left(\iota \right)=-\frac{1}{\iota }+\iota$, $\mathcal{F}\left(\iota \right)=\frac{1}{1-exp\left(\iota \right)},\mathcal{F}\left(\iota \right)=\frac{1}{exp\left(-\iota \right)-exp\left(\iota \right)}$ in Theorem 2.

In [22], Ćirić introduced one of the most generalized contractive conditions (so-called quasicontraction) in the context of metric spaces as follows:

Definition 12.

The self-mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ on metric space $\left(\mathcal{X},d\right)$ is called quasicontraction (in the sense of Ćirić) if there exists $\lambda \in [0,1)$ such that, for all $\xi ,\zeta \in \mathcal{X}$ holds true

$$d\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\le \lambda max\left\{d\left(\xi ,\zeta \right),d\left(\xi ,\mathcal{T}\xi \right),d\left(\zeta ,\mathcal{T}\zeta \right),d\left(\xi ,\mathcal{T}\zeta \right),d\left(\zeta ,\mathcal{T}\xi \right)\right\}.$$

(19)

In [22], Ćirić proved the following result:

Theorem 3.

Each quasicontraction $\mathcal{T}$ on a complete metric space $\left(\mathcal{X},d\right)$ has a unique fixed point (say) η. Moreover, for all $\xi \in \mathcal{X}$, the sequence ${\left\{{\mathcal{T}}^n\xi \right\}}_{n=0}^{+\infty },{\mathcal{T}}^0\xi =\xi$ converges to the fixed point η as $n\to +\infty$.

Finally, we formulate the following notion and an open question:

Definition 13.

Let $\left(\mathcal{X},d_{ml}\right)$ be a metric-like space and $\alpha :{\mathcal{X}}^2\to [0,+\infty )$. A mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is said to be a triangular $\left(\alpha ,\mathcal{F}\right)$-contraction mapping of Ćirić type, if there exists $\tau >0$ such that, for all $\xi ,\zeta \in \mathcal{X}$ with $d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)>0$ and $\alpha \left(\xi ,\zeta \right)\ge 1$ holds true:

$$\alpha \left(\xi ,\zeta \right)\left(\tau +\mathcal{F}\left(d_{ml}\left(\mathcal{T}\xi ,\mathcal{T}\zeta \right)\right)\right)\le \mathcal{F}\left(\mathcal{N}\left(\xi ,\zeta \right)\right),$$

(20)

where

$$\mathcal{N}\left(\xi ,\zeta \right)=max\left\{d_{ml}\left(\xi ,\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\xi \right),d_{ml}\left(\zeta ,\mathcal{T}\zeta \right),d_{ml}\left(\xi ,\mathcal{T}\zeta \right),d_{ml}\left(\zeta ,\mathcal{T}\xi \right)\right\},$$

$\mathcal{F}:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ is strictly increasing function satisfying only ($\mathcal{F}1$).

An open question: Prove or disprove the following claim: each triangular$\left(\alpha ,\mathcal{F}\right)$-contraction mapping$\mathcal{T}:\mathcal{X}\to \mathcal{X}$of Ćirić type defined on $0-d_{ml}-$ complete metric-like space$\left(\mathcal{X},d\right)$has a unique fixed point.