Common Fixed Points in a Partially Ordered Partial Metric Space

cited. In the �rst part of this paper, we prove some generalized versions of the result of Matthews in (Matthews, 1994) using diﬀerent types of conditions in partially ordered partial metric spaces for dominated self-mappings or in partial metric spaces for self-mappings. In the second part, using our results, we deduce a characterization of partial metric 0-completeness in terms of �xed point theory. is result extends the Subrahmanyam characterization of metric


Introduction
In the mathematical �eld of domain theory, attempts were made in order to equip semantics domain with a notion of distance. In particular, Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of data for networks, showing that the contraction mapping principle can be generalized to the partial metric context for applications in program veri�cation. Moreover, the existence of several connections between partial metrics and topological aspects of domain theory has been lately pointed by other authors as O'Neill [2], Bukatin and Scott [3], Bukatin and Shorina [4], Romaguera and Schellekens [5], and others (see also [6][7][8][9][10][11][12][13][14] and the references therein).
Aer the result of Matthews [1], the interest for �xed point theory developments in partial metric spaces has been constantly growing, and many authors presented signi�cant contributions in the directions of establishing partial metric versions of well-known �xed point theorems for the existence of �xed points, common �xed points, and coupled �xed points in classical metric spaces (see e.g., [15,16]). Obviously, we cannot cite all these papers but we give only a partial list .
Recently, Romaguera [50] proved that a partial metric space ( is 0-complete if and only if every -Caristi mapping on has a �xed point. In particular, the result of Romaguera extended Kirk's [51] characterization of metric completeness to a kind of complete partial metric spaces.
Successively, Karapinar in [36] extended the result of Caristi and Kirk [52] to partial metric spaces.
In the �rst part of this paper, following this research direction, we prove some generalized versions of the result of Matthews by using different types of conditions in ordered partial metric spaces for dominated self-mappings or in partial metric spaces for self-mappings. e notion of dominated mapping of economics, �nance, trade, and industry is also applied to approximate the unique solution of nonlinear functional equations. In the second part, using the results obtained in the �rst part, we deduce a characterization of partial metric 0-completeness in terms of �xed point theory. is result extends the Subrahmanyam [53] characterization of metric completeness. For other characterizations of metric completeness in terms of �xed point theory, the reader can see, for example, [54,55] and for partial metric completeness, [41].

Preliminaries
First, we recall some de�nitions and some properties of partial metric spaces that can be found in [1,2,40,48,50]. A partial metric on a nonempty set is a function [0 +∞ such that for all International Journal of Analysis A partial metric space is a pair ( ) such that is a nonempty set and is a partial metric on . It is clear that if ( ) , then from ( 1 ) and ( 2 ), it follows that . But if , ( ) may not be . A basic example of a partial metric space is the pair . Other examples of partial metric spaces which are interesting from a computational point of view can be found in [1].
Each partial metric on generates a topology on which has as a base the family of open -balls { ( ) , where for all and . If is a partial metric on , then the function [ ∞) given by . We say that ( ) is 0-complete if every 0-Cauchy sequence in converges, with respect to , to a point such that ( ) . On the other hand, the partial metric space (ℚ ∩ [ ∞) ), where ℚ denotes the set of rational numbers and the partial metric is given by ( ) max{ , provides an example of a 0-complete partial metric space which is not complete.
It is easy to see that every closed subset of a complete partial metric space is complete.

Main Results
Let ( ) be a partial metric space and be such that ⊂ . For every we consider the sequence { ⊂ de�ned by 1 for all and we say that { is a --sequence of the initial point (see [57] e following theorem is one of our main results, and it ensures the existence of a common �xed point for two selfmappings in the setting of partially ordered partial metric spaces. If max{ , )), , we obtain a contradiction and so max , , , en, we have , ≤ , 0 , ∀ and hence lim , = 0.
Fix 0 and we choose ) such that for all ). Let ) and we show that Clearly, (12) is true for = . Suppose that (12) for every 0 . By condition (iii), and are comparable for every and hence, by condition (5) with = and = , we deduce that , ≤ max , , for every 0 . Letting in the previous inequality and using Lemma 2, we obtain which implies , ) = 0, that is, = . us, we have shown that = = is a point of coincidence of and . If and are weakly compatible, then = = = . By condition (iii), = , that is, and are comparable. Using the contractive condition (5), we get , ≤ max , , , , , which implies = = and hence is a common �xed point of and .
We shall give a sufficient condition for the uniqueness of the common �xed point in eorem 5. Proof. Let be two common �xed points of and with . If and are comparable, then using the contractive condition (5), we deduce that = . If and are not comparable, then there exists 0 such that 0 ⪯ = , As is a -dominated mapping, we get that To continue, we obtain and hence and are comparable. Using the contractive condition (5)  Proof. Proceeding as in the proof of eorem 5, we get that and have a unique point of coincidence and, by Lemma 3, and have a unique common �xed point.
From eorem 7, we can deduce the following corollaries.

Corollary 8 (Banach type). Let
) be a partial metric space and two mappings such that . Assume that for all , where 0 ≤ < . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common �xed point.

Corollary 9 (Bianchini type). Let
) be a partial metric space and let be two mappings such that . Assume that ≤ max (30) for all , where 0 ≤ < . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common �xed point.

Corollary 10 (Reich type [58]). Let
) be a partial metric space and let be two mappings such that . Assume that for all , where 0 and < . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common �xed point.
e following example shows that there exist mappings that satisfy the contractive condition (28), but are not quasicontractions [59]. and . Obviously, is a partial metric on , but it is not a metric (since for and ). Clearly, is a 0-complete partial metric space. Let be de�ned by and for every . Take for every . First, we will check that and satisfy the contractive condition (28). If , then and (28) trivially holds. Let, for example, , then we have the following three cases: us, all the conditions of eorem 7 are satis�ed and the existence of a common �xed point of and (which is ) follows. e same conclusion cannot be obtained by the main results from [59]. Indeed, using , and then taking instead of , in (5), we obtain max max max 4 .
Since , the conclusion follows.
e following example shows that there exist mappings that satisfy the contractive condition (5), but do not satisfy the contractive condition (28).
Clearly, is a 0-complete partial metric space. Let be de�ned by and for each . As and have many common �xed points (each is a common �xed point), then it is immediate to show that and do not satisfy the contractive condition (28). If is ordered by then and satisfy the contractive condition (5) where Using eorem 5, we deduce that and have a common �xed point.

Completeness in Partial Metric Spaces and Fixed Points
In this section, we characterize those partial metric spaces for which every Bianchini mapping has a �xed point in the style of Subrahmanyam characterization of metric completeness. is will be done by means of the notion of 0-completeness which was introduced by Romaguera in [50]. Let be a partial metric space and be a mapping. We recall that is a Bianchini [60] In particular From the de�nition of , we deduce that satis�es the condition (ii). On the other hand, satis�es also the condition (i). In fact, (i) is veri�ed by assuming and , and noting that It is clear that has not �xed points since ′ ≠ , , , . us, the assumption that there is a 0-Cauchy sequence } which is not convergent in ( , ) leads to a contradiction to eorem 13 and thereby establishes the same.
If in eorem 13 we choose 0, ), by Corollary 9, we obtain the following characterization of 0-completeness for partial metric spaces.