Journal of Function Spaces

Journal of Function Spaces / 2020 / Article
Special Issue

Nonlinear and Variational Analysis and their Applications

View this Special Issue

Research Article | Open Access

Volume 2020 |Article ID 9851063 | https://doi.org/10.1155/2020/9851063

Thabet Abdeljawad, Kifayat Ullah, Junaid Ahmad, Manuel de la Sen, Junaid Khan, "Approximating Fixed Points of Operators Satisfying (RCSC) Condition in Banach Spaces", Journal of Function Spaces, vol. 2020, Article ID 9851063, 7 pages, 2020. https://doi.org/10.1155/2020/9851063

Approximating Fixed Points of Operators Satisfying (RCSC) Condition in Banach Spaces

Academic Editor: Yuanfang Ru
Received13 Jul 2020
Revised25 Aug 2020
Accepted02 Sep 2020
Published22 Sep 2020

Abstract

Let be a nonempty subset of a Banach space . A mapping is said to satisfy (RCSC) condition if each , . In this paper, we study, under some appropriate conditions, weak and strong convergence for this class of maps through iterates in uniformly convex Banach space. We also present a new example of mappings with condition (RCSC). We connect iteration and other well-known processes with this example to show the numerical efficiency of our results. The presented results improve and extend the corresponding results of the literature.

1. Introduction

will denote the set of all natural numbers throughout. In 2008, Suzuki [1] introduced a new class of mappings as follow. A self-map on a subset of a Banach space is said to satisfy condition if for all , we have

Obviously, when is nonexpansive mapping, that is, holds for all , then satisfies the condition. However, an example in [1] shows that there exists mappings, which satisfy the condition but not nonexpansive. A mapping with condition is often called Suzuki-type nonexpansive mapping. The class of Suzuki-type nonexpansive mappings is extensively studied by many authors (cf. [212] and others).

In 2012, motivated by Suzuki condition, Karapinar [13] suggested a new condition on mappings, the so-called (RCSC) condition (or Reich-Chatterjea-Suzuki condition). A self-map on a subset of a Banach space is said to satisfy the (RCSC) condition if for all , we have

The purpose of this work is to prove some weak and strong convergence results for this class of mappings through the iteration process [12] in the context of Banach spaces. We also give a numerical example to show the usefulness of our results. In this way, we extend and improve many well-known corresponding results of the current literature.

Approximating fixed points of nonlinear mappings played an important role and solved many problems [1420]. It is now well known that if is nonexpansive, then the sequence of Picard iterates may not converge to a fixed point of . To overcome such problems and to get better a rate of convergence, many iterative processes are available in the literature. The well-known iterative processes are the Mann [21], Ishikawa [22], Noor [23], Agarwal et al. [24], Abbas and Nazir [25], Thakur et al. [7], Ullah and Arshad [12], and so on. Let , , and be a self-map on a nonempty convex subset of a Banach space.

The Mann iteration process [21] is a sequence defined as follows:

The Ishikawa iteration process [22] is a sequence defined as follows:

The Noor iteration process [23] is a sequence defined as follows:

The iteration process [24] is a sequence defined as follows:

The Abbas and Nazir iteration process [25] is a sequence defined as follows:

The Thakur et al. iteration process [7] is a sequence defined as follows:

The iteration process [12] is a sequence defined as follows:

In this paper, we will present some weak and strong convergence results using the iteration process (9) for mappings with (RCSC) condition. Similar results for the processes (3)–(8) can be proved on the same line of proofs.

2. Preliminaries

is called a fixed point of a self-map on if . We will denote by throughout the set of all fixed points of . A Banach space is said to satisfy Opial condition [26] if and only if for each weakly convergent sequence with a weak limit , we have the following property:

A self-map on a subset of a Banach space is said to satisfy the condition [27] if there is nondecreasing function with the properties , for every , and for all .

Let be a nonempty subset of a Banach space and a bounded sequence in . For each , define (i)asymptotic radius of at by (ii)asymptotic radius of relative to by (iii)asymptotic center of relative to by

When the space is uniformly convex [28], then the set is always singleton. Notice also that the set is convex as well as nonempty provided that is weakly compact convex (see, e.g., [29, 30]).

Lemma 1. [13].
Let be a self-map on a subset of a Banach space. If satisfies the (RCSC) condition, then for all , the following holds:

The following facts are also needed.

Lemma 2. [13].
Let be a Banach space having Opial’s property, and . If satisfies the condition (RCSC), then the following condition holds:

The following lemma gives the structure of the fixed point set associated with a mapping satisfying (RCSC) condition.

Lemma 3. [13].
Let be a self-map on a subset of a Banach space. If satisfies the (RCSC) condition, then is closed. Moreover, if is strictly convex and is convex, then is also convex.

Lemma 4. [13].
Let be a self-map on a subset of a Banach space. If satisfies (RCSC) condition, then for all and , holds.

Lemma 5. [31].
Let for each and and be any two sequences in a uniformly convex Banach space such that , , and for some ; then, .

3. Main Results

We begin this section by proving a crucial lemma.

Lemma 6. Let be a self-map on a subset of a Banach space. Assume that satisfies the (RCSC) condition and let be a sequence generated by (9). If , then exists for each .

Proof. Let and . By Lemma 4, we have which implies that Hence, for all and . Thus, is bounded and nonincreasing, which implies that exists for each .

Now we give the necessary and sufficient condition for the existence of a fixed point for mapping with (RCSC) condition defined on a nonempty closed convex subset of a complete uniformly convex Banach space.

Theorem 7. Let be a self-map on a closed convex subset of a uniformly convex Banach space. Assume that satisfies the (RCSC) condition and let be a sequence generated by (9). Then, if and only if is bounded and .

Proof. Let be bounded and . Let . By Lemma 1, we have Hence, we conclude that . Since is uniformly convex, consists of a unique element. Thus, we have .
Conversely, suppose that and . By Lemma 6, exists and is bounded. Put From (13), we have By Lemma 4, we have From (14), we have From (17) and (19), we have From (20), we have Hence, By Lemma 5, we have

Now we can prove the following weak convergence theorem.

Theorem 8. Let be a self-map on a closed convex subset of a uniformly convex Banach space having Opial’s property. Assume that satisfies the (RCSC) condition with and let be a sequence generated by (9). Then, converges weakly to a fixed point of .

Proof. Since is uniformly convex, is reflexive. By Theorem 7, is bounded and for all . By the reflexivity, one can find a weakly convergent subsequence of with a weak limit say . By Lemma 2, we have . It is suffice to show that is the weak limit of . is not the weak limit of . Then, one can find another weakly convergent subsequence of with a weak limit such that . Again by Lemma 2, . By Lemma 6 together with Opial’s property, we have This is a contradiction. So, we have . Hence, is the weak limit of

Now we prove a strong convergence theorem as follows.

Theorem 9. Let be a self-map on a compact convex subset of a uniformly convex Banach space. Assume that satisfies the (RCSC) condition with and let be a sequence generated by (9). Then, converges strongly to a fixed point of .

Proof. By Theorem 7, for all . Since is compact and convex, we can find a strongly convergent subsequence of with a strong limit say . By Lemma 1, we have By the uniqueness of limits in Banach spaces, . By Lemma 6, exists and hence is the strong limit of .

Now we state the following theorem. Since the proof is elementary, we will not include the details.

Theorem 10. Let be a self-map on a closed convex subset of a uniformly convex Banach space. Assume that satisfies the (RCSC) condition with and let be a sequence generated by (9). Then, converges strongly to a fixed point of provided that .

The following convergence theorem is based on condition .

Theorem 11. Let be a self-map on a closed convex subset of a uniformly convex Banach space. Assume that satisfies the (RCSC) condition with and let be a sequence generated by (9). Then, converges strongly to a fixed point of provided that satisifes the condition .

Proof. By Theorem 7, it follows that . By the condition , we have . The conclusion follows from Theorem 10.

4. Example

In this section, we compare the rate of convergence of the iteration process with other iterations in the setting of mappings with (RCSC) condition.

Example 1. Let be endowed with the usual norm. Set if and if . We shall prove that satisifes the (RCSC) condition. The case when is trivial. We consider only the following three nontrivial cases.
When , then and . Using triangle inequality, we have When and , then and . Now Finally, when and , then and . Now Next, for and , but . Hence, does not satisfy the condition. Let , , and . The strong convergence of the (9), Thakur et al. (8), Abbas and Nazir (7), (6), Noor (5), Ishikawa (4), and Mann (3) iterates to a fixed point is given in Table 1.

Remark 12. From Table 1 and Figure 1, we see that the iteration process converges faster to than the others.


Thakur et al.Abbas and NazirNoorIshikawaMann

3333333
2.16252.19312.24562.38632.48512.53632.6500
2.02642.03732.06032.14922.23532.28762.4225
2.00432.00722.01482.05762.11412.15422.2746
2.00072.00142.00362.02232.05542.08272.1785
2.00012.00032.00092.00862.02692.04432.1160
22.00012.00022.00332.01302.02382.0754
222.00012.00132.00632.01232.0490
2222.00052.00312.00682.0318
2222.00022.00152.00372.0207
2222.00012.00072.00202.0134
22222.00032.00112.0088
22222.00022.00062.0057
22222.00012.00032.0037
222222.00022.0024
222222.00012.0016
2222222.0010
2222222.0007
2222222.0004
2222222.0003
2222222.0002
2222222.0001
2222222

Now using the above example, we make different choices of parameters and and intial points and also we get as our stopping criterion where is a fixed point of . The number of iterations for (9) to reach is compared with the leading three steps of Thakur et al. (8) and leading two steps of (6) iterations. The numbers in italic in Tables 24 show that iteration is better than the others. The "-" represents that the number of iterations exceeds 50.


Initial points (6)Thakur et al. (8) (9)

8
8
8
8
8
8


Initial points (6)Thakur et al. (8) (9)

13
14
14
14
14
14


Initial points (6)Thakur et al. (8) (9)

10
11
11
11
12
12

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The authors are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19.

References

  1. T. Suzuki, “Fixed point theorems and convergence theorems for some generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp. 1088–1095, 2008. View at: Publisher Site | Google Scholar
  2. A. Abkar and M. Eslamian, “A fixed point theorem for generalized nonexpansive multivalued mappings,” Fixed Point Theory, vol. 12, no. 2, pp. 241–246, 2011. View at: Google Scholar
  3. M. Basarir and A. Sahin, “On the strong and -convergence of S-iteration process for generalized nonexpansive mappings on CAT (0) spaces,” Thai Journal of Mathematics, vol. 12, pp. 549–559, 2014. View at: Google Scholar
  4. S. Dhompongsa, W. Inthakon, and A. Kaewkhao, “Edelstein's method and fixed point theorems for some generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 12–17, 2009. View at: Publisher Site | Google Scholar
  5. B. Nanjaras, B. Panyanak, and W. Phuengrattana, “Fixed point theorems and convergence theorems for Suzuki generalized nonexpansive mappings in CAT (0) spaces,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 25–31, 2010. View at: Publisher Site | Google Scholar
  6. W. Phuengrattana, “Approximating fixed points of Suzuki-generalized nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 5, no. 3, pp. 583–590, 2011. View at: Publisher Site | Google Scholar
  7. B. S. Thakur, D. Thakur, and M. Postolache, “A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings,” Applied Mathematics and Computation, vol. 275, pp. 147–155, 2016. View at: Publisher Site | Google Scholar
  8. I. Uddin and M. Imdad, “Convergence of SP-iteration for generalized nonexpansive mapping in Hadamard spaces,” Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 6, pp. 1595–1604, 2018. View at: Google Scholar
  9. I. Uddin and M. Imdad, “On certain convergence of S-iteration scheme in CAT (0) spaces,” Kuwait Journal of Science, vol. 42, no. 2, pp. 93–106, 2015. View at: Google Scholar
  10. I. Uddin and M. Imdad, “Some convergence theorems for a hybrid pair of generalized nonexpansive mappings in CAT (0) spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 3, pp. 447–457, 2015. View at: Google Scholar
  11. K. Ullah and M. Arshad, “New iteration process and numerical reckoning xed point in Banach spaces,” UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, vol. 79, no. 4, pp. 113–122, 2017. View at: Google Scholar
  12. K. Ullah and M. Arshad, “Numerical reckoning fixed points for Suzuki's generalized nonexpansive mappings via new iteration process,” Univerzitet u Nišu, vol. 32, no. 1, pp. 187–196, 2018. View at: Publisher Site | Google Scholar
  13. E. Karapinar, “Remarks on Suzuki (C)-condition,” in Dynamical Systems and Methods, A. Luo, J. Machado, and D. Baleanu, Eds., pp. 227–243, Springer, New York, NY, 2012. View at: Publisher Site | Google Scholar
  14. K. Ullah, F. Ayaz, and J. Ahmad, “Some convergence results of M iterative process in Banach spaces,” Asian-European Journal of Mathematics, vol. 2019, no. article 2150017, 2019. View at: Publisher Site | Google Scholar
  15. T. Abdeljawad, K. Ullah, and J. Ahmad, “On Picard–Krasnoselskii hybrid iteration process in Banach spaces,” Journal of Mathematics, vol. 2020, Article ID 2150748, 5 pages, 2020. View at: Publisher Site | Google Scholar
  16. K. Ullah, J. Ahmad, and M. de la Sen, “On generalized nonexpansive maps in Banach spaces,” Computation, vol. 8, no. 3, p. 61, 2020. View at: Publisher Site | Google Scholar
  17. T. Abdeljawad, K. Ullah, J. Ahmad, and N. Mlaiki, “Iterative approximations for a class of generalized nonexpansive operators in Banach spaces,” Discrete Dynamics in Nature and Society, vol. 2020, Article ID 4627260, 6 pages, 2020. View at: Publisher Site | Google Scholar
  18. K. Ullah, M. S. U. Khan, N. Muhammad, and J. Ahmad, “Approximation of endpoints for multivalued nonexpansive mappings in geodesic spaces,” Asian-European Journal of Mathematics, vol. 2019, no. article 2050141, 2019. View at: Publisher Site | Google Scholar
  19. K. Ullah, J. Ahmad, and N. Muhammad, “Approximation of endpoints for multi-valued mappings in metric spaces,” Journal of Linear and Toplogical Algebra, vol. 9, no. 2, pp. 129–137, 2020. View at: Google Scholar
  20. K. Ullah, J. Ahmad, M. de la Sen, and M. N. Khan, “Approximating fixed points of Reich–Suzuki type nonexpansive mappings in hyperbolic spaces,” Journal of Mathematics, vol. 2020, Article ID 2169652, 6 pages, 2020. View at: Publisher Site | Google Scholar
  21. W. R. Mann, “Mean value methods in iteration,” Proceedings of American Mathematical Society, vol. 4, no. 3, pp. 506–510, 1953. View at: Publisher Site | Google Scholar
  22. S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of American Mathematical Society, vol. 44, no. 1, pp. 147–150, 1974. View at: Publisher Site | Google Scholar
  23. M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000. View at: Publisher Site | Google Scholar
  24. R. P. Agarwal, D. O'Regon, and D. R. Sahu, “Iterative construction of fixed points of nearly asymtotically nonexpansive mappings,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 61–79, 2007. View at: Google Scholar
  25. M. Abbas and T. Nazir, “A new faster iteration process applied to constrained minimization and feasibility problems,” Math Vesnik, vol. 66, no. 2, pp. 223–234, 2014. View at: Google Scholar
  26. Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, no. 4, pp. 591–598, 1967. View at: Publisher Site | Google Scholar
  27. H. F. Senter and W. G. Dotson, “Approximating fixed points of nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 44, no. 2, pp. 375–380, 1974. View at: Publisher Site | Google Scholar
  28. J. A. Clarkson, “Uniformly convex spaces,” Transactions of the American Mathematical Society, vol. 40, no. 3, pp. 396–414, 1936. View at: Publisher Site | Google Scholar
  29. R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications Series, Topological Fixed Point Theory and Its Applications, vol. 6, Springer, New York, 2009.
  30. W. Takahashi, Nonlinear Functional Analysis, Yokohoma Publishers, Yokohoma, 2000.
  31. J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991. View at: Publisher Site | Google Scholar

Copyright © 2020 Thabet Abdeljawad 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views83
Downloads28
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.