Convergence of Generalized Quasi-Nonexpansive Mappings in Hyperbolic Space

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 ρ-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.


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][3][4][5].
The concept of hyperbolic space was introduced by Kohlenbach [6] in 2005, 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][10][11][12][13][14][15][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 ϱ-preserving was introduced by Al-Rawashdeh and Mehmood [23] which is generalized than the concept of monotone.
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 x, y, z, H ∈ X and α, β ∈ ½0, 1: Let X be a hyperbolic space and K be a nonempty subset of X and Γ : K ⟶ K be a mapping. According to [23], the underlying concepts are defined in hyperbolic space as follows: A mapping Γ is said to be nonexpansive if and quasi-nonexpansive provided FðΓÞ is nonempty and for each y ∈ FðΓÞ Let ϱ be a relation on X; a self-map Γ of X is said to be ϱ-preserving if Let ðX, d,≼Þ be a partially order hyperbolic space, and a self map Γ of X is said to be monotone nonexpansive if Γ is monotone and A quasi ϱ-preserving mapping which is also L.R.Q.N is called quasi ϱ-preserving L.R.Q.N.
Condition (S) A hyperbolic space X having a relation ϱ on it satisfying condition ðSÞ if every convergent sequence fx n g, x n ⟶ x where x ∈ X has a subsequence fx n k g such that xϱx n k for all k ∈ ℕ: Let X be a hyperbolic space and ϱ be the relation on X; ϱ is said to be compatible if for all x, y ∈ X, (a) xϱy implies ðx ⊕ zÞϱðy ⊕ zÞ, (b) xϱy implies αxϱαy for α ∈ ð0, 1Þ: Remark 1. In the above definitions, we consider only a relation which needs not to be a partial order relation necessarily. The condition S 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 S (see the last paragraph on page 4 of [23]).

Main Results
Let X be a hyperbolic space and K be a nonempty subset of X, I be the identity map, and Γ : K ⟶ K be a mapping. For x 0 ∈ K, and α, β ∈ ð0, 1Þ, let fx n g be a sequence with iterations given as Since Γ is L.R.Q.N, so for all n ∈ ℕ and y ∈ FðΓÞ, taking inf over y implies fdðx n , FðΓÞÞ ≥ 0g is a nonincreasing sequence and bounded as well. So 2 Journal of Function Spaces Now, we prove that fx n g is a Cauchy sequence. For a given ε > 0, there exists k ∈ ℕ, such that for all n ≥ k, For x ∈ FðΓÞ and all l, m ≥ k, we have by adding Taking inf over x implies fx n g is a Cauchy sequence. Since K is complete, so there exists x ∈ K, such that Next, we have to show that FðΓÞ is closed. Let x ∈ K be a limit point of FðΓÞ; then, there exists a sequence fx n g ⊆ FðΓÞ, and using condition ðSÞ, a subsequence fx n k g of fx n g converges to x and xϱx n k for all k ∈ ℕ: ð23Þ so x ∈ FðΓÞ.
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 X be Banach space, having a compatible relation ϱ on it satisfying condition ðSÞ. Let K be a closed and convex subset of X. Suppose Γ : K ⟶ K be a quasi ϱ -preserving L.R.Q.N mapping. If there exists some c 0 ∈ K such that c 0 ϱy for all y ∈ FðΓÞ, then, the sequence (10) converges to 3 Journal of Function Spaces All the results of [23] are the consequences of Theorem 2. Following proposition from [1] will be helpful to proof the next results.
Next, we will discuss the convergence of the Agarwal iteration process defined in [28] as where Proposition 10. If Γ is quasi ϱ-preserving, then, Γ β γ is also quasi ϱ-preserving.
Theorem 13. Let X be hyperbolic space, having a compatible relation ϱ on it satisfying condition ðSÞ. Let K be a closed and convex subset of X. Suppose Γ : K ⟶ K be a quasi ϱ-preserving L.R.Q.N mapping: If there exists some c 0 ∈ K such that c 0 ϱy for all y ∈ FðΓÞ, then, the sequence (53) converges to a fixed point of Γ in K if and only if Proof. As Γ is quasi ϱ-preserving L.R.Q.N mapping, by the Proposition 15, Γ β γ is also quasi ϱ-preserving L.R.Q.N mapping. For y ∈ FðΓÞ, and ðΓ β γ Þ n ðc 0 Þϱy for all n ∈ ℕ. By Theorem 2, we get conclusion.
Next, we will discuss the convergence of the iteration process (Abbas and Nazir) defined in [29] as where Proof. Let q ∈ FðΓÞ that is ΓðqÞ = q then Proposition 15. If Γ is quasi ϱ-preserving, then Γ α,β γ is also quasi ϱ-preserving.
The following theorem describes the necessary and sufficient conditions for convergence of iterative sequence (61) of quasi ϱ-preserving L.R.Q.N mappings.
Theorem 17. Let X be hyperbolic space, having a compatible relation ϱ on it satisfying condition ðSÞ. Let K be a closed and convex subset of X. Suppose Γ : K ⟶ K be a quasi ϱ-preserving L.R.Q.N mapping: If there exists some c 0 ∈ K such that c 0 ϱy for all y ∈ FðΓÞ, then, the sequence (61) converges to a fixed point of Γ in K if and only if Proof. As Γ is quasi ϱ-preserving L.R.Q.N mapping, by the is also quasi ϱ-preserving L.R.Q.N and ðΓ α,β γ Þ n ðc 0 Þϱy for all n ∈ ℕ. By Theorem 2, we get conclusion.
Next, we will discuss the convergence of the Noor iteration process defined in [30] as where Proof. Let q ∈ FðΓÞ that is ΓðqÞ = q then Now consider which implies The following theorem describes the necessary and sufficient conditions for convergence of iterative sequence (53) of quasi ϱ-preserving L.R.Q.N mappings.
Theorem 19. Let X be hyperbolic space, having a compatible relation ϱ on it satisfying condition ðSÞ. Let K be a closed and convex subset of X. Suppose Γ : K ⟶ K be a quasi ϱ-preserving L.R.Q.N mapping: If there exists some c 0 ∈ K such that c 0 ϱy for all y ∈ FðΓÞ, then, the sequence (12) converges to a fixed point of Γ in K if and only if lim n⟶∞ inf d x n , F Γ ð Þ ð Þ= 0: Proof. As Γ is quasi ρ-preserving L.R.Q.N mapping, by the Proposition 18, Γ α,β,γ is also quasi ρ-preserving L.R.Q.N and ðΓ α,β,γ Þ n ðc 0 Þϱy for all n ∈ ℕ. By Theorem 2, we get conclusion.

Conclusion
In the present article, the concept of monotone has been generalized to ϱ-preserving in the framework of hyperbolic space. We also constructed a nontrivial example to show that locally related quasi-nonexpansive mapping is not necessarily ϱ-preserving or ϱ-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.