New Contributions in Generalization -Metric Spaces to -Partial Metric Spaces with Some Results in Common Fixed Point Theorems

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 that However, , 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 , 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 [14].

Proof. Put in Corollary 2.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

Researchers would like to thank the Deanship of Scientific Research, Qassim University, for funding publication of this project.