Convergence of Mann and Ishikawa Iterative Processes for Some Contractions in Convex Generalized Metric Space

El Kouch, Youness; Mouline, Jamal

doi:https://doi.org/10.1155/2022/3168414

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2022 | Article ID 3168414 | https://doi.org/10.1155/2022/3168414

Convergence of Mann and Ishikawa Iterative Processes for Some Contractions in Convex Generalized Metric Space

Youness El Kouch¹and Jamal Mouline¹

Academic Editor: Devendra Kumar

Received05 Oct 2021

Accepted26 Jan 2022

Published14 Feb 2022

Abstract

In this paper, we consider a JS metric space endowed with convexity structure, which will allow us to examine and study convergence of Mann iteration and Ishikawa iteration for Banach type and Chatterjea type contractions defined on JS metric space.

1. Introduction

Convexity structure in normed spaces and vector topological spaces has stimulated mathematicians to enlarge and generalize this important notion to metric space and generalized metric spaces. This work has been achieved by different ways and various directions. In 1970 W. Takahashi [1] opened up a new line of investigation by introducing an abstract convexity structure in metric space and obtained fixed point theorems for nonexpansive mappings; his results generalize fixed point theorems previously proved by Browder [2] and Kirk [3].

In recent years, many papers studied Mann [4] and Ishikawa [5] processes in various ways for single and multivalued contractive mappings. For more details in this direction, we refer the reader to [6–13].

In 2015, Jleli and Samet [14] introduced a new generalization of metric space that also recovers dislocated metric spaces [15], b-metric spaces [16], and modular metric spaces with Fatou property [17, 18]. Since then, fixed point theory has been widely studied in this new framework, and the reader may refer to [19–22].

In [23], the authors have introduced a Mann type iteration method; they also obtained a result about strongly convergence of this iteration method to a fixed point of nonexpansive mappings on Banach spaces. They provide a result about strong convergence of the iteration method to a common fixed point of two mappings on uniform convex Banach spaces via a new iteration method based on Ishikawa iteration idea.

Motivated by above, in this paper, we consider a JS metric space endowed with convexity structure introduced by Takahashi; this will allow us to examine and study convergence of Mann iteration [4] and Ishikawa iteration [5] for Banach type and Chatterjea type contractions defined on JS metric space.

In the next, we recall the basic definitions which are used throughout in this paper. For every , let us define the set

Definition 1 (see [14]). We say that is a JS metric on if it satisfies the following conditions:
(D₁) For every , we have
(D₂) For every , we have
(D₃) There exists such that if , then

Also, we say the pair is a JS metric space.

We mention that convergent sequences and Cauchy sequences can be introduced in a similar manner as in metric space.

Definition 2 (see [14]). Let be a JS metric space. Let be a sequence in and . (i)We say that converges to , if (ii)We say that is Cauchy sequence if (iii) is said to be complete, if every Cauchy sequence in is convergent to some element in

In the following, we introduce convexity structure defined on JS metric space. Notice that properties and consequences may differ from those arising from convexity structure defined on ordinary metric spaces; this dissimilarity occurs because of absence of triangular inequality in JS metric space.

Definition 3. Let be a JS metric space, and . A mapping is said to be convex structure on , if for any and , the following inequality holds:

If is a JS metric space with a convex structure , then is called a convex JS metric space.

In the sequel, we are interested to study the convergence of Mann and Ishikawa iteration procedures for k-contraction mappings and Chatterjea mappings in the frame of JS metric space. We recall that (i)Mann iteration procedure [4] is defined by(ii)Ishikawa iteration procedure [5] is defined as followswhere and satisfy , for all .

2. Main Results

2.1. Convergence of Mann and Ishikawa Iterations for k-Contraction

We first need to state a basic lemma that can be found in [24] page 45.

Lemma 4. Let and be two bounded real sequences such that converges to . Then, .

Lemma 5. Let and be two positive sequences such that for all . Then, we have the following: (1)The sequence converges to some (2)If , then

Proof. (1)The sequence is positive; then, from (6), we get , for all . Hence, is decreasing and bounded from below. As a result, converges to some (2)By (1), the sequence has a limit . Now, suppose that , and by using Lemma 4, we get ; it follows that , which is a contradiction, and we conclude that .

Lemma 6. Let be a sequence in .
If and diverges, then .

Proof. For all and , we have , which yields .
Since , we get .
Hence, it follows that .

Definition 7 (Definition 3.1 in [14]). Let be a JS metric space and be a mapping. Let We say that is a k-contraction if

Remark 8. Let be a k-contraction on .
The ordering in inequality (7) is defined in extended real number system. Note that if we have for some elements , then .
For every , let

Theorem 9 (Theorem 3.3 in [14]). Suppose that the following conditions hold: (i) is complete(ii) is a k-contraction for some (iii)There exists such that Then, converges to , a fixed point of . Moreover, if is another fixed point of such that , then .

Our first essential main result is given as follows:

Theorem 10. Let be a complete convex JS metric space and a k-contraction for some . Suppose that there exists such that . If , then Mann iterative process defined in (4) converges to a fixed point of .

Proof. All conditions of Theorem 9 are satisfied, so .
Let and the sequence defined by and .
Set for all . The proof goes in two steps.
Step 1: we shall prove that is bounded.
For every , we have Now suppose that for every , . We have We conclude that By Theorem 9, we have and using , we get Hence, by (11), we obtain Therefore, is bounded.
Step 2: converges to 0 (i.e., converges to ).
From (3) and (7), we have Denote , and we obtain Since , we get .
Hence, by Lemma 5, it follows that .
Therefore, Mann iterative process converges to the fixed point .

Theorem 11. Let be a complete convex JS metric space and a k-contraction for some . Suppose that there exists such that . If diverges, then Mann iterative process defined in (4) converges to a fixed point of .

Proof. We proceed as in the proof of Theorem 10. By the first step, the sequence is bounded.
Also by (15), we have with , which can be rewritten as follows: Hence, inductively, we get Then, since , and , it follows by Lemma 6 that as . So, Mann iterative process converges to the fixed point , as desired.

Remark 12. If is an element such that and for all , then Mann iteration is simply reduced to Picard iteration. Indeed, for all , we have By (11), take , and we obtain ; hence,

Therefore, by (D₂), we obtain , for all . Also, we have ; thus, , for all .

In [14], Jleli and Samet showed that every b-metric space, with coefficient , is a JS metric space for , and every modular space with the Fatou property is a JS metric space for . Hence, the following Corollaries are immediate consequences of Theorems 10 and 11.

Corollary 13. Let be a complete convex b-metric space with constant and a k-contraction for some . If either or diverges, then Mann iterative process defined in (4) converges to a fixed point of .

Remark 14. The condition that there exists such that is omitted. Since for each and for each with , we have

Therefore, for each , we have .

Corollary 15. Let be a complete convex modular space and be a mapping. Suppose that the following conditions hold: (i) satisfies the Fatou property(ii)For some , we have(iii)There exists such that(iv) or divergesThen, Mann iterative process defined in (4) converges to a fixed point of .

In our next theorem, we give a convergence result of Ishikawa iteration for k-contraction considered in a complete convex JS metric space.

Theorem 16. Let be a complete convex JS metric space and a k-contraction for some . Suppose that there exists such that . If either or diverges, then Ishikawa iterative process defined in (5) converges to a fixed point of .

Proof. By Theorem 9, there exists some . Let and be the sequences defined in (5) and set for all . Now, we complete the proof in the following steps.
Step 1: we shall prove that is bounded.
For every , we have .
Now suppose that for every .
From (3) and (7), we have And similarly, Since and , we get It follows that . Thus, By Theorem 9, we have , and using , we get Hence, by (27), we obtain Then, is bounded as desired.
Step 2: converges to 0 (i.e., converges to ).
From (3) and (7), we have We have, for all , Set Now, by (30) and (31), we obtain Therefore, Now, if , then we get ; hence, by (33) and Lemma 5, it follows that .
If , then the same result follows by (34) and Lemma 6.
Consequently, Ishikawa iterative process converges to the fixed point .☐

We have the following results, which can be deduced from Theorem 16.

Corollary 17. Let be a complete convex b-metric space with constant and a k-contraction for some . If either or diverges, then Ishikawa iterative process defined in (4) converges to a fixed point of .

Corollary 18. Let be a complete convex modular space and be a mapping. Suppose that the following conditions hold: (i) satisfies the Fatou property(ii)For some , we have(iii)There exists such that(iv) or divergesThen, Ishikawa iterative process defined in (4) converges to a fixed point of .

Remark 19. If is an element such that and for all , then in Ishikawa iteration is simply reduced to Picard iteration of . Indeed, for all , we have By (27), we obtain and ; hence, Therefore, by (D₂), we obtain and , for all .

Example 1. Let and let be the JS metric defined by the following:

Define by ; then, is a complete convex JS metric space.

Now, consider the mapping given by

The mapping is a k-contraction with , and for every , we have . In addition, for all ; consequently, .

Hence, by Theorems 9, 11, and 16, has a fixed point that can be approximated by Picard iteration, Mann iteration with , and Ishikawa iteration with , and we take for initial value .

The convergence behaviors of the three iterative schemes are displayed in Table 1 and graphically in Figure 1.

2.2. Convergence of Mann and Ishikawa Iterations for Chatterjea Contraction

Definition 20. Let be a JS metric space, be a mapping, and . We say that is Chatterjea contraction if

In the next, we need the following result proved by El Kouch and Marhrani, in [19].

Theorem 21 (Theorem 3.9 in [19]). Let be a complete generalized metric space, , and let be a Chatterjea mapping.
If there exists a point such that , then the sequence converges to some . Moreover, if , then is a fixed point of , and for any fixed point of such that , we have .

Now, we give a convergence result of Mann iterative process for Chatterjea mapping in complete convex JS metric space.

Theorem 22. Let be a complete convex JS metric space and be a Chatterjea mapping. Let be the sequence defined by , . Suppose that the following conditions hold: (i)(ii)For all , (iii)converges to some satisfying (iv) or divergesThen, Mann iterative process defined in (4) converges to the fixed point .

Proof. All conditions of Theorem 21 are satisfied and . Set for all . The proof goes in three steps.
Step 1: set , and we will prove that We will prove the statement (42) by induction on .
For , we have for all , so we take .
Suppose that (42) is true for some , that is, Now, we will prove that it is true again for .
From (3), we obtain, for all , By successive application of the inequality (41), we have Since for all , we have . Thus, from (45), we get We have and ; hence, Using (47) and for , there exists such that for all , we have We take ; thus, from (48) and (46), we obtain, for all , From (49) and (43), we have, for all , and .
Let us now return to (44), and we obtain, for all , .
Hence, we conclude that the statement (42) is true.
Step 2: we shall prove that is bounded.
By step 1, we have, for all and every , Using Theorem 21, we have converges to and by , and we obtain Hence, by (50), we obtain Therefore, is bounded.
Step 3: converges to 0 (i.e., converges to ).
From (3), we have It follows from (41) that Hence, we get Set , and we obtain from (53) and (55) with .
Therefore, Now, if , then we get .
Hence, by (56) and Lemma 5, it follows that If , then by (57) and Lemma 6, we have the same result.
Consequently, Mann iterative process converges to the fixed point .☐

Theorem 23. Let be a complete convex JS metric space and be a Chatterjea mapping. Let and be the sequences defined in (5), . Suppose that the following conditions hold: (i)(ii)For all , and (iii)converges to some satisfying (iv) or divergesThen, Ishikawa iterative process defined in (4) converges to the fixed point .

Proof. By Theorem 21, we have . Let and be the sequences defined in (5) and set for all . Now, we complete the proof in the following steps.
Step 1: we prove that We will prove the statement (59) by induction on .
For , we have for all , so we take .
Suppose that (59) is true for some , that is, From (3), we obtain, for all , By (41), we get, for all , On the other hand, According to (49), there exists such that for all , Then, from (60), (63), and (64), we have, for all , Hence, from (65) and (62), we get, for all , We have and ; hence, Using (67), and for , there exists such that for all , we have If we take , then from (68) and (66), we obtain, for all , Now, by (61), (60), and (69), we obtain, for all , It follows that the statement (59) is true.
Step 2: we shall show that is bounded.
We have for every , there exists such that for all , Using Theorem 21, converges to , and by , we obtain Then, is bounded.
Step 3: converges to 0 (i.e., converges to ).
By (3) and (7), we have It follows from (41) that Hence, we get with .
Similarly, we have We have, for all , Set Now, by (73), (75), and (77), we obtain Therefore, Now, if , then we get ; hence, by (79) and Lemma 6, it follows that .
If , then the same result follows by (80) and Lemma 6.
Consequently, Ishikawa iterative process converges to the fixed point .☐

3. Applications

In this section, we apply Corollaries 13 and 17 in order to show the existence of solution of the Fredholm integral equation:

where , and is continuous function.

Remark 24. The convexity of the function implies the followings: (i)For all and , we have (ii)Particularly for , we have Let and let be the mapping defined by

Using Remark 24 above and by considering defined by , we can easily show that is a complete convex b-metric space with .

Theorem 25. Suppose that the following hypotheses hold: (i)There exists a continuous function such that for all and for all , we have (ii).Then, the integral Equation (81) has a unique solution on . Moreover, Mann and Ishikawa iteration processes converge to this solution for any sequences such that or diverges.

Proof. Define a self-map on by Therefore, Hence, is a k-contraction mapping on with . All conditions of Corollaries 13 and 17 are satisfied. Hence, has a unique fixed point. That is to say, there exists a unique solution to the integral Equation (81). Moreover, this solution could be approximated by Mann and Ishikawa iterative processes.☐

4. Conclusion

In this paper, we consider a JS metric space endowed with Takahashi convexity structure, and we establish convergence theorems for Mann iteration and Ishikawa iteration for Banach contraction and Chatterjea contraction. Also, we provide an example to validate the established theorems and give an application of our result to Fredholm integral equation. Finally, it remains an open problem to establish the Convergence of Mann iteration and Ishikawa iteration to fixed point for Kannan contraction.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2022 Youness El Kouch and Jamal Mouline. 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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