#### Abstract

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.

#### 1. Introduction

Let be a nonempty set. Call the subset of ,* almost-singleton* (in short:* asingleton*), provided implies and* singleton* if, in addition, is nonempty; note that, in this case, , for some . Take a* metric* over , as well as a self-map . (Here, for each couple , of nonempty sets, denotes the class of all* functions* from to ; when , we write in place of ). Denote ; 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 ), if(a01), for all .

Theorem 1. *Assume that is -contractive, for some . In addition, let be -complete. Then,*(i)* is a singleton, ;*(ii)* as , for each .*

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), , , , , ≤ , for all with , where is a function, and is a* relation* over . Note that, when is the* trivial* relation (i.e., ), a large list of such contractive maps is provided in Rhoades [2]. Further, when 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 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 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 is being assured in a “local” way. Further aspects will be delineated elsewhere.

#### 2. 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 system are standard. Some important ones are described below.

(A) Let be a nonempty set. By a* relation* over , we mean any nonempty part . For simplicity, we sometimes write as . Note that may be regarded as a mapping between and (= the class of all subsets in ). In fact, denote for : (the* section* of through ); then, the desired mapping representation is [, ].

Among the classes of relations to be used, the following ones (listed in an “increasing” scale) are important for us:(P0) is* amorphous*; that is, it has no specific properties at all;(P1) is an* order*; that is, it is* reflexive* [, ],* transitive* [ and imply ], and* antisymmetric* [ and imply ];(P2) is a* quasiorder*; that is, it is reflexive and transitive;(P3) is transitive (see above).

A basic ordered structure is ; here, is the set of natural numbers and is defined as if and only if , for some . For each , let 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 , 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 , over , define their* product* as(b01) if there exists with , . This allows us to introduce the powers of a relation as (b02), , . (Here, is the* identical relation* over ). The following properties will be useful in the sequel:

Given , let us say that is -*transitive*, if ; clearly,* transitive* is identical with* 2-transitive*. We may now complete the increasing scale above as(P4) is* finitely transitive*; that is, is -transitive for some ;(P5) is* locally finitely transitive*; that is, for each (effectively) denumerable subset of , there exists , such that the restriction to of is -transitive;(P6) is* trivial*; that is, ; hence, [, ].

Concerning these concepts, the following property will be useful. Call the sequence in , -*ascending*, if for all .

Lemma 2. *Let the -ascending sequence in and the natural number be such that *(b03)* is -transitive on .** Then, necessarily,
*

*Proof. *We will use the induction with respect to . First, by the choice of our sequence, ; whence, the case holds. Moreover, by definition, ; and this, along with the -transitive property, gives ; hence, the case of holds too. Suppose that this property holds for some ; we claim that it holds as well for . In fact, let be arbitrary fixed. Again by the choice of our sequence, , 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 a -convergence 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 or ; moreover, when no confusion can arise, we further simplify this notation as or , respectively. Also, any sequence with as will be referred to as a* subsequence* of . Given the sequence in and the point , we say that ,-*converges* to (written as: ) provided as ; that is,
The set of all such points will be denoted ; note that it is an asingleton, because is triangular symmetric; if 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 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 is -*semi-Cauchy*; that is,
But the reciprocal is not in general true.

The introduced concepts allow us to give a useful property.

Lemma 3. *The mapping is -Lipschitz, in the sense
**
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 be a (nonempty) relation over ; the triple will be referred to as a* relational metric space*. Further, take some . Call the subset of , -*almost-singleton* (in short: -*asingleton*) provided , and -*singleton* when, in addition, is nonempty. We have to determine circumstances under which is nonempty; and, if this holds, to establish whether is* fix*--*asingleton* (i.e., is -asingleton) or, equivalently, is* fix*--*singleton* (in the sense: is -singleton); to do this, we start from the working hypotheses:(b04) is -semi-progressive: ;(b05) is -increasing: implies .

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 ) if, for each , is -convergent.(2b)We say that is a* strong Picard operator* (modulo ) when, for each , is -convergent and .(2c)We say that is a* globally strong Picard operator* (modulo ) when it is a strong Picard operator (modulo ) and is fix--asingleton (hence, fix--singleton).

The sufficient (regularity) conditions for such properties are being founded on* ascending orbital* concepts (in short: (a-o)-concepts). Remember that the sequence in is called -*ascending*, if for all ; further, let us say that is -*orbital*, when it is a subsequence of , for some ; the intersection of these notions is just the precise one.(2d)Call , a-o,-*complete*, provided (for each (a-o)-sequence) -Cauchy -convergent.(2e)We say that is a-o,-*continuous*, if (=(a-o)-sequence and ) imply .(2f)Call , a-o,-*almost-self-closed*, if: whenever the (a-o)-sequence in and the point fulfill , there exists a subsequence of with , for all .

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: o-notions). The list of these is obtainable from the previous one; so, further details are not needed. Finally, when , the list of such notions is comparable with the one in Rus ([12], Ch 2, Section 2.2): because, in this case, .

#### 3. Meir-Keeler Contractions

Let be a relational metric space; and let be a self-map of , supposed to be -semi-progressive and -increasing. The basic directions and sufficient regularity conditions under which the problem of determining the fixed points ofis 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(c01) is* nonidentical*: is nonempty. Note that, by definition, the introduced relation writes (c02) if and only if and ; so, is* irreflexive* [ is false, for each ]. Denote for (c03), ,
,
. Then, let us introduce the functions (c04), , , , , , , .Note that, for each , we have
The former of these will be referred to as is* sufficient*; note that, by the properties of , we must have
And the latter of these means that is* diameter bounded*.

Given , we say that is* Meir-Keeler* -*contractive*, if(c05) implies , expressed as is strictly -nonexpansive;(c06)for all , : [, ] , expressed as has the Meir-Keeler property (modulo ). Note that, by the former of these, the Meir-Keeler property may be written as (c07)for all , : [, ] .

In the following, two basic examples of such contractions will be given.

(A) Let stand for the class of all with the (strong)* regressive* property: [; , for all ]. We say that is* Meir-Keeler admissible*, if(c08)for all , , : ; or, equivalently: for all , , : . Now, given , , call ,*-contractive*, if (c09), for all , .

Lemma 4. *Assume that is -contractive, where is Meir-Keeler admissible. Then, is Meir-Keeler -contractive.*

*Proof. *(i) Let be such that . The contractive condition, and regressiveness of , yield , so that, is strictly -nonexpansive.

(ii) Let be arbitrary fixed; and be the number assured by the Meir-Keeler admissible property of . Further, let be such that and . By the contractive condition and admissible property,
so that has the Meir-Keeler property (modulo ).

Some important classes of such functions are given below.(I)For any and any , put(c10), where ;(c11). By this very definition, we have the representation (for all ) From the regressive property of , these limit quantities are finite; precisely,

Call ,* Boyd-Wong admissible,* if(c12) (or, equivalently: ), for all . (This convention is related to the developments in Boyd and Wong [13]; we do not give details). In particular, is Boyd-Wong admissible provided it is upper semicontinuous at the right on :
Note that this is fulfilled when is continuous at the right on ; for, in such a case, , for all .(II)Call ,* Matkowski admissible* [14], provided(c13) is increasing and as , for all . (Here, stands for the th iterate of ). Note that the obtained class of functions is distinct from the above introduced one, as simple examples show.

Now, let us say that is* Boyd-Wong-Matkowski admissible* (abbreviated: BWM-admissible) if it is either Boyd-Wong admissible or Matkowski admissible. The following auxiliary fact will be useful.

Lemma 5. *Let be a BWM-admissible function. Then, is Meir-Keeler admissible (see above).*

*Proof (sketch). *The former of these is an immediate consequence of definition. And the second one is to be found in Jachymski [15].

(B) Let us say that is a pair of weak generalized altering functions in , if(c14) is increasing, and [; , for all ](c15)(): , whenever .

Here, given the sequence in and the point , we denoted (resp., ), if and (resp., ), for all large enough.

Given and the couple of functions in , let us say that is -*contractive*, provided(c16), for all , .

Lemma 6. *Suppose that is -contractive, for a pair of weak generalized altering functions in . Then, is Meir-Keeler -contractive (see above).*

*Proof. *(i) Let be such that . Then (as is sufficient), , so that (by the choice of our pair), ; wherefrom . This via ( = increasing) yields , so that is strictly -nonexpansive.

(ii) Assume by contradiction that does not have the Meir-Keeler property (modulo ); that is, for some ,
Taking a zero converging sequence in , we get a couple of sequences and in , so as
By the contractive condition (and = increasing), we get
or, equivalently,
By (16), , so that passing to as ,
But, from the hypothesis about , these relations are contradictory. This ends the argument.

#### 4. Main Result

Let be a relational metric space. Further, let be a self-map of , supposed to be -semi-progressive and -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 . Denote, for each ,
Clearly, , but the possibility of cannot be removed. This fact remains valid even if is* orbital admissible*, in the sense [ implies ], when the associated orbit 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 , let us say that is -*semirecurrent* at the orbital admissible , if for each , there exists such that . A global version of this convention is the following: call ,* finitely semirecurrent* if, for each orbital admissible , there exists , such that is -semirecurrent at .

Assume in the following that(d01) is finitely semirecurrent and nonidentical.

Our main result in this exposition is the following.

Theorem 7. *Assume that is Meir-Keeler -contractive, for some . In addition, let be (a-o, -complete; and one of the following conditions holds:*(i)* is (a-o, -continuous;*(ii)* is (a-o, -almost-self-closed and ;*(iii)* is (a-o, -almost-self-closed and is -contractive, for a certain Meir-Keeler admissible function ;*(iv)* is (a-o, -almost-self-closed and is -contractive, for a certain pair of weak generalized altering functions in .** Then is a globally strong Picard operator (modulo ).*

*Proof. *First, we check the fix--asingleton property. Let be such that ; and assume by contradiction that ; whence . From the very definitions above,
whence . This, via being strictly -nonexpansive, yields an evaluation like
which is contradictory; hence the claim follows. It remains now to establish the strong Picard property (modulo ). The argument will be divided into several steps.*Part 1.* We firstly assert that
Let be such that . As is strictly -nonexpansive, one has . On the other hand,
This, along with
gives the desired fact.*Part 2.* Take some ; and put . If for some , we are done, so, without loss, one may assume that, for each , (d02); hence, , . From the preceding part, we derive
so that the sequence is strictly descending. As a consequence, exists as an element of . Assume by contradiction that ; and let be the number given by the Meir-Keeler -contractive condition upon . By definition, there exists a rank such that implies ; hence (by a previous representation) . This, by the Meir-Keeler contractive condition we just quoted, yields (for the same ), ; contradiction. Hence, , so that
that is, (see above): is -semi-Cauchy.*Part 3.* Suppose that (d03)there exist such that , . Denoting , we thus have and , so that
By the introduced notations, , for all . This, along with as , yields , in contradiction with the initial choice of . Hence, our working hypothesis cannot hold; wherefrom
*Part 4*. As a consequence of this, the map is injective; hence, is orbital admissible. Let be the semirecurrence constant of at (assured by the choice of this relation). Further, let be arbitrary fixed; and be the number associated by the Meir-Keeler -contractive property; without loss, one may assume that . By the -semi-Cauchy property and triangular inequality, there exists a rank , such that
We claim that the following relation holds:
wherefrom, is -Cauchy. To do this, an induction argument upon will be used. The case is evident, by the preceding evaluation. Assume that it holds for all , where ; we must establish its validity for . As is -semirecurrent at , there exists such that ; note that the former of these yields (from the -increasing property of ), . Now, by the inductive hypothesis and (30),
This, along with the triangular inequality, gives us
wherefrom , so that (by the diameter boundedness property), . Taking the Meir-Keeler -contractive assumption imposed upon into account gives
so that by the triangular inequality (and (30) again),
and our claim follows.*Part 5*. As is (a-o, -complete, , for some (uniquely determined) . If there exists a sequence of ranks with [ as ] such that (hence, ) for all , then, as is a subsequence of , one gets . So, in the following, we may assume that the opposite alternative is true: (d04): . There are several cases to discuss.*Case 5a*. Suppose that is (a-o, -continuous. Then as . On the other hand, is a subsequence of ; whence ; and this yields (as is sufficient), .*Case 5b*. Suppose that is (a-o, -almost-self-closed. Put, for simplicity reasons, . By definition, there exists a subsequence of , such that , for all . Note that, as , one may arrange for , for all , so that, from (d04),
This, along with being as well a subsequence of , gives (via (27) and Lemma 3)
whence (by definition)

We now show that the assumption (i.e., ) yields a contradiction. Two alternatives must be treated.*Alter 1*. Suppose that . By the Meir-Keeler contractive condition,
so that, combining with the preceding relations, . This, along with (37) + (38), is impossible for any ; whence, .*Alter 2*. Suppose that . The above convergence properties of tell us that, for a certain rank , we must have
This, by the -Lipschitz property of , gives
wherefrom, ,. Combining these yields
Two subcases are now under discussion.*Alter 2a*. Suppose that is -contractive, for a certain Meir-Keeler admissible function . (The case was already clarified in a preceding step.) By (42) and this contractive property,
Passing to limit gives (by (37) above), ; contradiction; hence, .*Alter 2b*. Suppose that is -contractive, for a certain pair of weak generalized altering functions in . (As before, the case is clear, by a preceding step.) From this contractive condition,
or, equivalently (combining with (42) above),
Note that, as a consequence, , for all . Passing to limit as and taking (37) into account, yields . This, however, contradicts the choice of , so that . The proof is complete.

In particular, when 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.

#### 5. Further Aspects

Let in the following 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.

Before doing this, let us remark that the condition(e01) is locally finitely transitive and nonidentical appears as a particular case of (d01). On the other hand, (d01) is not deductible from (e01). In fact, (d01) has nothing to do with the points of (e02). So, even if the restriction of to is arbitrarily taken, (d01) may hold. On the other hand, (e01) cannot hold whenever admits a denumerable subset such that the restriction of 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.

Theorem 8. *Assume that is -semiprogressive, -increasing, and Meir-Keeler -contractive, for some . In addition, let be finitely semirecurrent nonidentical, be (a-o, -complete, and one of the conditions below holds:*(i1)* is (a-o, -continuous;*(i2)* is (a-o, -almost-self-closed and .** Then is a globally strong Picard operator (modulo ).*

The following particular cases of this result are to be noted.(1-1)Let be a function in ; and denote the associated relation: [ if and only if ]. Then, if we take and , 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 (i.e., is the* trivial relation* over ). Then, Theorem 8 is comparable with the main results in Włodarczyk and Plebaniak [19–22], 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 -semiprogressive, -increasing, and -contractive, for some and a certain Meir-Keeler admissible function . In addition, let be finitely semirecurrent nonidentical, be (a-o, -complete, and one of the conditions below holds:*(j1)* is (a-o, -continuous;*(j2)* is (a-o, -almost-self-closed.** Then is a globally strong Picard operator (modulo .*

The following particular cases of this result are to be noted.(2-1)Suppose that (= the trivial relation over ) and . Then, Theorem 9 is comparable with the main results in Włodarczyk et al. [23, 24], based on contractive type conditions like(e03), for all . (Here, 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.(2-2)Suppose that ; and is BWM-admissible (i.e., it is either Boyd-Wong admissible or Matkowski admissible). Then, if , Theorem 9 includes the Boyd-Wong result [13] when is Boyd-Wong admissible; and, respectively, the Matkowski’s result [14] when is Matkowski admissible. Moreover, when , Theorem 9 includes the result in Leader [25].(2-3)Suppose that is an order on . Then, Theorem 9 includes the results in Agarwal et al. [26]; see also O'Regan and Petruşel [27].

*Case 3. *As a final consequence of Theorem 7, we have

Theorem 10. *Assume that the self-map is -semiprogressive, -increasing, and -contractive, for a certain and some pair of generalized altering functions in . In addition, let be finitely semirecurrent nonidentical, be (a-o, -complete, and one of the conditions below holds:*(k1)* is (a-o, -continuous;*(k2)* is (a-o, -almost-self-closed.** Then is a globally strong Picard operator (modulo .*

The following particular cases of this result are to be noted.(3-1)Let , be a couple of functions in ; and , stand for the associated relations:
Then, if we take and , this result includes (cf. Lemma) the one in Berzig et al. [28], based on global contractive conditions like
referred to as is -*contractive*. In particular, when , 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.

(3-2) Let be a metric space; and be a self-map of . Given , let be a finite system of closed subsets of with(e04), for all (where ). Define a relation over as (e05);then, put . Clearly, is a self-map of ; and the relation is -semirecurrent at each orbital admissible point of . The corresponding version of Theorem 10 includes the related statement in Berzig et al. [28].

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, 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.

#### Acknowledgment

The author is very indebted to all referees of the paper, for a number of useful suggestions.