#### Abstract

In this paper, we introduce the notion of -partial metric spaces which is a generalization each of -metric spaces and partial-metric space. Also, we study and prove some topological properties, to know the convergence of the sequences and Cauchy sequence. Finally, we study a new common fixed point theorem in these spaces.

#### 1. Introduction

There are a large number of generalizations of the Banach contraction principle with different use forms of contractual terms in a variety of generalized metric spaces. For some of these, the circulares are obtained through contractual terms expressed in reasonable terms. Latif et al. [1] introduced the notion of -rational Geraghty contractive mappings in the setup of ordered generalized -metric spaces and investigated the existence of fixed points for such mappings. They also provided an example to illustrate the presented results and show that they are more general than some existing ones.

One such generalization is a partial-metric space introduced by Matthews [2] in 1994. Since in this space, a self-distance of an arbitrary point need not to be equal zero. Shahkoohi and Razani [3] introduced new classes of rational Geraghty contractive mappings in the setup of -metric spaces and the existence of some fixed point for such mappings in ordered -metric spaces. Bakhtin [4] and Bourbaki [5] introduced a notion of -metric spaces. Later on, Czerwik [6] generalized the Banach contraction mapping theorem by formally defining a -metric space and giving a postulate which was weaker than the condition of triangular inequality. After that, Fagin and Stockmeyer [7] defined some kind of relaxation in triangular inequality and called this new distance measure as the nonlinear elastic mathing. All these notions and applications pushed us to introduce the concept of extended -metric space by -metric space and partial metric space.

In 2007, Sedghi et al. [8] introduced -metric spaces which are the generalization of the notion of -metric spaces introduced by Dhage [9]. Also, Sedghi et al. proved some of the basic properties in -metric spaces. In 2012, Sedghi et al. [10] introduced -metric spaces and give some properties and fixed point theorem for a complete -metric space. Soliman and Zidan [11] introduced a new coupled fixed point theorem in a generalized metric space, and they utilized the same to study the stability for a system of set-valued functional equations. In 2015, Gupta and Deep [12] proved fixed point theorems for nonlinear contractive mappings in -metric spaces. On the other way, Chauhan and Gupta [13] introduced the notion of fuzzy cone -metric space and defined fuzzy cone -contractive mapping and proved Banach contraction theorem for a single mapping in the setting of fuzzy cone -metric space.

Recently, in 2019, Mustafa et al. [14] introduced the structure of -metric spaces which is a generalization of -metric space and gave some properties and fixed point theorem for this spaces.

#### 2. Preliminaries and Definitions

We begin the section with some basic definitions and concepts.

*Definition 1. *Let be a nonempty set and be a given real number. A function is called *-metric*, if it satisfies the following properties for each :

()

()

()

The pair is called a -metric space.

*Example 2. *Let with where . Define as
where , . Then, is a -metric space.

*Definition 3 (see [2]). *Let be a set. A function is said to be a partial metric spaces on a nonempty set , if for all , the following conditions hold:

() (nonnegativity and small self-distances)

() If then (indistancy implies equality)

() If (symmetry)

() (triangularity)

Hence, the function is called a -metric on and the pair is called a partial-metric space.

Mustafa and Sims [15] extended the Banach principle by introducing the notation of generalized metric spaces the so-called -metric spaces as follows.

*Definition 4. Let be a set. *A function is said to be generalized metric spaces on a *nonempty* set , if for all , the following conditions hold:

()

() and

() and

() , (symmetry in all three variables)

()

Hence, the function is called a -metric on and the pair is called a generalized metric space.

Mustafa and Sims [15] found some basic properties and examples of G-metric spaces.

After that, Zand and Nezhad [16] introduced the definition of -metric space by generalization and unification of both the partial-metric space and -metric space as follows:

*Definition 5. Let be a set. *A function is said to be a generalized partial metric spaces on a nonempty set , if for all , the following conditions hold:

() if

()

() , (symmetry in all three variables)

()

Hence, the function is called a -metric on and the pair is called a generalized partial metric space.

*Example 6 (see [16]). *Let and for all , then is a *-metric* space. Also, is not a -metric space.

Proposition 7. *Let is a -metric space, then for any and , it follows that
*(i)

*(ii)*

*(iii)*

*Sedghi et al. [8] introduced the notion of -metric which is a modification of the definition of -metric introduced by Dhage [9, 17], and they proved some basic properties in -metric spaces.*

*Definition 8 (see [8]). *Let be a set. A function is said to be a *-metric* spaces on a nonempty set , if for all , the following conditions hold:

()

()

() , (where is a permutation function)

Hence, the function is called a -metric on and the pair is called a generalized partial metric space.

*Example 9 (see [8]). *Let . Denote , for all . Since
Hence,
On the other way, Sedghi et al. introduced -metric spaces as follows.

*Definition 10 (see [10]). *Let be a set. A function is said to be *-metric* spaces on a nonempty set , if for all the following conditions hold:

()

()

() for all (rectangle inequality)

Hence, the function is called a -metric on and the pair is called an -metric space.

*Example 11 (see [10]). *Let and a norm on , then
is a -metric on .

*Remark 12. *Every *-metric is**-metric*, but in general, the converse is not true, see the following example.

*Example 1. *Let and a norm on , then *is**-metric on*, but it is not *-metric* because it is not symmetric.

*Example 2. *Let and is an ordinary metric on . Therefore, is an *-metric* on . Then, if the points connected by a line. Hence, the triangle when choose a point a mediating this triangle then the inequality
for all holds.

*Definition 13 (see [10]). *Let be a *S-metric* space. Then, for and , the *-open ball* and *-closed ball* of radius with centered at is

Proposition 14 (see [10]). *Let be a -metric space and :
*(1)

*If for every there exists such that , then the subset is called an open subset of*(2)

*A subset of is said to be -bounded if there exists such that for all*(3)

*A sequence in converges to if and only if as . That is, for each , there exists such that for all , and we denote this by*

Lemma 15 (see [10]). *Let be an -metric space, then the following is satisfied:
*(1)

*If and , then the ball is an open subset of*(2)

*If the sequence in converges to , then is unique*(3)

*If the sequence in converges to , then is a Cauchy sequence*

*Definition 16 (see [15]). *Two classes of the following mappings are
(1) is nondecreasing, continuous and (2) is nondecreasing, lower semi-continuous and

*Definition 17. Let be a partially ordered set. *Two maps are said to be weak increasing if and for all

Barakat and Zidan [18] proved a common fixed point theorem for weak contractive maps by using the concept of -metric spaces.

Theorem 18. *Let be a partially ordered set wit and be weakly increasing self mapping on a complete -partial metric space. Suppose that there exist and such that
*

*for all , where*

*where for with .*

*Then, of the following two cases, assume that one of the following cases is satisfied:*(a)

*If a nondecreasing sequence converges to implies for all*(b)

*or is continuous*

*Therefore, the maps or have a common fixed point.*

The present paper is aimed at introducing the notion of -partial metric spaces which is a generalization each of -metric spaces and partial-metric space. Also, we give some of the topological properties that are important in knowing the convergence of the sequences and Cauchy sequence. Finally, we study a new fixed point theory in this spaces.

#### 3. -Partial Metric Spaces and some Properties

We first introduce the concept of a -partial metric space or .

*Definition 19. Let be a set and . *A function is said to be a *-partial metric spaces* on a nonempty set , if for all , the following conditions hold:

then,

for all (rectangle inequality)

Hence, the function is called an -partial metric on and the pair is called an -partial metric space.

*Example 20. *Let and a norm on , then we have
is a -partial metric on .

*Remark 21. *From Example 20, we get every *-metric* is *-metric*, but the converse is not true at all.

*Definition 22. Let be a -partial metric space. *Then, for and , the *-open ball* and *-open closed of radius* with centered at *is*

Proposition 23. Let be a -partial metric space. *Then for and , the following statements are satisfying:
*(1)*If and , then *(2)*If , then their exist , such that *

*Proof. *(1)The proof is straightforward.(2)Let , then we haveAlso, we suppose
and , then we get
Therefore, , and so (2) holds.
Hence, .

*Definition 24. *Let be a *-partial metric space* and a sequence in . A point is said to be the limit of the sequence if
Hence, the sequence is -convergent to .

Therefore, if in a -partial metric space . Then, for any , there exists such that, for all , we have

*Definition 25. *A *-partial metric space* is called a *-partial asymmetric space if*

Lemma 26. be a -partial metric space. *If the sequence in converges to . Therefore, we get is unique.*

*Proof. *Let converges to and . Therefore, for each , there exist , then we have
If set , then for every . Also, we have a third condition of -partial metric
Hence, , but the converse is not necessarily true.

*Definition 27. Let be a -partial metric space. *Then, for a *sequence* and a point , the following are equivalent:
(1) is a convergent to (2)

Proposition 28. Let be a -partial metric space. *If then, we get
*(1)*A sequence is called a Cauchy sequence if for each , there exists such that**for each .*

*Definition 29. Let be a -partial metric space. *Then, is said to be complete if every Cauchy sequence is convergent.

*Definition 30. Let and let be two -partial metric space. *Also, we suppose a function
then is said to be - continuous at a point a if and only if, for given , there exists such that and the inequality
This is an indication
Hence, a function is - continuous on if and only if it -continuous at all .

*Definition 31. *The two classes of following mappings are defined is continuous, nondecreasing, and . is lower semicontinuous, nondecreasing, and .

#### 4. A Generalization of Common Point Theorems in -Partial Metric Spaces

Theorem 32. *Let be a partially ordered set, and be weakly increasing self -mapping on a complete -metric space with . Suppose that there exist and such that
for all , where
*

*Assume that one of the following cases is satisfied:*(a)

*if a nondecreasing sequence converges to implies for all*(b)

*or is continuous*

*Therefore, the maps or have a common fixed point.*

*Proof. *Suppose that is a fixed point of and . From 4.1 with , we have
where
Hence, we have

A contradiction. Therefore, . So, is common fixed point of and . Similarly, if is a fixed point of , then one can deduce that is also fixed point of .

If, we let be an arbitrary point of with , then the proof is finished, so we assume that .

Now, one can construct a sequence as follows:

Since and are comparable, then we may assume that , for every . If not, then for some . For all those , using (24), we obtain

Therefore

It means that and . Following the similar arguments, we obtain and hence becomes a common fixed point of and .

By taking for , now, we consider

Now if for some , then and from (32), we have

implying that is a contradiction. Therefore, for all ,

Similarly, we have

for all . Hence, we get

Also, is a nonincreasing sequence, then there exists , such that

Hence, by the lower semicontinuity of ,

Now, we claim that . By lower semicontinuity of , taking the upper limit as on either side of then, we get

this implies that , then we have

To show that is a -Cauchy sequence for each , and , we have

By taking the limit as to both sides of the above inequality and from (41), we have

It follows that is a -Cauchy sequence and by -completeness of , so there exist such that converges to as . Now, we will distinguish the cases and of this theorem. (a)Suppose is continuous, since , we obtain that

But , as a subsequence of . It follows that and from the beginning of the prove we get .The proof, assuming that is continuous, is similar to the above. (b)Suppose that and for and a nondecreasing sequence with in indicate that , . Therefore, from (24), we havewhere

By taking limit as , this implies

Hence,

This is a contradiction. Thus, we have

Corollary 33. *Let be a partially ordered set, and be weakly increasing self-mapping on a complete -metric space with . Suppose that there exist such that
*