Cyclic <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="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 31.0837 11.6718" width="31.0837pt"><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(.017,0,0,-0.017,11.737,0)"><path id="g190-46" d="M300 253H71L57 198H286L300 253Z"/></g><g transform="matrix(.017,0,0,-0.017,17.87,0)"><path id="g113-153" d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z"/></g></svg>-</span>Weak Contraction-Weak Nonexpansive Mappings and Some Fixed Point Theorems in Metric Spaces

Bakr, Sahar Mohamed Ali Abou

doi:https://doi.org/10.1155/2021/6642564

Abstract and Applied Analysis

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Abstract Conclusion Abbreviations Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6642564 | https://doi.org/10.1155/2021/6642564

Cyclic -Weak Contraction-Weak Nonexpansive Mappings and Some Fixed Point Theorems in Metric Spaces

Sahar Mohamed Ali Abou Bakr¹

Academic Editor: Narendra K. Govil

Received27 Jan 2021

Accepted07 May 2021

Published21 May 2021

Abstract

This paper introduces novel concepts of joint cyclic -weak contraction and joint cyclic -weak nonexpansive mappings and then proves the existence of a unique common fixed point of such mappings in case of complete and compact metric spaces, respectively, in particular, it proves the existence of a unique fixed point for both cyclic -weak contraction and cyclic -weak nonexpansive mappings, and hence, it also proves the existence of a unique fixed point for both cyclic -weak contraction and cyclic -weak nonexpansive mappings. The results of this research paper extend and generalize some fixed point theorems previously proved via the attached references.

1. Introduction, Notations, and Basic Definitions

In 1922, Banach [1] proved the highly important fixed point principle for contraction mappings; this principle has many applications in different branches of mathematics, including variational linear inequalities, optimization, differential equations, approximation theory, and in minimum norm problems.

Since then, exclusive interest has been shown by researchers and many research papers have been written to generalize this Banach contraction principle. Step by step, we have the following historical generalizations:

Let be a metric space and be a self mapping on , satisfying

In case of normed space, let be a normed space, be a subset of , and be a self-mapping on satisfying for some real numbers . Then, the class of all mappings defined by (1) or by (2) includes the following types of mappings.

The contraction type: [, ]. In 1922, Banach [1] initiated the field of fixed point theory. Indeed, his famous Banach contraction principle proved that every contraction mapping on a complete metric space has a ufp.

The nonexpansive type: . In 1965, Kirk [2] proved interesting fixed point theorems (fp may not be unique) concerning nonexpansive type provided that is bounded closed and convex subset of a Banach space .

The Kannan type: . In 1971 and 1975, Kannan and Wong [3, 4], respectively, enriched the fixed point theory by some results concerning such type.

The Chatterjea type: . In 1972, Chatterjea [5] proved that every Chatterjea type contraction on a complete metric space has a ufp.

The Hardy-Rogers type: [, and with ]. In 1973, Hardy and Rogers [6] proved that Hardy-Rogers type on a complete metric space has a ufp. In 2007, Sahar [7] proved that () this type has a ufp in weakly Cauchy normed space, and recently, in 2020 and 2021, Sahar [8, 9] proved the existence of a ufp in complete theta-cone metric spaces.

The general-Gregus type: [, , and ]. In 1973, Goebel et al. [10] proved the existence of fixed points of this type, where is assumed to be uniformly convex Banach space, is continuous, and is bounded closed and convex subset of .The Gregus type: [, , and ]. In 1980, Gregus [11] proved the existence of a ufp of this type provided that is a closed convex subset of a Banach space . His theorem is now known as the Gregus fixed point theorem.

The general-Gregus type: [, , and ]. In 1977/1980/1981, Gregus, Park, and Rhoades [12, 13] independently proved the existence of a ufp for such a type provided that is a compact metric space and is continuous.

In 2013, another extension of the Banach contraction principle and Gregus fixed point theorem was given by Sahar [14]; it states the following.

Theorem 1 (see [14]). Let be a closed convex and weakly Cauchy subset of a normed space and be a self-mapping satisfying for some real numbers with , , , and . Then, has a ufp.

We recall the following definition and classes of mappings.From now on, the classes , , and denote the following:

Definition 2. Let be a normed space. Then, a function from into is said to be lower semicontinuous if and only if for any real number , the level set is a closed convex subset of .

In 1997, Alber and Guerre-Delabriere [15] introduced the concept of weakly -contractive mappings and proved some fixed point theorems concerning them.

Definition 3. (see [15]). A self-mapping is said to be weakly -contractive if there is such that

Remark 4. Every contraction mapping is weakly -contractive with , where is a real number with . The converse is not true.

In 2001, Rhoades [16] delivered a study on the existence of fixed points of weakly -contractive mappings and gave further generalized version of Banach’s fixed point theorem. He showed that every weakly -contractive self-mapping on a complete metric space has a ufp.

In 2008, Dutta and Choudhury [17] generalized the concept of weakly -contractive mappings and proved the existence of fp for -weak contraction mappings.

Definition 5. A self-mapping is said to be a -weak contraction iff there are two mappings and such that the following is true: In 2011, Karapinar and Sadarangani [18, 19] gave some generalized fixed point theorems for cyclic -weak contraction type of mappings.

In 2012, Karapinar and Kumar [20] introduced the notion of cyclic weakly generalized Chatterjea mappings in metric spaces and proved corresponding fp theorems.

Definition 6. Let and be nonempty subsets of , and be a self mapping. Then is said to be a cyclic representation of with respect to iff and .

Definition 7. Let and be nonempty subsets of , and be a cyclic representation of with respect to . If there is a real number and there is a continuous function satisfying iff, then is said to be cyclic weakly Chatterjea type contraction iff satisfies Recall in [21] that a real valued function is defined by

Definition 8. Let be a cyclic representation of with respect to and be real constants such that (1)If , then is said to be cyclic generalized -contraction w.r.t (2)If , then is said to be cyclic generalized -nonexpansive w.r.t

Definition 9 (see [21]). Let be an element belonging to such that (1)If , then is said to be generalized cyclic weak contraction w.r.t .(2)If , then is said to be generalized cyclic weak non-expansive w.r.t .

In 2020, for these generalized types of contraction mappings, Sahar [21] gave the following result.

Theorem 10. If is a metric space, , , and are compact, and is continuous generalized cyclic -weak nonexpansive mapping w.r.t , then there is a single point such that . Moreover, for every , we have .

Fixed point theorems corresponding to these types of contraction mappings have many applications, not exclusively; see [22, 23] and also see the applications of the prior results given via the references mentioned in these research papers.

Motivated from the useful applications of the fixed point theorems corresponding to such types of mappings in variety fields and areas of mathematics, we introduce the following novel concepts and definitions.

Definition 11. Let be two self-mappings on a metric space and be real numbers.
Then, (1)A function is supposed to be given by(2)A function is supposed to be given by

Definition 12. Let be a self-map satisfying (1)If , then is said to be contraction type(2)If , then is said to be non-expansive type

Remark 13. Every Hardy-Roger mapping is contraction type and every general Gregus mapping is nonexpansive type. In fact, we observe that

Definition 14. Let be a cyclic representation of with respect to and be real constants such that (1)If , then is said to be cyclic contraction(2)If , then is said to be cyclic nonexpansive

Remark 15. The class of all mappings defined by (15) includes the class of all mappings defined by (13).

Definition 16. Let be a cyclic representation of with respect to , be real constants, and be such that (1)If , then is said to be cyclic -weak contraction w.r.t (2)If , then is said to be cyclic -weak nonexpansive w.r.t

Definition 17. Let be a cyclic representation of with respect to , , and be real constants such that (1)If , then is said to be cyclic -weak contraction(2)If , then is said to be cyclic -weak nonexpansive

Remark 18. The class of all mappings defined by (17) includes the class of all mappings defined by (16).

For more generalized types of contraction mappings we use the notion of class functions, see [24] and its included references.

Definition 19. A class functions is the set of all real valued continuous functions , satisfying the conditions (1) for all (2)If , then either or Now, we introduce the following main definitions:

Definition 20. Let be a cyclic representation of with respect to , be real constants, , and be such that (1)If , then is said to be cyclic -weak contraction(2)If , then is said to be cyclic -weak nonexpansive

Remark 21. The class of all -weak types of mappings includes all the previously mentioned classes. In fact; in particular replacing by for in Definition 20 ensures that the class of all mappings defined by (18) includes the class of all mappings defined by ((17)).

We also introduce the following. (1)If , then and are said to be joint cyclic--weak contraction types(2)If , then and are said to be joint cyclic--weak nonexpansive types

Definition 22. Let be a metric space with , be two self-mappings, and be real numbers satisfying (1)The cyclic condition: and (2)The following contractivity condition:where , , and .

This paper proves the existence of a unique common fixed point between any two joint cyclic -weak contraction mappings in complete metric spaces and proves the existence of a unique common fixed point of two joint cyclic -weak nonexpansive mappings in compact metric spaces. In particular, it proves the existence of a ufp of both cyclic -weak contraction and cyclic -weak nonexpansive mappings and hence proves the existence of a ufp of cyclic -weak contraction and weak nonexpansive mappings. These results extend and generalize some fixed point theorems previously proved and given in the attached references.

2. Main Results

The following fixed point theorem is for contraction mappings:

Theorem 23. Let be a complete metric space and be two closed subsets of which . If are continuous joint cyclic -weak contraction mappings on , , then there is only one point such that .

Proof. Let be an arbitrarily chosen element in . Then, is lying either in or in , if is in , then define , , , and then by induction define First, suppose that is an odd natural number. Then, Since is nondecreasing, we see that We have the following: (i)If , thenBack to the inequalities (23) with the help of inequalities (24), we have (ii)If , thenBack to the inequalities (23) with the help of inequalities (24), we have Let , clearly , we see that

Continuing this process, we see that

Now, we use the inequalities (29) to show that the sequence is Cauchy, let , . Then,

Taking the limit of the two sides of inequalities (31) as proves that . This proves that is Cauchy sequence. Since is complete, there is such that that is; . We notice that because they are subsequences of the same convergent sequence, thus because and are closed. We will prove that such a limit point is a common fixed point of and . In fact, we have thus,

Taking the limit as implies the following:

That is, , thus because ; hence, and is a fixed point of . On the other side, proves that . Therefore, accordingly and is a fixed point of . Now, we prove that such a common fixed point is unique. In fact, if there exists another point such that with , then we get

Hence, the following gives a contradiction:

This shows that , that is .

Corollary 24. Let be a complete metric space and be two closed subsets of which . If is continuous cyclic -weak contraction mapping, , then there is only one point such that .

Proof. Using Theorem 23 with completes the proof.

Corollary 25. Let be a complete metric space and be two closed subsets of which . If is continuous cyclic -weak contraction mapping, , then there is only one point such that . Moreover, for any .

Proof. Using Corollary (24) with completes the proof.

Furthermore, the following fixed point theorem is for nonexpansive mappings.

Theorem 26. Let be a compact metric space and be two closed subsets of which . If are continuous joint cyclic -weak nonexpansive mappings on , then and have common fixed points . Moreover, if , then they have a unique common fixed point.

Proof. Let be an arbitrarily chosen element in and consider the sequence defined by (20). Back to the inequalities (23) with the help of inequalities (24), we have Using the inequalities (27), we see that Consequently, we have Continuing gives Similarly, if is an even natural number, we get same inequalities as (41). These show that both of the two real valued sequences and are monotonic nonincreasing sequences; both are bounded below by ; then, the limit of each one exists and equals its infimum. On the other hand, the inequalities (41) prove that their infimums should coincide. Let be such coinciding infimums (limits). Then, Using the lower semicontinuity property of gives Taking the least upper limits on the two sides of the inequalities (22) as gives Thus, Since is class function, then either or . If , then . If while , then the following is a contradiction Hence, Therefore, There is and such that This gives Consequently, hence, and . To prove that such a fixed point is unique, we use analogous contradiction given by inequalities (37).

We have the following algorithm and convergent proposition:

Proposition 27. The iterated sequence defined by (20) converges strongly to the unique common fixed point of and .

Proof. Let be the unique common fixed point of and , in addition suppose that with . Then, for each , we have Hence, That is, Using equation (47) with the limiting approach as prove that hence, , since , we get , that is, .

Corollary 28. Let be compact metric space andbe two closed subsets of which. Ifis continuouscyclic-weak nonexpansive mapping on, , thenhas a ufp. Moreover, for any, we have.

Proof. Using Theorem 26 with completes the proof.

Corollary 29. Letbe compact metric space andbe two closed subsets of which. Ifis continuousgeneralized cyclic-weak nonexpansive mapping on, , thenhas a ufp. Moreover, for any, we have.

Proof. Using Theorem (24) with and completes the proof.

We have the following examples.

Example 1. Let, , and. Endowwith the metricand defineasThen, is cyclic -weak nonexpansive type with the unique fixed point , where , , and . Indeed, if and , then we have On the other hand, we noticed that , , and . Thus, ; hence, That is, is not weakly -contractive.
This mapping is also cyclic--weak nonexpansive type, where , , and . Indeed, if and , then we have On the other hand, we noticed that , , and . Thus, ; hence, That is, is not weakly -contractive.

3. Conclusion

This paper suggests two new theorems (Theorems 23 and 26); these theorems prove the existence of a common ufp of novel introduced joint cyclic -weak contraction and weak nonexpansive mappings in complete and compact metric spaces, respectively.

Their corollaries, Corollary 24 and Corollary 28 in particular, prove the existence of a ufp for both cyclic -weak contraction and cyclic -weak nonexpansive mappings in complete and compact metric spaces, respectively.

In addition, Corollary 25 and Corollary 29 prove the existence of a ufp for both cyclic -weak contraction and cyclic -weak nonexpansive mappings in complete and compact metric spaces, respectively. These results extend and generalize previously proved fixed point theorems given in the references.

Abbreviations

fp:	Fixed point
fps:	Fixed points
ufp:	Unique fixed point
w.r.t:	With respect to
iff:	If and only if.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author has no conflict of interests.

Authors’ Contributions

The sole author contributed to the article. The author read and approved the final manuscript. Sahar Mohamed Ali Abou Bakr, Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt. E-mail(s): [email protected]; [email protected]

Acknowledgments

Deep thanks and gratitude go to the collaboration and valuable comments recommended by the editors and the anonymous referees for this work to appear.

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Copyright © 2021 Sahar Mohamed Ali Abou Bakr. 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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