Fixed Point Theorems on Almost (φ,θ)-Contractions in Jleli-Samet Generalized Metric Spaces

Abstract

In this paper, we present extensions of some classic results regarding the existence and uniqueness of fixed points of operators fulfilling generalized contractive conditions defined by sums involving some functions with suitable properties in the setting of Jleli–Samet generalized metric spaces. As a consequence, some known results in the literature are obtained. Examples are provided to prove the usability of our developments.

1. Introduction

Since their introduction, metric spaces have proved to be the adequate environment for the development of a significant number of outcomes in various branches of mathematics, one of them being fixed point theory. Because of their importance both from a theoretical and applied point of view, this direction of study was one of the reasons why the setting of metric spaces has been generalized in various ways. The change of the first of its axioms was used by Hitzler and Seda [1] to introduce the dislocated metric spaces. The drop of the symmetry of the metric function was another way to extend this concept, by Wilson [2]. Generalizations related to the triangle inequality have produced metric-type spaces; in this respect, we refer here, for example, to b-metric spaces introduced by Czerwik [3] and Bakhtin [4], extended b-metric spaces by Kamran et al. [5], rectangular metric spaces by Branciari [6], or modular metric spaces by Nakano [7] and then Musielak and Orlicz [8]. Various fixed-point results have been developed in these metric structures. Jungck and Rhoades [9] studied weakly compatible maps in relation to dislocated metric spaces, while Hitzler [10] connected them to real-world applications. In [11], comparison functions have been adapted by Samreen et al. to extended b-metric spaces to design generalized contractive conditions. Common fixed points of nonlinear contractions defined by means of comparison functions are developed in b-metric spaces by Shatanawi [12], while Ali et al. [13] used this setting to solve Volterra integral inclusions. Budhia et al. [14] considered rectangular metric spaces to develop solutions to fractional-order functional differential equations. Banach principle type results in the framework of modular metric spaces were presented by Martínez-Moreno et al. [15], while Okeke et al. [16] used the same setting to study Reich and Geraghty type generalized contractions. In [17], Jleli and Samet came up with the idea of a new generalization of classic metric spaces, which strictly includes some of the extensions previously discovered. In the same work, they prove contractive principles in this setting. In [18], Karapınar et al. used a binary relation to develop some existence results in this generalized context, which was also considered by Senapati et al. [19], to extend Wardowski implicit contractive conditions. Altun and Samet [20] studied pseudo-Picard operators in the framework of Jleli–Samet metric spaces.

Our work aims to introduce existence and unique results regarding fixed points associated with generalized contractive conditions defined by means of comparison functions and continuous mappings of four variables, which additionally satisfy suitable conditions. The paper is organized as follows. Section 2 is dedicated to preliminary issues. Section 3 contains fixed-point theorems related to $(\phi ,\theta )$-contractions defined by means related to classic contractions. Section 4 has in view $(\phi ,\theta )$-contractions in the sense of Kannan, while Section 5 links them to the Ćirić type.

2. Preliminaries

In this part, we recollect some definitions and remarks needed to develop our original results. The setting chosen here is that of Jleli–Samet generalized spaces [17].

Definition 1

([17]). Let L be a nonempty set and consider the function $D:L\times L\to [0,\infty ]$. For every point $x_{*}\in L$, we define the set of all sequences ${\left\{x_n\right\}}_n\subset L$ that converge to $x_{*}$ in the “metric” D (or are D-convergent):

$$\mathcal{C}(D,L,x_{*})=\left\{{\left\{x_n\right\}}_n\subset L:\underset{n\to \infty }{lim}D(x_n,x_{*})=0\right\}.$$

We say that D is a $JS$-metric on L if the following axioms are satisfied:

$(D1)$ For all x, $y\in L$, the next implication holds true

$$D(x,y)=0⟹x=y.$$

$(D2)$ For all x, $y\in L$, the symmetry condition $D(x,y)=D(y,x)$ holds.

$(D3)$ There exists some constant $C>0$ such that for every x, $y\in L$ and for every sequence such that ${\left\{x_n\right\}}_n$$\in \mathcal{C}(D,L,x)$, the following inequality is accomplished

$$D(x,y)\le C\underset{n\to \infty }{lim\; sup}D(x_n,y).$$

The pair $(L,D)$ is called a JS-metric space. As proved in [17], this class of generalized metric spaces comprises those of classic metric spaces, b-metric spaces, dislocated metric spaces, or modular spaces with the Fatou property. On the other hand, it is wider than the reunion of these classes previously mentioned; in this respect, we insert the next example inspired by [18].

Example 1.

Let $X=\{0,p,q\}$ with p, $q\in {\mathbb{R}}^{*}$, $p\ne q$ and $D:X\times X\to [0,\infty ]$ defined as:

$$\left\{\begin{array}{c}D(0,0)=0;\hfill \\ D(0,p)=D(p,0)=D(p,p)=p;\hfill \\ D(0,q)=D(q,0)=D(p,q)=D(q,p)=q;\hfill \\ D(q,q)=\infty .\hfill \end{array}\right.$$

It is not hard to see that $(D1)$ and $(D2)$ are fulfilled by the definition of the metric D. If one searches for convergent sequences in this space, the answer is given by the stationary sequences at 0 from a specific rank $k_0$. Then choosing $C=1$ in the definition of the JS-metric spaces, it is clear that the inequality from $(D3)$ turns into equality, and it is true in all cases, for all x, $y\in X$. Moreover, the definition of D shows that it does not belong to the classes of metrics mentioned previously.

Remark 1.

It is easy to observe that every convergent sequence in $(L,D)$ has a unique limit. Unfortunately, the Jleli–Samet metrics do not necessarily fulfill the property of continuity since b-metric spaces are a subclass of it.

Regarding the convergence of sequences in these spaces and also the property of being D-Cauchy sequences, we recall the next two definitions.

Definition 2

([17]). Let $(L,D)$ be a JS-metric space and ${\left\{x_n\right\}}_n$ a sequence in L. We say that ${\left\{x_n\right\}}_n$ is a D-Cauchy sequence if

$$\underset{m,n\to \infty }{lim}D(x_m,x_n)=0.$$

$\left\{x_n\right\}$ is convergent to $x\in L$ if ${lim}_{n\to \infty }D(x_n,x)=0$.

It is obvious that any convergent sequence in the JS-metric space $(L,D)$ is D-Cauchy, but the converse is not true. A JS-space in which each D-Cauchy sequence in L is D-convergent to an element L is called D-complete.

To overcome the difficulties brought on by the lack of the triangle axiom, we shall focus on some hypotheses formulated by means of the sets:

$${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})=sup(\{D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^m{x}_{*}):n,m\in \mathbb{N},n,m\ge n_0\})$$

and

$$\delta (D,\mathsf{\Lambda },x_{*})=sup(\{D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^m{x}_{*}):n,m\in \mathbb{N}\}),$$

where $n_0\in \mathbb{N}$.

Moreover, a binary relation on L is needed, that is, a nonempty subset of the Cartesian product $L\times L$. Let us denote it by $\mathbf{B}$. For simplicity, we shall write $x\mathbf{B}y$, each time when $(x,y)\in \mathbf{B}$. It is known that such a relation is called a partial order if it is reflexive, transitive, and antisymmetric. If ⪯ is a partial order on L, then let $E_{⪯}=\{(x,y)\in :L\times L:x⪯y\}$. A binary relation $\mathbf{B}$, which is reflexive and transitive, is called preorder.

The notion of regularity in these generalized metric spaces is defined next.

Definition 3

([17]). We say that the pair $(L,D)$ is D-regular if the following condition holds: for every sequence ${\left\{x_n\right\}}_n\subset L$ satisfying $(x_n,x_{n+1})\in E_{⪯}$, for every n large enough, if ${\left\{x_n\right\}}_n$ is D-convergent to $x\in L$, then there exists a subsequence ${\left\{x_{n_q}\right\}}_q$ of ${\left\{x_n\right\}}_n$ such that $(x_{n_q},x)\in E_{⪯}$, for every q large enough.

Definition 4

([18]). For a given preorder $\mathbf{B}$, a sequence ${\left\{x_n\right\}}_n\subset L$ is $\mathbf{B}$-nondecreasing if $x_n\mathbf{B}{x}_{n+1}$ for all $n\in \mathbb{N}$.

Closely related to the preorder $\mathbf{B}$ are the adequate types of monotone.

Definition 5

([18]). The Jleli–Samet metric space $(L,D)$ is $\mathbf{B}$-nondecreasing-regular, where $\mathbf{B}$ is a preorder, if for every ${\left\{x_n\right\}}_n\in \mathcal{C}(D,L,z)$ that is $\mathbf{B}$-nondecreasing it happens that $x_n\mathbf{B}z$, for all $n\in \mathbb{N}$.

Definition 6

Let $(L,D)$ be a Jleli–Samet space endowed with a binary relation $\mathbf{B}$, which is also a preorder, and $\mathsf{\Lambda }:L\to L$. The operator Λ is called $\mathbf{B}$-nondecreasing if $x\mathbf{B}y$ implies $\mathsf{\Lambda }x\mathbf{B}\mathsf{\Lambda }y$ for all x, $y\in L$.

Obviously, each regular Jleli–Samet space is also $\mathbf{B}$-nondecreasing-regular, whatever the binary relation $\mathbf{B}$ (which is supposed to be a partial order since the notion of regularity is defined by such relations), but the converse is no longer true.

Definition 7

([18]). The Jleli–Samet metric space $(L,D)$ is $\mathbf{B}$-nondecreasing-complete if every ${\left\{x_n\right\}}_n$ which is D-Cauchy and $\mathbf{B}$-nondecreasing is D-convergent in L.

Every Jleli–Samet metric space that is complete is also $\mathbf{B}$-nondecreasing-complete, but the converse is not true, as the next example shows.

Example 2.

Let us consider $L=(0,1)$ endowed with the Euclidean metric $D(x,y)=\left|x-y\right|$, for all x, $y\in L$, which is clearly a Jleli–Samet space. A binary relation $\mathbf{B}$ to L can be defined as follows: $x\mathbf{B}y$ if and only if $\frac{1}{10}\le x\le y\le \frac{9}{10}$. A sequence ${\left\{x_n\right\}}_n$ that is $\mathbf{B}$-nondecreasing and D-Cauchy will be D-convergent to a limit l, $\frac{1}{10}\le l\le \frac{9}{10}$. The space is $\mathbf{B}$-nondecreasing-complete, but $(L,D)$ is not complete with respect to the Jleli–Samet metric.

To formulate the main outcomes of this work, we shall introduce some classes of functions that play a key role in the development of our results.

Definition 8

([21]). A map $\phi :[0,+\infty )\to [0,+\infty )$ is called a comparison function if:

• φ is monotone nondecreasing;
• ${lim}_{n\to +\infty }{\phi }^n(t)=0$, where the upper index n refers to the number of compositions of φ with itself.

Among the properties of comparison functions, we recall that $\phi (t)<t$, for all $t>0$, and $\phi (0)=0$. In addition to this, it is known that $\phi$ is continuous at $t=0$. More information about these functions can be found in [21].

Inspired by [22,23], we introduce $\mathsf{\Theta }$, the set of all continuous functions $\theta :{[0,+\infty )}^4\to [0,+\infty )$, such that

$$\left\{\begin{array}{cc}\theta (0,a,b,g)=0,\hfill & \mathrm{for}\mathrm{all}a,b,g\in [0,+\infty ),\hfill \\ \theta (a,b,0,g)=0,\hfill & \mathrm{for}\mathrm{all}a,b,g\in [0,+\infty ).\hfill \end{array}\right.$$

A technical lemma regarding the family $\mathsf{\Theta }$ can be proved.

Lemma 1.

Let $\left\{t_n\right\}$ be a sequence of positive numbers with ${lim}_{n\to \infty }{t}_n=0$ and let $\left\{a_{m,n}\right\}$ be a bounded sequence of positive numbers depending on two natural parameters. Then, the following relation is satisfied:

$$\underset{k\to \infty }{lim}\underset{m,n\ge k}{sup}\theta (a_{m,n},a_{n,m},t_n,t_m)=0,$$

for all $\theta \in \mathsf{\Theta }$.

3. Generalizations in the Sense of the Contraction Principle

In the following paragraphs, some preliminary issues will be presented to develop our results using the Jleli–Samet context. The generalized contraction we use is inspired by the work of Samet [17].

Lemma 2.

Let $(L,D)$ be a Jleli–Samet metric space and let $\mathsf{\Lambda }:L\to L$ be self-mapping. Presume that the next conditions are fulfilled:

(i)

There is $x_{*}\in L$, and $n_0\in \mathbb{N}$ for which we have ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$;

(ii)

There is a comparison function φ, and $\theta \in \mathsf{\Theta }$, such that:

$$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (D(x,y))+\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y))$$

(1)

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})=\{{\mathsf{\Lambda }}^n{x}_{*}:n\in \mathbb{N}\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set

$${\mathsf{\Omega }}_{n_0}=\{D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^m{x}_{*}):m,n\in \mathbb{N},m,n\ge n_0\};$$

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}.$

Then, the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-Cauchy.

Proof. 

First of all, we have to observe that for all $n\ge n_0$, we have:

$$\begin{array}{cc}\hfill & D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})\le \phi (D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}))\hfill \\ & +\theta (D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}),D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}))\hfill \\ \hfill & =\phi (D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})).\hfill \end{array}$$

Inductively, for all $n\ge n_0$, it follows:

$$\begin{array}{cc}\hfill D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})\le & {\phi }^{n-n_0}(D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}))\hfill \\ \hfill & \le {\phi }^{n-n_0}({\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})).\hfill \end{array}$$

It can be observed that:

$$\underset{n\to \infty }{lim}D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})=0.$$

From the definition, the next inequalities hold for all $k\ge n_0$,

$${\delta }_{k+1}(D,\mathsf{\Lambda },x_{*})\le {\delta }_k(D,\mathsf{\Lambda },x_{*})\le {\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty .$$

Let us take $m,n\ge k+1$, $m^{\prime }=m-1$, and $n^{\prime }=n-1$, with $m^{\prime }\ge n^{\prime }\ge k$. We are going to insert the information into our contractive inequality, and we are led to

$$\begin{array}{cc}\hfill D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})& =D({\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})\hfill \\ \hfill & \le \phi (D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }}{x}_{*}))+\theta (D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),\hfill \\ \hfill & D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})).\hfill \end{array}$$

Note that all numbers $D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*})$, $D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})$, $D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*})$, and $D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})$ are in ${\mathsf{\Omega }}_k$.

By the properties of $\phi$, we have:

$$\begin{array}{c}D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})\le \phi ({\delta }_k(D,\mathsf{\Lambda },x_{*}))+\theta (D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})).\hfill \end{array}$$

Our next step is to take the supremum, obtaining

$$\begin{array}{cc}\underset{m^{\prime },n^{\prime }\ge k}{sup}D({\mathsf{\Lambda }}^m{x}_{*},\hfill & {\mathsf{\Lambda }}^n{x}_{*})\le \underset{k}{sup}\phi ({\delta }_k(D,\mathsf{\Lambda },x_{*}))+\underset{m^{\prime },n^{\prime }\ge k}{sup}\theta (D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),\\ \hfill & D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})).\hfill \end{array}$$

Taking into account Lemma 1, it follows that:

$$\underset{k\to \infty }{lim}\underset{m^{\prime },n^{\prime }\ge k}{sup}\theta (a_{m^{\prime },n^{\prime }},a_{n^{\prime },m^{\prime }},t_{n^{\prime }},t_{m^{\prime }})=0.$$

Passing at the limit over k, the above inequality leads to:

$$\underset{k\to \infty }{lim}{\delta }_k=\underset{k\to \infty }{lim}(\underset{m,n\ge k}{sup}D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}))\le \underset{k\to \infty }{lim\; sup}\phi ({\delta }_k(D,\mathsf{\Lambda },x_{*})),$$

where ${\delta }_k={\delta }_k(D,\mathsf{\Lambda },x_{*})$. Since $\left\{{\delta }_k\right\}$ is monotone nonincreasing and lower bounded by 0, it converges to $l\in [0,{\delta }_{n_0}]$. Note that l is an accumulation point for ${\mathsf{\Omega }}_{n_0}$. Having in view the upper semi-continuity of $\phi$, we get that:

$$l\le \underset{k\to \infty }{lim\; sup}\phi ({\delta }_k(D,\mathsf{\Lambda },x_{*}))\le \phi (l).$$

The last inequality implies $\phi (l)=l$, and ${lim}_{k\to \infty }{\delta }_k=0$. We are now able to conclude that

$$\underset{m,n\to \infty }{lim}D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0,$$

therefore, $\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}$ is D-Cauchy. □

To provide a result on the existence of a fixed point of such generalized contractions in these kinds of spaces, the next definition is needed.

Definition 9

([18]). A self map $\mathsf{\Lambda }:L\to L$ is called $\mathbf{B}$-nondecreasing-continuous at $z\in L$ if ${\left\{\mathsf{\Lambda }{x}_n\right\}}_n\in \mathcal{C}(D,L,\mathsf{\Lambda }z)$ for all $\mathbf{B}$-nondecreasing sequences ${\left\{x_n\right\}}_n\in \mathcal{C}(D,L,z)$. This function is $\mathbf{B}$-nondecreasing-continuous if it is $\mathbf{B}$-nondecreasing-continuous at each point of L.

We are now in a position to introduce the first theorem regarding the existence and uniqueness of fixed points associated with the contractive condition previously introduced.

Theorem 1.

Let $(L,D)$ be a Jleli–Samet metric space endowed with a preorder $\mathbf{B}$ such that it is $\mathbf{B}$-nondecreasing-complete and $\mathsf{\Lambda }:L\to L$ a mapping which is $\mathbf{B}$-nondecreasing. Presume that the next conditions are fulfilled:

(i)

There is $x_{*}\in L$ such that $x_{*}\mathbf{B}\mathsf{\Lambda }{x}_{*}$, and $n_0\in \mathbb{N}$ for which we have ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$;

(ii)

There is a comparison function φ, and $\theta \in \mathsf{\Theta }$ such that:

$$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (D(x,y))+\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$;

(v)

Λ is $\mathbf{B}$-nondecreasing-continuous.

Then, the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point ω of Λ, and $D(\omega ,\omega )=0$. If there is another fixed point of Λ, ${\omega }^{\prime }$, then if $D(\omega ,{\omega }^{\prime })<\infty$ and $D({\omega }^{\prime },{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Proof. 

As previously proved, the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is $\mathbf{B}$-nondecreasing and Lemma 2 ensures its D-convergence to some $\omega \in L$, since the space is supposed to be $\mathbf{B}$-nondecreasing-complete. It can be easily proved that ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is $\mathbf{B}$-nondecreasing, and, by the $\mathbf{B}$-nondecreasing-continuity of $\mathsf{\Lambda }$, we obtain:

$${\left\{\mathsf{\Lambda }{x}_n\right\}}_n\stackrel{D}{\to }\mathsf{\Lambda }\omega .$$

In conclusion, $\mathsf{\Lambda }\omega =\omega$, so $\omega$ is a fixed point of $\mathsf{\Lambda }$.

Let us consider another fixed point of $\mathsf{\Lambda }$, denoted by ${\omega }^{\prime }$, and having $D(\omega ,{\omega }^{\prime })<\infty$ and $D({\omega }^{\prime },{\omega }^{\prime })<\infty$. In inequality (1), we obtain:

$$\begin{array}{cc}\hfill D(\omega ,{\omega }^{\prime })=& D(\mathsf{\Lambda }\omega ,\mathsf{\Lambda }{\omega }^{\prime })\hfill \\ \hfill & \le \phi (D(\omega ,{\omega }^{\prime }))+\theta (D({\omega }^{\prime },\mathsf{\Lambda }\omega ),\hfill \\ \hfill & D(\omega ,\mathsf{\Lambda }{\omega }^{\prime }),D(\omega ,\mathsf{\Lambda }\omega ),D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })).\hfill \end{array}$$

If we take into consideration property $(D3)$ from the metric axioms we can say that:

$$D(\omega ,\omega )\le C\underset{n\to \infty }{lim\; sup}D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0,$$

because ${\left\{x_n\right\}}_n\in \mathcal{C}(D,\mathsf{\Lambda },x_{*})$ so $D(\omega ,\omega )=0$, and the previous inequality leads to $D(\omega ,{\omega }^{\prime })\le \phi (D(\omega ,{\omega }^{\prime }))$, that is $D(\omega ,{\omega }^{\prime })=0$, and the uniqueness of the fixed point is, thus, proved. □

Theorem 2.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a self mapping which is $\mathbf{B}$-nondecreasing. Presume that the next conditions are fulfilled:

(i)

There is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$, for some $n_0\in \mathbb{N}$;

(ii)

There is a comparison function φ, and $\theta \in \mathsf{\Theta }$, such that

$$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (D(x,y))+\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. If $D(\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Proof. 

Consider the trivial preorder on L, i.e., $x\mathbf{B}y$, for all x, $y\in L$. Then, $\mathsf{\Lambda }$ becomes $\mathbf{B}$-nondecreasing and the space $(L,D)$ is $\mathbf{B}$-nondecreasing-complete. Since the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is $\mathbf{B}$-nondecreasing, and D-Cauchy, there exists $\omega \in L$ such that ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n\stackrel{D}{\to }\omega$. As $\omega \in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})$, we get that:

$$\begin{array}{cc}\hfill D({\mathsf{\Lambda }}^{n+1}{x}_{*},\mathsf{\Lambda }\omega )=& D(\mathsf{\Lambda }{\mathsf{\Lambda }}^n{x}_{*},\mathsf{\Lambda }\omega )\hfill \\ \hfill & \le \phi (D({\mathsf{\Lambda }}^n{x}_{*},\omega ))+\theta (D(\omega ,{\mathsf{\Lambda }}^{n+1}{x}_{*}),\hfill \\ \hfill & D({\mathsf{\Lambda }}^n{x}_{*},\mathsf{\Lambda }\omega ),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),D(\omega ,\mathsf{\Lambda }\omega )).\hfill \end{array}$$

Taking into account that $D({\mathsf{\Lambda }}^n{x}_{*},\omega )\stackrel{n}{\to }0$ and $\phi$ is continuous at $t=0$, it follows that:

$$\underset{n\to \infty }{lim}\phi (D({\mathsf{\Lambda }}^n{x}_{*},\omega ))=\phi (0)=0,$$

and, as a consequence:

$$\underset{n\to \infty }{lim}\phi (D({\mathsf{\Lambda }}^{n+1}{x}_{*},\mathsf{\Lambda }\omega ))=0.$$

Combining all these pieces, it follows that $D(\omega ,\mathsf{\Lambda }\omega )=0$, and $\mathsf{\Lambda }\omega =\omega$, so $\omega$ is a fixed point for $\mathsf{\Lambda }$. Let us consider another fixed point of $\mathsf{\Lambda }$, denoted by ${\omega }^{\prime }$, endowed with the properties $D(\omega ,{\omega }^{\prime })<\infty$, and $D(\omega ,\omega )=0$. Inequality (1) becomes

$$\begin{array}{cc}\hfill D(\omega ,{\omega }^{\prime })=& D(\mathsf{\Lambda }\omega ,\mathsf{\Lambda }{\omega }^{\prime })\hfill \\ \hfill & \le \phi (D(\omega ,{\omega }^{\prime }))+\theta (D({\omega }^{\prime },\mathsf{\Lambda }\omega ),D(\omega ,\mathsf{\Lambda }{\omega }^{\prime }),\hfill \\ & D(\omega ,\mathsf{\Lambda }\omega ),D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })),\hfill \end{array}$$

which compels $D(\omega ,{\omega }^{\prime })\le \phi (D(\omega ,{\omega }^{\prime }))$, that is $D(\omega ,{\omega }^{\prime })=0$, and $\omega ={\omega }^{\prime }$. □

The next example illustrates the application of these outcomes.

Example 3.

Let us start with the set $L=[0,1]$ endowed with the metric

$$D(x,y)=\left\{\begin{array}{cc}0,\hfill & x=y;\hfill \\ max\{x,y\},\hfill & x\ne y.\hfill \end{array}\right.$$

$(L,D)$ is a complete Jleli–Samet space, on which we define the operator

$$\mathsf{\Lambda }:L\to L,\mathsf{\Lambda }x=x-ln(1+x).$$

Consider now the comparison function

$$\phi :[0,\infty )\to [0,\infty ),\phi (x)=\left\{\begin{array}{cc}{\displaystyle \frac{x^2}{2}-\frac{x^3}{3},}\hfill & {\displaystyle \mathrm{if}x\le \frac{1}{6},}\hfill \\ {\displaystyle \frac{x}{6},}\hfill & {\displaystyle \mathrm{if}x\ge \frac{1}{6},}\hfill \end{array}\right.$$

and$\theta (t,u,v,w)=\frac{3}{2}{v}^{2}t$, and$\theta \in \mathsf{\Theta }$. We are going to show that Λ is a $(\phi ,\theta )$-contraction. For x, $y\in L$, $x>y$, it can be observed that:

$$\begin{array}{ccc}& & \phi (D(x,y))+\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y))\hfill \\ & =& \frac{x^2}{2}-\frac{x^3}{3}+\frac{3}{2}D{(x,\mathsf{\Lambda }x)}^{2}D(y,\mathsf{\Lambda }x)\hfill \\ & =& \frac{x^2}{2}-\frac{x^3}{3}+\frac{3}{2}{x}^2(max\{y,x-ln(1+x)\})\hfill \\ & \ge & \frac{x^2}{2}-\frac{x^3}{3}+\frac{3}{2}{x}^2(x-ln(x+1))\hfill \end{array}$$

Having in mind the inequality

$$\frac{x^2}{2}-\frac{x^3}{3}+\frac{x^4}{4}\ge x-ln(1+x),$$

it is proved that Λ is a $(\phi ,\theta )$-contraction. In conclusion, Λ is a $(\phi ,\theta )$-contraction which is not φ-contraction. Note that Λ is nondecreasing with respect to the usual order of real numbers. All conditions of Theorem 4 are now satisfied, so Λ has a unique fixed point, $x=0$. We emphasize that, for $x=1$ and $y=\frac{1}{2}$, the inequality $D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (D(x,y))$ is not true, so Λ is not a φ-contraction.

Theorems 1 and 2 generalize important results in the literature; in the next lines, we mention some corollaries in this regard.

For $\theta \equiv 0$, the existence of a fixed point for the weak contractions ($\phi$-contractions) is obtained.

Corollary 1.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping which is $\mathbf{B}$-nondecreasing. Suppose that the following conditions hold:

(i)

There exists $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$, for some $n_0\in \mathbb{N}$;

(ii)

There exists a comparison function φ such that:

$$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (D(x,y)),forallx,$$

$y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ D-converges to a fixed point of Λ, ω. If, additionally, $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, with $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

We feature this corollary with an adequate example.

Example 4.

If we take now $L=\left[0,\frac{\sqrt{5}-1}{2}\right]$, we can attach a metric D defined as

$$D(x,y)=\left\{\begin{array}{cc}0,\hfill & x=y;\hfill \\ max\{x,y\},\hfill & x\ne y.\hfill \end{array}\right.$$

It is obvious that D is a metric in the usual sense, and it is a Jleli–Samet metric as well. Moreover, $(L,D)$ is a complete metric space.

We can define now

$$\mathsf{\Lambda }:L\to L,\mathsf{\Lambda }x=x-ln(1+x),$$

and choose the genuine comparison function $\phi :[0,\infty )\to [0,\infty )$, $\phi (x)=\frac{x}{x+1}$. Λ fulfills the hypotheses of Corollary 1, since

$$x-ln(1+x)\le \frac{x}{1+x},forallx\in L,$$

and, for $x>y$, we have

$$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)=x-ln(1+x)\le \frac{x}{x+1}=\phi (D(x,y)).$$

Therefore, Λ has a fixed point.

Considering now $\theta (t,u,v,w)=\tau inf\{t,u,v,w\}$, $\tau$ a positive constant, t, u, v, $w\in [0,\infty )$, in Theorem 1, the next consequence follows.

Corollary 2.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ an $\mathbf{B}$-nondecreasing operator. Suppose that there is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$ and the following assumptions are satisfied:

(i)

There is a comparison function φ such that:

$$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (D(x,y))+\tau inf\{(D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y))\},$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(ii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. Furthermore, if $D(\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

By taking now $\theta (t,u,v,w)=\tau tuvw$, where $\tau >0$ is a constant, t, u, v, $w>0$, the next consequence of Theorem 1 follows.

Corollary 3.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping which is $\mathbf{B}$-nondecreasing. Suppose that there is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$ and the next conditions are accomplished:

(i)

There is φ a comparison function such that:

$$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (D(x,y))+\tau D(y,\mathsf{\Lambda }x)D(x,\mathsf{\Lambda }y)D(x,\mathsf{\Lambda }x)D(y,\mathsf{\Lambda }y),$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$.

(ii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is a sequence D-convergent to a fixed point of Λ, ω. If, in addition, $D(\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

When we consider the comparison function $\phi$ as a contraction, we are led to the result in the following lines.

Corollary 4.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping which is $\mathbf{B}$-nondecreasing. Suppose that there is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$, for some $n_0\in \mathbb{N}$, and that the following conditions hold:

(i)

There is $\theta \in \mathsf{\Theta }$ such that:

$$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \alpha D(x,y)+\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y))$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$, and $\alpha \in [0,1)$.

(ii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. Moreover, if $D(\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Furthermore, we consider $\theta (t,u,v,w)=\tau tuvw$, where $\tau >0$ is a constant, t, u, v, $w>0$.

Corollary 5.

Let $(L,D)$ be a complete Jleli–Samet space and the $\mathbf{B}$-nondecreasing $\mathsf{\Lambda }:L\to L$ mapping. Suppose that there is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$, for some $n_0\in \mathbb{N}$, and that:

(i)

$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \alpha D(x,y)+\tau D(y,\mathsf{\Lambda }x)D(x,\mathsf{\Lambda }y)D(x,\mathsf{\Lambda }x)D(y,\mathsf{\Lambda }y))$, for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$, and $\alpha \in [0,1)$;

(ii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. Furthermore, if $D(\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in L$ is another fixed point of Λ that meets the condition $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Consider now a genuine comparison function $\phi (t)=\frac{t}{t+1}$, $t\in [0,\infty )$, we are led to the result in the next lines.

Corollary 6.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping which is $\mathbf{B}$-nondecreasing. Suppose that there is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$, for some $n_0\in \mathbb{N}$, and that the following conditions hold:

(i)

There is $\theta \in \mathsf{\Theta }$ such that

$${\displaystyle D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \frac{D(x,y)}{D(x,y)+1}+\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y))}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(ii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. Moreover, if $D(\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Considering the high degree of generality brought about by the JS-metric spaces, the above theorems allow us to obtain, as consequences, some known results in the literature for the settings of b-metric spaces, Hitzler–Seda metric spaces, or modular spaces having Fatou’s property.

Corollary 7.

Let $(L,d)$ be a complete b-metric space and $\mathsf{\Lambda }:L\to L$ a self mapping which is $\mathbf{B}$-nondecreasing. Presume that the next conditions are fulfilled:

(i)

There is $x_{*}\in L$ such that ${\delta }_{n_0}(d,\mathsf{\Lambda },x_{*})<\infty$, for some $n_0\in \mathbb{N}$;

(ii)

There is a comparison function φ, and $\theta \in \mathsf{\Theta }$ such that

$$d(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (d(x,y))+\theta (d(y,\mathsf{\Lambda }x),d(x,\mathsf{\Lambda }y),d(x,\mathsf{\Lambda }x),d(y,\mathsf{\Lambda }y)),$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }d({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$d({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is convergent to a fixed point of Λ, ω. If $d(\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $d(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Corollary 8.

Let $(L,D)$ be a complete dislocated space in the sense of Hitzler and Seda and let $\mathsf{\Lambda }:L\to L$ be a self mapping which is $\mathbf{B}$-nondecreasing. Suppose that the next conditions are accomplished:

(i)

There is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$, for some $n_0\in \mathbb{N}$;

(ii)

There is a comparison function φ, and $\theta \in \mathsf{\Theta }$ such that

$$D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (D(x,y))+\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. If $D(\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Corollary 9.

Let $(X_{\rho },\rho )$ be a complete modular space having the Fatou’s property and consider $\mathsf{\Lambda }:X_{\rho }\to X_{\rho }$ a self-mapping which is $\mathbf{B}$-nondecreasing. Suppose that the following assertions are verified:

(i)

There is $x_{*}\in X_{\rho }$ such that ${\delta }_{n_0}(\rho ,\mathsf{\Lambda },x_{*})<\infty$, for some $n_0\in \mathbb{N}$;

(ii)

There is a comparison function φ, and $\theta \in \mathsf{\Theta }$ such that

$$\rho (\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi (\rho (x,y))+\theta (\rho (y,\mathsf{\Lambda }x),\rho (x,\mathsf{\Lambda }y),\rho (x,\mathsf{\Lambda }x),\rho (y,\mathsf{\Lambda }y)),$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in X_{\rho }:{lim}_{n\to \infty }\rho ({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$\rho ({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is ρ-convergent to a fixed point of Λ, ω. If $\rho (\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in X_{\rho }$ is another fixed point of Λ with $\rho (\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

We emphasize that from these three corollaries, interesting outcomes can be obtained by considering various choices for $\phi$, or $\theta$, as seen previously.

4. Generalization in the Sense of Kannan

To develop some new results regarding the existence of fixed points, we aim to combine the idea developed by Kannan, with comparison functions and theta functions. The first result refers to the property of the Picard sequence associated with an adequate generalized contraction of being D-Cauchy.

Lemma 3.

Let $(L,D)$ be a Jleli–Samet metric space and let $\mathsf{\Lambda }:L\to L$ be self-mapping. Presume that the following conditions are accomplished:

(i)

There exists $x_{*}\in L$, and $n_0\in \mathbb{N}$ for which ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$;

(ii)

There is a comparison function φ and $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{ccc}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & \phi \left(\frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{2}\right)\hfill \\ & & +\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then, ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is a D-Cauchy sequence.

Proof. 

First, we prove that:

$$\underset{n\to \infty }{lim}D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})=0.$$

Using the contractive inequality, we get that:

$$\begin{array}{c}\hfill D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})\le \phi \left(\frac{D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})+D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})}{2}\right)\\ \hfill +\theta (D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}),D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})).\end{array}$$

Hence, for $n\ge n_0$, we have:

$$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})\le \phi \left(\frac{D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})+D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})}{2}\right).$$

If we denote ${\alpha }_n=D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})$, the above inequality becomes:

$${\alpha }_n\le \phi \left(\frac{{\alpha }_{n-1}+{\alpha }_n}{2}\right).$$

Since $\phi$ is a comparison function, $\phi (t)<t$ for all $t>0$. It follows that ${\alpha }_n<\frac{{\alpha }_{n-1}+{\alpha }_n}{2}$, that is

$${\alpha }_n<{\alpha }_{n-1},\mathrm{for}\mathrm{all}n\in {\mathbb{N}}^{*},$$

therefore ${\left\{{\alpha }_n\right\}}_n$ is a decreasing sequence of positive numbers. Let w be its limit. Observe that w is an accumulation point of ${\mathsf{\Omega }}_{n_0}$. It can be written as

$$w=\underset{n\to \infty }{lim}{\alpha }_n\le \underset{n\to \infty }{lim\; sup}\phi \left(\frac{{\alpha }_{n-1}+{\alpha }_n}{2}\right)\le \phi \left(\underset{n\to \infty }{lim}\frac{{\alpha }_{n-1}+{\alpha }_n}{2}\right)=\phi (w),$$

and it follows that $w=0$, and ${lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_0,{\mathsf{\Lambda }}^{n+1}{x}_0)=0$. Consider m, $n\ge k+1$, $m^{\prime }=m-1$ and $n^{\prime }=n-1$, with $m^{\prime }\ge n^{\prime }\ge k$. From the contractive inequality it follows that:

$$\begin{array}{cc}\hfill D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})& =D({\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})\hfill \\ \hfill & \le \phi \left(\frac{D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*})+D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}))}{2}\right)\hfill \\ \hfill & +\theta (D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}),\hfill \\ & D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})).\hfill \end{array}$$

Observe that $D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*})$, $D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})$, $D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*})$, $D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})$ belong to ${\mathsf{\Omega }}_k$. Moreover, using the properties of $\phi$ we have:

$$\begin{array}{cc}\hfill D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})& \le \phi \left(\frac{{\delta }_k(D,\mathsf{\Lambda },x_{*})+{\delta }_{k+1}(D,\mathsf{\Lambda },x_{*})}{2}\right)+\theta (D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),\hfill \\ & D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})).\hfill \end{array}$$

Let us take the supremum in the above inequality:

$$\begin{array}{cc}\hfill \underset{m,n\ge k}{sup}D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})& \le \phi \left(\frac{{\delta }_k(D,\mathsf{\Lambda },x_{*})+{\delta }_{k+1}(D,\mathsf{\Lambda },x_{*})}{2}\right)\hfill \\ \hfill & +\underset{m^{\prime },n^{\prime }\ge k}{sup}\theta (D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}),\hfill \\ & D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})).\hfill \end{array}$$

Using Lemma 1, our inequality leads to

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim}{\delta }_k& =& \underset{k\to \infty }{lim}(\underset{m,n\ge k}{sup}D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}))\hfill \\ & \le & \underset{k\to \infty }{lim\; sup}\phi \left(\frac{{\delta }_k(D,\mathsf{\Lambda },x_{*})+{\delta }_{k+1}(D,\mathsf{\Lambda },x_{*})}{2}\right),\hfill \end{array}$$

where ${\delta }_k={\delta }_k(D,\mathsf{\Lambda },x_{*})$. Again, ${\left\{{\delta }_k\right\}}_k$ is monotone decreasing and lower bounded by 0, therefore it converges to some $l\in [0,{\delta }_{n_0}]$. Note that l is an accumulation point for ${\mathsf{\Omega }}_{n_0}$ and $\phi$ is upper semi-continuous in such points, therefore we are led to:

$$l\le \underset{k\to \infty }{lim\; sup}\phi \left(\frac{{\delta }_k(D,\mathsf{\Lambda },x_{*})+{\delta }_{k+1}(D,\mathsf{\Lambda },x_{*})}{2}\right)\le \phi (l).$$

The last inequality implies $\phi (l)=l$ and ${lim}_{k\to \infty }{\delta }_k=0$. We have established that ${lim}_{m,n\to \infty }D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, that is ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is a D-Cauchy sequence. □

Having such a result for $(\phi ,\theta )$-Kannan type contractions, we are now in a position to provide an existence and uniqueness theorem.

Theorem 3.

Let $(L,D)$ be a Jleli–Samet metric space that is $\mathbf{B}$-nondecreasing-complete relative to the preorder $\mathbf{B}$ and $\mathsf{\Lambda }:L\to L$ a mapping. Presume that:

(i)

There exists $x_{*}\in L$ such that $x_{*}\mathbf{B}\mathsf{\Lambda }{x}_{*}$, and ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is a comparison function φ and $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{ccc}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & \phi \left(\frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{2}\right)\hfill \\ & & +\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$;

(v)

Λ is $\mathbf{B}$-nondecreasing-continuous and $\mathbf{B}$-nondecreasing.

Then, ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point ω of Λ, and $D(\omega ,\omega )=0$. Moreover, if ${\omega }^{\prime }$ is a fixed point of Λ, for which $D(\omega ,{\omega }^{\prime })<\infty$, and $D({\omega }^{\prime },{\omega }^{\prime })=0$, then $\omega ={\omega }^{\prime }$.

Proof. 

The iterated sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is $\mathbf{B}$-nondecreasing for inductive reasons, and it follows that it is D-convergent to $\omega \in L$, since the space is supposed to be $\mathbf{B}$-nondecreasing-complete. Using the fact that $\mathsf{\Lambda }$ is $\mathbf{B}$-nondecreasing-continuous, we obtain:

$${\left\{\mathsf{\Lambda }{x}_n\right\}}_n\stackrel{D}{\to }\mathsf{\Lambda }\omega .$$

In conclusion, $\mathsf{\Lambda }\omega =\omega$ and $\omega$ is a fixed point of $\mathsf{\Lambda }$.

Using property $(D3)$ from the Jleli–Samet metric axioms, it follows that:

$$D(\omega ,\omega )\le C\underset{n\to \infty }{lim\; sup}D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0,$$

because ${\left\{x_n\right\}}_n\in \mathcal{C}(D,L,x_{*})$, and $D(\omega ,\omega )=0$.

Denote by ${\omega }^{\prime }$ another fixed point of the operator $\mathsf{\Lambda }$, so that $D(\omega ,{\omega }^{\prime })<\infty$. We obtain:

$$\begin{array}{ccc}\hfill D(\omega ,{\omega }^{\prime })& =& D(\mathsf{\Lambda }\omega ,\mathsf{\Lambda }{\omega }^{\prime })\hfill \\ & \le & \phi \left(\frac{D(\omega ,\mathsf{\Lambda }\omega )+D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })}{2}\right)\hfill \\ & & +\theta (D({\omega }^{\prime },\mathsf{\Lambda }\omega ),D(\omega ,\mathsf{\Lambda }{\omega }^{\prime }),D(\omega ,\mathsf{\Lambda }\omega ),D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime }))\hfill \end{array}$$

that is $D(\omega ,{\omega }^{\prime })\le 0$, compelling that $D(\omega ,{\omega }^{\prime })=0$ and $\omega ={\omega }^{\prime }$. □

Theorem 4.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$. In addition to this, presume that the following conditions hold:

(i)

There is $x_{*}\in L$, ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is a comparison function φ and $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{ccc}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & \phi \left(\frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{2}\right)\hfill \\ & & +\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. If $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, satisfying $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Proof. 

We emphasize that by considering the trivial preorder on L, i.e., $x\mathbf{B}y$ for all x, $y\in L$, the operator $\mathsf{\Lambda }$ becomes $\mathbf{B}$-nondecreasing and the space $(L,D)$ is $\mathbf{B}$-nondecreasing-complete. The Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is $\mathbf{B}$-nondecreasing and D-Cauchy. Since $(L,D)$ is complete, there exists $\omega \in L$ such that ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n\stackrel{D}{\to }\omega$. Note that $\omega \in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})$, therefore we obtain:

$$\begin{array}{cc}\hfill D({\mathsf{\Lambda }}^{n+1}{x}_{*},\mathsf{\Lambda }\omega )=& D(\mathsf{\Lambda }{\mathsf{\Lambda }}^n{x}_{*},\mathsf{\Lambda }\omega )\hfill \\ \hfill & \le \phi \left(\frac{D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})+D(\omega ,\mathsf{\Lambda }\omega )}{2}\right)+\theta (D(\omega ,{\mathsf{\Lambda }}^{n+1}{x}_{*}),\hfill \\ \hfill & D({\mathsf{\Lambda }}^n{x}_{*},\mathsf{\Lambda }\omega ),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),D(\omega ,\mathsf{\Lambda }\omega )).\hfill \end{array}$$

Taking into consideration the properties of $\phi$, the above inequality becomes:

$$\begin{array}{cc}\hfill D({\mathsf{\Lambda }}^{n+1}{x}_{*},\mathsf{\Lambda }\omega )& \le \frac{D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})+D(\omega ,\mathsf{\Lambda }\omega )}{2}+\theta (D(\omega ,{\mathsf{\Lambda }}^{n+1}{x}_{*}),\hfill \\ \hfill & D({\mathsf{\Lambda }}^n{x}_{*},\mathsf{\Lambda }\omega ),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),D(\omega ,\mathsf{\Lambda }\omega )).\hfill \end{array}$$

Since $\theta \in \mathsf{\Theta }$ and ${lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_0,{\mathsf{\Lambda }}^{n+1}{x}_0)=0$, by the use of Lemma 1 it follows that

$$D(\omega ,\mathsf{\Lambda }\omega )\le \frac{D(\omega ,\mathsf{\Lambda }\omega )}{2},$$

and it is obvious now that $D(\omega ,\mathsf{\Lambda }\omega )=0$, that is $\mathsf{\Lambda }\omega =\omega$.

Let us consider another fixed point of $\mathsf{\Lambda }$, denoted by ${\omega }^{\prime }$, which satisfies $D(\omega ,{\omega }^{\prime })<\infty$ and $D(\omega ,\omega )=0$. The contractive condition leads to the following:

$$\begin{array}{cc}\hfill D(\omega ,{\omega }^{\prime })=& D(\mathsf{\Lambda }\omega ,\mathsf{\Lambda }{\omega }^{\prime })\hfill \\ \hfill & \le \phi \left(\frac{D(\omega ,\mathsf{\Lambda }\omega )+D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })}{2}\right)+\theta (D({\omega }^{\prime },\mathsf{\Lambda }\omega ),\hfill \\ \hfill & D(\omega ,\mathsf{\Lambda }{\omega }^{\prime }),D(\omega ,\mathsf{\Lambda }\omega ),D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })),\hfill \end{array}$$

from which follows directly $D(\omega ,{\omega }^{\prime })=0$, because $D(\omega ,\mathsf{\Lambda }\omega )=D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })=0$. In conclusion, $\omega ={\omega }^{\prime }$ and the theorem is completely proved. □

Taking into account now some particular choices of the auxiliary functions $\theta$ and $\phi$, Theorem 4 provides some known results in the literature.

First, for $\theta \equiv 0$, the next consequence arises.

Corollary 10.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping. In addition to this, presume that the following conditions hold:

(i)

There is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is φ a comparison function such that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi \left(\frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{2}\right),\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. If, additionally, $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ which satisfies the relations $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

For $\theta (t,u,v,w)=\tau tuvw$, where $\tau$ is a positive constant, t, u, v, $w>0$, Theorem 3 provides the consequence in the next lines.

Corollary 11.

Let $(L,D)$ be a complete Jleli–Samet space and a $\mathbf{B}$-nondeacreasing operator $\mathsf{\Lambda }:L\to L$. Presume that:

(i)

There exists $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is φ a comparison function so that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi \left(\frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{2}\right)\\ \hfill +\tau inf\{(D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y))\},\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is a D-convergent sequence, to a fixed point of Λ, ω. Moreover, if $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, for which $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Corollary 12.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping. Suppose that the next conditions are accomplished:

(i)

There exists $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is φ a comparison function such that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi \left(\frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{2}\right)\\ \hfill +\tau D(y,\mathsf{\Lambda }x)D(x,\mathsf{\Lambda }y)D(x,\mathsf{\Lambda }x)D(y,\mathsf{\Lambda }y),\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. If $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, so that $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Let us now consider as a comparison function in Theorem 4 a classic contraction.

Corollary 13.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping. Suppose that the next hypotheses are fulfilled:

(i)

There is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{cc}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \alpha \frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{2}+& \theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),\hfill \\ \hfill & D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y))\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$ and $\alpha \in [0,1)$;

(iii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. Assuming that $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then necessarily $\omega ={\omega }^{\prime }$.

Corollary 14.

Let $(L,D)$ be a complete Jleli–Samet space and and operator $\mathsf{\Lambda }:L\to L$. Presume that:

(i)

There exists $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is $\alpha \in [0,1)$ so that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \alpha ·\frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{2}+\tau D(y,\mathsf{\Lambda }x)D(x,\mathsf{\Lambda }y)D(x,\mathsf{\Lambda }x)D(y,\mathsf{\Lambda }y),\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is a D-convergent sequence, to a fixed point of Λ, ω. Moreover, if $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, for which $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Corollary 15.

Let $(L,D)$ be a complete Jleli–Samet space and a operator $\mathsf{\Lambda }:L\to L$. Presume that:

(i)

There exists $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is φ a comparison function so that:

$$\begin{array}{ccc}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & {\displaystyle \frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)+2}}\hfill \\ & & +\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is a D-convergent sequence, to a fixed point of Λ, ω. Moreover, if $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, for which $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Considering here some subclasses of the JS-class of metric spaces, we get the next corollaries.

Corollary 16.

Let $(L,d)$ be a complete b-metric space and $\mathsf{\Lambda }:L\to L$. In addition to this, presume that the following conditions hold:

(i)

There is $x_{*}\in L$, ${\delta }_{n_0}(d,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is a comparison function φ and $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{ccc}\hfill d(\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & \phi \left(\frac{d(x,\mathsf{\Lambda }x)+d(y,\mathsf{\Lambda }y)}{2}\right)+\theta (d(y,\mathsf{\Lambda }x),d(x,\mathsf{\Lambda }y),d(x,\mathsf{\Lambda }x),d(y,\mathsf{\Lambda }y)),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }d({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$d({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is convergent to a fixed point of Λ, ω. If $d(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, satisfying $d({\omega }^{\prime },{\omega }^{\prime })=0$, $d(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Corollary 17.

Let $(L,D)$ be a complete Hitzler–Seda space and $\mathsf{\Lambda }:L\to L$. In addition to this, presume that the following conditions hold:

(i)

There is $x_{*}\in L$, ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is a comparison function φ and $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{ccc}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & \phi \left(\frac{D(x,\mathsf{\Lambda }x)+D(y,\mathsf{\Lambda }y)}{2}\right)+\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. If $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, satisfying $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Corollary 18.

Let $(X_{\rho },\rho )$ be a complete modular space having the Fatou’s property and $\mathsf{\Lambda }:X_{\rho }\to X_{\rho }$. In addition to this, presume that the following conditions hold:

(i)

There is $x_{*}\in X_{\rho }$, ${\delta }_{n_0}(\rho ,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is a comparison function φ and $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{ccc}\hfill \rho (\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & \phi \left(\frac{\rho (x,\mathsf{\Lambda }x)+\rho (y,\mathsf{\Lambda }y)}{2}\right)+\theta (\rho (y,\mathsf{\Lambda }x),\rho (x,\mathsf{\Lambda }y),\rho (x,\mathsf{\Lambda }x),\rho (y,\mathsf{\Lambda }y)),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in X_{\rho }:{lim}_{n\to \infty }\rho ({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$\rho ({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is ρ-convergent to a fixed point of Λ, ω. If $\rho (\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in X_{\rho }$ is another fixed point of Λ, satisfying $\rho ({\omega }^{\prime },{\omega }^{\prime })=0$, $\rho (\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

5. Generalizations in the Sense of Ćirić

We now develop some results in the direction of Ćirić, in this setting introduced by Jleli and Samet.

Lemma 4.

Let $(L,D)$ be a Jleli–Samet metric space and let $\mathsf{\Lambda }:L\to L$ be a self-mapping. Suppose the next hypotheses are accomplished:

(i)

There exists $x_{*}\in L$, and $n_0\in \mathbb{N}$, so that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$;

(ii)

There is a comparison function φ and $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{ccc}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & \phi \left(max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}\right)\hfill \\ & & +\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$ for all $n\in \mathbb{N}$.

Then, the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-Cauchy.

Proof. 

The inequality

$$\begin{array}{c}D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})\le \phi (max\{D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}),D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}),\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})\})+\theta (D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}),D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})),n\in \mathbb{N},\hfill \end{array}$$

leads to

$$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})\le \phi \left(max\{D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})\}\right).$$

Suppose that there is $n_1\in \mathbb{N}$ such that

$$max\{D({\mathsf{\Lambda }}^{n_1-1}{x}_{*},{\mathsf{\Lambda }}^{n_1}{x}_{*}),D({\mathsf{\Lambda }}^{n_1}{x}_{*},{\mathsf{\Lambda }}^{n_1+1}{x}_{*})\}=D({\mathsf{\Lambda }}^{n_1}{x}_{*},{\mathsf{\Lambda }}^{n_1+1}{x}_{*}).$$

Then, by the properties of $\phi$, it follows that $D({\mathsf{\Lambda }}^{n_1}{x}_{*},{\mathsf{\Lambda }}^{n_1+1}{x}_{*})=0$. It follows that ${\mathsf{\Lambda }}^{n_1}{x}_{*}={\mathsf{\Lambda }}^{n_1+1}{x}_{*}=\mathsf{\Lambda }{\mathsf{\Lambda }}^{n_1}{x}_{*}$, that is ${\mathsf{\Lambda }}^{n_1}{x}_{*}$ is a fixed point of $\mathsf{\Lambda }$. Otherwise, we get:

$$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})\le \phi (D({\mathsf{\Lambda }}^{n-1}{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})),$$

which compels

$$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})\le {\phi }^{n-n_0}(D({\mathsf{\Lambda }}^{n_0}{x}_{*},{\mathsf{\Lambda }}^{n_0+1}{x}_{*})).$$

Passing through the limit over n, we conclude that

$$\underset{n\to \infty }{lim}D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})=0.$$

Let us take m, $n\ge k+1$, $m^{\prime }=m-1$ and $n^{\prime }=n-1$, with $m^{\prime }\ge n^{\prime }\ge k$. The above inequality becomes:

$$\begin{array}{cc}\hfill D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=& D({\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})\le \phi (max\{D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }}{x}_{*}),\hfill \\ \hfill & D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})\})\hfill \\ & +\theta (D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}),\hfill \\ & D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})).\hfill \end{array}$$

It can be observed that:

$$\begin{array}{c}D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})\in {\mathsf{\Omega }}_k.\hfill \end{array}$$

Taking the supremum in the above inequality, we get

$$\begin{array}{ccc}\hfill \underset{m,n\ge k}{sup}D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})& \le & \phi ({\delta }_k(D,\mathsf{\Lambda },x_{*}))\hfill \\ & & +\underset{m^{\prime },n^{\prime }\ge k}{sup}\theta (D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*}),\hfill \\ & & D({\mathsf{\Lambda }}^{n^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{n^{\prime }+1}{x}_{*}),D({\mathsf{\Lambda }}^{m^{\prime }}{x}_{*},{\mathsf{\Lambda }}^{m^{\prime }+1}{x}_{*})).\hfill \end{array}$$

Using Lemma 1 for the right side of the contractive inequality, it follows that:

$$\underset{k\to \infty }{lim}(\underset{k\to \infty }{sup}D({\mathsf{\Lambda }}^m{x}_{*},{\mathsf{\Lambda }}^n{x}_{*}))\le \underset{k\to \infty }{lim\; sup}\phi ({\delta }_k(D,\mathsf{\Lambda },x_{*})).$$

Taking advantage of the fact that ${\left\{{\delta }_k\right\}}_k$ is a nonincreasing sequence of positive numbers, it converges to $l\ge 0$, and

$$l\le lim\; sup\phi ({\delta }_k)\le \phi (lim\; sup{\delta }_k)=\phi (l),$$

which gives us $l=0$. In conclusion, ${lim}_{m,n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^m{x}_{*})=0$, and the proof is completed. □

Another two results can be proved now.

Theorem 5.

Let $(L,D)$ be a Jleli–Samet metric space that is $\mathbf{B}$-nondecreasing-complete relative to the preorder $\mathbf{B}$ and $\mathsf{\Lambda }:L\to L$ a $\mathbf{B}$-nondecreasing continuous operator. Suppose that the next hypotheses are fulfilled:

(i)

There is $x_{*}\in L$ such that $x_{*}\mathbf{B}\mathsf{\Lambda }{x}_{*}$ and ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is φ a comparison function and $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{ccc}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & \phi \left(max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}\right)\hfill \\ & & +\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then, the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a point ω of Λ, and $D(\omega ,\omega )=0$. If ${\omega }^{\prime }$ is another fixed point of Λ, with $D(\omega ,{\omega }^{\prime })<\infty$ and $D({\omega }^{\prime },{\omega }^{\prime })=0$, then $\omega ={\omega }^{\prime }$.

Proof. 

By an inductive argumentation, ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is a $\mathbf{B}$-nondecreasing sequence, and Lemma 4 imposes its D-convergence to a point $\omega$ in L, since the space is $\mathbf{B}$-nondecreasing-complete. Using the fact that $\mathsf{\Lambda }$ is $\mathbf{B}$-nondecreasing-continuous, ${\left\{\mathsf{\Lambda }{x}_n\right\}}_n\stackrel{D}{\to }\mathsf{\Lambda }\omega$. So, $\mathsf{\Lambda }\omega =\omega$ and $\omega$ is a fixed point of $\mathsf{\Lambda }$.

The last property $(D3)$ from the metric axioms ensures us that:

$$D(\omega ,\omega )\le C\underset{n\to \infty }{lim\; sup}D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0,$$

as ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n\in \mathcal{C}(D,\mathsf{\Lambda },x_{*})$, that is $D(\omega ,\omega )=0$.

Suppose ${\omega }^{\prime }$ is another fixed point of $\mathsf{\Lambda }$, with $D(\omega ,{\omega }^{\prime })<\infty$ and $D({\omega }^{\prime },{\omega }^{\prime })=0$. In the contractive inequality, we obtain the following.

$$\begin{array}{cc}\hfill D(\omega ,{\omega }^{\prime })=& D(\mathsf{\Lambda }\omega ,\mathsf{\Lambda }{\omega }^{\prime })\hfill \\ \hfill & \le \phi \left(max\{D(\omega ,{\omega }^{\prime }),D(\omega ,\mathsf{\Lambda }\omega ),D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })\}\right)\hfill \\ & +\theta (D({\omega }^{\prime },\mathsf{\Lambda }\omega ),D(\omega ,\mathsf{\Lambda }{\omega }^{\prime }),D(\omega ,\mathsf{\Lambda }\omega ),D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })),\hfill \end{array}$$

that is $D(\omega ,{\omega }^{\prime })\le \phi (D(\omega ,{\omega }^{\prime }))$, so implies $D(\omega ,{\omega }^{\prime })=0$ and $\omega ={\omega }^{\prime }$. □

Theorem 6.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping. Suppose that the following conditions hold:

(i)

There is $x_0\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_0)<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is φ a comparison function and $\theta \in {\mathsf{\Theta }}^{\prime }$ such that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi \left(max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}\right)\\ \hfill +\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y))\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on $[0,+\infty )$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$ for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. If, additionally, $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, with $D(\omega ,{\omega }^{\prime })<\infty$ and $D({\omega }^{\prime },{\omega }^{\prime })=0$, then $\omega ={\omega }^{\prime }$.

Proof. 

Working with the trivial preorder, ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is $\mathbf{B}$-nondecreasing and D-Cauchy, therefore, there exists $\omega \in L$, the limit of the Picard sequence. Clearly, $\omega \in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})$, and the contractive condition leads to

$$\begin{array}{cc}\hfill D({\mathsf{\Lambda }}^{n+1}{x}_{*},\mathsf{\Lambda }\omega )=D(\mathsf{\Lambda }{\mathsf{\Lambda }}^n{x}_{*},\mathsf{\Lambda }\omega )\le & \phi (max\{D({\mathsf{\Lambda }}^n{x}_{*},\omega ),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),\hfill \\ & D(\omega ,\mathsf{\Lambda }\omega )\})+\theta (D(\omega ,{\mathsf{\Lambda }}^{n+1}{x}_{*}),\hfill \\ & D({\mathsf{\Lambda }}^n{x}_{*},\mathsf{\Lambda }\omega ),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),D(\omega ,\mathsf{\Lambda }\omega )).\hfill \end{array}$$

Taking into account that ${lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*})=0$, we may pass through the superior limit over $n\to \infty$. As $\phi$ is upper semi-continuous, Lemma 1 imposes:

$$\begin{array}{ccc}\hfill D(\omega ,\mathsf{\Lambda }\omega )& \le & \underset{n\to \infty }{lim\; sup}\phi (max\{D({\mathsf{\Lambda }}^n{x}_{*},\omega ),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),D(\omega ,\mathsf{\Lambda }\omega )\})\hfill \\ & \le & \phi (\underset{n\to \infty }{lim\; sup}(max\{D({\mathsf{\Lambda }}^n{x}_{*},\omega ),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),D(\omega ,\mathsf{\Lambda }\omega )\})).\hfill \end{array}$$

It can be observed that ${lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=D(\omega ,\omega )$, and the inequality becomes:

$$D(\omega ,\mathsf{\Lambda }\omega )\le \phi (D(\omega ,\mathsf{\Lambda }\omega )).$$

By the properties of $\phi$, we get $D(\omega ,\mathsf{\Lambda }\omega )=0$ and $\omega =\mathsf{\Lambda }\omega$. As a remark, in the above relation, we used the fact that

$$\underset{n\to \infty }{lim}(max\{D({\mathsf{\Lambda }}^n{x}_{*},\omega ),D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^{n+1}{x}_{*}),D(\omega ,\mathsf{\Lambda }\omega )\})=D(\omega ,\mathsf{\Lambda }\omega ).$$

Let us consider another fixed point of $\mathsf{\Lambda }$, denoted by ${\omega }^{\prime }$, that satisfies the above properties. By the inequality

$$\begin{array}{cc}\hfill D(\omega ,{\omega }^{\prime })=& D(\mathsf{\Lambda }\omega ,\mathsf{\Lambda }{\omega }^{\prime })\le \phi \left(max\{D(\omega ,{\omega }^{\prime }),D(\omega ,\mathsf{\Lambda }\omega ),D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })\}\right)\hfill \\ \hfill & +\theta (D({\omega }^{\prime },\mathsf{\Lambda }\omega ),D(\omega ,\mathsf{\Lambda }{\omega }^{\prime }),D(\omega ,\mathsf{\Lambda }\omega ),D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })),\hfill \end{array}$$

it follows that $D(\omega ,{\omega }^{\prime })\le \phi (D(\omega ,{\omega }^{\prime }))$ because $D(\omega ,\mathsf{\Lambda }\omega )=D({\omega }^{\prime },\mathsf{\Lambda }{\omega }^{\prime })=0$. In conclusion, $D(\omega ,{\omega }^{\prime })=0$ so $\omega ={\omega }^{\prime }$. □

Given proper choices of functions that appear in the theorems above, Theorem 6 recovers some known results in research.

If the $\theta$-term becomes 0, the following consequence arises.

Corollary 19.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping. Suppose that the following conditions hold:

(i)

There is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is φ a comparison function such that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi \left(max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}\right)\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. If, additionally, $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ which satisfies the relations $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

For $\theta (t,u,v,w)=\tau tuvw$, where $\tau$ is a positive constant, t, u, v, $w>0$, Theorem 6 gives the following consequence.

Corollary 20.

Let $(L,D)$ be a complete Jleli–Samet space and an operator $\mathsf{\Lambda }:L\to L$. Presume that:

(i)

There exists $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is φ a comparison function so that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \phi \left(max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}\right)\\ \hfill +\tau inf\{(D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y))\},\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is a D-convergent sequence, to a fixed point of Λ, ω. Moreover, if $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, for which $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

Corollary 21.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping. Suppose that the next conditions are accomplished:

(i)

There exists $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is φ a comparison function such that:

$$\begin{array}{ccc}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)& \le & \phi \left(max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}\right)\hfill \\ & & +\tau D(y,\mathsf{\Lambda }x)D(x,\mathsf{\Lambda }y)D(x,\mathsf{\Lambda }x)D(y,\mathsf{\Lambda }y),\hfill \end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

φ is upper semi-continuous on the accumulation points of the set ${\mathsf{\Omega }}_{n_0}$;

(iv)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. If $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, so that $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

We could now consider as a comparison function in Theorem 6 a classic contraction.

Corollary 22.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping. Suppose that the next hypotheses are fulfilled:

(i)

There is $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \alpha max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}\\ \hfill +\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$ and $\alpha \in [0,1)$;

(iii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. Assuming that $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then necessarily $\omega ={\omega }^{\prime }$.

Corollary 23.

Let $(L,D)$ be a complete Jleli–Samet space and an operator $\mathsf{\Lambda }:L\to L$. Presume that:

(i)

There exists $x_{*}\in L$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$ for some $n_0\in \mathbb{N}$;

(ii)

There is $\alpha \in [0,1)$ so that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \alpha max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}\\ \hfill +\tau D(y,\mathsf{\Lambda }x)D(x,\mathsf{\Lambda }y)D(x,\mathsf{\Lambda }x)D(y,\mathsf{\Lambda }y),\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is a D-convergent sequence, to a fixed point of Λ, ω. Moreover, if $D(\omega ,\omega )<\infty$ and ${\omega }^{\prime }\in L$ is another fixed point of Λ, for which $D({\omega }^{\prime },{\omega }^{\prime })=0$, $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

If we set the function $\phi (t)=\frac{t}{t+1}$, $t\in [0,\infty )$, we can give the next consequence.

Corollary 24.

Let $(L,D)$ be a complete Jleli–Samet space and $\mathsf{\Lambda }:L\to L$ a mapping. Presume that the following conditions are fulfilled:

(i)

There is $x_{*}\in X$ such that ${\delta }_{n_0}(D,\mathsf{\Lambda },x_{*})<\infty$, for some $n_0\in \mathbb{N}$;

(ii)

There is $\theta \in \mathsf{\Theta }$ such that:

$$\begin{array}{c}\hfill D(\mathsf{\Lambda }x,\mathsf{\Lambda }y)\le \frac{max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}}{max\{D(x,y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)\}+1}\\ \hfill +\theta (D(y,\mathsf{\Lambda }x),D(x,\mathsf{\Lambda }y),D(x,\mathsf{\Lambda }x),D(y,\mathsf{\Lambda }y)),\end{array}$$

for all x, $y\in {\mathcal{O}}_{\mathsf{\Lambda }}^{{}^{\prime }}(x_{*})={\mathcal{O}}_{\mathsf{\Lambda }}(x_{*})\cup \{\omega \in L:{lim}_{n\to \infty }D({\mathsf{\Lambda }}^n{x}_{*},\omega )=0\}$;

(iii)

$D({\mathsf{\Lambda }}^n{x}_{*},{\mathsf{\Lambda }}^n{x}_{*})=0$, for all $n\in \mathbb{N}$.

Then the Picard sequence ${\left\{{\mathsf{\Lambda }}^n{x}_{*}\right\}}_n$ is D-convergent to a fixed point of Λ, ω. Moreover, if $D(\omega ,\omega )<\infty$, and ${\omega }^{\prime }\in L$ is another fixed point of Λ with $D(\omega ,{\omega }^{\prime })<\infty$, then $\omega ={\omega }^{\prime }$.

As in the case of the other two sections, results on the existence and uniqueness of fixed points in the setting of b-metric spaces, dislocated metric spaces or modular spaces with the Fatou property can be presented.

6. Conclusions

In this paper, we proved existence and uniqueness fixed point results in the Jleli–Samet generalized metric spaces endowed with a partial order. We have used generalized contractive conditions defined by means of comparison functions with adequate continuity properties, and continuous functions of four variables with additional suitable properties. These results extend well-known fixed point theorems such as the Banach contraction principle on JS spaces, Kannan-type contractions, or $\phi$-contraction theorems. The general nature of these metrics allowed us to obtain all these outcomes in spaces such as b-metric spaces, Hitzler–Seda spaces, or modular spaces with the Fatou property.