Strong Convergence of a New Hybrid Iterative Scheme for Nonexpensive Mappings and Applications

Jia, Jie; Shabbir, Khurram; Ahmad, Khushdil; Shah, Nehad Ali; Botmart, Thongchai

doi:https://doi.org/10.1155/2022/4855173

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Research Article | Open Access

Volume 2022 | Article ID 4855173 | https://doi.org/10.1155/2022/4855173

Strong Convergence of a New Hybrid Iterative Scheme for Nonexpensive Mappings and Applications

Jie Jia,¹Khurram Shabbir,²Khushdil Ahmad,²Nehad Ali Shah,³and Thongchai Botmart⁴

Academic Editor: Santosh Kumar

Received27 Oct 2021

Revised02 Jan 2022

Accepted01 Feb 2022

Published19 Mar 2022

Abstract

In the article, we have proposed a new type of hybrid iterative scheme which is a hybrid of Picard and Thakur et al. repetitive schemes. This new hybrid iterative scheme converges faster than all leading schemes like Picard-S^∗ hybrid, Picard-S, Picard-Ishikawa hybrid, Picard-Mann hybrid, Thakur et al. and Abbas and Nazir, S-iterative, Ishikawa and Mann iterative schemes for contraction mapping. By using the Picard-Thakur hybrid iterative scheme, we can find the solution of delay differential equations and also prove some convergence results for nonexpansive mapping in a uniformly convex Banach space.

1. Introduction

In this article, the set of all positive integers is denoted by . Let denote the nonempty convex subset of a normed space and be its convex subset, and is called contraction mapping if for all and . If , then, the mapping is called nonexpansive mapping. An element is said to be a fixed point of if , and the set of fixed points of is denoted by

In 1890, Picard [1] presented an iterative scheme for approximating the fixed point which is defined by the sequence as

The Krasnoselskii [2] iterative sequence is defined as where .

In 1953, Mann [3] proposed an iterative scheme which is defined as where

In 1974, Ishikawa [4] gave the concept of the two-step iterative scheme and the sequence of this iterative is defined as where .

In 2007, Agarwal et al. [5] introduced a more generalized form of the Ishikawa iterative scheme and they called it the -iterative scheme and the sequence of the iterative scheme is defined as where .

In 2016, Sahu et al. [6] and Thakur et al. [7] proposed a new scheme which converges faster than all the existing schemes. The iterative sequence of this scheme is defined as where , and .

Thakur et al. [7] proposed another iterative scheme which converges faster than all the above schemes and the iterative sequence of Thakur et al. is defined as where .

Recently, Lamba and Panwar [8] introduced a new three-step iteration process for Susuzki’s nonexpansive mapping and called it the Ap iterative scheme whose rate of convergence is faster than those of the leading schemes. The sequence of the Ap iterative scheme is defined as where .

Many physical problems of engineering and applied sciences are mostly constructed in the form of fixed point equations. In the existence of a fixed point equation involving an operator, is guaranteed but the exact solution is not possible. We can only approximate the solution which becomes very relevant and this necessitated various iterative schemes [9–14]. Also, the iterative schemes are used for solving different problems like minimization, equilibrium, viscosity approximation, and many more problems in different spaces [15–18].

The Picard iterative scheme is the simplest iteration to estimate the solution of a fixed point equation. Chidume [19] introduced some basic results on this iterative scheme. Chidume generalized and improved the existing results of the fixed point equation in [20]. Okeke and Abbas [21] proved the convergence and almost -stability of Mann-type and Ishikawa-type random iterative schemes.

In 2013, Khan [22] proposed the Picard-Mann hybrid iterative scheme. The sequence of this scheme is defined as where .

In 2017, Okeke and Abbas [23] proposed the Picard-Krasnoselskii hybrid iterative scheme and the sequence of this iterative scheme is defined as where .

In 2019, Okeke [24] proposed the Picard-Ishikawa hybrid iterative scheme and the sequence of this iteration defined as where and .

Recently, Srivastava [25] introduced a new type of hybrid iterative scheme from Picard and -iteration (Picars- hybrid iterative scheme). The sequence of the scheme is defined as where and .

Also Lamba and Panwar [26] introduced another hybrid scheme from Picard and -iteration (Picard- hybrid iterative scheme) and the sequence of the scheme is defined as where , and .

With the motivation towards the usage of hybridization of iterative schemes, we proposed another type of hybrid scheme which is the Picard-Thakur hybrid iterative scheme, defined by the sequence as where and .

Rhoades [27] commented on the convergence of two iterative schemes that converges to a certain fixed point is as follows:

Let and be the two fixed point iterative schemes that converge to a certain fixed point of a given operator . The sequence is better than if

2. Preliminaries

Berinde and Takens [10] gave the following definitions.

Definition 1 (see [10]). Let and be the two sequences of the real number converging to and , respectively. Suppose that (i)If , then, faster than (ii)If , then, the rate of convergence of both sequences are the same

Definition 2 (see [10]). Let and be the two sequences of a fixed point iterative scheme, both converges to a fixed point for a given operator and are two sequences of positive numbers. Suppose that the error estimates, are available and converge to zero. If converges faster than , then, converges faster than Most of the literature on the iterative schemes deals with the convergence rate and some analyzes its stability [28–34]. For proving the results, we need the following lemma.

Lemma 3 (see [35]). Let be a sequence with If and then,

Definition 4 (see [36]). Let be a subset of Banach space which is nonempty closed and convex. A mapping is demiclosed w.r.t. , if for each sequence in and , converges weakly at and converges strongly at .

Definition 5 (see [37]). A Banach space is said to satisfy Opial’s condition if for any sequence , implies that for all with .

Lemma 6 (see [38]). Let be a uniformly convex Banach space and . Let , be the two sequences such that , , and hold for some , then

Lemma 7 (see [36]). Let be a nonexpansive mapping with Opial’s property. If and , then, , i.e., is demiclosed at zero, where is the identity mapping on .

Proposition 8 (see [39]). Let be a subset of Banach space and a nonexpansive mapping. Then, for all

Senter and Dotson [40] introduced the concept of condition (I) which is defined as

Definition 9. Let be a self-mapping on which is said to satisfy condition (I), if there is an increasing function with and for all such that where .

In this article, we proposed a new hybrid iterative scheme which converges faster than Mann [3], Ishikawa [4], -iteration [5], Abbas et al. [9], Thakur et al. [7], Picard-Mann hybrid [22], Picard-Krasnoselskii [23], Picard-Ishikawa [24], and Picard- hybrid iterative schemes [25]. Recently, Srivastava [25] already proved that the Picard- hybrid iterative scheme converges faster than all of the above iterative schemes. Therefore, we show that our hybrid iterative scheme converges faster than all the leading schemes. We find the solution of delay differential equations using our proposed hybrid iterative scheme while in last section, we prove some results of this scheme for nonexpansive mapping in the uniformly convex Banach space.

3. Convergence Analysis

This section deals with the rate of convergence of the Picard-Thakur hybrid iterative scheme (14) with Picard- (12), Picard-Ishikawa (11), Picard-Mann (9), and Thakur et al. (6).

Proposition 10. Assume that be a nonempty closed convex subset of a normed space and let be a contraction mapping. Suppose that the iterative schemes (12), (11), (10), (9), and (6) converge to the same fixed point of where , and are sequences in such that , and for some and Then, the Picard-Thakur hybrid iterative scheme (14) converges faster than all the other schemes.

Proof. Let be a fixed point of an operator . Using the definition of contractive mapping and the Thakur et al. iterative scheme (6), we have Let Now, for (14), Also, Let Then, Thus, converges faster than , i.e., the Picard-Thakur iterative scheme converges faster than the Thakur iterative scheme. Similarly, the inequality proved in Proposition 3.1 of the Picard- hybrid iterative scheme [25] is as follows: Then, Thus, converges faster than ., i.e., the Picard-Thakur iterative scheme converges faster than the Picard- iterative scheme. Similarly, we can show that Picard-Thakur hybrid iterative scheme (14) converges faster than (11), (10), and (9).

Next, we gave an example to show that the Picard-Thakur hybrid iterative scheme (14) converges faster than the Picard- hybrid, Picard-Ishikawa hybrid, Picard-Mann hybrid, and Thakur iterative schemes.

Example 11. Let where be an operator defined by

Choose for each with an initial value is nonexpansive mapping. All the iterative schemes converge to the fixed point . Clearly, in the Table 1 and Figure 1, we can see that the Picard-Thakur hybrid iterative scheme (14) converges faster than the schemes discussed above.

4. Application: Delay Differential Equations

In this section, we can find the solution of the delay differential equation by using our proposed iterative scheme.

Let the space of all continuous real-valued functions be denoted by on closed interval endowed with the Chebyshev norm and defined as , and it is clear that in [41] that () is a Banach space. Now, consider the following delay differential equation with initial condition

By the solution of the above delay differential equation, we mean a function satisfying (30) and (31).

Assume that the following conditions are satisfied. (1)(2)(3)(4)There exists such that(5)

Now, we construct (30) and (31) by the integral equation as

The following result is the generalization of the result of Coman et al. [42].

Theorem 12. Let the conditions to be satisfied. Then, (30) and (31) have unique solution and

Now, by using the Picard-Thakur hybrid iterative scheme (14), we prove the following result.

Theorem 13. Let the conditions be satisfied. Then, (30) and (31) have a unique solution and the Picard-Thakurb hybrid iterative scheme (14) converges to .

Proof. Let be a sequence generated by the Picard-Thakur hybrid iterative scheme (14) for an operator defined by Let be a fixed point of . Now, we prove that as . It is easy to see that as for each .
Now, for each , we have Now, As For as Putting (42) in (41), we get Putting (43) in (40), we get Putting (44) and (43) in (39), we get Putting (45) in (38), we get Putting (46) and (40) in (37), we get Let , and by using condition , we have Putting (48) in (36), we have Again, using condition , we get Let and . So, the conditions of Lemma 3 are satisfied. Hence,

5. Convergence Results for Nonexpansive Mapping

Lemma 14. Let be a nonempty closed and convex subset of uniformly convex Banach space and be a nonexpansive mapping. If be a sequence generated by Picard-Thakur hybrid iterative scheme (14) and , then, exists.

Proof. Let , and is nonexpansive then Also, Similarly, Now, This shows that is a decreasing sequence and bounded below Hence, exists.

Lemma 15. Let and be as in Lemma 14. Let be a sequence defined by Picard-Thakur hybrid iterative scheme (14) with Then, .

Proof. As from the above Lemma 14, exists for each Suppose that for some , we have As from (53), (52), and (51), we have Taking as of (58), (57), and (56), we get Since is nonexpansive, we have From (63) and (61), we get Now, from (53), we have Taking as , we have So, from (67) and (59), we have From (68), (62), and (55) and applying Lemma 6, we get

Theorem 16. Let , , be as in Lemma 14. Let be the uniformly convex Banach space which satisfies Opial’s condition; then, converges weakly to a fixed point of

Proof. Let ; then, by Lemma 14, exists. Now, we show that has a unique weak subsequential limit in .
Let and be two subsequences of and , be the weak limits of the subsequences of , respectively. From Lemma 15, and is demiclosed at zero. By Lemma 7.
Therefore, we get For , we follow the same manner.
From Lemma 14, we know that exists.
For uniqueness, supposing that then, by using Opial’s condition, This is a contradiction, so . Hence, converges weakly to

Theorem 17. Let , , be as in Lemma 14. Then, converges to a point of if and only if or , where

Proof. If the sequence then, it is oblivious that or .
Conversely, assume that . From Lemma 14,
exists, . Therefore, by assumption, Now, to show, the sequence is cauchy in . As , for given there exists such that Particularly, . Therefore, there is such that Now, for This shows that the sequence is cauchy in . As , so, is a point in such that . Now, gives that

Theorem 18. Let , , be as in Lemma 14. Then, converges strongly to

Proof. By Lemma 15, we have Since, is compact, then, let be a subsequence of which converges strongly to , for some . By Proposition 8, we have Letting we get Also, by Lemma 14, exists. Thus, converges strongly to .

Now, by using condition (I), we prove the strong convergence result.

Theorem 19. Let , be as in Lemma 14. Let be a uniformly convex Banach space which is satisfying condition (I). Then, the sequence defined by the Picard-Thakur hybrid iterative scheme (14) converges strongly to

Proof. As by Lemma 15, we have By condition (I) and (78), we get Since is an increasing function satisfying
Hence, we have Since all the conditions of Theorem 17 are satisfied, therefore, we can say that converges strongly to

6. Conclusion

In this paper, we present a new hybrid scheme of Picard and Thakur et al. We discuss the convergence of this scheme to the iterative scheme of Mann, Ishikawa, Picard-Mann, Picard-Ishikawa, Picard-, and Thakur et al. We showed the convergence of Picard-Thakur hybrid iterative with other iterative schemes on graphs and gave application to delay differential equations. We also generalize and extend various results for nonexpansive mapping in a uniformly convex Banach space including [7, 24, 25, 43].

Data Availability

All data required for this research is included within this paper.

Conflicts of Interest

The authors declare that they do not have any competing interests.

Authors’ Contributions

Jie Jia analyzed the results and used a software to compare the results, Khurram Shabbir proposed the problem and supervised this work, Khushdil Ahmad wrote the first version of this paper, Nehad Ali Shah verified the results and wrote the final version of this paper, and Thongchai Botmart prepared the example sketch and the plots and arranged the funding for this paper. Jie Jia and Nehad Ali Shah are the first co-authors and contributed equally in this work.

Acknowledgments

This research work is supported by the Higher Education Commission (Islamabad) through the National Research Program for Universities, Grant no. 7359/Punjab/NRPU/R$D/HEC/2017.

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Copyright © 2022 Jie Jia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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