Fixed Point Approximation of Monotone Nonexpansive Mappings in Hyperbolic Spaces

Fixed points of monotone 
 
 α
 
 -nonexpansive and generalized 
 
 β
 
 -nonexpansive mappings have been approximated in Banach space. Our purpose is to approximate the fixed points for the above mappings in hyperbolic space. We prove the existence and convergence results using some iteration processes.


Introduction
In 1965, Browder [1], Göhde [2], and Kirk [3] started working in the approximation of fixed point for nonexpansive mappings. Firstly, Browder obtained fixed point theorem for nonexpansive mapping on a subset of a Hilbert space that is closed bounded and convex. Soon after, Browder [1] and Göhde [2] generalized the same result from a Hilbert space to a uniformly convex Banach space. Kirk [3] utilized normal structure property in a reflexive Banach space to sum up the similar results. Recently, Dehici and Najeh [4] and Tan and Cho [5] approximated fixed point result for nonexpansive mappings in Banach space and Hilbert space.
Fixed point theory in partially ordered metric spaces has been initiated by Ran and Reurings [6] for finding application to matrix equation. Nieto and Lopez [7] extended their result for nondecreasing mapping and presented an application to differential equations. Recently, Song et al. [8] extended the notion of α-nonexpansive mapping to monotone α-nonexpansive mapping in order Banach spaces and obtained some existence and convergence theorem for the Mann iteration (see also [9] and the reference therein). Motivated by the work of Suzuki [10], Aoyama and Kohsaka [11], Dehaish and Khamsi [9], and Song et al. [8], Pant and Shukla obtained existence results in ordered Banach space for a wider class of nonexpansive mappings [12,13]. There are many mathematicians who worked on weak and strong convergence of nonexpansive mappings and its generalizations by using one step, two step, and multistep iteration process ( [8,14,15]). We obtain existence results in partial ordered hyperbolic space for monotone generalized α-nonexpansive and monotone generalized β-nonexpansive map. Particularly, in Section 3, some auxiliary results and existence theorems for monotone α-nonexpansive mappings in ordered hyperbolic spaces are presented. In Section 4, we presented numerical examples and graphical representation. In Section 5, we obtained some existence results for monotone generalized β-nonexpansive mappings in ordered hyperbolic spaces.

Preliminaries
In 1976, the concept of Δ -convergence was given by Lim [14].
Lim [14] initiated the idea that in a metric space, Δ -convergence is possible. This concept is adapted for CAT(0) spaces by Kirk and Panyanak [16], and they have indicated that in numerous Banach space, outcomes comprising weak convergence were having exactly accurate analogs in this manner.
T is called to be contractive if k ∈ ð0, 1, and T is called nonexpansive mapping, if k = 1, that is, Suzuki [10] introduced an interesting generalized nonexpansive mapping as follows.
Suzuki type generalized nonexpansive mapping is another name of self map T holding condition (3).
Many generalizations of nonexpansive mapping have been introduced in the literature (see [17][18][19]). Aoyama and Kohsaka [11] defined a new type of nonexpansive mapping that satisfies the condition (3) known as α-nonexpansive mapping as follows.
Theorem 6 [11]. Let W be a nonempty closed convex subset of a uniformly convex Banach space H and T : W ⟶ W be an α -nonexpansive mapping. Then, F ðT Þ is nonempty iff there exists ξ ∈ W such that fT n ðξÞg is bounded.
Since the hyperbolic spaces contain all normed linear spaces and their convex subsets, so uniformly convex Banach space is contained in hyperbolic metric space so it is natural to generalize the above result to hyperbolic metric space.
Definition 7 [16]. A bounded sequence ðξ n Þ in H is known as Δ-converge to an element ξ ∈ H, if ξ is a unique asymptotic centre of each subsequence fξ n k g of fξ n g: In this section, following definitions and lemma are stated in [9]. Definition 8. Let W be a nonempty set of a hyperbolic metric space ðH, σÞ. A map τ : W ⟶ ½0,∞Þ is said to be a type function, if there exists a bounded sequence fu n g in H such that It is known that each bounded sequence generates a unique type function. Journal of Function Spaces such that Definition 10. A hyperbolic space ðH, σÞ known as uniformly convex, if for every r > 0 and ε > 0, for any a ∈ H: Definition 11. Let W be a nonempty subset of a hyperbolic space H and fξ υ g be a bounded sequence in H: Then, for every ξ ∈ H, define (i) Asymptotic radius of fξ υ g at ξ by (ii) Asymptotic radius of the sequence fξ υ g relative to the above supposed set W by (iii) Asymptotic centre of the sequence fξ υ g relative to the above supposed set W by Note that AðW, fξ υ gÞ ≠ ∅. Further, AðW, fξ υ gÞ has exactly one point if H is uniformly convex.
From now to onward, we will suppose that the ordered intervals are convex and closed, and they are also contained in ordered hyperbolic space ðH,⪯,FÞ; these are described as follows:

Monotone α-Nonexpansive Mappings
In this section, we will use the following iteration introduced by Kalsoom et al. [25].
Definition 12. Let W be a nonempty closed convex subset of an ordered hyperbolic metric space H. A self map T on W is monotone α-nonexpansive mapping, if T is monotone and for some α < 1, for all ξ, η ∈ W with ξ⪯η: Lemma 13. Let W be a nonempty closed convex subset of an ordered hyperbolic metric space H. A self map T on W is monotone α-nonexpansive mapping; then, (i) T is monotone quasi.

Journal of Function Spaces
This completes the proof. ☐ ☐ Definition 14. An ordered hyperbolic metric space ðH, σ,⪯Þ is said to be uniformly convex, if for an arbitrary h ∈ H,r > 0 and ε > 0, Now, we utilize iteration processes for monotone α -nonexpansive mappings.

Lemma 15.
Let ðH, σ,⪯Þ be a uniformly convex partially ordered hyperbolic metric space (in short, UCPOHMS) and W a nonempty closed convex subset of H: Let T : W ⟶ W be a monotone mapping and ξ 1 ∈ W be such that ξ 1 ⪯T ξ 1 ½ or Tξ 1 ⪯ξ 1 : Then, for sequence fξ υ g defined by (14), we have Theorem 16. Let W be a nonempty closed convex subset of a UCPOHMS ðH, σ,⪯Þ and T : W ⟶ W be a monotone α -nonexpansive mapping. Assume that there exists ξ 1 ∈ W such that ξ 1 ⪯T ξ 1 and fξ υ g defined in (14) is a bounded sequence with ξ υ ≤ η for all η ∈ W such that Then, FðT Þ ≠ ∅: Proof. Let fξ υ g defined by (14) be a bounded sequence such that Then, there exists a subsequence fξ υ q g such that By Lemma 15, we have Define for all q ∈ ℕ: Clearly, for every q ∈ ℕ, W q is closed convex. As u ∈ W q , it shows that W q ≠ ∅: Define Then, W ∞ is a closed convex subset of W: Let u ∈ W ∞ : Then, As we know, T is a mapping which is monotone; then, for q ∈ ℕ, Then, there exists a unique point z ∈ W ∞ such that By definition of type function, By using Lemma 13, we get From the boundedness of the sequence fξ υ q g and lim q⟶∞ σðξ υ q , T ξ υ q Þ = 0, we have lim sup  (14) is a bounded sequence with ξ υ ≤ η for all η ∈ W such that Then, FðT Þ ≠ ∅: Theorem 18. Let ðH, σ,⪯Þ be a UCPOHMS, C a closed convex cone, and T : C ⟶ C a monotone α-nonexpansive mapping. Assume that ξ 1 = 0 and fξ υ g defined by (14) is a bounded sequence with ξ υ ≤ η for all η ∈ W such that Then, FðT Þ ≠ ∅: Proof. With the help of definition of partial order ⪯, we know that ξ 1 = 0⪯T 0 = T ξ 1 ; then, the proof is directly from Theorem 16.

☐ ☐
We now prove some convergence results for monotone α -nonexpansive mappings.
Proof. By the Theorem 18, we have fξ υ g is bounded. So, there exists a subsequence fξ υ q g of fξ υ gΔ-converges to some p ∈ W such that In the next step, we prove there exists a unique Δ-limit in FðT Þ corresponding to each Δ-convergent subsequence of fξ υ g. Consider fξ υ g has two subsequences fξ υ q g and fξ υ r g which are Δ-convergent to l and m, respectively. Then, fξ υ g is bounded and 5 Journal of Function Spaces which concludes that l ∈ FðT Þ: Let τ : W ⟶ ½0,∞Þ is a type function which is generated by fξ υ q g: Then, By Lemma 13, we infer as By uniqueness of element l and definition of Δ-convergence, we conclude that Similarly, one can easily show that By continuity of τ and definition of Δ-convergence, we get Then, fξ υ g converges strongly to a fixed point of T .
Proof. By Theorem 18, there exists a subsequence fξ υ q g of f ξ υ g which converges strongly to a point η ∈ W: From Lemma 15, we get By Theorem 17, FðT Þ ≠ ∅ and fξ υ g is bounded, and Without loss of generality, we get On the other hand, by Lemma 13, we derive as By boundedness of fξ υ q g, we have and hence, Therefore,

Mann iteration process
In 1953, Mann proposed an iteration, namely, Mann iteration, for calculation of a fixed point for a nonexpansive mapping T , defined as for each υ ≥ 1 and fβ υ g ⊂ ð0, 1Þ.

Ishikawa iteration process
In 1974, Ishikawa proposed the two-step iteration process as follows: where fα υ g and fβ υ g are in ð0, 1Þ.

Agarwal iteration process
In 2007, Agarwal et al. introduced the three-step iteration as follows: where fα υ g and fβ υ g are in ð0, 1Þ.

Abbas and Nazir iteration process
In 2014, Abbas and Nazir introduced the three-step iteration as follows: where fα υ g, fβ υ g, and fγ υ g are in ð0, 1Þ.

Thakur iteration process
In 2016, Thakur et al. proposed the three-step iteration as follows: where fα υ g and fβ υ g are in ð0, 1Þ. Two qualities fastness and stability play a vital role in iteration process to be performed. In [32], Rhoades mentioned that for the increasing functions, Ishikawa [27] iteration process is faster than Mann iteration process [26] but in the case of decreasing function, condition is reverse. In [29], Agarwal et al. proved that their iteration process was more stable than the previous ones. In [31], Thakur iteration process was considered faster convergent than all the abovementioned iteration processes.
Recently, Kalsoom et al. [25] introduced a new iteration process and proved it to be the fastest convergent than all. The following example is given to support this claim.
In Table 1, we discussed the convergence behavior of some iteration processes. It is clear that all iterations approach to 0 which is the fixed point of mapping T : In this case, Figures 1-3 show that Kalsoom et al. iteration process converges faster to the fixed point as compared the other iterations.

Monotone Generalized β-Nonexpansive Mappings
In this section, we define monotone generalized β-nonexpansive mapping which generalizes the results of Pandey and Shukla [13] in hyperbolic spaces. Now we will define monotone generalized β-nonexpansive mappings in hyperbolic space with nontrivial example.
Definition 23. Let W be a nonempty subset of an ordered hyperbolic metric space ðH, σ,⪯Þ. A mapping T : W ⟶ W is said to be monotone generalized β-nonexpansive mapping, if T is monotone and there exists β ∈ ½0, 1Þ such that for all ξ, η ∈ W with ξ⪯η.
Proposition 24. Every monotone nonexpansive mapping is monotone generalized β-nonexpansive mapping, but converse is not true.
Proof. By putting β = 0 in ð69Þ, we have which shows that monotone generalized β-nonexpansive mapping reduces to monotone nonexpansive mapping satisfying the condition (3). The following example will prove the converse statement does not hold. ☐ ☐ implies and so T does not hold condition (3). Again, ξ ∈ ð2, 3 and η = 4, and T does not satisfy condition (3). Nevertheless, T is generalized β-nonexpansive mapping with β ≥ 1/3: Proposition 25. Let W be a nonempty subset of an ordered hyperbolic metric space ðH, σ,⪯Þ and T : W ⟶ W a monotone generalized β-nonexpansive mapping which has a fixed point η ∈ W with ξ⪯η: Then, T can be referred as a monotone quasi-nonexpansive mapping.
☐ ☐ Proposition 26. Let W be a nonempty subset of an ordered hyperbolic metric space ðH, σ,⪯Þ. If T : W ⟶ W is monotone generalized β-nonexpansive mapping, then FðT Þ is closed. Furthermore, if H is strictly convex, then W is convex and FðT Þ is also convex.
In Table 2, we discussed the convergence behavior of some iteration processes. It is clear that all iterations approach to 4 which is the fixed point of T . In this case, Figures 4-6 show that Kalsoom et al. iteration process converges faster to the fixed point as compared the other iterations.

Conclusion
It concludes that we have approximated fixed point results of monotone α and generalized β-nonexpansive mappings in hyperbolic spaces. Moreover, we proved some numerical applications and presented the graphical representations by using different iteration processes.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.