<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="14.1865pt" version="1.1" viewBox="-0.0657574 -13.9141 37.6938 14.1865" width="37.6938pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-84" d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"/></g><g transform="matrix(.012,0,0,-0.012,8.1,-7.578)"><path id="g50-43" d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"/></g><g transform="matrix(.012,0,0,-0.012,15.439,-7.578)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g><g transform="matrix(.017,0,0,-0.017,23.322,0)"><path id="g190-46" d="M300 253H71L57 198H286L300 253Z"/></g><g transform="matrix(.017,0,0,-0.017,29.454,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Partial Metric Spaces with some Results in Common Fixed Point Theorems

Zidan, A. M.

doi:https://doi.org/10.1155/2021/5586936

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Fixed Point Theory and Applications for Function Spaces

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Research Article | Open Access

Volume 2021 | Article ID 5586936 | https://doi.org/10.1155/2021/5586936

-Partial Metric Spaces with some Results in Common Fixed Point Theorems

A. M. Zidan^1,2

Academic Editor: Zoran Mitrovic

Received10 Feb 2021

Accepted03 May 2021

Published15 May 2021

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 continuousTherefore, 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 thatfor 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 continuousTherefore, 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 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 continuousTherefore, the maps or have a common fixed point.

Proof. Put in Theorem 32.

Corollary 34. Let be a partially ordered set, and be weakly increasing self-mapping on a complete -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 continuousTherefore, the maps or have a common fixed point.

Proof. A corollary is -partial metric spaces version of Theorem 18.

Corollary 35. Let be a partially ordered set, and be weakly increasing self-mapping on a complete -metric space. Suppose that there exist 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 continuousTherefore, the maps or have a common fixed point.

Proof. Put in Corollary 34.

Remark 36. It will be interesting to find more applications to our current paper in other fields, see [11, 19–29].

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

The author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under grant number G.R.P./45/42.

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Copyright © 2021 A. M. Zidan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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