Common Fixed Point Theorems via Inverse <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.3443pt" id="M1" height="15.7437pt" version="1.1" viewBox="-0.0657574 -11.3994 28.4565 15.7437" width="28.4565pt"><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><g transform="matrix(.012,0,0,-0.012,11.377,4.134)"><path id="g50-108" d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 643L128 642C165 641 171 640 162 602L24 -5L31 -12C51 -4 81 4 110 9L145 181C154 195 180 220 205 242C233 170 263 106 291 53C318 1 335 -12 360 -12C383 -12 425 0 483 82L463 105C437 79 411 57 400 57C388 57 378 68 355 110C329 156 288 244 268 298C299 333 355 376 405 376C426 376 440 373 448 368C453 365 465 368 468 373C477 386 485 405 485 418Z"/></g><g transform="matrix(.017,0,0,-0.017,18.171,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g></svg> Class Functions in Metric Spaces

Zhou, Mi; Liu, Xiao-Lan; Hojat Ansari, Arslan; Jain, Mukesh Kumar; Deng, Jia

doi:https://doi.org/10.1155/2021/6648993

Journal of Mathematics

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Recent Advances in Fixed Point Theory in Abstract Spaces

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Research Article | Open Access

Volume 2021 | Article ID 6648993 | https://doi.org/10.1155/2021/6648993

Common Fixed Point Theorems via Inverse Class Functions in Metric Spaces

Mi Zhou,¹Xiao-Lan Liu,^2,3,4Arslan Hojat Ansari,^5,6Mukesh Kumar Jain,⁷and Jia Deng²

Academic Editor: Naeem Saleem

Received10 Dec 2020

Revised07 Jan 2021

Accepted31 Mar 2021

Published21 Apr 2021

Abstract

In this paper, we firstly introduce a new notion of inverse class functions which extends the notion of inverse class functions introduced by Saleem et al., 2018. Secondly, some common fixed point theorems are stated under some compatible conditions such as weak semicompatible of type , weak semicompatibility, and conditional semicompatibility in metric spaces. Moreover, we introduce a new kind of compatibility called compatibility which is weaker than property and also present a common fixed point theorem in metric spaces via inverse class functions. Some examples are provided to support our results.

1. Introduction and Preliminaries

As a follow-up work of A.H. Ansari’s research on fixed point (or common fixed point) theory via auxiliary C-class functions, very recently, Saleem et al. [1] introduced the new concept of inverse C-class functions and obtained some corresponding fixed point theorems under certain weak compatibility assumption via inverse C-class functions. In 1976, Jungck [2] defined the concept of commutative maps and initiated the study of the existence of a common fixed point of such maps in metric spaces. After which, Sessa [3] introduced the weak version of commuting maps called weak commuting maps. Next, Jungck [4, 5] provided some generalizations of weak commuting maps by providing the notions of compatible maps and compatible maps of type . Minor relaxations of compatible of type are introduced by Pathak and Khan [6], which are well known as compatible and compatible (see [6], for more details).

Singh et al. [7] proposed the notion of compatibility of type by making a minor modification of compatibility of type . By splitting the concept of compatibility of type , Singh et al. [7] also gave some relaxations of compatibility type of which are known as compatibility of type and compatibility of type .

Definition 1. (see [7]). Two self-maps and of a metric space are said to be compatible of type , if , whenever is a sequence in such that , for some . Similarly, two self-maps and of a metric space are said to be compatible of type , if , whenever is a sequence in such that , for some .
It is easy to see that compatibility of type implies both compatibility of type and compatibility of type ; however, compatibility or compatibility of type does not imply compatibility of type .
In 1994, Pant [8] introduced the following definition.

Definition 2. (see [8]). Two self-maps and defined on a metric space are said to be weakly commuting, if there exists a real number such that , for all .
Note that ; then, and are weakly commuting.
In 1997, Pathak et al. [9] introduced the notions of weak commuting of type and weak commuting of type as follows.

Definition 3. (see [9]). Two self-maps and of a metric space are said to be weakly commuting of type , if there exists a real number such that , for all .

Definition 4. (see [9]). Two self-maps and of a metric space are said to be weakly commuting of type , if there exists a real number such that , for all .
It is noted that compatible maps and are also weakly commuting of type and weakly commuting of type . Moreover, we can find suitable examples which show that weakly commuting mappings and weakly commuting of type (or ) are independent concepts (see examples of [9, 10]).
In 2008, Gopal et al. [10] introduced the notions of absorbing and absorbing stated as follows.

Definition 5. (see [10]). Let and be two self-maps of a metric space ; then, is said to be -absorbing, if there exists a real number such that for all . Similarly, let and be two self-maps of a metric space ; then, is said to be -absorbing, if there exists a real number such that for all .
Jungck and Rhoades [11], in 1998, introduced the concept of weak compatibility which is weaker than the concept of compatibility.
Another generalization of compatible maps called semicompatible maps was firstly introduced by Cho et al. [12] under the setting of topological spaces in which a pair of self-maps are called to be semicompatible if condition implies that ; for sequence in and , whenever , , and then , as , hold. However, Singh and Jain [13] redefined this concept by using condition only stated as follows.

Definition 6. (see [13]). A pair of self-maps of a metric space is said to be semicompatible, if holds whenever is a sequence in such that for some .
It follows that if is semicompatible and , then . It is also noted that if the pair is semicompatible, then it is weak compatible; however, the converse is not true. Further, the semicompatibility of the pair does not imply the semicompatibility of the pair (see Example 3.2 in [13]).
Now, we make a minor modification of semicompatibility to introduce the notion of semicompatible of type as follows.

Definition 7. A pair of self-maps of a metric space is said to be semicompatible of type , if and hold whenever is a sequence in such that for some .
It is obvious that semicompatibility of type implies semicompatibility; however, the converse is not true.
Recently, Saluja et al. [14, 15] introduced the weak semicompatible maps and conditional semicompatible maps and obtained corresponding fixed point theorems (see [14–16], for more details).

Definition 8. (see [14]). A pair of self-maps of a metric space is said to be weakly semicompatible, if or , whenever is a sequence in such that for some .

Definition 9. (see [15]). A pair of self-maps of a metric space is said to be conditionally semicompatible; if whenever the set of sequences satisfying is nonempty, then there exists at least a sequence satisfying such that and .
It is obvious that semicompatibility of type implies weak semicompatibility. From the definition itself, it is clear that if a pair of self-maps is semicompatible of type , then it is necessarily conditionally semicompatible; however, the conditionally semicompatible maps are not necessarily semicompatible of type .

Example 1. Let and be the usual metric on . Define as follows:Let us consider the sequence ; we haveHowever, if we take , we have thatThus, the pair is conditional semicompatible.
Finally, we introduce a new kind of compatibility of a pair of self-maps called compatible firstly proposed by Jain et al. [17] as follows.

Definition 10. Let be a self-map defined on satisfying for some sequence and . Then, a pair of self-maps defined on is called compatible if .

Example 2. Let , , , and . Take . Since with and , then pair is compatible. However, and .
It is obvious that compatibility of a pair self-maps implies . property of a pair of self-maps by taking self-map as an identity map.
Let , , , and (identity function on ). Take . Here, . Hence, pair self-maps satisfy property.
Now, we introduce one more example of compatibility including four maps as follows.

Example 3. If with the usual metric. Define byChoose , where when , then , , , and .
Since and , then the pair is compatible. Next, since and , then the pair is compatible. Further, since and , then the pair is compatible. Finally, since and , then the pair is compatible.
Ansari, in 2014, firstly [18], introduced the concept of class functions and proved some fixed point theorems via class functions (see [19, 20] for more details).

Definition 11. (see [18]). A mapping is called a class function if it is continuous and the following axioms hold:(1) for all (2) implies that or Denote the family of class functions by .

Example 4. (see [18]). The following functions are elements of , for all :(1), implies (2), for some , implies (3), for some , implies or (4), for some , implies or (5), for , implies (6), , for , implies (7), for , implies or (8), implies (9), where is continuous, implies (10), implies (11), implies , where is a continuous function such that if and only if (12), implies , where is a continuous function such that for all (13), implies (14), implies (15), implies , where is a continuous function such that and , for Afterward, by the motivation of class functions, Saleem et al. [1, 21] introduced a new notion of inverse class functions as follows.

Definition 12. (see [1]). A mapping is called an inverse class function if it is continuous and the following axioms hold:(1) for all (2) implies that or Denote the family of inverse class functions by .

Example 5. (see [1]). The following functions are elements of , for all :(1) implies (2), for some implies (3), for some , implies or (4), for some , implies (5), implies , where is an upper semicontinuous function such that and , for Motivated by the above definition, we now define inverse class functions as follows.

Definition 13. A mapping is called an inverse class function if it is continuous and the following axioms hold:(1) for all and some (2) implies that or Denote the family of inverse class functions by . Every inverse class function and inverse class function are equivalent when ; however, an inverse class function may not be an inverse class function.

Example 6. A mapping is defined by for all . Then, clearly, is an inverse class function for , but it is not an inverse class function.

Example 7. The following functions are elements of , for all :(1) implies for some and (2) implies for and some (3) implies or for and some (4) implies for some and some (5), implies , where is an upper semicontinuous function such that and , for and some

Definition 14. Let denote the class of functions which satisfy the following conditions:(a) is continuous and increasing with (b)

Definition 15 (see [22]). A function is said to be ultra-altering distance function if is nondecreasing and continuous; , for all and . Denote the class of ultra-altering distance functions by .

Lemma 1. Every sequence in metric space will be Cauchy if there exists such that , for all .
The aim of this presented paper is to provide some common fixed point theorems under several compatible conditions mentioned above via inverse class functions, which extend, generalize, and improve the existing results in the literature. Some examples are provided to illustrate the validity of our results.

2. Main Results

Theorem 1. Let be a complete metric space and a let pair of self-maps be semicompatible, satisfying the following assumptions: Here, and , for all . Moreover, , with , , , , , for some and , . If the pair is compatible of type , then and have a unique common fixed point in .

Proof. Let be any point in . Since , there exists such thatContinuing this way, we can construct a sequence in satisfyingBy the assumption of , we haveBy the monotonicity of , we haveAgain, from the triangle inequality, that is, , we havewhich further yields thatSince , , then ; from Lemma 1, it follows that sequence is a Cauchy sequence. Since is complete, there exists a point such thatNow, we will show that is a common fixed point of and .
Since the pair is semicompatible, we haveSince and are compatible of type , it follows thatNow, by the definition of and assumption , we obtainTaking the limit as in above inequality, it follows thatwhich implies thatAgain, it follows from the definition of and assumption thatTaking the limit as in above inequality, we conclude thatSince and , it follows thatwhich implies thatFrom the definition of class functions, we haveBy the definitions of and , we have that , which proves that , that is, is a common fixed point of and .
Next, we will prove the uniqueness of common fixed point of and in .
Suppose that is another common fixed point of and , that is .
From above argument, it may be concluded thatSince and , it follows thatwhich implies thatFrom the definition of class functions, we haveBy the definitions of and , we have that , which proves that .

Remark 1. (i)The conclusion of Theorem 1 still holds under the assumptions of semicompatibility of the pair and compatibility of type .(ii)If the inequality in assumption is replaced bythe conclusion still holds.
Here is an illustrated example to support the validity of Theorem 1 as follows.

Example 8. Let be a usual metric space. We define and on as follows:To verify that the pair is semicompatible as well as compatible of type , we take any sequence ; then, and . Define by , with and , for all . Then, for all , we haveIt is verified that assumption holds true provided by , and such that , , , and .
Also, it is obvious that assumption holds true, for all .
Hence, functions and satisfy all conditions of Theorem 1 with are common fixed point.
Taking as the identity map in Theorem 1, we have the following fixed point theorem.

Corollary 1. Let be a self-map defined on a complete metric space satisfying the following assumption:where and , for all . Moreover, with , , , , for some and , . Then, has a unique fixed point in .
Since semicompatibility of type implies semicompatibility, we can obtain the following theorem.

Theorem 2. Let be a complete metric space and let a pair of self-maps be semicompatible of type satisfying the following assumptions: where and , for all . Moreover, with , , , , , for some and , . If the pair is either compatible of type or compatible of type , then and have a unique common fixed point in .
Now, we will present a common fixed point theorem under weakly semicompatible condition via -class functions as follows.

Theorem 3. Let be a complete metric space and let a pair of self-maps be weakly semicompatible and weakly commuting type of satisfying the following assumptions: Here, and , for all . Moreover, , , for some with , , , , and , . If the pair is either compatible of type or compatible of type , then and have a unique common fixed point in .

Proof. Let be any point in . Since , there exists such thatContinuing this way, we can have a sequence in satisfyingFrom the discussion of Theorem 1, we have that there exists a point such thatNow consider the following possibilities: Case 1. ( and are compatible of type ). Since and are weakly semicompatible, it follows that Firstly, we take Since and are compatible of type , it yields that Hence, the conclusion can directly follow from the proof of Theorem 1. Secondly, we take Since and are compatible of type , it yields that Again since and are weakly commuting of type , which implies Taking limit as in above inequality, we have From the monotonicity of and assumption , we have Taking limit as in above inequality, we have Since and , it follows that which implies that From the definition of class functions, we have By the definitions of and , we have that , which proves that . Again, from the definition of and assumption , we have Together with , , from the above inequality, it follows that , which proves that . Hence, is a common fixed point of and . Case 2 ( and are compatible of type ).Since and are weakly compatible, it follows thatFirstly, we takeSince and are compatible of type , it yields thatTherefore, the conclusion directly follows from Theorem 1 and (i) of Remark 1.
Secondly, we takeSince and are compatible of type , it yields thatAlso and are weakly commuting of type , which impliesTaking limit as in the above inequality, we haveFrom the monotonicity of and assumption , we haveTaking limit as in the above inequality, we haveSince and , it follows thatwhich implies thatFrom the definition of class functions, we haveBy the definitions of and , we have that , which proves that .
Again, from the monotonicity of and assumption , we haveTaking limit as in above inequality and by the definition of , we haveSince , it yields that or , which further implies ; hence, is a common fixed point of and .
Finally, the uniqueness of the common fixed point of and can be directly obtained from the second half proof of Theorem 1.

Remark 2. If and are assumed to be weakly commuting of type , the conclusion of Theorem 3 still holds true.
Now, we provide an example to verify the validity of Theorem 3 as follows.

Example 9. Let be a usual metric space and . We define and on as follows: for all and for all . It is easy to check that .
To show that the pair of mappings is weak semicompatible and compatible of type , we take any sequence , then . Hence, and . It also can be observed that pair satisfy weak commuting of type for . To satisfy condition , we choose with and , for all . Then, for all , we haveHere are three possible cases as follows: Case (1): if , taking , and satisfying , and , then we have which proves that condition holds true. Case (2): if , taking , and satisfying , and , then we have which shows that condition holds true. Case (3): if , taking , and satisfying , and , then we havewhich shows that condition holds true.
Hence, the pair satisfies all conditions of Theorems 3, with 1 being the unique common fixed point.
Next, we will present some common fixed point theorems under conditional semicompatibility as follows.

Theorem 4. Let be a complete metric space and a pair of self-maps be conditional semicompatible satisfying the following assumptions: Here, and , for all . Moreover, , , for some with , , , , and , . If is absorbing or is absorbing, then and have a unique common fixed point in .

Proof. Similar to the first part of the proof in Theorems 1 or 3, we can construct a sequence in such thatAgain since and are conditional semicompatible and (nonempty), then there exists a sequence satisfying such that and . Case 1: is absorbing. Since is absorbing, this yields that Taking limit as in the above inequality, it follows that Since , then there exists in such that . Next, we will show that . By the definition of and assumption , we have Together with , , from above inequality, it follows that or . Again, since is absorbing, it yields which implies that or . Now, we will show . By the definition of , and assumption , we have Together with , , from the above inequality, it follows that or . Therefore, is a common fixed point of and . Case 2: is absorbing.Since is absorbing, this yieldsTaking limit as in the above inequality, it follows thatNext, we will show .
By the definition of and assumption , we haveTogether with , , from the above inequality, it follows that or . Therefore, is a common fixed point of and .
The uniqueness of common fixed point of and can be directly obtained by following the same argument as Theorem 1.

Remark 3. The results of Theorem 4 still hold by replacing condition with condition stated as follows. : .Here, , for all . Moreover, with , , , for some and , .
Moreover, is said to be an altering distance function (see [19]), which satisfies is continuous and increasing; . Denote the class of altering distance functions by .
Now, we provide an example to verify the validity of Theorem 4 as follows.

Example 10. Let , with and let be the usual metric on . Define as follows:It is clear that .
Taking , where as , we have and . Therefore, (nonempty). Then, we have a sequence , where as , for which and . Moreover, and . Therefore, the pair is conditional semicompatible.
For , then and . Therefore, and satisfy with . Also, for , then for all real number . Therefore, and satisfy , for all and ; that is, is absorbing with .
To satisfy condition , we choose with and , for all . Then, for all we haveHere are three possible cases: Case (1): if , we have It is obvious that holds true by taking , and satisfying , , , and . Case (2): if , we have It is obvious that holds true by taking and . Case (3): if , we haveIt is obvious that holds true by taking , and satisfying , , , and .
To sum up, condition holds true by taking , and satisfying , , , and .
Therefore, and satisfy all the conditions of Theorems 4 with 5 being the unique common fixed point.

Theorem 5. Let be a complete metric space and let a pair of self-maps be conditional semicompatible satisfying the following assumptions: Here, and , for all . Moreover, , , for some with , , , , and , . If pair is weak commuting either of type or of type , then and have a unique common fixed point in .

Proof. From the part of the proof in Theorems 1 or 3, we can construct a sequence in such that .
Again since and are conditional semicompatible and (nonempty), then there exists a sequence satisfying such that and . Case 1: pair is weak commuting of type . Since pair is weak commuting of type , it yields Taking limit as in the above inequality, it follows that Now, we will show that . By the definition of and assumption , we have Together with , , from the above inequality, it follows that or . Again, since pair is weak commuting of type , then we have which implies . Now, we will show . By the definition of and assumption , we have Since , , and , it follows that which implies that From the definition of class functions, we have By the definitions of and , we have that , which proves that . Therefore, is a common fixed point of and . Case 2: pair is weak commuting of type .Since pair is weak commuting of type , it yieldsTaking limit as in the above inequality, it follows thatNow, we will show that .
By the definition of and assumption , we haveTaking limit as in the above inequality, we haveSince , , it follows thatwhich implies thatFrom the definition of class functions, we haveBy the definitions of and , we have that , which proves that .
Since , there exists such that . Now, we will show that .
By the definition of , , and assumption , we haveTogether with , , from above inequality, it follows that or .
Since pair is weak commuting of type , it yieldswhich yields .
Next, we will show .
By the definition of and assumption , we haveSince , , , and , it follows thatwhich implies thatFrom the definition of class functions, we haveBy the definitions of and , we have that , which proves that . Therefore, is a common fixed point of and .
The uniqueness of common fixed point of and is also obtained by following the same proof of Theorem 2.

Remark 4. The conclusions of Theorem 5 still hold true by replacing condition with condition stated as before.
Now, we will provide an example to verify the validity of Theorem 5 as follows.

Example 11. Let with the usual metric and . We define and on as follows:It is clear that .
To show that the pair of self-maps is conditional semicompatible, we take any sequence , then (nonempty). Now, we choose another sequence , then and . It can be observed that the pair satisfies weak commuting of type in intervals and for all real numbers. Also, the pair satisfies weak commuting of type at with . To verify assumption , we define with , , for all . For , we haveIt is obvious that holds true by taking , , , and such that , , , and . It is also clear that the above inequality holds true for all and .
Therefore, and satisfy all the conditions of Theorem 5 with being the unique common fixed point.
Finally, we will present a common fixed point theorem under compatibility as follows.

Theorem 6. Let be a complete metric space and let be self-maps satisfying the following assumptions: , Here, , for all . Moreover, with , , , for some and , . If the pair is compatible and is compatible, then have a unique common fixed point in .

Proof. Let . Since , then there exists such that . For and , there exists such that . Continuing this way, we can construct the sequence in such that and .
By the assumption of and properties of and , we haveBy the similar procedure demonstrated in Theorem 1, we can get .
From Lemma 1, it follows that sequence is a Cauchy sequence. Since is complete, there exists a point such thatNext, we will show that is a common fixed point of .
Since the pair is compatible and the pair is compatible and by equation (102), we have the following outcomes:By the assumption of and properties of and , we haveLetting in the above inequality, with equation (103), we haveSince , , we havewhich implies thatFrom the definition of class functions, we haveBy the definitions of and , we have that , which proves that .
We also haveLetting in the above inequality, with equation (104), we haveSince , , we havewhich implies thatFrom the definition of class functions, we haveBy the definitions of and , we have that , which proves that .
Now, we will prove . If not, by the assumption of and properties of and , we haveLetting in the above inequality, we havewhich implies ; that is, .
In the same manner, we can also get .
Hence, is a common fixed point of .
The uniqueness of can also be obtained by the similar way stated in Theorem 1.
Now, we present the following example to support the validity of Theorem 6.

Example 12. Let with the usual metric. We define on as follows:It is clear that and .
To show that pair is compatible and pair is compatible, we take ; then, , , , with , and .
Hence, the pair is compatible and pair is compatible.
To verify assumption , we define with , where and satisfying , and . Define , for all . Then, for all , we haveIt is easy to verify that for all satisfying with , , , and ; that is, assumption holds true.
Then, , and T satisfy all the conditions of Theorem 6; moreover, 0 is the unique common fixed point of , and T.

3. Conclusion

In this paper, concepts of semicompatibility of type (A), –compatibility are introduced, in which –compatibility is weaker than (E.A) property. We also give a brief discussion on the relation between new notions and other existing types of compatibility. Motivated by the notion of inverse C-class functions, a distinct concept of inverse class functions is introduced which extends the notion of inverse C-class functions introduced by Saleem et al. [1, 21]. Moreover, some common fixed point theorems are stated under some compatible conditions such as semicompatibility, semicompatibility of type (A), weak semicompatibility, conditional semicompatibility, and –compatibility in metric spaces via inverse class functions which are a valuable supplement to the common fixed point theory.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally and significantly to the writing of this article. All the authors read and approved the manuscript.

Acknowledgments

The authors thank the colleagues for their proofreading and other helpful suggestions. Xiao-lan Liu was partially supported by National Natural Science Foundation of China (no. 11872043), Sichuan Science and Technology Program (no. 2019YJ0541), Zigong Science and Technology Program (no. 2020YGJC03), Scientific Research Project of Sichuan University of Science and Engineering (nos. 2017RCL54, 2019RC42, and 2019RC08), the Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (no. 2020WYJ01), Opening Project of Sichuan Province University Key Laboratory of Bridge Non-Destruction Detecting and Engineering Computing (no. 2019QZJ03), the Open Fund Project of Artificial Intelligence Key Laboratory of Sichuan Province (no. 2018RYJ02), and 2020 Graduate Innovation Project of Sichuan Univaersity of Science and Engineering (no. y2020078).

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Copyright © 2021 Mi Zhou 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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