#### Abstract

In this article, we consider a wider class of nonexpansive mappings (locally related quasi-nonexpansive) than monotone nonexpansive mappings. We obtained the convergence of fixed point for quasi -preserving locally related quasi-nonexpansive mappings in hyperbolic space. An iterative process is also used to obtain the convergence results for this mapping. Fixed point is approximated numerically in a nontrivial example by using Matlab.

#### 1. Introduction

A first attempt to enrich metric spaces with convexity was essentially due to Takahashi [1] and is known as convex metric spaces. The class of convex metric spaces is general in nature and has constant curvature. Fixed point theory has many developments regarding convex metric spaces; for example, see [2–5].

The concept of hyperbolic space was introduced by Kohlenbach [6] in , which is more general than the concept of hyperbolic space in [7] and more restrictive than the hyperbolic space defined in [8]. This definition is different from Takahashi’s notion of convex metric space in the sense that every convex subset of a hyperbolic space is itself a hyperbolic space. These spaces are nonlinear in nature and more general than normed spaces. Fixed point theorems for nonexpansive mappings in hyperbolic space have been studied in [7, 9–16]. The existence of fixed point for nonexpansive mapping was initiated by Browder [17], Kirk [18], and Göhde [19] independently in 1965. In 1967, Diaz and Metcalf [20] gave an idea about quasi-nonexpansive mapping. Example of quasi-nonexpansive mappings was given by Doston [21] in 1972 which was not nonexpansive. Fixed point results for monotone nonexpansive mappings were presented by Bachar and Khamsi [22] in 2015. In 2019, the concept of -preserving was introduced by Al-Rawashdeh and Mehmood [23] which is generalized than the concept of monotone.

In this article, we will generalize the results of [23] in hyperbolic space. A nontrivial example is also given in which Picard [24], Mann [23, 25], Ishikawa [26, 27], Agarwal [28], Abbas and Nazir [29], and Noor [30] iteration schemes are used to approximate the fixed point.

#### 2. Preliminaries

A hyperbolic space [6] is a metric space , together with a mapping which satisfies the following for all and (1)(2)(3)(4).

If a space satisfies only Condition 1, it coincides with the convex metric space introduced by Takahashi [1].

Throughout this article, we consider for all and

Let be a hyperbolic space and be a nonempty subset of and be a mapping. According to [23], the underlying concepts are defined in hyperbolic space as follows: A mapping is said to be *nonexpansive* if
and quasi-nonexpansive provided is nonempty and for each

Let be a relation on ; a self-map of is said to be -*preserving* if

Let be a partially order hyperbolic space, and a self map of is said to be *monotone nonexpansive* if is monotone and

A mapping is said to be (i)Locally related quasi-nonexpansive (abbreviated as L.R.Q.N) provided is nonempty, and for each (ii)Quasi -preserving provided is nonempty, and for each

A quasi -preserving mapping which is also L.R.Q.N is called *quasi*-*preserving* L.R.Q.N.

Condition () A hyperbolic space having a relation on it satisfying condition if every convergent sequence where has a subsequence such that for all

Let be a hyperbolic space and be the relation on ; is said to be *compatible* if for all (a) implies ,(b) implies for

*Remark 1. *In the above definitions, we consider only a relation which needs not to be a partial order relation necessarily. The condition is utilized in Theorems 2, 5, 9, 13, 17, and 19 which is moderate than the conditions already been considered in the literature [31]. The condition
mentioned in each Theorems 2, 5, 9, 13, 17, and 19 is based on the condition (see the last paragraph on page 4 of [23]).

#### 3. Main Results

Let be a hyperbolic space and be a nonempty subset of , be the identity map, and be a mapping. For and , let be a sequence with iterations given as

Theorem 2. *Let be hyperbolic space, having a compatible relation on it satisfying condition . Let be a closed and convex subset of . Suppose be a quasi -preserving L.R.Q.N self map of If there exists some such that for all then, (10) converges to a fixed point of in if and only if
*

*Proof. *If converges to some fixed point of , then holds. Conversely, suppose holds. As is quasi -preserving and , we have . Similarly,

Since is L.R.Q.N, so for all and taking over implies is a nonincreasing sequence and bounded as well. So

Now, we prove that is a Cauchy sequence. For a given there exists such that for all

For and all we have by adding

Taking over implies is a Cauchy sequence. Since is complete, so there exists , such that

Next, we have to show that is closed. Let be a limit point of ; then, there exists a sequence , and using condition , a subsequence of converges to and

Consider so .

*Example 1. *Let be a hyperbolic space and be the relation defined as

Let be defined as

Let and be a mapping defined by

As is the fixed point of and implies and that is where we have which shows that is L.R.Q.N.

Next since but does not hold, which shows that is not -preserving.

Also implies so is quasi -preserving. As hence, is not nonexpansive. This example shows that L.R.Q.N mapping is not necessarily -preserving or -preserving nonexpansive.

*Example 2. *Let be a hyperbolic space and be the relation defined as

Let be defined as

Let and be a mapping defined by

As is the fixed point of , and implies and , this shows that is L.R.Q.N.

There are many examples in literature which shows that hyperbolic spaces are more general than Banach spaces for detail [1], so we have the following corollary which is Theorem 2.6 of [23].

Corollary 3. *Let be Banach space, having a compatible relation on it satisfying condition . Let be a closed and convex subset of . Suppose be a quasi -preserving L.R.Q.N mapping. If there exists some such that for all , then, the sequence (10) converges to a fixed point of in if and only if
*

All the results of [23] are the consequences of Theorem 2.

Following proposition from [1] will be helpful to proof the next results.

Proposition 4. *Let be a Takahashi convex structure on metric space . If and , then
*(1)* and ,*(2)*,*(3)* and *

Theorem 5. *Let be hyperbolic space, having a compatible relation on it satisfying condition . Let be a closed and convex subset of . Suppose be a quasi -preserving L.R.Q.N mapping If there exists some such that for all then sequence (11) converges to a fixed point of in if and only if
*

*Proof. *Let , then
so Suppose that is then from (11)
implies and Hence,

Now, we show that is quasi -preserving. Suppose then and by the compatibility of

Also implies

Finally, we prove that is L.R.Q.N. For this let then

Using Theorem 2, we get the result.

*Remark 6. *If is -preserving, then is also -preserving.

*Proof. *As is -preserving, implies . Using the compatibility of we get
which implies

Now, we discuss the convergence of the iterative scheme (12) given as

Proposition 7. *For , , whenever is -preserving nonexpansive and for *

*Proof. *Let that is then
so

For the other inclusion, suppose that is and consider

which gives and therefore implies Hence,

Proposition 8. *If is quasi -preserving L.R.Q.N mapping, then is also quasi -preserving L.R.Q.N mapping.*

*Proof. *Firstly we show that if is quasi -preserving, then is quasi -preserving. For this, let and such that implies Also by Theorem 5, is quasi -preserving, ; therefore, Further and by using the compatibility of , we get

Hence, is quasi -preserving.

Let and such that then as is -preserving. Consider

Theorem 9. *Let be hyperbolic space, having a compatible relation on it satisfying condition . Let be a closed and convex subset of . Suppose be a quasi -preserving L.R.Q.N mapping If there exists some such that for all then the sequence (12) converges to a fixed point of in if and only if
*

*Proof. *As is quasi-preserving L.R.Q.N mapping, by the Proposition 8, is also quasi -preserving L.R.Q.N mapping. For and for all . By Theorem 2, we get conclusion.

Next, we will discuss the convergence of the Agarwal iteration process defined in [28] as where

Proposition 10. *If is quasi -preserving, then, is also quasi -preserving.*

*Proof. *Suppose is quasi -preservin,g then for and such that implies , and Then, by using the compatibility of , we get
Hence, is quasi -preserving.

Proposition 11. *If is quasi -preserving L.R.Q.N mapping, then is also quasi -preserving L.R.Q.N mapping.*

*Proof. *Let and such that . Consider
Hence, is also quasi -preserving L.R.Q.N mapping.

Proposition 12. *For *

*Proof. *Let that is then
so

The following theorem describes the necessary and sufficient conditions for convergence of iterative sequence (53) of quasi -preserving L.R.Q.N mappings.

Theorem 13. *Let be hyperbolic space, having a compatible relation on it satisfying condition . Let be a closed and convex subset of . Suppose be a quasi -preserving L.R.Q.N mapping If there exists some such that for all then, the sequence (53) converges to a fixed point of in if and only if
*

*Proof. *As is quasi -preserving L.R.Q.N mapping, by the Proposition 15, is also quasi -preserving L.R.Q.N mapping. For and for all . By Theorem 2, we get conclusion.

Next, we will discuss the convergence of the iteration process (Abbas and Nazir) defined in [29] as where

Proposition 14. *For *

*Proof. *Let that is then
so

Proposition 15. *If is quasi-preserving, then is also quasi -preserving.*

*Proof. *Suppose is quasi -preserving, then, by Proposition 10 for and are also quasi -preserving. Suppose for and such that implies , and Then, by using the compatibility of , we get

Hence, is quasi -preserving.

Proposition 16. *If is quasi -preserving L.R.Q.N mapping, then is also quasi -preserving L.R.Q.N mapping.*

*Proof. *Let and such that , then, as is -preserving. Consider

Hence, is also quasi -preserving L.R.Q.N mapping.

The following theorem describes the necessary and sufficient conditions for convergence of iterative sequence (61) of quasi -preserving L.R.Q.N mappings.

Theorem 17. *Let be hyperbolic space, having a compatible relation on it satisfying condition . Let be a closed and convex subset of . Suppose be a quasi -preserving L.R.Q.N mapping If there exists some such that for all then, the sequence (61) converges to a fixed point of in if and only if
*

*Proof. *As is quasi -preserving L.R.Q.N mapping, by the Proposition 15, is also quasi -preserving L.R.Q.N mapping. For and for all . By Theorem 2, we get conclusion.

Next, we will discuss the convergence of the Noor iteration process defined in [30] as where

Proposition 18. *For and .*

*Proof. *Let that is then
so

Now consider which implies

The following theorem describes the necessary and sufficient conditions for convergence of iterative sequence (53) of quasi -preserving L.R.Q.N mappings.

Theorem 19. *Let be hyperbolic space, having a compatible relation on it satisfying condition . Let be a closed and convex subset of . Suppose be a quasi -preserving L.R.Q.N mapping If there exists some such that for all , then, the sequence (12) converges to a fixed point of in if and only if
*

*Proof. *As is quasi -preserving L.R.Q.N mapping, by the Proposition 18, is also quasi -preserving L.R.Q.N mapping. For and for all . By Theorem 2, we get conclusion.

All iterations converges to

Table 1 provides the rate of convergence of Mann, Ishikawa, Agarwal, Noor and Abbas iterations for the mapping given in Example 2. For initial values are and

Using different values for and , we can see that in Table 2 Abbas iteration not only converges faster but also stable than other iterations.

*Remark 20. *The numerical Example 2 validates the existence of L.R.Q.N mappings for hyperbolic spaces in the perspective of different iterations with the help of Table 3 and Figure 1.

#### 4. Conclusion

In the present article, the concept of monotone has been generalized to -preserving in the framework of hyperbolic space. We also constructed a nontrivial example to show that locally related quasi-nonexpansive mapping is not necessarily -preserving or -preserving nonexpansive and approximate the fixed point numerically and compare the convergence result of different iterations with Abbas iteration by using Matlab.

#### Data Availability

No data were used to submit this work.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

All authors contributed equally.