A Modified Iterative Approach for Fixed Point Problem in Hadamard Spaces

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Introduction
Iterative algorithms and fixed point problems are key concepts in numerical analysis and optimization that offer a powerful and flexible framework for solving diverse mathematical problems in computer science, engineering, and industry.Their applications continue to grow and motivate the development of new and more efficient algorithms and techniques that can tackle emerging challenges and applications.Therefore, metric fixed point theory has become a vital instrument for verifying procedures and algorithms using iterative schemes and functional equations in current emerging sciences, such as the field of artificial intelligence [1] and logic programming [2].The subject has been studied for a long time using different principles of contraction [3].Its usefulness mostly hinges on the availability of solutions to mathematical problems generated from systems engineering [4] and computer science [5].Because of its novel development as a confluence of analysis [6,7] and geometry [8], the theory of fixed points has also become a powerful and vital instrument for the study of nonlinear problems [9].As such, a choice between several distinct iteration approaches must be made, taking important aspects into account.For example, simplicity and convergence speed are the two main factors that determine whether one iteration approach is more effective than the others.In situations like this, the following problems unavoidably come up: Which iteration method is speeding up convergence among these?It is thus shown in this article that our proposed iteration scheme converges faster than modified Picard, Picard-S, and Picard-Mann iterations.
One type of problem that can be addressed by iterative algorithms is the fixed point problem, which involves finding a point that remains unchanged when a mapping is iteratively applied to it.Specifically, given a nonlinear mapping ð R n ⟶ R n , we seek a fixed point x ⋆ in R n such that x ⋆ = ð x ⋆ .If such a fixed point exists, it can be found by running an iterative algorithm that generates a sequence of points x k that converges to x ⋆ , for example, by iterating the following updated rule: x k+1 = g x k , where g x = x − λð x for some scalar λ > 0. This description is known as the fixed point iteration or the Picard iteration method.Here and what follows, ℵ, d is a ℂAT 0 space, C is nonempty closed convex subset of ℵ, d , Q = F ð is a set of fixed points of the mapping ð, and N is a set of natural numbers.Therefore, the following points are important for further development of this research.Let ð be a selfmapping defined on C ; then, ð is said to be as follows: (4) Total asymptotically nonexpansive [10], if there exist non-negative sequences ζ m , Q m , and m ≥ 1 with ζ m , Q m ⟶ 0, as m ⟶ ∞ and strictly increasing and continuous function ϕ 0,∞ ⟶ 0,∞ with ϕ 0 = 0 such that The last condition (4) contained the aforementioned conditions (1-3) such as ζ m = P m − 1 , Q m = 0 , ∀m ≥ 1, and ϕ t = t, t ≥ 0. Additionally, every asymptotically nonexpansive mapping is an L-Lipschitzation mapping with L = sup m∈ N P m .
The modified Picard-S hybrid iterative process ℏ m introduced in [23] is defined as follows: Another such iterative scheme introduced in [24] is stated as follows: All of the aforementioned researchers focused on achieving a better convergence rate by minimizing the time needed to run their proposed iteration scheme.Taking motivation from the above discussion, we propose a novel modified four-step iterative algorithm as follows:

Preliminaries and Lemmas
This section contains some well-known concepts and results that are often used in this article.Note: throughout the article, we use ℵ for nonempty set, ℵ, d for metric space, and N for set of natural numbers.
Lemma 1 (see [25]).Let û, v, w ∈ ℵ, d and t ∈ 0, 1 ; then, Consider a bounded sequence ûm in ℵ, d and for Then, the asymptotic radius r ûm is defined as and the asymptotic center A ûm of ûm is given by Note that A ûm has exactly one point in ℵ, d .If û is the distinct asymptotic center for each subsequence ẑm of ûm in ℵ, d , then this sequence Δ-converges to û ∈ ℵ, d .Lemma 2 (see [26]).Consider the bounded sequence ûm ∈ ℵ, d .If A ûm = p and ẑm is a subsequence of ûm such that A ẑm = ẑ and d ûm , ẑ converges, then p = ẑ.

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Journal of Function Spaces Karapinar et al. [27] demonstrated that the above result can be derived using the fixed point existence theorem and demiclosedness principle for those satisfying in ℵ, d .Lemma 3 (see [27]).Let C ⊂ ℵ, d and self-mapping ð C ⟶ C be a total asymptotically nonexpansive and uniformly continuous mapping.Moreover, if the set of fixed points F ð is convex and closed, then ð has a fixed point.
Lemma 4 (see [27]).Consider a self-mapping ð on a complete metric space ℵ, d and let ð C ⟶ C be a total asymptotically nonexpansive mapping that is uniformly continuous.Then, it follows that lim m⟶∞ d ℏ m , ðℏ m = 0 and lim m⟶∞ ℏ m = q imply that ðq = q.

Main Result
Theorem 7. Let C be a closed convex and bounded subset of ℵ, d .Consider ð C ⟶ C be a total asymptotically nonexpansive, which is uniformly L-Lipschitzian.Moreover, let I m , Ψ m , and m ≥ 1 be non-negative sequences with I m , Ψ m ⟶ 0, as m ⟶ ∞ and strictly increasing continuous function ϕ 0,∞ ⟶ 0,∞ with ϕ 0 = 0 satisfying the following conditions: Then, the sequence ℏ m generated by (13) Δ-converges to an element of ℵ.
Proof.Let us use Lemma 3, which implies that F ð ≠ ∅.
Here, on the first hand, we will prove that lim m⟶∞ d ℏ m , p exists for any p ∈ F ð , where ℏ m is defined by (13), and let p ∈ F ð ; then, we have Moreover, for each m ∈ N, we have Finally, we obtain where 3 Journal of Function Spaces and as stated earlier Also, using Lemma 6 as well as the inequalities ( 26) and ( 27), we compute that the limit lim m⟶∞ d ℏ m , p exists.
Further, we will show that lim m⟶∞ d ℏ m , ðℏ m = 0; therefore, we consider and from (24), we have According to the definition of ð, we get [24].
Then, from ( 30) and ( 31), we obtain Similarly, we compute therefore, by taking lim m⟶∞ inf on both sides, we obtain Continuing in this way, we obtain the following from expressions (30) and (34): Next, applying lim m⟶∞ sup on both sides, we get Similarly, using ( 29) and (33), we obtain Then, applying lim sup lim as well as using ( 36) and (37), we get Next, by making use of ( 29), ( 33), (41), and Lemma 5, we obtain Similarly, by making use of ( 24)-( 26), we obtain Next, by applying the lim inf on both sides and using (24), we obtain Continuing in this way, we apply lim m⟶∞ sup on the both sides and we use ( 42) and (43) to get In the next step, we apply lim sup Journal of Function Spaces and use By applying Lemma 5, we obtain Next, by using lim inf on both sides, we obtain and using (25), we get Let us apply lim sup on both sides to get Moreover, Since ð m is I m , Ψ m , ϕ is a total asymptotically nonexpansive mapping, therefore By taking limit m ⟶ ∞ and using Next, by taking lim m ⟶ ∞ and using (51), we obtain Hence, we obtain the following: and using (42), ( 60), (62), and (64), we get Since ð is nonexpansive and uniformly L-Lipschitzian, therefore we obtain Journal of Function Spaces  Senter and Dotson [30] defined condition I for mapping ð C ⟶ C by following the same steps as followed by Thakur et al. [19].Hence, we were able to obtain the following result.Theorem 9. Considering ð satisfies condition (I) and ℵ, d , C, (A1), (A2), (A3), σ m , and I m be the same as in Theorem 7, then ℏ m defined by (13) converge to a point of F ð .

Application
In this section, we compare the numerical outcomes of the existing algorithms (2), (5), and (9) with our proposed algo-rithm (13).We ensure the fast convergence for our proposed iterative scheme (13) by considering Examples 1 and 2.
It was proved in [23] that such a class of mappings is a total asymptotically nonexpansive mapping.Particular conditions satisfied by the mapping ð were discussed in [24] by using the initial point ℏ = 0 5 and setting the stopping criteria ℏ − 2 ≤ 10 −15 74  We apply iterative schemes (2), ( 5), (9), and (13).Hence, the corresponding numerical values are provided in Table 1 and Table 2, and their graphical comparison is provided in Figures 1 and 2, respectively.Then, it is obvious that ð is continuous uniform L-Lipschitzian and F ð = 1.Moreover, it was proved in [23] that this class of mappings is a total asymptotically nonexpansive.The graphical comparison using discussed iterative schemes is provided in Figure 3.We consider the following choice of sequences: We apply iterative schemes (2), ( 5), (9), and (13).Hence, the corresponding numerical values are provided in Table 3, and the graphical comparison is provided in Figure 3.However, the graphical comparison of the speed of convergence among the proposed and existing algorithms is provided in Figure 4.

Conclusion
In this research article, we proposed a modern iterative algorithm and used it to obtain numerical results.These results proved that the proposed method is effective and can accelerate the convergence rate of existing methods for tackling the fixed point problems of the total asymptotically nonexpansive mapping.This research provides both theoretical and practical contributions to the study of fixed point theory and iterative algorithms in the Hadamard spaces.For future work, algorithm can be further modified to obtain better rate of convergence for different classes of mapping [32].
lim m⟶∞ sup d a m , p ≤ J 55 Next, using (51) and (53), we obtain J = lim m⟶∞ sup d a m , p = lim m⟶∞ sup d ð m j m , p , d ð m m , p ≤ d j m , p + I m R 1 d j m , p + Ψ m ≤ 1 + I m R 1 d j m , p + Ψ m , 56 and applying lim sup on both sides, we get lim m⟶∞ sup d ð m j m , p ≤ lim m⟶∞ sup d j m , p , 57 J ≤ lim m⟶∞ sup d j m , p 58

lim m⟶∞ sup d j m , p ≤ J 59
By using (45), (55), (57), and Lemma 5, we get lim m⟶∞ d j m , ð m j m = 0 60 and a subsequence z m of ℏ m with A z m = x exists; then, Lemmas 3 and 4 will imply that there exists a subsequence y m of z m such that y m Δ-converges to y ∈ C and y ∈ F ð , respectively.Next, we verify that W Δ ℏ m contains only one point.For this, let z m be a subsequence of ℏ m with A z m = x and A ℏ m = ℏ .We see that x = y and y ∈ F ð .Finally, since d ℏ m , y converges therefore by Lemma 2, we obtain ℏ = y ∈ F ð .This shows that W Δ ℏ m = ℏ .Theorem 8. Consider ℵ, d , ð, C, (A1), (A2), (A3), σ m , and I m be the same as defined in Theorem 7.Then, ℏ m defined in (13) converges strongly to a fixed point of lim m⟶∞ inf d ℏ m , F ð m = 0, 69 where d ℏ, F ð = inf d ℏ, p : p ∈ F ð .

Table 1 :
Numerical values of the sequence σ l ; ζ l with the initial point (0.5).

Table 2 :
Numerical values of the sequence σ m ; ζ m with the initial point (0.5).

Table 3 :
Numerical values of the sequence σ n ; ζ n with the initial point (50).