Abstract
The role of iterative algorithms is vital in exploring the diverse domains of science and has proven to be a powerful tool for solving complex computational problems in the most trending branches of computer science. Taking motivation from this fact, we develop and apply a modified four-step iterative algorithm to solve the fixed point problem in the Hadamard spaces using a total asymptotic nonexpansive mapping. MATLAB R2018b is used for numerical experiments to ensure a better convergence rate of the proposed iterative algorithm with existing results.
1. 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 , we seek a fixed point in such that . If such a fixed point exists, it can be found by running an iterative algorithm that generates a sequence of points that converges to , for example, by iterating the following updated rule: , where for some scalar . This description is known as the fixed point iteration or the Picard iteration method. Here and what follows, is a space, is nonempty closed convex subset of , is a set of fixed points of the mapping , and is a set of natural numbers. Therefore, the following points are important for further development of this research. Let be a self-mapping defined on ; then, is said to be as follows: (1)Nonexpansive, if (2)Asymptotically nonexpansive, if for the sequence with such that for all and for all integers (3)Uniformly L-Lipschitzian, if for constant , and (4)Total asymptotically nonexpansive [10], if there exist non-negative sequences , , and with , as and strictly increasing and continuous function with such that
The last condition (4) contained the aforementioned conditions (1-3) such as , , and . Additionally, every asymptotically nonexpansive mapping is an L-Lipschitzation mapping with .
Mann [11], Ishikawa [12], and Halpern [13] are the fundamental iterative algorithms to approximate the fixed points of nonexpansive mappings. Following them, several new iterative algorithms were developed by Noor [14], Agarwal et al. [15], Garodia et al. [16], Abbas and Nazir [17], and Garodia and Uddin [18]. More specifically, the following iterative algorithm was defined using total asymptotically nonexpansive mappings in [19]: where is a nonempty bounded closed and convex subset in a complete space and . For further development in this direction, we refer the interested reader to [20–22].
The modified Picard-S hybrid iterative process 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:
2. 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 for nonempty set, for metric space, and for set of natural numbers.
Lemma 1 (see [25]). Let and ; then,
Consider a bounded sequence in and for Then, the asymptotic radius is defined as
and the asymptotic center of is given by
Note that has exactly one point in .
If is the distinct asymptotic center for each subsequence of in , then this sequence -converges to .
Lemma 2 (see [26]). Consider the bounded sequence . If and is a subsequence of such that and converges, then .
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 .
Lemma 3 (see [27]). Let and self-mapping be a total asymptotically nonexpansive and uniformly continuous mapping. Moreover, if the set of fixed points is convex and closed, then has a fixed point.
Lemma 4 (see [27]). Consider a self-mapping on a complete metric space and let be a total asymptotically nonexpansive mapping that is uniformly continuous. Then, it follows that and imply that .
Lemma 5 (see [28]). Let be a metric space and , where are sequences in and assume that is a sequence in such that , , and . Then,
Lemma 6 (see [29]). Let the positive number sequences , , and be such that If and , then exists. However, if there exists a subsequence such that as , then .
3. Main Result
Theorem 7. Let be a closed convex and bounded subset of . Consider be a total asymptotically nonexpansive, which is uniformly L-Lipschitzian. Moreover, let , , and be non-negative sequences with , as and strictly increasing continuous function with satisfying the following conditions:
A1. and
A2. For constants with for each
A3. for each and is a constant
Then, the sequence generated by (13) -converges to an element of .
Proof. Let us use Lemma 3, which implies that . Here, on the first hand, we will prove that exists for any , where is defined by (13), and let ; then, we have
Moreover, for each , we have
Similarly,
Finally, we obtain
where
and as stated earlier
Also, using Lemma 6 as well as the inequalities (26) and (27), we compute that the limit exists.
Further, we will show that ; therefore, we consider
and from (24), we have
According to the definition of , we get [24].
Then, from (30) and (31), we obtain
Similarly, we compute
Since
therefore, by taking on both sides, we obtain
Continuing in this way, we obtain the following from expressions (30) and (34):
Next, applying on both sides, we get
Similarly, using (29) and (33), we obtain
Then, applying 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 on both sides
and using (24), we obtain
Continuing in this way, we apply on the both sides
and we use (42) and (43) to get
In the next step, we apply and use
By applying Lemma 5, we obtain
Next, by using on both sides, we obtain
and using (25), we get
Let us apply on both sides to get
Next, using (51) and (53), we obtain
and applying on both sides, we get
Moreover,
By using (45), (55), (57), and Lemma 5, we get
Since is is a total asymptotically nonexpansive mapping, therefore
By taking limit and using
we obtain
Next, by taking and using (51), we obtain
Hence, we obtain the following:
Then, by taking limit and using (42), (60), (62), and (64), we get
Since is nonexpansive and uniformly L-Lipschitzian, therefore we obtain
Let and a subsequence of with exists; then, Lemmas 3 and 4 will imply that there exists a subsequence of such that -converges to and , respectively. Next, we verify that contains only one point. For this, let be a subsequence of with and . We see that and . Finally, since converges therefore by Lemma 2, we obtain . This shows that .
Theorem 8. Consider , , (A1), (A2), (A3), , and be the same as defined in Theorem 7. Then, defined in (13) converges strongly to a fixed point of where .
Senter and Dotson [30] defined condition for mapping by following the same steps as followed by Thakur et al. [19]. Hence, we were able to obtain the following result.
Theorem 9. Considering satisfies condition (I) and , (A1), (A2), (A3), , and be the same as in Theorem 7, then defined by (13) converge to a point of .
By following to Karapinar et al. [27], the concept of semicompact mapping was introduced in [31]. Thereafter, we state the following theorem.
Theorem 10. Considering satisfies property of semicompact and , , (A1), (A2), (A3), , and be the same as in Theorem 7, then, defined by (13) converges to a point of .
Remark 11. If we choose , then algorithms ((5)), (9), and (13) reduce to the following: respectively.
4. Application
In this section, we compare the numerical outcomes of the existing algorithms (2), (5), and (9) with our proposed algorithm (13). We ensure the fast convergence for our proposed iterative scheme (13) by considering Examples 1 and 2.
Example 1. Let , , and be a self-mapping as follows:
It was proved in [23] that such a class of mappings is a total asymptotically nonexpansive mapping. Particular conditions satisfied by the mapping were discussed in [24] by using the initial point and setting the stopping criteria
We consider the following sequences:
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.
Example 2. Let , , and be a self-mapping Then, it is obvious that is continuous uniform L-Lipschitzian and . 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.
5. 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].
Data Availability
No underlying data was collected or produced in this study.
Conflicts of Interest
The authors declare no conflict of interest/competing interests.
Authors’ Contributions
All have equally contributed to this manuscript in all stages, from conceptualization to the write-up of the final draft.
Acknowledgments
The authors extend the appreciation to the Deanship of Postgraduate Studies and Scientific Research at Majmaah University for funding this research work through the project number R-2024-959.