#### Abstract

In this paper, we introduce the notion of -partial metric spaces which is a generalization of S-metric spaces and partial-metric spaces. 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 common fixed point theorems in this 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. Some of the circulares are obtained through contractual terms expressed in reasonable terms. Shahkoohi and Razani [1] introduced a new class of rational Geraghty contractive mappings in the setup of b-metric spaces and proved some fixed point for such mappings in ordered b-metric spaces which are investigated. One such generalization is a partial-metric space which was introduced by Matthews [2] in 1994, since, in this space, a self-distance of an arbitrary point need not be equal zero. Also, in 2007, Sedghi et al. [3] introduced -metric spaces which are generalization of the notion of -metric spaces introduced by Dhage [4]. Also, Sedghi et al. proved some basic properties in -metric spaces. In 2012, Sedghi et al. [5] introduced -metric spaces and gave some properties and fixed point theorem for a complete S-metric space.

Recently, in 2019, Z. Mustafa et al. [6] introduced the structure of -metric spaces which generalization of -metric space and gave some properties and fixed point theorem for these spaces.

#### 2. Preliminaries and Definitions

We begin the section with some basic definitions and concepts.

Definition 1. (see [2]). Let be a set. A function is said to be a partial metric spaces on a nonempty set ; for all , the following conditions are held: (nonnegativity and small self-distances) If , then (indistancy implies equality) (symmetry) (triangularity)Hence, the function is called an -metric on and the pair is called a partial metric space.

Example 1. (see [7, 8]). Let and given by , for all ; then, is partial metric space.

Definition 2. Let be a partial metric space; then, one has the following:(i)A sequence in converges to a point if and only if (ii)A sequence in is Cauchy if exists and is finite(iii) is complete if every Cauchy sequence in converges to a point , that is,

Lemma 1 (see [2]). Let be a partial metric space. Then, is complete if and only if the metric space is complete. Furthermore, if and only if

In [9], Romaguera introduced the definitions of 0-complete partial metric spaces and 0-Cauchy sequence. Also, he obtained a characterization of completeness for partial metric space using the notion of 0-completeness.

Mustafa and Sims [10] extended Banach principle and introduced the notation of generalized metric spaces, so-called -metric spaces, as follows.

Definition 3. Let be a set. A function is said to be a generalized metric spaces on a nonempty set , for all , and the following conditions hold: and , and (symmetry in all three variables)Hence, the function is called an -metric on and the pair is called an generalized metric space.

To find some basic properties and examples of G-metric spaces in Mustafa and Sims [10].

After that, Zand and Nezhad [11] introduced definition of -metric space by generalization and unification of both partial metric space and -metric space as follows.

Definition 4. Let be a set. A function is said to be a generalized partial metric spaces on a nonempty set , for all , and the following conditions hold: if (symmetry in all three variables)Hence, the function is called an -metric on and the pair is called an generalized partial metric space.

Example 2. (see [11]). Let and for all , then is a -metric space. Also, is not a -metric space.

Proposition 1. Let is a -metric space, then, for any and , it follows that(i)(ii)(iii)

Shaban Sedghi et al. [3] introduced the notion of -metric which is a modification of the definition of -metric introduced by Dhage [4, 12], and they proved some basic properties in -metric spaces.

Definition 5. (see [3]). Let be a set. A function is said to be a -metric spaces on a nonempty set , for all , and the following conditions hold: (where is a permutation function)Hence, the function is called an -metric on , and the pair is called an generalized partial metric space.

Example 3. (see [3]). Let . Denote , for all . SinceHence,

On other way, Sedghi et al. introduced -metric spaces as follows.

Definition 6. (see [5]). Let be a set. A function is said to be a -metric spaces on a nonempty set , for all , and the following conditions hold: for all (rectangle inequality)Hence, the function is called an S-metric on and the pair is called an S-partial metric space.

Example 4. (see [5]). Let and a norm on ; then,is a -metric on .

Remark 1. Every -metric is -metric, but in general, the converse is not true, see the following example.

Example 5. Let and a norm on ; then, is -metric on , but it is not -metric because it is not symmetric.

Example 6. Let and is an ordinary metric on . Therefore, is an -metric on . Then, if the points are connected by a line, hence, we have a triangle and if we choose a point a mediating this triangle, then the inequalityfor all holds.

Definition 7. (see [5]). Let be a S-metric space. Then, for and , the -open ball and -closed ball of radius with centered at are

Proposition 2 (see [5]). 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 2 (see [5]). Let be an -metric space; then, the following satisfy:(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 8. (see [10]). Two classes of the following mappings are(1) is nondecreasing, continuous, and (2) is nondecreasing, lower semicontinuous, and

Definition 9. Let be a partially ordered set. Two maps are said to be weakly increasing if and , for all .

Barakat and Zidan [13] proved a common fixed point theorem for weak contractive maps by using the concept of -metric spaces.

Theorem 1. Let be a partially ordered set with and as weakly increasing self-mapping on a complete -partial metric space. Suppose that there exist and such thatfor all , wherewhere , for , with .

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.

#### 3. S-Partial Metric Spaces and Some Properties

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

Definition 10. Let be a set. A function is said to be a -partial metric space on a nonempty set , 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 7. Let and a norm on ; then, we havewhich is a -partial metric on .

Remark 2. From Example 7, we get every -metric is -metric, but the converse is not true at all, see the following example.

Definition 11. Let be a -partial metric space. Then, for and , the -open ball and -open closed of radius with centered at are

Proposition 3. Let be a -partial metric space. Then, for and , the following statements are satisfied:(1)If and , then (2)If , then their exist , such that

Proof. (1)The proof is straightforward.(2)Let ; then, we have Also, we suppose and ; then, we obtainTherefore, , and so (2) holds.Hence, .

Definition 12. Let be a -partial metric space and a sequence in . A point is said to be the limit of the sequence ifHence, the sequence is -convergent to .
Therefore, if in a S-partial metric space , then, for any , there exists such that, for all , we have

Definition 13. A -partial metric space is called a -partial asymmetric space if

Lemma 3. 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 haveIf set , then, for every , we have a third condition of -partial metric:Hence, , but the converse is not necessarily true.

Definition 14. Let be a -partial metric space. Then, for a sequence and a point , the following are equivalent:(1) is a convergent to (2)

Proposition 4. Let be a -partial metric space. If we get a sequence which is called a Cauchy sequence if, for each , there exists such that

Proof. for each , if the sequence converges to .

Definition 15. Let be a -partial metric space. Then, is said to be complete if every Cauchy sequence is convergent.

Definition 16. Let , , and be two -partial metric space. Also, we suppose a functionand then, is said to be -continuous at a point; if and only if, for given , there exists such that and the inequalityThis is an indication:Hence, a function is -continuous on if and only if it is -continuous at all .

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

Theorem 2. Let be a partially ordered set with and as weakly increasing self-mapping on a complete -partial metric space. Suppose that there exist and such thatfor 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 (24), with , we havewhereHence, we havewhich is 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 obtainTherefore,It means that and . Following the similar arguments, we obtain , and hence, becomes a common fixed point of and .
By taking for ., now, we considerNow, if , for some , then , and from (32), we havewhich implies that is a contradiction. Therefore, for all ,Similarly, we havefor all . Hence, we obtainAlso, is a nonincreasing sequence; then, there exists such thatHence, by the lower semicontinuity of ,Now, we claim that . By lower semicontinuity of , taking the upper limit as on either side ofthen we obtainand this implies that ; then, we haveTo show that is a -Cauchy sequence for each , and , we haveBy taking the limit as on both sides of the above inequality and from (41), we haveIt 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 thatHowever, , 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 above.(b)Suppose that , and for and a nondecreasing sequence with in , we indicate that , . Therefore, from (24), we havewhereTaking limit as impliesHence,This a contradiction. Thus, we have

Corollary 1. Let be a partially ordered set with and as weakly increasing self-mapping on a complete -partial metric space. Suppose that there exists such thatfor 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. Put in Theorem 2.

Corollary 2. Let be a partially ordered set with and as weakly increasing self-mapping on a complete -partial metric space. Suppose that there exist and such thatfor all , wherewhere for with .

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. A corollary is -partial metric spaces version of Theorem 1.

Corollary 3. Let be a partially ordered set with and as weakly increasing self-mapping on a complete -metric space. Suppose that there exist such thatfor all , where