#### Abstract

Some fixed point results are given for a class of functional contractions acting on (reflexive) triangular symmetric spaces. Technical connections with the corresponding theories over (standard) metric and partial metric spaces are also being established.

#### 1. Introduction

Let be a nonempty set. By a symmetric over we will mean any map with, .The couple will be referred to as a symmetric space. The introduction of such structures goes back to Wilson ; for its “modern” aspects, we refer to Hicks and Rhoades .

Call the symmetric , triangular if, ,and reflexive-triangular, provided it fulfills (the stronger condition), .The class of such particular (symmetric) spaces has multiple connections with the one of partial metric spaces, due to Matthews . For, as we will see in the following, the fixed point theory for functional contractive maps in (reflexive) triangular symmetric spaces is a common root of both corresponding theories in standard metric spaces and partial metric spaces. This ultimately tells us that, for most of the function contractions taken from the list in Rhoades , any such theory over partial metric spaces is nothing but a clone of the corresponding one developed for standard metric spaces. Further aspects will be delineated elsewhere.

#### 2. Preliminaries

Let be a symmetric space, where is triangular. Call the subset of , -singleton provided .

(A) We introduce a -convergence and a -Cauchy structure on under the lines in Romaguera  (related to the context of partial metric spaces; cf. Section 4). 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 as ; note that it is a -singleton, because is triangular. If is nonempty, then is called-convergent. We stress that the concept (b01) does not match the standard requirements in Kasahara , because, for the constant sequence , we do not have if . Further, call the sequence , -Cauchy when as , ; that is,. As is triangular, any -convergent sequence is -Cauchy too, but the reciprocal is not in general true. Let us say that is 0-complete, if each -Cauchy sequence is -convergent.

(B) Call the sequence , -semi-Cauchy provided, as.

Clearly, each -Cauchy sequence is -semi-Cauchy, but not conversely. The following auxiliary statement about such objects is useful for us.

Lemma 1. Let be a -semi-Cauchy sequence in that is not -Cauchy. There exist then , , and a couple of rank-sequences , with

Proof. As is not -Cauchy, there exists, via (b02), an with Having this precise, denote, for each , As a consequence, the couple of rank-sequences , fulfills (1). On the other hand, letting the index be such that it is clear that (2) holds too. Finally, by the triangular property, and this establishes the case , of (3). Combining with yields the case , of the same. The remaining situations are deductible in a similar way.

Note finally that the exposed facts do not exhaust the whole completeness theory of such objects applicable to partial metric spaces. Some complementary aspects involving these last objects may be found in Oltra and Valero .

Let stand for the class of all functions from to itself. For any , and any in , put; where .By this very definition, we have the representation Clearly, the quantity in the right member may be infinite. A basic situation when this cannot hold may be described as follows. Call , normal when . Note that, under such a property, one has The following consequence of this will be useful. Given the sequence in and , define (as ) provided [, ] and .

Lemma 2. Let the function be normal, and let be arbitrarily fixed. Then,(i), for each sequence with ,(ii)there exists a sequence with and .

Proof. (i) Given , there exists a rank such that , for all ; hence It suffices taking the infimum over in this relation to get the desired fact.
(ii) From (b04), , so, by the definition of infimum, This, in the case of , gives the written conclusion with , for, as a direct consequence of (10), one has . Suppose now that . Again from (b04), This, along with the definition of supremum, tells us that there must be some in with . Taking a sequence in with , there exists a corresponding sequence in with and , hence the conclusion.

Call the normal , right limit normal if  (or,  equivalently:  ), .(The last assertion is a consequence of (9) above.) In particular, the normal function is right limit normal, whenever it is usc at the right on :, for each .Note that this property is fulfilled when is continuous at the right on , for, in such a case, , for all . Another interesting example is furnished by Lemma 2. Let us say that the normal function is Geraghty normal provided (cf. Geraghty ) imply .

Lemma 3. Let the normal function be Geraghty normal. Then, is necessarily right limit normal.

Proof. Suppose that the normal function is not right limit normal. From (10), there exists some with . Combining with Lemma 2, there exists a sequence with and , whence ; that is, is not Geraghty normal.

Remark 4. The reciprocal of this is not in general true. In fact, for the (continuous) right limit normal function and the sequence in , we have , but, evidently, .

#### 3. Main Result

Let be a symmetric space, with, in addition, is triangular and is 0-complete.

Further, let be a selfmap of . Call , -fixed if and only if ; the class of all such elements will be denoted as . Technically speaking, the points in question are obtained by a limit process as follows. Let us say that is a Picard point (modulo) if (i) is -convergent, and (ii) each point of is in . If this happens for each , then is referred to as a Picard operator (modulo ); if (in addition) is -singleton, then is called a global Picard operator (modulo ); compare Rus [9, Chapter 2, Section 2.2].

Now, concrete circumstances guaranteeing such properties involve (in addition to (c01)) contractive selfmaps with the -asymptotic property:, .Precisely, denote for : and fix . Given , we say that is-contractive, if, .The main result of this note is the following.

Theorem 5. Suppose (under (c02)) that is -contractive, for some right limit normal function . Then, is a global Picard operator (modulo ).

Proof. Let be arbitrary fixed. By this very choice, In addition (from the triangular property) so that , which tells us that . On the other hand, again from the choice of our data and the triangular property, Combining with the contractive condition yields (for either choice of ) wherefrom , so that is -singleton. It remains now to establish the Picard property. Fix some , and put ; note that by (c02), is -semi-Cauchy.
(I) We claim that is -Cauchy. Suppose this is not true. By Lemma 1, there exist , , and a couple of rank-sequences , , with the properties (1)–(3). For simplicity, we will write (for ), , in place of , , respectively. By the contractive condition,
Denote . From (1), , for all ; moreover, (3) yields as . So, passing to limit as in (19) one gets (via Lemma 2) , contradiction, so that our assertion follows.
(II) As is 0-complete, this yields as , for some . We claim that is an element of . Suppose not: that is, . By the previously mentioned properties of , there exists such that, for all ,
This gives (again for all ) By the contractive condition, we then have (for either choice of ) Passing to limit as yields , contradiction. Hence, is an element of , and the proof is complete.

#### 4. Reflexive-Triangular Case

Now, it remains to determine concrete circumstances under which is -asymptotic. Let be a symmetric space, with is reflexive-triangular and is 0-complete.

Further, let be a selfmap of ; and fix .

Lemma 6. Suppose that is -contractive, for some right limit normal function . Then, is -asymptotic.

Proof. By definition, we have On the other hand, by the reflexive-triangular property, So, by simply combining these, Fix some , and put . From the contractive condition, . As is normal, this yields In particular, is descending; hence, exists in . Assume that . By Lemma 2, we must have ; contradiction. Hence, , and the conclusion follows.

Now, by simply combining this with Theorem 5, we have (under (d01)) the following.

Theorem 7. Suppose that is -contractive, for some right limit normal function . Then, is a global Picard operator (modulo ).

A basic particular case of this result is the following. Call the symmetric on an almost partial metric provided it is reflexive-triangular and is sufficient).

Note that, in such a case, Assume in the following that is almost partial metric and is 0-complete.

Theorem 8. Let the selfmap be -contractive, for some right limit normal function . Then,

Proof. By Theorem 7, we have (taking (27)+(28) into account) and, moreover, (30) holds. It remains to establish that . For each , we must have (by (30)) , which means ; hence (as = sufficient) . The proof is complete.

Now, let us give two important examples of such objects.

(A) Clearly, each (standard) metric on is an almost partial metric. Then, Theorem 8 is just the main result in Jachymski ; see also Cho et al. . In fact, its argument mimics the one in those papers. The only “specific” fact to be underlined is related to the reflexive-triangular property of our symmetric .

(B) According to Matthews , call the symmetric , a partial metric provided it is reflexive-triangular and  ( is strongly sufficient),   (Matthews property).

Note that, by the reflexive-triangular property, one has (with ) and this, along with (d04), yields (d02); that is, each partial metric is an almost partial metric. Hence, Theorem 8 is applicable to such objects; its corresponding form is just the main result in Romaguera ; see also Altun et al. . It is to be stressed here that the Matthews property (d05) was not used in the quoted statement. This forces us to conclude that this property is not effective in most fixed point results based on such contractive conditions. On the other hand, the argument used here is, practically, a clone of that developed for the standard metric setting. Hence—at least for such results—it cannot get us new insights for the considered matter. Clearly, the introduction of an additional order structure on does not change this conclusion. Hence, the results in the area due to Altun and Erduran  are but formal copies of the ones (in standard metric spaces) due to Agarwal et al. . This is also true for the common fixed points question, when, for example, the results in Shobkolaei et al.  or Karapınar and Shatanawi  are but a translation of the ones (in standard metric spaces) due to Jachymski . Finally, we may ask whether this reduction scheme comprises as well the class of contraction maps in general complete partial metric spaces taken as in Ilić et al. . Formally, such results are not reducible to the previous ones. But, from a technical perspective, this is possible; see Turinici  for details.

#### 5. Triangular Symmetrics

Let be a symmetric space, taken as in (c01), and let be a selfmap of .

Lemma 9. Suppose that is -contractive, for some right limit normal function . Then, is -asymptotic.

The argument is based on the evaluation (23); see also Zhu et al. .

Now, by simply combining this with Theorem 5, we have (under (c01)) the following.

Theorem 10. Suppose that is -contractive, for some right limit normal function . Then, is a global Picard operator (modulo ).

A basic particular case of this result is to be stated as below. Call the symmetric , a weak almost partial metric provided it is triangular and sufficient (i.e., (d02) holds). Note that, in such a case, relations (27) and (28) are still retainable. Assume in the following that is a weak almost partial metric and is 0-complete.

Theorem 11. Let the selfmap be -contractive, for some right limit normal function . Then, conclusions of Theorem 8 are holding.

The proof mimics the one of Theorem 8 (if one takes Theorem 10 as starting point), so, it will be omitted.

Now, let us give two important examples of such objects.

(A) Clearly, each (standard) metric on is a weak almost partial metric. Then, Theorem 11 is comparable with the main result in Jachymski .

(B) Remember that the symmetric is called a partial metric provided it is reflexive-triangular and (d04) + (d05) hold. As before, (32) tells us (via (d04)) that each partial metric is a weak almost partial metric; hence, Theorem 11 is applicable to such objects. In particular, when is linear (, , for some ), one recovers the Banach type fixed point result in Aage and Salunke , which, in turn, includes the one in Valero . On the other hand, Lemma 3 tells us that Theorem 11 includes as well a related fixed point statement due to Dukić et al. ; see also Golubović et al. ; moreover, by Remark 4, the converse inclusion is not in general true. It is to be stressed here that the Matthews property (d05) was not effectively used in the quoted statement; in addition, the (stronger) reflexive-triangular property of was replaced by the triangular property of the same. As before, the argument used here is, practically, a clone of that developed in the standard metric setting; whence, the results we just quoted are technically deductible from the one in Boyd and Wong . Further developments of these results may be performed under the lines in Turinici .

#### Acknowledgment

The author is very indebted to both referees, for a number of useful suggestions.