Approximation of the Fixed Point for Uniﬁed Three-Step Iterative Algorithm with Convergence Analysis in Busemann Spaces

: In this manuscript, a new three-step iterative scheme to approximate ﬁxed points in the setting of Busemann spaces is introduced. The proposed algorithms unify and extend most of the existing iterative schemes. Thereafter, by making consequent use of this method, strong and ∆ convergence results of mappings that satisfy the condition ( E µ ) in the framework of uniformly convex Busemann space are obtained. Our results generalize several existing results in the same direction.


Introduction
Throughout this paper, R, R + , and f ℘ denote the set of all real numbers, positive real numbers, and fixed points of the mapping ℘, respectively.
The fixed point theory is considered one of the most powerful analytical techniques in mathematics, especially in nonlinear analysis, where it plays a prominent role in algorithm technology. The purpose of investing in algorithms is to obtain the best algorithms with a faster convergence rate, because the lower the convergence rate, the faster the speed of obtaining the solution. This is probably the drawback of using the iterative methods.
It should be noted that the Mann iteration converges faster than the Ishikawa iteration for the class of Zamfirescu operators [1], and hence the convergence behavior of proclaimed and empirically proven faster iterative schemes need not always be faster. There was extensive literature on proclaimed new and faster iteration schemes in ancient times. Some of the iteration schemes are undoubtedly better versions of previously existed iteration schemes, whereas a few are only the special cases. There are more than twenty iteration schemes in the present literature. Our analysis's focal objective is to unify the existing results in the framework of Busemann spaces (see [2] for the precise definitions and properties of Busemann spaces). This analysis has a special significance in terms of unification, and numerous researchers have intensively investigated various aspects of it.
Recall that a metric space (B, ∂) is called a geodesic path (or simply a geodesic [17] is a convex; that is, the metric of Busemann space is convex. In a Busemann space the geodesic joining any two points is unique. Proposition 1 ([25]). In such spaces, the hypotheses below hold: Busemann spaces are also hyperbolic spaces, which were introduced by Kohlenbach [26]. Further, B is said to be uniquely geodesic [17] if there is exactly one geodesic joining  and for each , ∈ B. Definition 4 ([17]). Suppose that B is a uniquely geodesic space and γ ([α, β]) is a geodesic segment joining  and and α ∈ [0, 1]. Then, In the sequel, the notation [, ] is used for geodesic segment γ ([α, β]) and ε is denoted by (1 − α) ⊕ α . A subset κ ⊆ B is said to be geodesically convex if κ includes every geodesic segment joining any two of its points. Let B be a geodesic metric space and ℘ : B → R. We say that ℘ is convex if for every geodesic path γ : [α, β] → B, the map ℘ • γ : [α, β] → is a convex. It is known that if ℘ : B → R is a convex function and ℘ : B → R is an increasing convex function, then ℘ • ℘ : B → R is convex.
We now introduce our algorithm.
Let B be a complete Busemann space, B s be a nonempty convex subset of B and ℘ : B s → B s be a mapping. For any υ 0 ∈ B s ,
in the standard three-step iteration scheme, we obtain the SP iterative scheme [28].
in the standard three-step iteration scheme, we obtain the Picard-S iterative scheme [29].
in the standard three-step iteration scheme, we obtain the CR iterative scheme [30].
in the standard three-step iteration scheme, we obtain the Abbas and Nazir iterative scheme [31].
in the standard three-step iteration scheme, we obtain the P iterative scheme [32].
in the standard three-step iteration scheme, we obtain the D iterative scheme [33].
in the standard three-step iteration scheme, we obtain the Mann iterative scheme [34].
in the standard three-step iteration scheme, we obtain the Ishikawa iterative scheme [35].

Preliminaries
In this section, we present some relevant and essential definitions, lemmas, and theorems needed in the sequel.

Definition 5 ([36]
). The Busemann space B is called uniformly convex if for any ζ > 0 and ∈ (0, 2], there exists a map δ such that for every three points α, where m denotes the midpoint of any geodesic segment [, ] (i.e., m = 1 2  ⊕ 1 2 ) and inf{δ : ζ > 0}. A mapping ℘ : (0, ∞) × (0, 2] → (0, 1] is called a modulus of uniform convexity, for ℘(η, ) := δ and for a given η > 0, ∈ (0, 2]. Henceforth, the uniform convexity modulus with a decreasing modulus concerning η (for a fixed ) is termed as the uniform convexity monotone modulus. The subsequent lemmas and geometric properties, which are instrumental throughout the discussion to learn about essential terms of Busemann spaces, are necessary to achieve our significant findings and are as follows: Lemma 1. If ℘ is a mapping satisfying condition (E ) and has a fixed point then it is a quasinonexpansive mapping.
Let B s be a nonempty closed convex subset of a Busemann space B, and let { η } be a bounded sequence in B. For  ∈ B, we set The asymptotic radius of ζ({ η } is given by It is known that, in a Busemann space, A({ η }) consists of exactly one point [37]. Recall that a bounded sequence { η } ∈ B is said to be regular [38] (v) Suzuki generalized nonexpansive if it verifies Condition (E ).
Theorem 1. Let C s be a nonempty bounded, closed and convex subset of a complete CAT(0) space C. If ℘ : C s → C s is a generalized nonexpansive mapping, then ℘ has a fixed point in C s . Moreover, f ℘ is closed and convex.

Main Results
We begin this section with the proof of the following lemmas: Lemma 7. Let B s be a nonempty closed convex subset of a complete Busemann space B, and let ℘ : B s → B s be a mapping satisfying condition (E µ ). For an arbitrary chosen υ 0 ∈ B s , let the sequence {υ η } be generated by a standard three-step iteration algorithm with the condition
From standard three-step iteration algorithm, we have Using the value of ∂( η , υ * ), we have On substituting Also, it is given that we have This implies that {∂(υ η , υ * )} is bounded and non-increasing for all υ * ∈ f ℘ . Hence, lim n→∞ ∂(υ η , υ * ) exists, as required. Lemma 8. Let B s be a nonempty closed convex subset of complete Busemann space B, and ℘ : B s → B s be a mapping satisfying condition (E µ ). For an arbitrary chosen υ 0 ∈ B s , let the sequence {υ η } be generated by a standard three-step iteration algorithm. Then, f ℘ is nonempty if and only if {υ η } is bounded and lim n→∞ ∂(℘υ η , υ η ) = 0 for a unique asymptotic center.
Proof. Since f ℘ = ∅, let υ * ∈ f ℘ and z ∈ B s . Using Lemma 7, there is an existence of lim n→∞ ∂(υ η , υ * ), which confirms the boundedness of {υ η }. Assuming lim n→∞ ∂(υ η , υ * ) = r, on combining this result with the values of ∂( η , υ * ) and ∂( η , υ * ) of Lemma 7 On the other hand, by using the value of ∂( η , υ * ) of Lemma 7, we have by the above-mentioned standard three-step iteration algorithm, This implies that, This implies that, and hence, we have As a consequence, the uniqueness of the asymptotic center ensures that υ is a fixed point of ℘ so this concludes the proof. Now, we state and prove our main theorems in this section.
Theorem 2. Let B s be a nonempty closed convex subset of a compete Busemann space B and ℘ : B s → B s be a mapping satisfying condition (E µ ). For an arbitrary chosen υ 0 ∈ B s , assume that {υ η } is a sequence generated by a standard three-step iteration algorithm. Then f ℘ = ∅ and {υ η } ∆−converges to a fixed point of ℘.
Theorem 3. Let B s be a nonempty closed convex and complete Busemann space B, and ℘ : B s → B s be a mapping verification condition (E µ ). For an arbitrary chosen υ 0 ∈ B s , assume that {υ η } is a sequence generated by a standard three-step iteration algorithm. Then {υ η } converges strongly to a fixed point of ℘.
Since υ is the unique asymptotic center of {υ η }, it follows that υ = υ, which is a contradiction. Hence, {υ η } converges strongly to a fixed point of ℘.

Conclusions
The extension of the linear version of fixed point results to nonlinear domains has its own significance. To achieve the objective of replacing a linear domain with a nonlinear one, Takahashi [40] introduced the notion of a convex metric space and studied fixed point results of nonexpansive mappings in this direction. Since the standard three-step iteration scheme unifies various existing iteration schemes for different values of ε η , σ η , τ i η , κ i η , ω i η , and ι i η for i = 0, 1, 2, existing results of the standard three-step iteration scheme including strong and ∆ − convergence results in the setting of Busemann spaces satisfying condition E are generalized.
Author Contributions: All authors have equally contributed to this work. All authors have read and agreed to the published version of the manuscript.
Funding: This work received no external funding.