#### 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 Banachâ€™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, *