Monotone Picard Maps in Relational Metric Spaces

Abstract

The 2008 fixed point result on rs-relational metric spaces due to Jachymski is equivalent with the classical 1922 Banach Contraction Principle. This is also valid for the 1961 statement in metric spaces due to Edelstein, or the 2005 fixed point result in quasi-ordered metric spaces obtained by Nieto and Rodriguez-Lopez.

1. Introduction

Let X be a nonempty set. Call the subset Y of X, almost singleton (in short: asingleton), provided [$y_1,y_2\in Y$ implies $y_1=y_2$]; and singleton if, in addition, Y is nonempty; note that in this case, $Y=\left\{y\right\}$, for some $y\in X$. Take a metric $d:X\times X\to R_{+}:=[0,\infty ]$ over X; as well as a selfmap $T\in \mathcal{F}\left(X\right)$. [Here, for each couple $(A,B)$ of nonempty sets, $\mathcal{F}(A,B)$ denotes the class of all functions from A to B; when $A=B$, we write $\mathcal{F}\left(A\right)$ in place of $\mathcal{F}(A,A)$]. Denote $\mathrm{Fix}\left(T\right)=\{x\in X;x=Tx\}$; each point of this set is referred to as fixed under T. The basic existence and uniqueness criteria involving such points are to be discussed along with some precise concepts. These, essentially, are suggested by the developments in Rus ([1], Ch 2, Sect 2.2); and consist of three (distinct) stages, described as below.

(St-1) The uniqueness context is ultimately based on the general convention

• (uni) We say that T is fix-asingleton, if $\mathrm{Fix}\left(T\right)$ is an asingleton;
  and fix-singleton, if $\mathrm{Fix}\left(T\right)$ is a singleton.

(St-2) The convergence context consists of two metrical concepts:

• (pic-1) We say that T is a Picard operator (modulo d) when, for each $x\in X$, the sequence $T^{N}x:=(T^{n}x;n\ge 0)$ is d-Cauchy; and global Picard operator (modulo d) when, in addition, T is fix-asingleton
• (pic-2) We say that T is a strongly Picard operator (modulo d) when, for each $x\in X$, the sequence $T^{N}x$ is d-convergent with ${lim}_n\left(T^{n}x\right)\in \mathrm{Fix}\left(T\right)$; and global strongly Picard operator (modulo d) when, in addition, T is fix-asingleton (or, equivalently: fix-singleton).

(St-3) The contractive setting consists of certain predicative type constructions involving the data $(X,d;T)$; and has the general structure

• (contr) ($\forall x,y\in X$): ${\Pi }_1(d;x,y)$⟹ ${\Pi }_2(d;T^{N}x,T^{N}y)$;

where ${\Pi }_1$, ${\Pi }_2$ are propositional constructions.

By combining these three stages we get the immense majority of results belonging to Metrical Fixed Point Theory. The simplest one in this area may be stated as follows. Let us define the concepts

• (Con-1) We say that T is $(d;\alpha )$-contractive (where $\alpha \in R_{+}$), provided $d(Tx,Ty)\le \alpha d(x,y)$, for all $x,y\in X$.
• (Con-2) We say that T is $(d;0,1)$-contractive, provided it is $(d;\alpha )$-contractive, for some $\alpha \in [0,1]$.

Theorem 1. 

Let the metric space $(X,d)$ and the selfmap T of X be such that

• $(X,T)$ is d-Banach: X is d-complete and T is $(d;0,1)$-contractive.

Then, T is global strongly Picard (modulo d).

Technically speaking, the proof of this result is essentially provided in the classical 1922 statement due to Banach [2]; so, it is natural to name it the Banach Contraction Principle (in short: (B-cp)).

This principle has found a multitude of applications in operator equations theory; so, it was the subject of many extensions. Among these, we quote the implicit relational way of enlarging (B-cp), based on set-implicit contractive conditions like

• (a01) $\left(d\right(Tx,Ty),d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(Tx,y\left)\right)\in \mathcal{M}$,
  for all $x,y\in X$, $x\mathcal{R}y$;

where $\mathcal{M}$ is a subset of $R_{+}^6$ and $\mathcal{R}\subseteq X\times X$ is a relation over X. In particular, when $\mathcal{R}=X\times X$, a basic variant of the general contractive property above is

• (a02) $\left(d\right(Tx,Ty),d(x,y\left)\right)\in \mathcal{M}$, for all $x,y\in X$;

where $\mathcal{M}\subseteq R_{+}^2$ is a (nonempty) subset. The classical examples in this direction are the ones due to Meir and Keeler [3], Cirić [4], and Matkowski [5]. Some other classical contributions in the area belong to Boyd and Wong [6], Leader [7], Matkowski [8], and Reich [9]; for different extensions of these, we refer to the survey papers by Rhoades [10], Park [11], or Collaco and E Silva [12]. Further, when

• (a03) $\mathcal{R}$ is a quasi-order (reflexive and transitive relation) over X,

a first result of this type is the 1986 one obtained by Turinici [13]. Two decades later, this fixed point statement was re-discovered by Ran and Reurings [14], Nieto and Rodriguez-Lopez [15], and Agarwal et al. [16]; and, since then, the number of papers devoted to such results has increased rapidly. Finally, under

• (a04) $\mathcal{R}$ is amorphous (i.e., with no specific properties) over X

a first couple of results was obtained in 2008 by Jachymski [17] over graph metric spaces, and in 2012 by Samet and Turinici [18] on relational metric spaces. In fact, as shown in Roldán and Shahzad [19], these approaches are, practically, identical.

Having these precise, it is our first aim in the following to establish—in a genuine relational context—a couple of functional extensions of Jachymski result by reducing them to (a couple of) simpler asymptotic statements obtainable from the 2003 developments in Kirk [20]. Then, passing to the linear case, we show that the rs-relational version of the quoted Jachymski result is nothing else than an equivalent version of the Banach Contraction Principle. In addition, we stress that this equivalence comprises as well as the 1961 fixed point result in metric spaces due to Edelstein [21], and the 2005 related statement in quasi-ordered metric spaces due to Nieto and Rodriguez-Lopez [15]. Finally, a quasi-order version of the 2008 Suzuki (parametric) conditional fixed point principle [22] is provided.

2. Preliminaries

Let X be a nonempty set. Take a metric $d(.,.)$ on X; and let $\nabla \subseteq X\times X$ be a (nonempty) relation over the same; the triple $(X,d,\nabla )$ will be referred to as a relational metric space. Call the subset Y of X, $(\nabla )$-almost-singleton (in short: $(\nabla )$-asingleton) provided [$y_1,y_2\in Y$, $y_1\nabla y_2$⟹$y_1=y_2$]; and $(\nabla )$-singleton when, in addition, Y is nonempty. Given the sequence $\left(z_n\right)$ in X, define the property

• $\left(z_n\right)$ is $(\nabla )$-ascending when $z_i\nabla z_{i+1}$, for each index i.

Furthermore, given the relation $\mathcal{S}$ on X, the sequence $\left(z_n\right)$ in X and the point $z\in X$, it will be useful to introduce the boundedness type relations

• (bd-1) $\left(z_n\right)\mathcal{S}z$ when ($z_n\mathcal{S}z$, for all n),
• (bd-2) $\left(z_n\right)\mathcal{S}\mathcal{S}z$ when $\left(w_n\right)\mathcal{S}z$, for some subsequence $\left(w_n\right)$ of $\left(z_n\right)$.

Finally, let T be a selfmap of X. We have to determine circumstances under which $\mathrm{Fix}\left(T\right)$ be nonempty. To do this, we start from the basic hypothesis

• (incr) T is $(\nabla )$-increasing ($x\nabla y$ implies $Tx\nabla Ty$).

Note that the natural condition to be added here is

• (s-prog) T is $(\nabla )$-semi-progressive ($X(T,\nabla ):=\{x\in X;x\nabla Tx\}\ne \varnothing$).

But, even if this fails, the posed problem is meaningful, in a vacuous manner.

The basic directions under which our investigations are conducted are described in the list below, comparable with the one in Turinici [23]:

(rpic-0) We say that T is fix-$(\nabla )$-asingleton, when $\mathrm{Fix}\left(T\right)$ is $(\nabla )$-asingleton; and fix-$(\nabla )$-singleton when $\mathrm{Fix}\left(T\right)$ is $(\nabla )$-singleton.

(rpic-1) We say that T is a Picard operator (modulo $(d,\nabla )$) if, for each $x\in X(T,\nabla )$, $(T^{n}x;n\ge 0)$ is d-Cauchy; and a global Picard operator (modulo $(d,\nabla )$) if, in addition, T is fix-$(\nabla )$-asingleton.

(rpic-2) We say that T is a strongly Picard operator (modulo $(d,\nabla )$) when, for each $x\in X(T,\nabla )$, $(T^{n}x;n\ge 0)$ is d-convergent with ${lim}_n\left(T^{n}x\right)$ belonging to $\mathrm{Fix}\left(T\right)$; and global strongly Picard operator (modulo $(d,\nabla )$) when, in addition, T is fix-$(\nabla )$-asingleton (or, equivalently: fix-$(\nabla )$-singleton).

In particular, when $(\nabla )$ is identical with $X\times X$ (the trivial relation over X) we introduce the generic definition:

• any concept (modulo $(d,X\times X)$) is referred to as a concept (modulo d).

This, applied to the last convention above, means

(rpic-2a) We say that T is a strongly Picard operator (modulo d) when, for each $x\in X$, $(T^{n}x;n\ge 0)$ is d-convergent with ${lim}_n\left(T^{n}x\right)$ belonging to $\mathrm{Fix}\left(T\right)$; and global strongly Picard operator (modulo d) when, in addition, T is fix-asingleton.

The sufficient (regularity) conditions for such properties are based on the (already introduced) ascending concept. We start their list with

(reg-1) Call X, $(\nabla ,d)$-complete provided (for each $(\nabla )$-ascending sequence): d-Cauchy ⟹ d-convergent

(reg-2) Call $(\nabla )$, d-almost-selfclosed when $\left(z_n\right)$ is $(\nabla )$-ascending and $z_n\stackrel{\mathit{d}}{⟶}z$ imply $\left(z_n\right)\nabla \nabla z$.

The next regularity condition to be used necessitates some preliminaries. Define the relation $(\perp )$ on X as

• (perp) ($x,y\in X$): $x\perp y$ iff $x=y$ or $x\nabla y$ or $y\nabla x$
  (the reflexive symmetric cover of $(\nabla )$);

clearly, $(\perp )$ is the minimal reflexive symmetric relation that includes $(\nabla )$. Given $x,y\in X$ and $k\ge 2$, any element $A=(z_1,...,z_k)\in X^k$ with $z_1=x$, $z_k=y$, and ($z_i\perp z_{i+1}$, $i\in \{1,...,k-1\}$), will be referred to as a k-dimensional $(\perp )$-chain between x and y; in this case, $k=dim\left(A\right)$ (the dimension of A) and $\Lambda \left(A\right)=d(z_1,z_2)+...+d(z_{k-1},z_k)$ is the length of A; the class of all these chains will be denoted as $C_k(x,y;\perp )$. Furthermore, put $C(x,y;\perp )=\cup \{C_k(x,y;\perp );k\ge 2\}$; any element of it will be referred to as a $(\perp )$-chain in X joining x and y. Now, the announced condition may be described as

(reg-3) Call X, $(\perp )$-connected provided $C(x,y;\perp )$ is nonempty, $\forall x,y\in X$.

As an essential completion of these developments, we have to introduce the contractive conditions to be used. Denote by ${\mathcal{F}}_0\left(R_{+}\right)$ the class of all functions $\phi \in \mathcal{F}\left(R_{+}\right)$ with $\phi \left(0\right)=0$; then, let ${\mathcal{F}}_0(in,re)\left(R_{+}\right)$ stand for the class of all functions $\phi \in {\mathcal{F}}_0\left(R_{+}\right)$ that are increasing and regressive ($\phi \left(t\right)<t$, $\forall t\in R_{+}^0$). The following basic properties of these functions will be used:

• (M-ad) $\phi$ is Matkowski admissible [8], when each sequence $(t_n;n\ge 0)$ in $R_{+}^0$ with ($t_{n+1}\le \phi \left(t_n\right)$, $\forall n$) fulfills ${lim}_n{t}_n=0$; the class of all these will be denoted as ${\mathcal{F}}_0(in,re;Mat)\left(R_{+}\right)$.
• (B-ad) $\phi$ is Browder admissible [24], when each sequence $(t_n;n\ge 0)$ in $R_{+}^0$ with ($t_{n+1}\le \phi \left(t_n\right)$, $\forall n$) fulfills ${\sum }_n{t}_n<\infty$; the class of all these will be denoted as ${\mathcal{F}}_0(in,re;Bro)\left(R_{+}\right)$.

It is clear by these definitions that ${\mathcal{F}}_0(in,re;Bro)\left(R_{+}\right)\subseteq {\mathcal{F}}_0(in,re;Mat)\left(R_{+}\right)$; but the converse inclusion is not, in general, true.

Now, for the arbitrary fixed $\phi \in {\mathcal{F}}_0\left(R_{+}\right)$, define the concept

• (nabla-phi) T is $(d,\nabla ;\phi )$-contractive:
  $d(Tx,Ty)\le \phi \left(d\right(x,y\left)\right)$, $\forall x,y\in X$, $x\nabla y$.

Note that, by the properties of the metric $d(.,.)$, this condition yields:

• (perp-phi) T is $(d,\perp ;\phi )$-contractive:
  $d(Tx,Ty)\le \phi \left(d\right(x,y\left)\right)$, $\forall x,y\in X$, $x\perp y$.

In particular, when

• $\phi$ is linear: $\phi \left(t\right)=\lambda t$, $t\in R_{+}$, for some $\lambda \ge 0$,

it will be useful for us to introduce the simplifying conventions:

• (lin-1) T is $(d,\nabla ;\phi )$-contractive is written as: T is $(d,\nabla ;\lambda )$-contractive.
• (lin-2) T is $(d,\perp ;\phi )$-contractive is written as: T is $(d,\perp ;\lambda )$-contractive.

Note that, under the choice $\lambda =1$, we have

• (nex-1) T is $(d,\nabla ;1)$-contractive means ($d(Tx,Ty)\le d(x,y)$, $x,y\in X$, $x\nabla y$); referred to as: T is $(d,\nabla )$-nonexpansive.
• (nex-2) T is $(d,\perp ;1)$-contractive means ($d(Tx,Ty)\le d(x,y)$, $x,y\in X$, $x\perp y$); referred to as: T is $(d,\perp )$-nonexpansive.

Finally, the following meta-convention is needed. Given the generic contraction principle (CP-gen), denote

• ((CP-gen))=the universe of (CP-gen); that is
  the class of all contraction principles deductible from (CP-gen).

In this case, given the contraction principles (CP-1) and (CP-2),

• (incl) ((CP-1))⊆ ((CP-2)) means: (CP-1) is deductible from (CP-2).
• (eq) ((CP-1)) = ((CP-2)) means: (CP-1) is equivalent with (CP-2).

Two basic examples of this type suggested by Kirk’s 2003 paper [20] are given below. Some preliminary conventions are needed.

(PC-1) Given the selfmap T of X, define the concept

• (s-asy) T is strongly d-asymptotic: ${\sum }_{n}d(T^{n}x,T^{n}y)<\infty$, $\forall x,y\in X$.

Remark 1. 

The following inclusion relative to this concept is valid:

• (T is strongly d-asymptotic) implies (T is fix-asingleton).

In fact, letting $z_1,z_2\in \mathrm{Fix}\left(T\right)$, we must have (by the imposed condition)

• $d(z_1,z_2)={lim}_{n}d(T^n{z}_1,T^n{z}_2)=0$; whence, $z_1=z_2$.

(PC-2) Given the structure $(X,d,\nabla )$ and the selfmap T of X, define

• T is left $(\nabla ,d)$-continuous when: for each sequence $\left(u_n\right)$ in X and each $u\in X$, we have: ($u_n\stackrel{\mathit{d}}{⟶}u$ and $\left(u_n\right)\nabla u$) imply $T{u}_n\stackrel{\mathit{d}}{⟶}Tu$.

Remark 2. 

A basic situation when this property holds is given by the inclusion

• (T is $(d,\nabla )$-nonexpansive) implies (T is left $(\nabla ,d)$-continuous).

In fact, let the sequence $\left(u_n\right)$ in X and the point $u\in X$ be as in the premise above. From the nonexpansive condition

• $d(T{u}_n,Tu)\le d(u_n,u)$, for all n;

and this, along with the convergence property, gives our desired fact.

(PC-3) Given a couple $(d,e)$ of metrics on X, let us introduce the concept

• d and e are linearly equivalent: $\alpha d\le e\le \beta d$, for some $\alpha ,\beta >0$, $\alpha \le \beta$.

We may now formulate the first statement in this series, referred to as Kirk asymptotic fixed point theorem in metric spaces (in short: (K-asy-ms)).

Theorem 2. 

Let the metric space $(X,d)$ and the selfmap T of X be such that $(X,T)$ is d-Kirk, in the sense:

• T is strongly d-asymptotic, X is d-complete, and T is d-continuous.

Then, T is global strongly Picard (modulo d).

Proof. 

By a preceding remark, T is fix-asingleton. It thus remains to establish that T is strongly Picard (modulo d).

Take some $x_0\in X$, and put $(x_n=T^n{x}_0;n\ge 0)$. From the strongly d-asymptotic property, we have (with $x=x_0$, $y=x_1$)

• ${\sum }_{n}d(x_n,x_{n+1})<\infty$; wherefrom, $(x_n;n\ge 0)$ is d-Cauchy.

As X is d-complete, $x_n\stackrel{\mathit{d}}{⟶}z$, for some $z\in X$. This, along with T being d-continuous, tells us that $(y_n=T{x}_n=x_{n+1};n\ge 0)$ fulfills ${lim}_n{y}_n=Tz$. On the other hand, evidently, ${lim}_n{y}_n=z$, so (via d=separated), $z=Tz$, as desired. □

Remark 3. 

By the above proof, it follows that ((B-cp)) ⊆ ((K-asy-ms)); just note that, from the contractive condition in (B-cp) it follows that $(X,T)$ is d-Kirk. Concerning the reciprocal inclusion ((K-asy-ms)) ⊆ ((B-cp)), let the couple $(d,e)$ of metrics on X be linearly equivalent. By our very definition, it is clear that

• ($(X,T)$ is d-Kirk) is equivalent with ($(X,T)$ is e-Kirk).

On the other hand, a relation like

• ($(X,T)$ is d-Banach) is equivalent with ($(X,T)$ is e-Banach)

is not, in general, true. This suggests that the reciprocal inclusion may not hold.

The second statement in this series, referred to as Kirk’s asymptotic fixed point theorem in relational metric spaces (in short: (K-asy-rms)) is described below.

Theorem 3. 

Let the relational metric space $(X,d,\nabla )$ and the selfmap T of X be such that $(X,T)$ is $(\nabla ,d)$-Kirk, in the sense:

• (i) X is $(\nabla ,d)$-complete and $(\nabla )$ is d-almost-selfclosed.
• (ii) T is strongly d-asymptotic, $(\nabla )$-increasing,
  $(\nabla )$-semi-progressive, and left $(\nabla ,d)$-continuous.

Then, T is global strongly Picard (modulo d).

Proof. 

By a preceding remark, T is fix-asingleton. It thus remains to establish that T is strongly Picard (modulo d); and this, via (T is strongly d-asymptotic), is fulfilled as soon as $\mathrm{Fix}\left(T\right)$ is nonempty.

Take some $x_0\in X(T,\nabla )$; and put $(x_n=T^n{x}_0;n\ge 0)$; clearly, $\left(x_n\right)$ is $(\nabla )$-ascending. From the strongly d-asymptotic property (with $x=x_0$, $y=x_1$)

• ${\sum }_{n}d(x_n,x_{n+1})<\infty$; wherefrom, $\left(x_n\right)$ is $(\nabla )$-ascending, d-Cauchy.

As X is $(\nabla ,d)$-complete, $x_n\stackrel{\mathit{d}}{⟶}z$, for some $z\in X$. Furthermore, as $(\nabla )$ is d-almost-selfclosed, there exists a subsequence $(u_n:=x_{i\left(n\right)};n\ge 0)$ of $(x_n;n\ge 0)$ (hence, $u_n\stackrel{\mathit{d}}{⟶}z$), such that $\left(u_n\right)\nabla z$. This, in view of (T is left $(\nabla ,d)$-continuous), gives $T{u}_n\stackrel{\mathit{d}}{⟶}Tz$. On the other hand, $(T{u}_n=x_{i\left(n\right)+1};n\ge 0)$ is a subsequence of $(x_n;n\ge 0)$; so that $T{u}_n\stackrel{\mathit{d}}{⟶}z$. Combining with d=separated, yields $d(z,Tz)=0$; whence, $z=Tz$; as desired. □

Remark 4. 

By a simple verification of the above conditions, it follows that

• ((K-asy-ms)) ⊆ ((K-asy-rms)); hence, ((B-cp)) ⊆ ((K-asy-rms));

just take $(\nabla )$ as $X\times X$ in the relational statement to verify this. The question of the reciprocal inclusion ((K–asy-rms)) ⊆ ((B-ms)) being also true is open; we conjecture that the answer is negative.

3. Functional Contractions

Let $(X,d)$ be a metric space. Further, let $(\nabla )$ be a relation over X, and $(\perp )$ stand for its reflexive symmetric cover; the quadruple $(X,d,\nabla ,\perp )$ will then be called a bi-relational metric space.

The following statement, referred to as Jachymski functional contraction principle in bi-relational metric spaces (in short: (J-fct-2rms)) is our starting point.

Theorem 4. 

Let the structure $(X,d,\nabla ,\perp )$ and the selfmap T of X be such that

• (c01) X is $(\perp )$-connected and T is $(\nabla )$-increasing,
• (c02) T is $(d,\nabla ;\phi )$-contractive, for some $\phi \in {\mathcal{F}}_0(in,re;Bro)\left(R_{+}\right)$.

Further, assume that one of the alternatives below is holding

• (c03) X is d-complete and T is d-continuous,
• (c04) X is $(\nabla ,d)$-complete, $(\nabla )$ is d-almost-selfclosed,
  and T is $(\nabla )$-semi-progressive.

Then, T is global strongly Picard (modulo d).

Technically speaking, we have a couple of results:

• (Res-1) (J-fct-2rms-cont)= the (c03) part of (J-fct-2rms).
• (Res-2) (J-fct-2rms-sclo)= the (c04) part of (J-fct-2rms).

The former of these, (J-fct-2rms-cont), is a function relational extension of a statement in Ran and Reurings [14], established on ordered metric spaces. The latter one, (J-fct-2rms-sclo), may be viewed as a function version of the 2008 statement in Jachymski ([17], Theorem 3.2), established over graph metric spaces. Note that, according to the developments in Roldán and Shahzad [19], the graph setting may be directly converted into a relational one.

Concerning the status of these results, the following is valid.

Proposition 1. 

Under these conventions, we have

• ((J-fct-2rms-cont)) ⊆ ((K-asy-ms)), ((J-fct-2rms-sclo)) ⊆ ((K-asy-rms)).

Proof. 

There are three steps to be passed.

Step 1. First, we deduce the strongly d-asymptotic property of T.

Let $x,y\in X$ be arbitrary fixed. As X is $(\perp )$-connected, there exists a k-dimensional $(\perp )$-chain $A=(z_1,...,z_k)\in X^k$ (where $k\ge 2$), joining x and y. Further, as T is $(\nabla )$-increasing (hence, $(\perp )$-increasing), one has

• ($\forall n$): $T^n{z}_i\perp T^n{z}_{i+1},\forall i\in \{1,...,k-1\}$; so, $T^n\left(A\right)=(T^n{z}_1,...,T^n{z}_k)\in X^k$ is a k-dimensional $(\perp )$-chain joining $T^{n}x$ and $T^{n}y$.

Moreover, as T is $(d,\nabla ;\phi )$-contractive (hence, $(d,\perp ;\phi )$-contractive)

• ($\forall n$): $d(T^n{z}_i,T^n{z}_{i+1})\le {\phi }^n\left(d(z_i,z_{i+1})\right),\forall i\in \{1,...,k-1\}$;
  so that: $d(T^{n}x,T^{n}y)\le {\sum }_{i=1}^{k-1}{\phi }^n\left(d(z_i,z_{i+1})\right)$.

As $\phi$ is Browder admissible, the described property follows.

Step 2. Suppose that (c03) is accepted. Then, $(X,T)$ is d-Kirk; and the desired conclusion of (J-fct-2rms-cont) is a direct consequence of (K-asy-ms).

Step 3. Suppose that (c04) holds. By the choice of $\phi$ (and a previous fact)

• (T is $(d,\nabla ;\phi )$-contractive) ⟹ (T is $(d,\nabla )$-nonexpansive)
  ⟹ T is left $(\nabla ,d)$-continuous.

Then, $(X,T)$ is $(\nabla ,d)$-Kirk; and the conclusion of (J-fct-2rms-sclo) is a direct consequence of (K-asy-rms). □

A natural problem is that of the functional result above being retained when the function $\phi \in {\mathcal{F}}_0(in,re)\left(R_{+}\right)$ is Matkowski admissible. Note that, in the genuine relational context, this is not, in general, possible. However, when $(\nabla )$ is a quasi-order (reflexive transitive relation) some positive answers to this problem are available. A basic statement of this type is that obtained in Agarwal et al. [16]. Further aspects of implicit nature may be found in Turinici [25], and the references therein.

4. Linear rs-Relational Case

Let $(X,d)$ be a metric space. Further, take a relation $(\nabla )$ on X, and let $(\perp )$ stand for its reflexive symmetric cover; then, $(X,d,\nabla ,\perp )$ will be referred to as a bi-relational metric space.

As precise, the couple of functional results [(J-fct-2rms-cont), (J-fct-2rms-sclo)] is reducible to the couple of simpler results [(K-asy-ms),(K-asy-rms)]. However, since none of these last principles is deductible from (B-cp), it follows that a similar conclusion is to be derived for our initial couple of results. This remains valid even if one passes to their linear versions:

• (lin-1) (J-lin-2rms-cont)=the result (J-fct-2rms-cont) where $\phi$=linear;
• (lin-2) (J-lin-2rms-sclo)=the result (J-fct-2rms-sclo) where $\phi$=linear.

However, in the particular case of

• $(\nabla )$ is reflexive symmetric (hence, identical with $(\perp )$)

the reduction to (B-cp) of the second statement above is possible, over the triple $(X,d,\perp )$, referred to as an rs-relational metric space. In fact, this reduction process involves a weaker form of the underlying statement. The precise formulation of this result, referred to as Jachymski linear contraction principle on rs-relational metric spaces (in short: (J-lin-rms)) is given below.

Theorem 5. 

Let the rs-relational metric space $(X,d,\perp )$ and the selfmap T of X be taken as (under the proposed conventions)

• (d01) X is $(\perp )$-connected and T is $(\perp )$-increasing;
• (d02) X is $(\perp ,d)$-complete and $(\perp )$ is d-almost-selfclosed;
• (d03) T is $(d,\perp ;\lambda )$-contractive, for some $\lambda \in [0,1]$.

Then, T is global strongly Picard (modulo d).

Concerning the status of this principle, the following is valid.

Proposition 2. 

Under the precise context, we have

• ((B-cp)) ⊆ ((J-lin-rms)) ⊆ ((B-cp)).

Hence, the statements (B-cp) and (J-lin-rms) are mutually equivalent.

Proof. 

The first inclusion is clear: just take $(\perp )$ as $X\times X$ in (J-lin-rms) to verify this. It remains now to establish that the second inclusion holds too. Let the conditions of (J-lin-rms) be accepted.

We introduce a pseudometric $e:X\times X\to R_{+}$ as: for each $x,y\in X$,

• $e(x,y)$ = the infimum of all $\Lambda \left(A\right)=d(z_1,z_2)+ ... +d(z_{k-1},z_k)$,
• where $A=(z_1,...,z_k)\in X^k$ (for $k\ge 2$) is a $(\perp )$-chain between x and y.

The definition is consistent, in view of (X is $(\perp )$-connected).

(I) Clearly, e is reflexive [$e(x,x)=0$, $\forall x\in X$], triangular [$e(x,z)\le e(x,y)+e(y,z)$, $\forall x,y,z\in X$], and symmetric [$e(y,y)=e(y,x)$, $\forall x,y\in X$]. In addition, the triangular property of d gives

• $d(x,y)\le \Lambda \left(A\right)=d(z_1,z_2)+...+d(z_{k-1},z_k)$,
  for any $(\perp )$-chain $A=(z_1,...,z_k)\in X^k$ (where $k\ge 2$) between x and y.

So, passing to infimum, yields

• (sub) $d(x,y)\le e(x,y)$, $\forall x,y\in X$ (meaning: d is subordinated to e).

Note that, as a direct consequence, e is sufficient [$e(x,y)=0$⟹$x=y$]; hence, it is a (standard) metric on X. Finally, by the very definition of e, one has

• (perp-id) $d(x,y)\ge e(x,y)$ (hence $d(x,y)=e(x,y)$), whenever $x\perp y$.

(II) We claim that X is e-complete. Let $\left(x_n\right)$ be an e-Cauchy sequence in X. By definition, there exists a strictly ascending sequence of ranks $\left(k\right(n);n\ge 0)$, with

• ($\forall n$): $k\left(n\right)<m$⟹$e(x_{k\left(n\right)},x_m)<2^{-n}$.

Denoting $(y_n:=x_{k\left(n\right)},n\ge 0)$, we therefore have ($e(y_n,y_{n+1})<2^{-n}$, $\forall n$). Moreover, by the imposed e-Cauchy property, $\left(x_n\right)$ is e-convergent iff so is $\left(y_n\right)$. To establish this last property, one may proceed as follows. As $e(y_0,y_1)<2^{-0}$, there exists (for the starting rank $p\left(0\right)=0$) a $(\perp )$-chain $(z_{p\left(0\right)},...,z_{p\left(1\right)})$ between $y_0$ and $y_1$ (hence $p\left(1\right)-p\left(0\right)\ge 1$, $z_{p\left(0\right)}=y_0$, $z_{p\left(1\right)}=y_1$), such that

• $d(z_{p\left(0\right)},z_{p\left(0\right)+1})+...+d(z_{p\left(1\right)-1},z_{p\left(1\right)})<2^{-0}$.

Further, as $e(y_1,y_2)<2^{-1}$, there exists a $(\perp )$-chain $(z_{p\left(1\right)},...,z_{p\left(2\right)})$ between $y_1$ and $y_2$ (hence $p\left(2\right)-p\left(1\right)\ge 1$, $z_{p\left(1\right)}=y_1$, $z_{p\left(2\right)}=y_2$), such that

• $d(z_{p\left(1\right)},z_{p\left(1\right)+1})+...+d(z_{p\left(2\right)-1},z_{p\left(2\right)})<2^{-1}$;

and so on. The procedure may continue indefinitely; it gives us a $(\perp )$-ascending sequence $(z_n;n\ge 0)$ in X with (according to (perp-id))

• (de-Ca) ${\sum }_{n}e(z_n,z_{n+1})={\sum }_{n}d(z_n,z_{n+1})<{\sum }_n{2}^{-n}<\infty$;
  so that: $\left(z_n\right)$ is both e-Cauchy and d-Cauchy.

By the second property above, one gets (as X is $(\perp ,d)$-complete)

• (d-conv) $z_n\stackrel{\mathit{d}}{⟶}z$ as $n\to \infty$, for some $z\in X$.

Combining with ($(\perp )$ is d-almost-selfclosed), there must be a subsequence $(t_n:=z_{q\left(n\right)};n\ge 0)$ of $(z_n;n\ge 0)$ with ($t_n\perp z$, $\forall n$). This firstly gives (by (d-conv)) $t_n\stackrel{\mathit{d}}{⟶}z$ as $n\to \infty$. Secondly, (again via (perp-id))

• ($e(t_n,z)=d(t_n,z)$, $\forall n$); so that: $t_n\stackrel{\mathit{e}}{⟶}z$ as $n\to \infty$
  (if we remember the above d-convergence property of $\left(t_n\right)$).

On the other hand, as already noted in (de-Ca), $\left(z_n\right)$ is e-Cauchy. Adding the e-convergence property of $\left(t_n\right)$ gives $z_n\stackrel{\mathit{e}}{⟶}z$ as $n\to \infty$; wherefrom (as $z_{p\left(n\right)}=y_n,n\ge 0$), $y_n\stackrel{\mathit{e}}{⟶}z$ as $n\to \infty$; and our claim follows.

(III) Given $x,y\in X$, let $(z_1,...,z_k)\in X^k$ (where $k\ge 2$) be a $(\perp )$-chain connecting them (possible, in view of X is $(\perp )$-connected). As T is $(\perp )$-increasing, $(T{z}_1,...,T{z}_k)\in X^k$ is a $(\perp )$-chain between $Tx$ and $Ty$. Taking the contractive condition into account, gives

• $e(Tx,Ty)\le {\sum }_{i=1}^{k-1}d(T{z}_i,T{z}_{i+1})\le \lambda {\sum }_{i=1}^{k-1}d(z_i,z_{i+1})$,

for all such $(\perp )$-chains; wherefrom, passing to infimum, $e(Tx,Ty)\le \lambda e(x,y)$. This, by the arbitrariness of $(x,y)$, tells us that T is $(e;\lambda )$-contractive.

(IV) Summing up, (B-cp) is applicable to $(X,e)$ and T; so that, by its conclusion

• T is fix-asingleton and strongly Picard (modulo e).

The second part of this tells us that

• for each $x_0\in X$, there exists $z\in \mathrm{Fix}\left(T\right)$ such that the iterative sequence $(x_n=T^n{x}_0;n\ge 0)$ fulfills $e(x_n,z)\to 0$ as $n\to \infty$; so that (as d is subordinated to e) we necessarily have $d(x_n,z)\to 0$ as $n\to \infty$.

Putting these together, we are finished. □

In the following, two basic particular cases of this result are given.

(Part-Case-1) Let $(X,d)$ be a metric space. Given $\epsilon >0$, let $\left[\epsilon \right]$ stand for the relation over X introduced as

• ($x,y\in X$): $x\left[\epsilon \right]y$ iff $d(x,y)<\epsilon$; clearly, $\left[\epsilon \right]$ is reflexive and symmetric.

Some basic properties of this relation are contained in

Proposition 3. 

Under the introduced setting, the following assertions hold

• (Aser-1) For each selfmap T of X and each $\lambda \in [0,1]$:
  T is $(d,[\epsilon ];\lambda )$-contractive implies T is $\left[\epsilon \right]$-increasing
• (Aser-2) X is $\left(\right[\epsilon ],d)$-complete iff X is d-complete
• (Aser-3) $\left[\epsilon \right]$ is d-almost-selfclosed.

Proof. 

(i) Let $x,y\in X$ be such that $x\left[\epsilon \right]y$; that is: $d(x,y)<\epsilon$. By the contractive property, we derive

• $d(Tx,Ty)\le \lambda d(x,y)\le \lambda \epsilon <\epsilon$; meaning: $Tx\left[\epsilon \right]Ty$;

and the conclusion follows.

(ii) The right-to-left inclusion is clear. For the converse inclusion, let $\left(x_n\right)$ be a d-Cauchy sequence in X. From the imposed condition, we have that, for our fixed $\epsilon >0$, there must be some index $k=k\left(\epsilon \right)$, with

• $d(x_n,x_m)<\epsilon$ (hence: $x_n\left[\epsilon \right]x_m$), whenever $k\le n\le m$.

The translated subsequence $(y_n=x_{k+n};n\ge 0)$ of $(x_n;n\ge 0)$ is thus $\left[\epsilon \right]$-ascending and d-Cauchy. By the starting hypothesis, $\left(y_n\right)$ is d-convergent; and then, as $\left(x_n\right)$ is d-Cauchy, we have that $\left(x_n\right)$ is d-convergent; as desired.

(iii) Let $\left(x_n\right)$ be a sequence in X with

• $\left(x_n\right)$ is $\left[\epsilon \right]$-ascending and $x_n\stackrel{\mathit{d}}{⟶}z$ as $n\to \infty$.

By the convergence property, $\left(x_n\right)$ is d-Cauchy. According to definition, there must be some index $q=q\left(\epsilon \right)$, such that

• $q\le m\le n$ implies $d(x_m,x_n)<\epsilon /2$.

Passing to limit as $n\to \infty$, gives

• ($\forall m\ge q$): $d(x_m,z)\le \epsilon /2<\epsilon$; hence, $x_m\left[\epsilon \right]z$.

But then, the translated subsequence $(y_n=x_{q+n};n\ge 0)$ of $(x_n;n\ge 0)$ fulfills $\left(y_n\right)\left[\epsilon \right]z$; and, from this, all is clear. □

Putting these together, the following statement, referred to as Edelstein contraction principle (in short: (E-cp)) is entering into our discussion.

Theorem 6. 

Let the metric space $(X,d)$, the number $\epsilon >0$, and the selfmap T of X be taken as (under the proposed conventions)

• (d04) X is $\left[\epsilon \right]$-connected, d-complete
• (d05) T is $(d,[\epsilon ];\lambda )$-contractive, for some $\lambda \in [0,1]$.

Then, T is global strongly Picard (modulo d).

Technically speaking, this statement is nothing else than the 1961 one formulated by Edelstein [21]. So, we may ask what is our motivation in rephrasing it. An appropriate answer to this problem is related to the existential status of Edelstein contraction principle, expressed as below.

Proposition 4. 

Under the precise context,

• ((B-cp)) ⊆ ((E-cp)) ⊆ ((J-lin-rms)) ⊆ ((B-cp)).

Hence, (B-cp), (E-cp), and (J-lin-rms) are mutually equivalent.

Proof. 

There are two steps to be passed.

Step 1. Clearly, ((E-cp)) ⊆ ((J-lin-rms)); just take $(\perp )$ as identical with $\left[\epsilon \right]$ in (J-lin-rms) to verify this. Moreover, by a preceding result, ((J-lin-rms)) ⊆ ((B-cp)).

Step 2. It remains now to establish that ((B-cp)) ⊆ ((E-cp)). Let the premises of (B-cp) be admitted:

• X is d-complete and T is $(d;\lambda )$-contractive, for some $\lambda \in [0,1]$.

Note that, as a direct consequence of these,

• T is strongly d-asymptotic: ${\sum }_{n}d(T^{n}x,T^{n}y)<\infty$, $\forall x,y\in X$.

So, to complete the argument, it will suffice proving that $\mathrm{Fix}\left(T\right)$ is nonempty. Fix some $x_0\in X$; and let the number $\rho >0$ be such that

• $d(x_0,T{x}_0)\le (1-\lambda )\rho$ (possible, if $\rho >0$ is large enough).

Then, denote $Y:=\{y\in X;d(x_0,y)\le \rho \}$. By the properties of d, one has

• (p-1) Y is d-bounded, in the sense: $\mathrm{diam}\left(Y\right)\le 2\rho$;
  hence, Y is (trivially) $\left[\epsilon \right]$-connected, for each $\epsilon >2\rho$
• (p-2) Y is d-closed (hence, d-complete)
• (p-3) $T\left(Y\right)\subseteq Y$ and T is $(d;\lambda )$-contractive on Y.

Summing up, (E-cp) is applicable to $(Y,d)$ and T; wherefrom, T (restricted to Y) is global strongly Picard (modulo d); hence, necessarily, $\mathrm{Fix}\left(T\right)\ne \varnothing$. □

(Part-Case-2) Let $(X,d)$ be a metric space; and $(\le )$ be a quasi-order (reflexive and transitive relation) over it; the triple $(X,d,\le )$ will be referred to as a quasi-ordered metric space. Denote

• ($x,y\in X$): $x<>y$ iff either $x\le y$ or $y\le x$ (x and y are comparable).

Clearly, this relation is reflexive and symmetric; but not, in general, transitive.

The following statement, referred to as Nieto–Lopez linear contraction principle on quasi-ordered metric spaces; in short: (NL-lin-qms) is coming into our discussion.

Theorem 7. 

Let the rs-relational metric space $(X,d,<>)$ and the selfmap T of X be taken as (under the proposed conventions)

• (d05) X is $(<>)$-connected and T is $(<>)$-increasing
• (d06) X is $(<>,d)$-complete and $(<>)$ is d-almost-selfclosed
• (d07) T is $(d,<>;\lambda )$-contractive, for some $\lambda \in [0,1]$.

Then, T is global strongly Picard (modulo d).

Concerning the status of this principle, the following is valid.

Proposition 5. 

Under the precise context, we have

• ((B-cp)) ⊆ ((NL-lin-qms)) ⊆ ((J-lin-rms)) ⊆ ((B-cp)).

Hence, (B-cp), (NL-lin-qms), and (J-lin-rms) are mutually equivalent.

Proof. 

The first inclusion is clear; just take $(<>)$ be identical with $X\times X$ in (NL-lin-qms) to verify this. Further, the second inclusion is again clear, by simply taking $(\perp )$ as $(<>)$ in (J-lin-rms). Finally, the third inclusion was already obtained in a preceding place. Putting these together, we are finished. □

Concerning the involved concepts, the following observations are useful

• (Obs-1) X is $(<>)$-connected when,
  for each $x,y\in X$, $\{x,y\}$ has lower and upper bounds
• (Obs-2) T is $(<>)$-increasing, whenever
  it is $(\le )$-monotone (in the sense: $(\le )$-increasing or $(\le )$-decreasing).

Note that, under these particular choices of our data, (NL-lin-qms) reduces to the 2005 result in Nieto and Rodriguez-Lopez [15]. In addition, it is worth noting that the regularity condition imposed by the quoted authors

• (s-prog) T is $(<>)$-semi-progressive
  ($X(T,<>):=\{x\in X;x<>Tx\}$ is nonempty)

is not necessary in our context. Further aspects of technical nature may be found in Turinici ([26], Section 24) and the references therein.

5. Parametric Relational Extensions

Let $(X,d)$ be a metric space. Define the function $F\in \mathcal{F}\left(R_{+}\right)$, as

• $F\left(t\right)=1$, if $0\le t\le (\sqrt{5}-1)/2$
  $F\left(t\right)=(1-t)t^{-2}$, if $(\sqrt{5}-1)/2\le t\le 2^{-1/2}$
  $F\left(t\right)={(1+t)}^{-1}$, if $2^{-1/2}\le t<\infty$;

clearly, F is continuous, decreasing, and $F\left(R_{+}\right)=[0,1]$. Given the selfmap $T:X\to X$, let us introduce the parametric relational contractive concepts

• (F;al) T is conditional $(F;\alpha )$-contractive (where $\alpha \in R_{+}$), provided
  ($x,y\in X$, $F\left(\alpha \right)d(x,Tx)\le d(x,y)$) implies $d(Tx,Ty)\le \alpha d(x,y)$
• (F;0,1) T is conditional $(F;0,1)$-contractive, provided
  the preceding condition holds, for some $\alpha \in [0,1]$.

(A) The following 2008 result in Suzuki [22], referred to as Suzuki parametric contraction principle (in short: (S-p-cp)) is our starting point.

Theorem 8. 

Let the metric space $(X,d)$ and the selfmap T of X be such that

• X is d-complete and T is conditional $(F;0,1)$-contractive.

Then, T is global strongly Picard (modulo d).

Note that, by the structure of our contractive condition, ((B-cp)) ⊆ ((S-p-cp)); but, the converse inclusion ((S-p-cp)) ⊆ ((B-cp)) is not available. For various extensions of this result, we refer to Altun and Erduran [27], Kikkawa and Suzuki [28], or Popescu [29]. Note that, in all these statements, the premise of our parametric contractive property (F;al) is “asymmetric” with respect to the couple $(x,y)$; so, it is natural to ask whether an alternate parametric condition is available, with a “dual” information about the variable y. It is our aim in the following to show that a positive answer to this is possible, within the class of quasi-ordered double metric spaces. Some other aspects will be discussed elsewhere.

(B) Let X be a nonempty set and $(d,e)$ be a couple of metrics on it; we then say that $(X,d,e)$ is a double metric space. As usual, denote by $\left(\stackrel{\mathit{d}}{⟶}\right)$ and $\left(\stackrel{\mathit{e}}{⟶}\right)$ the attached convergence structures on X; note that these are separated. Further, take a quasi-order $(\le )$ on X; and let $(<>)$ stand for its associated comparison relation.

Having these precise, let the selfmap $T:X\to X$ be given. As usual, we have to determine existence and uniqueness criteria concerning its fixed points set, $\mathrm{Fix}\left(T\right)$. To do this, we start from the basic hypothesis

• (incr) T is $(\le )$-increasing ($x\le y$ implies $Tx\le Ty$).

Note that the natural condition to be added here is

• (s-pro) T is $(\le )$-semi-progressive ($X(T,\le ):=\{x\in X;x\le Tx\}\ne \varnothing$).

But, even if this fails, our developments are working in a vacuous manner.

The basic directions under which the investigations be conducted are described in the list below.

(qpic-0) We say that T is fix-$(\le )$-asingleton, when $\mathrm{Fix}\left(T\right)$ is $(\le )$-asingleton; and fix-$(\le )$-singleton when $\mathrm{Fix}\left(T\right)$ is $(\le )$-singleton.

(qpic-1) We say that T is a Picard operator (modulo $(d,\le )$) if, for each $x\in X(T,\le )$, $(T^{n}x;n\ge 0)$ is d-Cauchy; and a global Picard operator (modulo $(d,\le )$) if, in addition, T is fix-$(\le )$-asingleton.

(qpic-2) We say that T is a strongly Picard operator (modulo $(e,\le )$) when, for each $x\in X(T,\le )$, $(T^{n}x;n\ge 0)$ is e-convergent with $e-{lim}_n\left(T^{n}x\right)$ belonging to $\mathrm{Fix}\left(T\right)$; and global strongly Picard operator (modulo $(e,\le )$) when, in addition, T is fix-$(\le )$-asingleton (or, equivalently: fix-$(\le )$-singleton).

The sufficient (regularity) conditions for such properties are being founded on the ascending concept we just introduced.

(Reg-1) Call X, $(\le ,d,e)$-complete provided (for each $(\le )$-ascending sequence): d-Cauchy ⟹ e-convergent.

(Reg-2) Call $(\le )$, e-selfclosed provided: $\left(z_n\right)$ is $(\le )$-ascending and $z_n\stackrel{\mathit{e}}{⟶}z$ imply $\left(z_n\right)\le z$.

(Reg-3) Call d, first variable left $(\le ,e)$-continuous when: ($\left(x_n\right)$ is $(\le )$-ascending, $x_n\stackrel{\mathit{e}}{⟶}z$, and $\left(x_n\right)\le x$) imply ($d(x_n,y)\to d(x,y)$, for all $y\in X$).

As a completion of these, we must formulate the conditional type metrical contractive property to be used. Define the function $G\in \mathcal{F}\left(R_{+}\right)$ as

• ($G\left(t\right)={(1+t)}^{-1}$, $t\in R_{+}$);
  note that G is continuous, decreasing, and $G\left(R_{+}\right)=[0,1]$.

Given $T\in \mathcal{F}\left(X\right)$, let us introduce the comparison parametric contractions

• (G;comp;al) T is conditional $(G;<>;\alpha )$-contractive (where $\alpha >0$), if ($x,y\in X$, $x<>y$, $G\left(\alpha \right)max\left\{d\right(x,Tx),d(y,Tx\left)\right\}\le d(x,y)$)
  implies $d(Tx,Ty)\le \alpha d(x,y)$
• (G;comp;0,1) T is conditional $(G;<>;0,1)$-contractive, when the preceding condition holds for at least one $\alpha \in [0,1]$.

(C) Under these preliminaries, we may now formulate the announced statement, referred to as: Suzuki parametric contraction principle in quasi-ordered double metric spaces (in short: (S-p-cp-q2ms)).

Theorem 9. 

Assume that the quasi-ordered double metric space $(X,d,e,\le )$ and the selfmap T of X are such that

• (e01) X is $(\le ,d,e)$-complete, and T is conditional $(G;<>;0,1)$-contractive
• (e02) $(\le )$ is e-selfclosed, and d is first variable left $(\le ,e)$-continuous.

Then T is global strongly Picard (modulo $(e,\le )$).

Proof. 

Let $\alpha \in [0,1]$ be the number appearing in the conditional $(G;<>;0,1)$-contractive property of T. There are three steps to be passed.

(I) We firstly show that T is fix-$(\le )$-asingleton. Let $x^{*},u^{*}\in \mathrm{Fix}\left(T\right)$ be such that $x^{*}\le u^{*}$. We have

• $d(x^{*},T{x}^{*})=0$, $d(u^{*},T{x}^{*})=d(u^{*},x^{*})$; hence (as $0<G\left(\alpha \right)\le 1$)
  $G\left(\alpha \right)max\{d(x^{*},T{x}^{*}),d(u^{*},T{x}^{*})\}\le d(x^{*},u^{*})$.

This, via contractive condition, yields $d(x^{*},u^{*})\le \alpha d(x^{*},u^{*})$; so that (as d=metric) $x^{*}=u^{*}$. It remains to show that T is strongly Picard (modulo $(e,\le )$). Let $x_0\in X(T,\le )$ be arbitrary fixed; and put $(x_n=T^{n}x;n\ge 0)$; clearly, $\left(x_n\right)$ is $(\le )$-increasing. By the definitions above (and $0<G\left(\alpha \right)\le 1$)

• ($\forall n$): $x_n\le x_{n+1},G\left(\alpha \right)max\{d(x_n,T{x}_n),d(x_{n+1},T{x}_n)\}\le d(x_n,x_{n+1})$;
  so (from the contractive condition), $d(x_{n+1},x_{n+2})\le \alpha d(x_n,x_{n+1})$.

Note that (in view of $0<\alpha <1$), this yields

• (e03) ${\sum }_{n}d(x_n,T{x}_n)={\sum }_{n}d(x_n,x_{n+1})\le \left({\sum }_n{\alpha }^n\right)d(x_0,x_1)<\infty$;

so that $\left(x_n\right)$ is d-Cauchy. As X is $(\le ,d,e)$-complete and $(\le )$ is e-selfclosed,

• $x_n\stackrel{\mathit{e}}{⟶}z$ and $\left(x_n\right)\le z$, for some $z\in X$.

(II) Suppose that our sequence is such that

• for each n, there exists $m>n$ with $x_m=z$.

It follows that a subsequence $(x_{p\left(n\right)};n\ge 0)$ of $\left(x_n\right)$ exists with ($x_{p\left(n\right)}=z$, $\forall n$). This, along with ($x_{p\left(n\right)+1}=Tz$, $\forall n$), gives (as $\left(\stackrel{\mathit{e}}{⟶}\right)$ is separated) $z=Tz$.

(III) Assume in the following that the opposite situation holds:

• there exists h such that: $x_n\ne z$, for all $n\ge h$.

Fix $k\ge h$; and put $x_k=u$; clearly, $u\le x_n\le z$, for all $n\ge k$. From (e03), $d(x_n,T{x}_n)\to 0$ as $n\to \infty$; moreover, as d is first variable left $(\le ,e)$-continuous, $d(x_n,u)\to d(z,u)>0$. Combining these with $\alpha \in [0,1]$, there must be some rank $p\ge k$ such that $d(x_n,T{x}_n)\le \alpha d(x_n,u)$, $\forall n\ge p$. On the other hand (for the same ranks) $d(u,T{x}_n)\le d(u,x_n)+d(x_n,T{x}_n)\le (1+\alpha )d(u,x_n)$; hence, summing up,

• $x_n\ge u,G\left(\alpha \right)max\{d(x_n,T{x}_n),d(u,T{x}_n)\}\le d(x_n,u)$, $\forall n\ge p$.

These, by the contractive condition, give $d(T{x}_n,Tu)\le \alpha d(x_n,u)$, $\forall n\ge p$; so that (passing to limit as $n\to \infty$), $d(z,Tu)\le \alpha d(z,u)$. By the triangle inequality $d(u,Tu)\le d(u,z)+d(z,Tu)\le (1+\alpha )d(u,z)$; wherefrom (putting these together)

• $u\le z$, $G\left(\alpha \right)max\left\{d\right(u,Tu),d(z,Tu\left)\right\}\le d(u,z)$;

so that (by the same contractive condition), $d(Tu,Tz)\le \alpha d(u,z)$. Taking into account the adopted notation, we have $d(T{x}_k,Tz)\le \alpha d(x_k,z)$, $\forall k\ge h$. Passing to limit as $k\to \infty$, one derives $z=Tz$; and the following conclusions. □

In particular, when $(\le )$ is the trivial quasi-order of X, the obtained result extends, in a partial way, (S-p-cp); this refers to the contractive condition in (S-p-cp-q2ms) having a dual character with respect to the involved variables. As before, by the parametric nature of our contractive condition, a comparison between (S-p-cp-qms) and (B-cp) is not available. Further aspects will be discussed elsewhere.

6. Conclusions

The main findings of this paper are the following:

(F-1) The 2008 fixed point result on rs-relational metric spaces due to Jachymski [17] is equivalent with the classical Banach Contraction Principle [2].

(F-2) This is also valid for the 1961 statement in metric spaces due to Edelstein [21], or the 2005 fixed point result in quasi-ordered metric spaces obtained by Nieto and Rodriguez-Lopez [15].

(F-3) The basic 2008 parametric contraction principle established by Suzuki may be extended to quasi-ordered double metric spaces.

From a methodological perspective, (F-1) is to be viewed as a practical illustration of the Roldán-Shahzad meta theorem (in short: (RS-meta)):

Theorem 10. 

The fixed point theory over graph metric spaces is completely reducible to the fixed point theory over relational metric spaces.

Unfortunately, this basic result—described in detail along the 2017 outstanding paper due to Roldán and Shahzad [19]—did not produce a reconsideration of the graph methods to be used. Perhaps, the present paper—in combination with another ones to be written—will generate, by means of (RS-meta), a critical mass for the long awaited unification of these theories.