Contractive Maps in Locally Transitive Relational Metric Spaces

Some fixed point results are given for a class of Meir-Keeler contractive maps acting on metric spaces endowed with locally transitive relations. Technical connections with the related statements due to Berzig et al. (2014) are also being discussed.


Introduction
Let be a nonempty set. Call the subset of , almostsingleton (in short: asingleton), provided 1 , 2 ∈ implies 1 = 2 and singleton if, in addition, is nonempty; note that, in this case, = { }, for some ∈ . Take a metric : × → + := [0, ∞[ over , as well as a self-map ∈ F( ). (Here, for each couple , of nonempty sets, F( , ) denotes the class of all functions from to ; when = , we write F( ) in place of F( , )). Denote Fix( ) = { ∈ ; = }; each point of this set is referred to as fixed under . Concerning the existence and uniqueness of such points, a basic result is the 1922 one due to Banach [1]. Call the self-map , ( ; )-contractive (where ≥ 0), if (a01) ( , ) ≤ ( , ), for all , ∈ . This result (referred to as: Banach's fixed point theorem) found some basic applications to the operator equations theory. As a consequence, a multitude of extensions for it were proposed. Here, we will be interested in the relational way of enlarging Theorem 1, based on contractive conditions like (a02) ( ( , ), ( , ), ( , ), ( , ), ( , ), ( , )) ≤ 0, for all , ∈ with R , where : 6 + → is a function, and R is a relation over . Note that, when R is the trivial relation (i.e., R = × ), a large list of such contractive maps is provided in Rhoades [2]. Further, when R is an order on , a first result is the 1986 one obtained by Turinici [3], in the realm of ordered metrizable uniform spaces. Two decades after, this fixed point statement was rediscovered (in the ordered metrical setting) by Ran and Reurings [4]; see also Nieto and Rodríguez-López [5]; and, since then, the number of such results increased rapidly. On the other hand, when R is an amorphous relation over , an appropriate statement of this type is the 2012 one due to Samet and Turinici [6]. The "intermediary" particular case of R being finitely transitive was recently obtained by Berzig and Karapınar [7], under a class of ( , )-contractive conditions suggested by Popescu [8]. It is our aim in the following to give further extensions of these results, when (i) the contractive conditions are taken after the model in Meir and Keeler [9]; (ii) the finite transitivity of R is being assured in a "local" way.
Further aspects will be delineated elsewhere.

Preliminaries
Throughout this exposition, the ambient axiomatic system is Zermelo-Fraenkel's (abbreviated ZF). In fact, the reduced system (ZF-AC + DC) will suffice; here, (AC) stands for the Axiom of Choice and (DC) for the Dependent Choice Principle. The notations and basic facts to be used in this reduced 2 The Scientific World Journal system are standard. Some important ones are described below.
(A) Let be a nonempty set. By a relation over , we mean any nonempty part R ⊆ × . For simplicity, we sometimes write ( , ) ∈ R as R . Note that R may be regarded as a mapping between and P( ) (= the class of all subsets in ). In fact, denote for ∈ : ( , R) = { ∈ ; R } (the section of R through ); then, the desired mapping representation is [R( ) = ( , R), ∈ ].
Among the classes of relations to be used, the following ones (listed in an "increasing" scale) are important for us: (P0) R is amorphous; that is, it has no specific properties at all; ], transitive [ R and R imply R ], and antisymmetric [ R and R imply = ]; (P2) R is a quasiorder; that is, it is reflexive and transitive; (P3) R is transitive (see above).
A basic ordered structure is ( , ≤); here, = {0, 1, . . .} is the set of natural numbers and (≤) is defined as ≤ if and only if + = , for some ∈ . For each ∈ (1, ≤), let ( , >) := {0, . . . , − 1} stand for the initial interval (in ) induced by . Any set with ∼ (in the sense: there exists a bijection from to ) will be referred to as effectively denumerable. In addition, given some natural number ≥ 1, any set with ∼ ( , >) will be said to be -finite; when is generic here, we say that is finite. Finally, the (nonempty) set is called (at most) denumerable if and only if it is either effectively denumerable or finite.
Given the relations R, S over , define their product R ∘ S as This allows us to introduce the powers of a relation R as (Here, I = {( , ); ∈ } is the identical relation over ).
The following properties will be useful in the sequel: Given ∈ (2, ≤), let us say that R is -transitive, if R ⊆ R; clearly, transitive is identical with 2-transitive. We may now complete the increasing scale above as (P4) R is finitely transitive; that is, R is -transitive for some ≥ 2; (P5) R is locally finitely transitive; that is, for each (effectively) denumerable subset of , there exists = ( ) ≥ 2, such that the restriction to of R istransitive; Concerning these concepts, the following property will be useful. Call the sequence ( ; ≥ 0) in , R-ascending, if R +1 for all ≥ 0.

Lemma 2.
Let the R-ascending sequence ( ; ≥ 0) in and the natural number ≥ 2 be such that Then, necessarily, Proof. We will use the induction with respect to . First, by the choice of our sequence, ( , +1 ) ∈ R; whence, the case = 0 holds. Moreover, by definition, ( , + ) ∈ R ; and this, along with the -transitive property, gives ( , + ) ∈ R; hence, the case of = 1 holds too. Suppose that this property holds for some ≥ 1; we claim that it holds as well for + 1.
In fact, let ≥ 0 be arbitrary fixed. Again by the choice of our sequence, ( +1+ ( −1) , +1+( +1)( −1) ) ∈ R −1 , so that, by the inductive hypothesis (and properties of relational product): and this, along with the -transitive condition, gives ( , The proof is thereby complete. (B) Let ( , ) be a metric space. We introduce aconvergence and -Cauchy structure on as follows. By a sequence in , we mean any mapping : → . For simplicity reasons, it will be useful to denote it as ( ( ); ≥ 0) or ( ; ≥ 0); moreover, when no confusion can arise, we further simplify this notation as ( ( )) or ( ), respectively. Also, any sequence ( := ( ) ; ≥ 0) with ( ) → ∞ as → ∞ will be referred to as a subsequence of ( ; ≥ 0). Given the sequence ( ) in and the point ∈ , we say that ( ), -converges to (written as: → ) provided ( , ) → 0 as → ∞; that is, The set of all such points will be denoted lim ( ); note that it is an asingleton, because is triangular symmetric; if lim ( ) is nonempty, then ( ) is called -convergent.
We stress that the introduced convergence concept (→) does match the standard requirements in Kasahara [10]. Further, call the sequence ( ), -Cauchy when ( , ) → 0 as , → ∞, < ; that is, As is triangular symmetric, any -convergent sequence is -Cauchy too; but, the reciprocal is not in general true. Concerning this aspect, note that any -Cauchy sequence ( ; ≥ 0) is -semi-Cauchy; that is, But the reciprocal is not in general true. The introduced concepts allow us to give a useful property.
The Scientific World Journal As a consequence, this map is -continuous; that is, The proof is immediate, by the usual properties of the ambient metric (⋅, ⋅); we do not give details.
(C) Let ( , ) be a metric space; and let R ⊆ × be a (nonempty) relation over ; the triple ( , , R) will be referred to as a relational metric space. Further, take some ∈ F( ). Call the subset of , R-almost-singleton (in short: R-asingleton) provided 1 , 2 ∈ , 1 R 2 ⇒ 1 = 2 and R-singleton when, in addition, is nonempty. We have to determine circumstances under which Fix( ) is nonempty; and, if this holds, to establish whether is fix-R-asingleton (i.e., Fix( ) is R-asingleton) or, equivalently, is fix-Rsingleton (in the sense: Fix( ) is R-singleton); to do this, we start from the working hypotheses: The basic directions under which the investigations be conducted are described by the list below, comparable with the one in Turinici [11]: (2a) We say that is a Picard operator (modulo ( , R)) if, for each ∈ ( , R), ( ; ≥ 0) is -convergent.
(2c) We say that is a globally strong Picard operator (modulo ( , R)) when it is a strong Picard operator (modulo ( , R)) and is fix-R-asingleton (hence, fix-R-singleton).
The sufficient (regularity) conditions for such properties are being founded on ascending orbital concepts (in short: (ao)-concepts). Remember that the sequence ( ; ≥ 0) in is called R-ascending, if R +1 for all ≥ 0; further, let us say that ( ; ≥ 0) is -orbital, when it is a subsequence of ( ; ≥ 0), for some ∈ ; the intersection of these notions is just the precise one. When the orbital properties are ignored, these conventions give us ascending notions (in short: a-notions). On the other hand, when the ascending properties are ignored, the same conventions give us orbital notions (in short: onotions). The list of these is obtainable from the previous one; so, further details are not needed. Finally, when R = × , the list of such notions is comparable with the one in Rus ([12], Ch 2, Section 2.2): because, in this case, ( , R) = .

Meir-Keeler Contractions
Let ( , , R) be a relational metric space; and let be a self-map of , supposed to be R-semi-progressive and R-increasing. The basic directions and sufficient regularity conditions under which the problem of determining the fixed points of is to be solved were already listed. As a completion of them, we must formulate the specific metrical contractive conditions upon our data. These, essentially, consist in a "relational" variant of the Meir-Keeler condition [9]. Assume that Note that, by definition, the introduced relation writes Note that, for each ∈ G, we have The former of these will be referred to as is sufficient; note that, by the properties of , we must have , ∈ ,R ⇒ ( , ) > 0.
And the latter of these means that is diameter bounded.  The Scientific World Journal Note that, by the former of these, the Meir-Keeler property may be written as In the following, two basic examples of such contractions will be given.
Some important classes of such functions are given below.
(This convention is related to the developments in Boyd and Wong [13]; we do not give details). In particular, ∈ F( )( + ) is Boyd-Wong admissible provided it is upper semicontinuous at the right on 0 + : (or, equivalently : Λ + ( ) ≤ ( )) , Note that this is fulfilled when is continuous at the right on Proof (sketch). The former of these is an immediate consequence of definition. And the second one is to be found in Jachymski [15]. Here, given the sequence ( ; ≥ 0) in and the point ∈ , we denoted → + (resp., → + +), if → and ≥ (resp., > ), for all ≥ 0 large enough.

Main Result
Let ( , , R) be a relational metric space. Further, let be a self-map of , supposed to be R-semi-progressive and R-increasing. The basic directions and regularity conditions under which the problem of determining the fixed points of is to be solved, were already listed; and the contractive type framework was settled. It remains now to precise the regularity conditions upon R. Denote, for each ∈ ( , R), Clearly, 1 ∈ spec( ), but the possibility of spec( ) = {1} cannot be removed. This fact remains valid even if ∈ ( , R) is orbital admissible, in the sense [ ̸ = implies ̸ = ], when the associated orbit := { ; ≥ 0} is effectively denumerable. But for the developments below, it is necessary that these spectral subsets of should have a finite Hausdorff-Pompeiu distance to ; hence, in particular, these must be infinite. Precisely, given ≥ 1, let us say that R is -semirecurrent at the orbital admissible ∈ ( , R), if for each ∈ (1, ≤), there exists ∈ spec( ) such that ≤ < + .
A global version of this convention is the following: call R, finitely semirecurrent if, for each orbital admissible ∈ ( , R), there exists ( ) ∈ (1, ≤), such that R is ( )semirecurrent at .
Assume in the following that (d01) R is finitely semirecurrent and nonidentical.
Our main result in this exposition is the following.
Then is a globally strong Picard operator (modulo ( , R)).
In particular, when R is transitive, this result is comparable with the one in Turinici [11]. Note that further extensions of these facts are possible, in the realm of triangular symmetric spaces, taken as in Hicks and Rhoades [16]; or, in the setting of partial metric spaces, introduced under the lines in Matthews [17]; we will discuss them elsewhere.

Further Aspects
Let in the following ( , , R) be a relational metric space; and let be a self-map of . Technically speaking, Theorem 7 that we just exposed consists of three substatements; according to the alternatives of our main result we already listed. For both practical and theoretical reasons, it would be useful to evidentiate them; further aspects involving the obtained facts are also discussed.
So, even if the restriction of R to ( , R) is arbitrarily taken, (d01) may hold. On the other hand, (e01) cannot hold whenever ( , R) admits a denumerable subset such that the restriction of R to is not finitely transitive; and this proves our assertion.
We may now pass to the particular cases of Theorem 7 with practical interest. Case 1. As a direct consequence of Theorem 7, we get the following.
The following particular cases of this result are to be noted. . Then, if we take R := S and = 1 , the alternative (i1) of Theorem 8 includes the related statement in Berzig and Rus [18]. By the previous remark, this inclusion is-at least from a technical viewpoint-effective, but, from a logical perspective, it is possible that the converse inclusion be also true. Finally, the alternative (i2) of Theorem 8 seems to be new.
(1-2) Suppose that R = × (i.e., R is the trivial relation over ). Then, Theorem 8 is comparable with the main results in Włodarczyk and Plebaniak [19][20][21][22], based on contractive type conditions involving generalized pseudodistances. However, none of these is reducible to the remaining ones; we do not give details.

Case 2.
As another consequence of Theorem 7, we have the following statement (with practical value).

Theorem 9.
Assume that is R-semiprogressive, R-increasing, and ( , R; , )-contractive, for some ∈ G and a certain Meir-Keeler admissible function ∈ F( )( + ). In addition, let R be finitely semirecurrent nonidentical, be (a-o, )complete, and one of the conditions below holds: Then is a globally strong Picard operator (modulo ( , R)).
The following particular cases of this result are to be noted.
(Here, CB( ) is the class of all (nonempty) closed bounded subsets of .) Clearly, this condition is stronger than the one we already used in Theorem 9. On the other hand, (e03) is written in terms of generalized pseudodistances. Hence, direct inclusions between these results are not in general available; we do not give details.
Case 3. As a final consequence of Theorem 7, we have Theorem 10. Assume that the self-map is R-semiprogressive, R-increasing, and ( , R; , ( , ))-contractive, for a certain ∈ G and some pair ( , ) of generalized altering functions in F( + ). In addition, let R be finitely semirecurrent nonidentical, be (a-o, )-complete, and one of the conditions below holds: Then is a globally strong Picard operator (modulo ( , R)).
The following particular cases of this result are to be noted.
referred to as is ( , )-contractive. In particular, when = 1 , this last result reduces to the one in Berzig and Karapınar [7]; which, in turn, extends the one due to Samet et al. [29]; hence, so does Theorem 10 above.
It is to be stressed that this last construction may be also attached to the setting of Case 2. Then, the corresponding version of Theorem 9 extends in a direct way some basic results in Kirk et al. [30].
Finally, we should remark that none of these particular theorems may be viewed as a genuine extension for the fixed point statement due to Samet and Turinici [6]; because, in the quoted paper, R is not subjected to any kind of (local or global) transitive type requirements. Further aspects (involving the same general setting) may be found in Berzig [31].

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.