New Approach for Common Fixed Point Theorems via <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 11.6374 11.6718" width="11.6374pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-68" d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z"/></g></svg>-Class Functions in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-7.2259pt" id="M2" height="18.6253pt" version="1.1" viewBox="-0.0657574 -11.3994 19.6824 18.6253" width="19.6824pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g><g transform="matrix(.012,0,0,-0.012,11.652,4.134)"><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></svg>-Metric Spaces

Ansari, A. H.; Barakat, M. A.; Aydi, H.

doi:https://doi.org/10.1155/2017/2624569

Journal of Function Spaces

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Research Article | Open Access

Volume 2017 | Article ID 2624569 | https://doi.org/10.1155/2017/2624569

New Approach for Common Fixed Point Theorems via -Class Functions in -Metric Spaces

A. H. Ansari,¹M. A. Barakat,^2,3and H. Aydi^4,5

Academic Editor: Tomonari Suzuki

Received14 Dec 2016

Accepted24 Jan 2017

Published16 Feb 2017

Abstract

We prove a new approach for some common fixed point results in complete -metric spaces for weakly increasing self-mappings satisfying -contractions via the concept of -class functions. An example is also provided.

1. Introduction and Mathematical Preliminaries

In 1922, Banach [1] proved his classical theorem which asserts suitable conditions ensuring the existence and uniqueness of fixed point of the underlying mapping. Over the last several decades, this theorem has been generalized and improved in various spaces (e.g., [2–6]). In 1994, Matthews [7] introduced the notion of a partial metric space and established the Banach contraction theorem in the class of partial metric spaces. Notably, in partial metric spaces, the distance from a point to itself need not be zero. In recent years, several authors proved variant (common) fixed point theorems in partial metric spaces. For more details, see [8–17].

For the sake of completeness, we recall the definition of a partial metric space (in short PMS) which runs as follows.

Definition 1 (see [7]). A partial metric on a nonempty set is a function , , such that for all , , , .A partial metric space is a pair such that is a nonempty set and is a partial metric on .

On the other hand, the notation of generalized metric spaces (in short -metric spaces) was introduced by Mustafa and Sims [18] who presented and improved the Banach contraction principle in the class of -metric spaces. The definition of a -metric space is introduced as follows.

Definition 2 (see [18]). Let be a nonempty set. Suppose that satisfies(a) if ,(b),(c),(d) (symmetry in all three variables),(e) Then is called a -metric on and is called a -metric space.

Recently, as a unification between partial metric spaces and -metric spaces, Zand and Nezhad [19] defined the concept of a -metric space in the following way.

Definition 3 (see [19]). Let be a nonempty set. Suppose that satisfies(a) if ;(b);(c) (symmetry in all three variables);(d). Then is called a -metric on and is called a -metric space.

Example 4 (see [19]). Let and define for all . Then is a -metric space. Note that is not a -metric space.

Proposition 5 (see [19]). Let be a -metric space. Then for any and , one has(i);(ii);(iii);(iv).

Proposition 6 (see [19]). Every -metric space defines a metric space , where for all

Definition 7 (see [19]). Let be a -metric space. A sequence is -convergent to if .
We may write the above as
Thus if in a -metric space , then for any , there exists such that for all .

Proposition 8 (see [19]). Let be a -metric space. Take a sequence in and a point . The following are equivalent:(i) is -convergent to ;(ii) as ;(iii) as .

Definition 9 (see [19]). Let be a -metric space.(i)A sequence is called a -Cauchy if and only if exists (and is finite).(ii)A -metric space is said to be -complete if and only if every -Cauchy sequence in is -convergent to ; that is,

Take is continuous, nondecreasing, and , and is lower semicontinuous, nondecreasing, and .

Definition 10 (see [20]). Let be a partially ordered set. Two maps are said to be weakly increasing if and for all .

Definition 11. Let be a -metric space endowed with a partial order . Let and be in . is said to be regular if and is nondecreasing; then for all .

Lemma 12 (see [21]). Let be a -metric space. One has the following.(i)If , then .(ii)If , then

We rewrite the continuity of mappings in -metric spaces.

Definition 13. Let be a -metric space and let be a given mapping. One says that is continuous at if for every sequence converging to in , the sequence converges to in . If is continuous at each point , then one says that is continuous on .

Ansari [22] introduced the class of -functions which covers a large class of contractive conditions.

Definition 14 (see [22]). A mapping is called a -function if it is continuous and satisfies the following axioms:(1) for all ;(2) implies that either or .

Mention that any -function verifies . We denote by the set of -class functions.

Example 15 (see [22]). The following functions are elements of . For all , consider(1);(2), where ;(3), where ;(4), where is continuous;(5), where is a continuous function such that .

Abbas et al. [20] proved the following result.

Theorem 16 (see [20]). Let be a partially ordered set and let and be weakly increasing self-mappings on a complete -metric space Suppose that there exist and such thatfor all comparable , where where for with . Assume either or is continuous, or is regular. Then and have a common fixed point.

Very recently, Barakat and Zidan [13] extended Theorem 16 to the class of -metric spaces where a general contractive condition is considered.

Theorem 17 (see [13]). Let be a partially ordered set. Let and be weakly increasing self-mappings on a complete -metric space satisfyingfor all comparable , where , , andAssume either or is continuous or is regular. Then and have a common fixed point.

In this paper, we prove a common fixed point result in complete -metric spaces for weakly increasing self-mappings satisfying -contractions via the concept of -class functions. Some corollaries are also presented for particular cases of the -function. For a given -function, Theorem 17 is reached.

2. Main Results

First, we introduce an auxiliary lemma as follows.

Lemma 18. Let be a -metric space and let be a sequence in such that is decreasing andIf is not a -Cauchy sequence, then there exist an and of positive integers such that the following sequences , , , , and tend to when .

Proof. Assume that is not a -Cauchy sequence. So there exist , and and of positive integers such that for all . Then By taking the limit in above inequalities and using (6), we getOn the other hand Letting , again using (6) and (9), we obtain Similarly, we can prove that the remaining sequences tend to as .

Now, we state and prove our main result in the following way.

Theorem 19. Let be a partially ordered set. Let and be weakly increasing self-mappings on a complete -metric space Assume there exist and such thatfor all comparable , where Suppose that one of the following two cases is satisfied:(i) or is continuous;(ii) is regular. Then the maps and have a common fixed point.

Proof. Assume that is a fixed point of . Taking in (12), we havewhere Hence we get We deduce By a property of the -class , we get or . The functions and are in , so ; that is, ; that is, is a common fixed point of and . Now, if is a fixed point of , similarly we get that is also fixed point of . Let be an arbitrary point of . The pair is weakly increasing, so we construct a sequence in as follows: We have for all .
Now, suppose that for some . Then ; that is, is a fixed point of . Proceeding similarly, we get that is a fixed point of .
From now on, we suppose that for every . Since and are comparable, by (12),where If for some , then . Using (18), we have By a property of , this implies that or , which is a contradiction. Therefore, for all , . Similarly, we may show that for all . We deduce that for all So the sequence is decreasing. Then there exists , such that . We claim that . We haveRecall that As , by continuity of , , and , we get By a property of , we get or ; that is, . We conclude thatWe shall show that is a -Cauchy sequence. Suppose that is not a -Cauchy sequences. By (12), where By taking the limit as , from Lemma 18, we have Hence . We deduce that or , which is a contradiction. This proves that is a -Cauchy sequence and hence is a -Cauchy sequence. By -completeness of , there exists such that converges to as .
Now, we will distinguish the cases (i) and (ii) of Theorem 19.
(i) Without loss of generality, suppose that is continuous. Since , we obtain that . But as (as a subsequence of ), it follows that . From the beginning of the proof, we get
The case that is continuous is treated similarly.
(ii) Suppose that is regular. We know that sequence is nondecreasing and in ; then by regularity of , for all . By (12) where By taking the limit as , we have . Thus Similarly, we may get and so

Now, we provide some corollaries from our obtained result given by Theorem 19. First, putting in Theorem 19, we obtain the following.

Corollary 20. Let be a partially ordered set. Let and be weakly increasing self-mappings on a complete -metric space satisfyingfor all comparable , where , , and Assume either or is continuous, or is regular. Then and have a common fixed point.

Proceeding as Theorem 19, we have the following.

Corollary 21. Let be a partially ordered set. Let and be weakly increasing self-mappings on a complete -metric space Assume that that there exist and such thatfor all comparable , where where for with Assume either or is continuous or is regular. Then and have a common fixed point.

The above corollary is the -metric space version of Theorem 16 via the -class function , except that is taken in addition to the fact that it is continuous in (with respect to the conditions on in Theorem 16). If we set in Corollary 21, we get the following.

Corollary 22. Let be a partially ordered set. Let and be weakly increasing self-mappings on a complete -metric space satisfyingfor all comparable , where , , and where for with . Assume either or is continuous or is regular. Then and have a common fixed point.

Taking with () in Theorem 19, we state the following.

Corollary 23. Let be a partially ordered set. Let and be weakly increasing self-mappings on a complete -metric space satisfyingfor all comparable . Assume either or is continuous or is regular. Then and have a common fixed point.

Proceeding similarly as Theorem 19, we have the following.

Corollary 24. Let be a partially ordered set. Let and be weakly increasing self-mappings on a complete -metric space satisfying for all comparable , where and .
Assume either or is continuous or is regular. Then and have a common fixed point.

Taking in Theorem 19, we have the following.

Corollary 25. Let be a partially ordered set. Let and be weakly increasing self-mappings on a complete -metric space satisfyingfor all comparable , where Assume either or is continuous or is regular. Then and have a common fixed point.

We provide the following example illustrating Theorem 19.

Example 26. Let for all . Let be a set endowed with the partial order . Let be a -metric on , given as and It is clear that is continuous on and the pair is weakly increasing. Take and for all . We shall prove that, for all , we have First, for we have Now, we will discuss the following two cases.
Case 1. Suppose that . We haveMoreover ThusTherefore Thus (42) holds.
Case 2. Suppose that ; henceMoreoverThereforeThusthat is, (42) holds.
Second, for Let us discuss the following two cases.
Case 1. Suppose that . HenceMoreover ThereforeWe deducethat is, (42) holds.
Case 2. Suppose that . ThenMoreoverThereforeThus Then (42) holds. All the conditions of Theorem 19 are satisfied and is the common fixed point for and .
On the other hand, the contractive condition of Theorem 17 is not satisfied. Indeed, for and , we have

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Copyright © 2017 A. H. Ansari et al. 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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