New Fixed Point Theorems for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 24.5728 16.7875" width="24.5728pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-230" d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"/></g><g transform="matrix(.017,0,0,-0.017,8.547,0)"><path id="g190-46" d="M300 253H71L57 198H286L300 253Z"/></g><g transform="matrix(.017,0,0,-0.017,14.68,0)"><path id="g113-243" d="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"/></g></svg>-</span>Contraction on Rectangular <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="M2" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><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>Metric Spaces

Kari, Abdelkarim; Rossafi, Mohamed; Marhrani, El Miloudi; Aamri, Mohamed

doi:https://doi.org/10.1155/2020/8833214

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2020 | Article ID 8833214 | https://doi.org/10.1155/2020/8833214

New Fixed Point Theorems for -Contraction on Rectangular -Metric Spaces

Abdelkarim Kari,¹Mohamed Rossafi,²El Miloudi Marhrani,¹and Mohamed Aamri¹

Academic Editor: Alberto Fiorenza

Received13 Aug 2020

Revised17 Sept 2020

Accepted04 Oct 2020

Published20 Oct 2020

Abstract

The Banach contraction principle is the most celebrated fixed point theorem and has been generalized in various directions. In this paper, inspired by the concept of -contraction in metric spaces, introduced by Zheng et al., we present the notion of -contraction in -rectangular metric spaces and study the existence and uniqueness of a fixed point for the mappings in this space. Our results improve many existing results.

1. Introduction

The Banach contraction principle is a fundamental result in fixed point theory [1]. Due to its importance and simplicity, several authors have obtained many interesting extensions and generalizations of the Banach contraction principle (see [2–4]).

Many generalizations of the concept of metric spaces have been defined, and some fixed point theorems were proven in these spaces. In particular, -metric spaces were introduced by Bakhtin [5] and Czerwik [6] as a generalization of metric spaces. Many mathematicians worked on this interesting space. For more, the reader can refer to [7–10].

In 2000, generalized metric spaces were introduced by Branciari [11], in such a way that triangle inequality is replaced by the quadrilateral inequality for all pairwise distinct points , and . Any metric space is a generalized metric space, but in general, generalized metric space might not be a metric space. Various fixed point results were established on such spaces (see [12–17] and references therein).

Recently, George et al. [10] announced the notion of -rectangular metric space; many authors initiated and studied many existing fixed point theorems in such spaces (see [18–23]).

Very recently, Zheng et al. [24] introduced a new concept of -contractions and established some fixed point results for such mappings in complete metric spaces and generalized the results of Brower and Kannan. For more works related to theta-contractions, see [25–27].

In this paper, we introduce a new notion of generalized -contractions and establish some fixed point results for such mappings in complete -rectangular metric spaces. The results presented in the paper extend the corresponding results of Kannan [3] and Reich [4] on -rectangular metric spaces. Also, we derive some useful corollaries of these results.

2. Preliminaries

Definition 1 (see [7]). Let be a nonempty set and be a given real number and let be a mapping such that for all and all distinct points each distinct from and : (1) if only if ; (2) ; and (3) .

Then is called a -rectangular metric space.

Example 2 (see [19]). Let , where and . Define as follows:

Then is a -rectangular metric space with coefficient .

Lemma 3 (see [20]). Let be a -rectangular metric space.(a)Suppose that sequences and in are such that and as with , and for all Then, we have (b)if and is a Cauchy sequence in with for any converging to then for all

Lemma 4. Let be a -rectangular metric space and let be a sequence in such thatIf is not a Cauchy sequence, then there exist and two sequences and of positive integers such that

Proof. If is not a Cauchy sequence, then there exist and two sequences and of positive integers such thatfor all positive integers . By the -rectangular inequality, we haveTaking the upper and lower limits as in (5) and using (2) (4), we obtainUsing the -rectangular inequality again, we haveTaking the upper and lower limits as in (7) and using (2) and (4), we obtainUsing the -rectangular inequality again, we haveTaking the upper and lower limits as in (9) and using (2) and (4), we obtainUsing the -rectangular inequality again, we haveTaking the upper and lower limits as in (11) and (12) and using (2) (6), we obtain

The following definition was given by Ding et al. in [13].

Definition 5 (see [13]). Let be the family of all functions such that is increasing; for each sequence ; ; and is continuous.

In [21] Radenovic et al. presented the concept of -contractions on metric spaces.

Definition 6 (see [21]). Let be the family of all functions , such that is nondecreasing; for each , and is continuous.

Lemma 7 (see [21]). If , then , and for all .

Definition 8 (see [21]). Let be a metric space and be a mapping.
is said to be a -contraction if there exist and such that for any where

In [27], Zheng et al. proved the following nice result.

Theorem 9 (see [21]). Let be a complete metric space and let be a -contraction. Then, has a unique fixed point.

3. Main Result

In this paper, using the idea introduced by Zheng et al., we present the concept -contraction in -rectangular metric spaces, and we prove some fixed point results for such spaces.

Definition 10. Let be a -rectangular metric space with parameter space and be a mapping.(1) is said to be a -contraction if there exist and such thatwhere(2) is said to be a -contraction if there exist and such thatwhere(3) is said to be a -Kannan-type contraction if there exist and such that we have(4) is said to be a -Reich-type contraction if there exist and such that we have

Theorem 11. Let be a complete -rectangular metric space and let be an -contraction, i.e., there exist and such that for any , we have

Then, has a unique fixed point.

Proof. Let be an arbitrary point in and define a sequence byfor all If there exists such that , then the proof is finished.
We can suppose that for all
Substituting and , from (22), for all , we havewhereIf by (24), we havewhich is a contradiction. Hence, Thus,Repeating this step, we conclude thatFrom (27) and using , we getTherefore, is a monotone strictly decreasing sequence of nonnegative real numbers. Consequently, there exists such thatNow, we claim that . Arguing by contradiction, we assume that Since is a nonnegative decreasing sequence, then we haveBy property of , we getBy letting in inequality (32), we obtainIt is a contradiction. Therefore,Next, we shall prove thatWe assume that for every , . Indeed, suppose that for some with and using (29), we haveContinuing this process, we can thatIt is a contradiction. Therefore, for every ,
Applying (22) with and , we havewhereSo, we getTake and Thus, by (40), one can writeBy we getBy (36), we haveIt implies thatTherefore, the sequence is a nonnegative decreasing sequence of real numbers. Thus, there exists such thatBy (34) assume that , we haveTaking the in (40) and using the property of , we obtainTherefore,By , we getIt is a contradiction. Therefore,Next, we shall prove that is a Cauchy sequence, i.e., for all . Suppose to the contrary. By Lemma 4 Then, there is an such that for an integer there exists two sequences and such that [i)] [ii)] [iii)] and [vi)]
Now, using (i), (ii), and (34), we conclude thatNow, applying (22) with and , we obtainLetting the above inequality and using , (51) and (iv), we obtainTherefore,Since is increasing, we getwhich is a contradiction. Then,Hence, is a Cauchy sequence in . By completeness of there exists such thatNow, we show that ; arguing by contradiction, we assume thatSince as for all , then from Lemma 3, we conclude thatNow, applying (22) with and , we havewhereTherefore,By letting in inequality (62), using (59) and , we obtainBy we getwhich implies thatwhich is a contradiction. Hence, .
Uniqueness: now, suppose that are two fixed points of such that . Therefore, we haveApplying (22) with and , we havewhereTherefore, we havewhich implies thatwhich is a contradiction. Therefore, .

Corollary 12. Let be a complete -rectangular metric space and be the given mapping. Suppose that there exist and such that for any we have

Then, has a unique fixed point.

Example 13. Let , where and
Define as follows:

Then, is a -rectangular metric space with coefficient . However, we have the following: (1) is not a metric space, as . (2) is not a rectangular metric space, as .

Define mapping by

Evidently, . Let . It is obvious that and

Consider the following possibilities:(1), . Then,

On the other hand,

Hence,

On the other hand,

which implies that(2)If or . Then,

which implies that

Hence, condition (22) is satisfied. Therefore, has a unique fixed point .

Theorem 14. Let be a complete -rectangular metric space and be a mapping. Suppose that there exist and such that for any where

Then, has a unique fixed point.

Proof. Let be an arbitrary point in and define a sequence byfor all If there exists such that , then the proof is finished.
We can suppose that for all
Substituting and , from (81), for all , we haveAs in the proof of Theorem 11, we conclude thatIf for some , , it follows from (84), , and using Lemma 7 we getIt implies thatwhich is a contradiction. Hence,
Therefore,Since is increasing, soTherefore, is a monotone strictly decreasing sequence of nonnegative real numbers. Consequently, there exists such thatNow, we claim that . Arguing by contradiction, we assume that Since is a nonnegative decreasing sequence, we haveThus, we haveBy letting in inequality (92), using , we obtainIt is a contradiction. Therefore,Next, we shall prove thatWe assume that for every Indeed, suppose that for some with , so we have .
By (89), we getContinuing this process, we can thatIt is a contradiction. Therefore,Applying (81) with and , we havewhereTherefore,which implies thatTake and . By (102), we haveAgain by (89), we getTherefore,Then, the sequence is monotone nonincreasing, so it converges to some such thatBy (94) assume that, we haveTaking the in (101) and using , , and Lemma 7, we obtainwhich implies thatTherefore,which is a contradiction. Therefore,Next, we shall prove that is a Cauchy sequence, i.e., for all . Suppose to the contrary, then there is an such that for an integer , there exist two sequences and , such that [i)] [i)] [iii)] and [vi)]
Since is a -contraction, applying (81) with and , we haveAs in the proof of Theorem 11, we haveBy letting in inequality (112) and using , , , vi), (114) and Lemma 7, we obtainTherefore,It is a contradiction. Therefore,Hence, is a Cauchy sequence in . By completeness of , there exists in such thatNow, we show that arguing by contradiction, we assume thatAs in the proof of Theorem 11, we conclude thatSince T is a -contraction, applying (81) with and , we conclude thatwhereThis implies thatBy letting in inequality (123) and using , (120) and Lemma 7, we obtainAs is increasing, then we deduced thatTherefore, . It is a contradiction. So, . Thus, has a fixed point.
Uniqueness: let fix where . Then, fromApplying (81) with and , we havewhereTherefore, we haveThis implies that . It is a contradiction. Therefore, .

Following from Theorem 14, we obtain the fixed point theorems for the -Kannan-type contraction and the -Reich-type contraction. The results presented in the paper improve and extend the corresponding results due to the Kannan-type contraction and Reich-type contraction on rectangular -metric space.

Theorem 15. Let be a complete -rectangular metric space and be a Kannan-type contraction, then has a unique fix.

Proof. Since is a Kannan-type contraction, then there exist and such that

Therefore, is -contraction. As in the proof of Theorem 14 we conclude that has a unique fixed point.

Theorem 16. Let be a complete -rectangular metric space and be a Reich-type contraction. Then, has a unique fixed point.

Proof. Since is a Reich-type contraction, then there exist and such that

Therefore, is a -contraction. As in the proof of Theorem 14 we conclude that has a unique fixed point.

Corollary 17. Let be a complete -rectangular metric space and be a Kannan type mapping, i.e., there exists such that for all ,

Then, has a unique fixed point.

Proof. Let for all , and for all . Clearly and . We prove that is a -Kannan-type contraction. Indeed,

As in the proof of Theorem 15, has a unique fixed point

Corollary 18. Let be a complete -rectangular metric space and be a Reich-type mapping, i.e., there exists such that for all ,

Then, has a unique fixed point.

Proof. Let for all and for all .
We prove that is a -Reich-type contraction. Indeed,As in the proof of Theorem 16, has a unique fixed point

Corollary 19. (Theorem 11 Let be a complete -rectangular metric space and be a mapping. Suppose that there exist and such that for any where

Then, has a unique fixed point.

Proof. By taking , with , obvious , then we conclude that is a -contraction. As in the proof of Theorem 14, has a unique fixed point.

Very recently, Kari et al. in [8] proved the result (Theorem 1) on -complete rectangular -metric spaces. In this paper, we prove this result in complete rectangular -metric spaces.

Corollary 20. Let be a complete -rectangular metric space with parameter , and let be self-mapping on . If for all , we havewhere , for . Then, has a unique fixed point.

Proof. We prove that is a -contraction. Indeed,As in the proof of Theorem 14, has a unique fixed point.

Example 21. Let , where and . Define as follows:Then, is a -rectangular metric space with coefficient . However we have the following: [1)]is not a metric space, as . [2)] is not a-metric space for, as . [3)] is not a rectangular metric space, as . Define mapping by

Then, . Let . It is obvious that and

Consider the following possibilities:

Case 1. with and assume that .

Therefore,

On the other hand,

Since , then

which implies that

Case 2. , or

Therefore, , , then .

In this case, consider two possibilities:(1) then Therefore,

So, we have

On the other hand,

Since then

This implies that(2) then Therefore,

So, we have

On the other hand,

Since then

This implies that

Hence, condition (81) is satisfied. Therefore, has a unique fixed point .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2020 Abdelkarim Kari 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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