Research Article | Open Access

# A Study on Common Fixed Point of Joint Generalized Cyclic Weak Nonexpansive Mappings and Generalized Cyclic Contractions in Quasi Metric Spaces

**Academic Editor:**Alberto Fiorenza

#### Abstract

This manuscript proves the existence of a single common fixed point of two joint generalized cyclic weak nonexpansive mappings, where and are compact sets. The result in particular demonstrated a single fixed point of generalized cyclic weak nonexpansive mappings, without the assumption of a single contracting point. Additionally, it introduces new types of generalized cyclic contraction mappings and describes the existence of a single fixed point of them in -metric spaces. Finally, the presented results establish a simpler convergence theorem for a sequence of generalized cyclic contraction mappings, extend, and generalize some of the previous reported results.

#### 1. Introduction and Preliminaries

In 1922, Banach [1] introduced the concept of contraction mapping and formulated his famous Banach contraction principle, namely, the contraction mapping defined on complete metric space has a unique fixed point.

*Definition 1. *A self mapping on a metric space is said to be contraction iff there is a real number such that

Since 1922 generalizations of the Banach contraction principle have historical achievements.

In 1962, Edelstein [2] introduced the concept of contractive mapping and proved the existence of single (fp) of contractive mapping defined on compact metric space.

*Definition 2. *A self mapping on a metric space is said to be contractive iff fulfills the condition:

In 1972, Chatterjea [3] introduced the Chatterjea type mapping and showed that every Chatterjea mapping on a complete metric space has a single (fp).

*Definition 3. *Let be a metric space. If is a self mapping on , then is said to be Chatterjea contraction type iff there is real number such that

In 1979, Browder [4] introduced the concept of a nonexpansive mapping and proved the existence of (fp) for nonexpansive mapping defined on bounded closed convex subset of uniformly convex Banach space.

*Definition 4. *A self mapping on a metric space is said to be nonexpansive iff fulfills the condition:

In 1997, Alber and Guerre-Delabriere [5] highlighted the weak contraction type of mapping and proved the existence of (fp) for this mapping in a Hilbert space setting.

In 2001, Rhoades [6] showed analogous results in the case of -weak contraction mapping on metric spaces.

*Definition 5. *A self mapping on a metric space is said to be *-*weak contraction iff it fulfills the condition:
where is a continuous nondecreasing function .

In 2003, Kirk et al. [7] generalized the above (fp) theorems for mappings satisfying cyclical contractive conditions in various settings.

*Definition 6. *Let be a metric space, be a positive integer, be a nonempty closed subsets of , and . Then, a self mapping on is said to satisfy cyclical contractive condition iff there is a constant such that:

In 2007, Elshobaky [8] generalized the Banach contraction principle for type contraction mapping.

*Definition 7. *Let be a metric space, be a self mapping on , and satisfy the condition*:.*(1)If , then is said to be generalized contraction(2)If , then is said to be generalized nonexpansive

In 2008, Dutta and Choudhury [8] proved the existence of (fp) for - weak contraction mappings.

*Definition 8. *Let be a metric space. If is a self mapping on , then is said to be weak contraction on iff there are two nondecreasing mappings , is continuous, *iff**and**iff* such that the following is true:

In 2009, Choudhury [9] introduced the weakly Chatterjea contractive type mappings and proved that every weakly Chatterjea contractive mapping on a complete metric space has single (fp).

*Definition 9. *If is a self mapping on a metric space , then is said to be weakly *-*contractive (or weak Chatterjea type contraction) iff
where is a continuous function satisfying iff .

In 2009, Bakr [10] proved the existence of (fp) for generalized nonexpansive mappings with one contracting point.

*Definition 10. *Let be a normed space, be bounded closed subset of , , , and . Then, the point is said to be contracting point with respect to iff there is satisfying:

This author showed that if is a normed space in which every bounded sequence has a weakly convergent subsequence and the parallelogram law hold in , is a subset of which containing -contracting point, and is a -generalized-nonexpansive mappings from into itself, then for every -contracting point of , , the following sequence converges strongly to the best approximation element of in .

In 2011, Karapinar [11] gave another generalization for cyclic weak -contraction.

*Definition 11. *Let be a metric space, be a positive integer, be nonempty closed subsets of , and . Then a self mapping on is said to be cyclic weak contraction iff there is a continuous nondecreasing mapping with *iff* such that:
(1) is cyclic representation of with respect to where .

At the same year, Karapinar and Sadarangani [12] gave (fp) for generalized cyclic -weak contraction mappings in complete metric spaces.

*Definition 12. *Let be a metric space, be a positive integer, be nonempty closed subsets of , , are two nondecreasing mappings, is continuous, *iff*, and *iff**. If* is a self mapping on , then, is said to be generalized cyclic *-*weak contraction mapping iff the following are true:
(1) is cyclic representation of with respect to (2), where

In 2012, Karapinar and Sadarangani [13] enhanced the corrigendum (fp) theory for cyclic -weak contraction mapping. Additionally, Karapinar et al. [14] gave some (fp) for generalized -weak contraction mapping.

At the same year, (2012), Karapinar and Kumar [15] introduced the notion of cyclic weakly Chatterjea type contraction in metric spaces and proved corresponding (fp) theorems.

*Definition 13. **Let* be a metric space, be a positive integer, be nonempty subsets of , , and be a self mapping on . Then, is said to be cyclic weakly Chatterjea type contraction iff:
(1) is cyclic representation of with respect to (2)there are a real number and a continuous function iff such that

In 2013, Bakr [16] discussed the concepts of -ntype and -ctype of mappings and proved the existence of single (fp) for both -ntype and -ctype on closed convex weakly Cauchy subset of normed space.

*Definition 14. *Let be a subset of a normed space and be a self mapping on satisfying*:.*(1)If and , then is said to be -contraction type mapping(2)If and , then is said to be -nonexpansive type mapping

In this paper, we are introducing the following:.

Let be a self mapping on a metric space and be three real numbers. Then, we define the real valued function by.

*Definition 15. **Let* be two self mappings on a metric space and be three real numbers. Then, we define the real valued function *by*

*Definition 16. **Let* be a metric space; and be a self mapping on with
(1) and and(2)there are real constants (i)If , then is said to be cyclic generalized contraction w.r.t .(ii)If , then is said to be cyclic generalized nonexpansive w.r.t .

Include the definition of lower semicontinuity of the function.

*Definition 17. *Let be a linear space. Then, a function from is said to be lower semicontinuous if for any real number , the set is closed convex subset of .

*Definition 18. **Let* be a self mapping on a metric space such that
where is lower semicontinuous nondecreasing function .
(1)If , then is said to be generalized cyclic weak contraction w.r.t .(2)If , then is said to be generalized cyclic weak nonexpansive w.r.t .

Additionally, we introduce the following fascinating concept of joint mappings:.

*Definition 19. *Let be a metric space with , be two self mappings, and :
(1)The cyclic condition1: and (2)The contractivity condition2:
where is lower semicontinuous nondecreasing function .
(i)If , then and are said to be joint generalized cyclic weak contraction type(ii)If , then and are said to be joint generalized cyclic weak nonexpansive type

In 1993, Czerwik [17] verified the principle in a wider class of metric spaces, namely, quasi metric space, these spaces sometimes are called -metric spaces, for some other types of quasi metric spaces one can see [18].

*Definition 20. *Let be a real number, , be a mapping . Then, is called -metric space iff
(1) iff (2).(3)

In 2017, Bakr [19] adjusted (fp) theorem by using both -metric spaces and generalized contraction type of mappings.

*Definition 21. *Let be a real number, and be *a**-*metric space. Then, the mapping is said to be *-*ctype mapping iff it satisfies:
where are some real numbers with and .

Now, the generalized cyclic contraction type mapping in -metric spaces is defined as follows:.

*Definition 22. *Let be a real number, and be a -metric space. Then, a self mapping on is said to be generalized cyclic contraction *w.r.t* iff satisfies the following:
(1) and and(2)there are real constants and

The purposes of this manuscript are designed as follows:

Theorem 23 together with Proposition 24 proves the existence of a single common (fp) for two joint generalized cyclic weak nonexpansive type mappings that have compact sets . The result in particular presented a single (fp) for generalized cyclic weak nonexpansive mapping, without the assumption of single contracting point, this result showed in Corollary 25.

Additionally, we introduce a new type of generalized cyclic contraction mapping. Theorem 29 described the existence of a single (fp) of such mappings in -metric spaces.

Finally, Theorem 31 established a simpler convergence theorem for a sequence of generalized cyclic contraction mappings. These results are extensions and generalizations of previous results reported in [10, 11, 17, 19, 20].

Since every contraction mapping is nonexpansive and both are cyclic w.r.t . We noticed the following: (i)The class of all generalized nonexpansive mappings is wider than that of generalized contraction type(ii)The class of all generalized cyclic contraction mappings is wider than that of generalized contraction type(iii)The class of all generalized cyclic nonexpansive mappings is wider than that of generalized nonexpansive type(iv)The class of all generalized cyclic nonexpansive mappings is wider than that of generalized cyclic contraction type(v)The class of all generalized cyclic weak nonexpansive mappings is wider than that of generalized cyclic weak contraction type(vi)The class of all generalized weak contraction is wider than that of weak contraction type(vii)The class of all joint generalized cyclic weak nonexpansive types is wider than that of joint generalized cyclic weak contraction types(viii)If are continuous self mappings on , then the restriction of the mapping is continuous (ix)If are two compact subsets of the metric space and are continuous self mappings on , then the restriction of the mapping attains its infimum as well as its supremum at some points in

#### 2. Main Results

##### 2.1. Common Fixed Point of Continuous Joint Generalized Cyclic Weak Nonexpansive Mappings

We have the following generalizations:

Theorem 23. *Let be a metric space and be two compact subsets of which . If are two continuous joint generalized cyclic weak nonexpansive type of mappings on , then there is single point such that .*

*Proof. *Let be an arbitrarily chosen element in . Then is either in or in . If is in , then define , , , and then by induction define:
First, suppose is an odd natural number. Then
Hence
Thus,
Therefore,
Hence
Continuing this process gives
Second, by a similar method when is an even natural number, we obtain
Hence, the two sequences and are monotonic nonincreasing and bounded below by with the same infimum, if this infimum is , then
Using the properties of ,
Taking least upper limits on two sides of the inequalities (25) as gives
that is . Thus , this gives . Hence,
This implies that
Since attains its infimum on , there are and such that
This gives
that is
since all are nonnegative real numbers, clearly
and we have
Notice that the converse is also true, if , then is clear. On the other side, this showed that is a common (fp) of and . If there exists another point such that , then we get
This shows that , that is .

Proposition 24. *The sequence defined iteratively by the induction* (24) *and by previous theorem notations is convergent to the unique common (fp) of**and*.

*Proof. *Let be the common (fp) of and and with , then
On limiting approach as , equation (35) proves that:
Hence, , since , we get , that is, .

Corollary 25. *If**is metric space,*, *, and**are compact, and**is continuous generalized cyclic**weak nonexpansive mapping w.r.t**, then there is single point**such that**. Moreover, for every**, we have*.

###### 2.1.1. Example

Let , , and . Define as:

It is clear that is cyclic with respect to the representation of . Endow with the metric . The mapping is having the unique (fp) and it is generalized cyclic weak nonexpansive w.r.t , where is defined by . In fact, if and , then we have

Using this, we have

##### 2.2. Generalized Results for Generalized abc;r Contraction.

Next, we generalize results for generalized contraction [19] to those of generalized cyclic contraction in -metric spaces. We have the following important lemmas:

Lemma 26. * Letbe generalized cycliccontraction w.r.tof the-metric space. Then for everyand any natural number, we have:*(1)

*The inequalities are true*(2)

*There are two natural numbers and with and*

*Proof. *Let , (similar conclusion if ). Then , using the contractivity condition of gives
hence,
For , we have , thus
Because , hence
Using the inequalities (52) and (54) gives
For , we have , and then