Fixed Point Approximation of Monotone Nonexpansive Mappings in Hyperbolic Spaces

Kalsoom, Amna; Saleem, Naeem; Işık, Hüseyin; Al-Shami, Tareq M.; Bibi, Amna; Khan, Hafsa

doi:https://doi.org/10.1155/2021/3243020

Journal of Function Spaces

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

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Nonlinear and Variational Analysis and their Applications 2021

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Research Article | Open Access

Volume 2021 | Article ID 3243020 | https://doi.org/10.1155/2021/3243020

Fixed Point Approximation of Monotone Nonexpansive Mappings in Hyperbolic Spaces

Amna Kalsoom,¹Naeem Saleem,²Hüseyin Işık,³Tareq M. Al-Shami,⁴Amna Bibi,¹and Hafsa Khan¹

Academic Editor: Mustafa Avci

Received10 May 2021

Revised01 Jul 2021

Accepted13 Jul 2021

Published10 Aug 2021

Abstract

Fixed points of monotone -nonexpansive and generalized -nonexpansive mappings have been approximated in Banach space. Our purpose is to approximate the fixed points for the above mappings in hyperbolic space. We prove the existence and convergence results using some iteration processes.

1. Introduction

In 1965, Browder [1], Göhde [2], and Kirk [3] started working in the approximation of fixed point for nonexpansive mappings. Firstly, Browder obtained fixed point theorem for nonexpansive mapping on a subset of a Hilbert space that is closed bounded and convex. Soon after, Browder [1] and Göhde [2] generalized the same result from a Hilbert space to a uniformly convex Banach space. Kirk [3] utilized normal structure property in a reflexive Banach space to sum up the similar results. Recently, Dehici and Najeh [4] and Tan and Cho [5] approximated fixed point result for nonexpansive mappings in Banach space and Hilbert space.

Fixed point theory in partially ordered metric spaces has been initiated by Ran and Reurings [6] for finding application to matrix equation. Nieto and Lopez [7] extended their result for nondecreasing mapping and presented an application to differential equations. Recently, Song et al. [8] extended the notion of -nonexpansive mapping to monotone -nonexpansive mapping in order Banach spaces and obtained some existence and convergence theorem for the Mann iteration (see also [9] and the reference therein). Motivated by the work of Suzuki [10], Aoyama and Kohsaka [11], Dehaish and Khamsi [9], and Song et al. [8], Pant and Shukla obtained existence results in ordered Banach space for a wider class of nonexpansive mappings [12, 13]. There are many mathematicians who worked on weak and strong convergence of nonexpansive mappings and its generalizations by using one step, two step, and multistep iteration process ([8, 14, 15]). We obtain existence results in partial ordered hyperbolic space for monotone generalized -nonexpansive and monotone generalized -nonexpansive map. Particularly, in Section 3, some auxiliary results and existence theorems for monotone -nonexpansive mappings in ordered hyperbolic spaces are presented. In Section 4, we presented numerical examples and graphical representation. In Section 5, we obtained some existence results for monotone generalized -nonexpansive mappings in ordered hyperbolic spaces.

2. Preliminaries

In the concept of -convergence was given by Lim [14].

Lim [14] initiated the idea that in a metric space, -convergence is possible. This concept is adapted for CAT(0) spaces by Kirk and Panyanak [16], and they have indicated that in numerous Banach space, outcomes comprising weak convergence were having exactly accurate analogs in this manner.

Definition 1. A self map on is known as Lipschitz, if there exists such that is called to be contractive if and is called nonexpansive mapping, if that is, Suzuki [10] introduced an interesting generalized nonexpansive mapping as follows.

Definition 2. A self map on is said to satisfy condition (3) if for all ,

Suzuki type generalized nonexpansive mapping is another name of self map holding condition (3).

Many generalizations of nonexpansive mapping have been introduced in the literature (see [17–19]). Aoyama and Kohsaka [11] defined a new type of nonexpansive mapping that satisfies the condition (3) known as -nonexpansive mapping as follows.

Definition 3. Let be a nonempty subset of a Banach space . A self map on can be referred -nonexpansive mapping, if for all and ,

The concept of monotone nonexpansive mappings was introduced in 2015 by Bachar and Khamsi [20], and they studied common approximate fixed point of monotone nonexpansive semigroup. To determine some order fixed points, Dehaish and Khamsi [9] proposed weak convergence theorems of the Mann iteration process for monotone nonexpansive mappings in uniformly convex ordered Banach spaces.

Definition 4. Let be a self map on ; then, is said to be (1)monotone [20] if for all with ;(2)monotone nonexpansive [20] if is monotone andfor all with ; (3)monotone quasi-nonexpansive [8] if is monotone andwhere and , and is the set of fixed points.

Definition 5 [21]. is called a hyperbolic space, if is a metric space and is a function holding (i);(ii);(iii);(iv),for all and

Here, (i) defines the convexity in metric space that was first considered by Takahashi [22], (ii) provides a unique geodesic between any two elements and of by convexity map that is the space of hyperbolic type in the sense of Goebel and Kirk [23], (iii) provides symmetry along the direction of geodesic, and (iv) defines negative curvature or hyperbolicity of metric space was first considered by Itoh [24].

Theorem 6 [11]. Let be a nonempty closed convex subset of a uniformly convex Banach space and be an -nonexpansive mapping. Then, F is nonempty iff there exists such that is bounded.

Since the hyperbolic spaces contain all normed linear spaces and their convex subsets, so uniformly convex Banach space is contained in hyperbolic metric space so it is natural to generalize the above result to hyperbolic metric space.

Definition 7 [16]. A bounded sequence in is known as -converge to an element if is a unique asymptotic centre of each subsequence of

In this section, following definitions and lemma are stated in [9].

Definition 8. Let be a nonempty set of a hyperbolic metric space . A map is said to be a type function, if there exists a bounded sequence in such that

It is known that each bounded sequence generates a unique type function.

Lemma 9. Let be a uniformly convex hyperbolic metric space and a nonempty closed convex subset of . Let be a type function. Then, is continuous. Furthermore, there exists a unique minimum point such that

Definition 10. A hyperbolic space known as uniformly convex, if for every and , for any

Definition 11. Let be a nonempty subset of a hyperbolic space and be a bounded sequence in Then, for every , define (i)Asymptotic radius of at by(ii)Asymptotic radius of the sequence relative to the above supposed set by(iii)Asymptotic centre of the sequence relative to the above supposed set by

Note that . Further, has exactly one point if is uniformly convex.

From now to onward, we will suppose that the ordered intervals are convex and closed, and they are also contained in ordered hyperbolic space ; these are described as follows:

3. Monotone -Nonexpansive Mappings

In this section, we will use the following iteration introduced by Kalsoom et al. [25]. where and are in .

Now we define monotone -nonexpansive mappings in partially ordered hyperbolic metric space as follows.

Definition 12. Let be a nonempty closed convex subset of an ordered hyperbolic metric space . A self map on is monotone -nonexpansive mapping, if is monotone and for some , for all with

Lemma 13. Let be a nonempty closed convex subset of an ordered hyperbolic metric space . A self map on is monotone -nonexpansive mapping; then, (i) is monotone quasi.(ii)For all with

Proof. To prove (i), it is followed by definition of monotone -nonexpansive mappings that implies
Hence, is monotone quasi for and
Now we will prove (ii), and if then we have This completes the proof.☐☐

Definition 14. An ordered hyperbolic metric space is said to be uniformly convex, if for an arbitrary and ,

Now, we utilize iteration processes for monotone -nonexpansive mappings.

Lemma 15. Let be a uniformly convex partially ordered hyperbolic metric space (in short, UCPOHMS) and a nonempty closed convex subset of Let be a monotone mapping and be such that Then, for sequence defined by (14), we have (a) (or );(b) provided -converges to a point .

Theorem 16. Let be a nonempty closed convex subset of a UCPOHMS and be a monotone -nonexpansive mapping. Assume that there exists such that and defined in (14) is a bounded sequence with for all such that Then,

Proof. Let defined by (14) be a bounded sequence such that Then, there exists a subsequence such that By Lemma 15, we have Define for all Clearly, for every is closed convex. As , it shows that Define Then, is a closed convex subset of Let Then, As we know, is a mapping which is monotone; then, for , which implies that Let a type function generated by such that Then, there exists a unique point such that By definition of type function, By using Lemma 13, we get From the boundedness of the sequence and we have implies It shows that , and hence, ☐☐

Theorem 17. Let be a nonempty closed convex subset of a UCPOHMS and a monotone -nonexpansive mapping. Assume that there exists such that and defined by (14) is a bounded sequence with for all such that Then,

Theorem 18. Let be a UCPOHMS, a closed convex cone, and a monotone -nonexpansive mapping. Assume that and defined by (14) is a bounded sequence with for all such that Then,

Proof. With the help of definition of partial order we know that ; then, the proof is directly from Theorem 16.☐☐

We now prove some convergence results for monotone -nonexpansive mappings.

Lemma 19 [9]. Suppose be a UC hyperbolic space and also monotone modulus of uniform convexity , and suppose and a sequence such that . If the sequences and are in , in such a way that , and , then .

Theorem 20. Let be a nonempty closed convex subset of a UCPOHMS and a monotone -nonexpansive mapping. Suppose there exists a sequence defined by (14) with and Then, (1) is bounded(2) and exists for all (3)

Proof. Suppose that and , and as we know that is monotone, then , and so It follows from Lemma 15. which gives that It shows that is bounded. On the other hand, by using Lemma 13, we get and so Then, the sequence is nonincreasing and bounded sequence; hence, (i) and (ii) proved. So exists for all and . Suppose that As is monotone quasi, and hence, It concludes from Lemma 19 that ☐☐

Theorem 21. Let be a nonempty closed convex subset of a UCPOHMS and a monotone -nonexpansive mapping. Suppose there exists a sequence defined by (14) with and Then, -converges to a fixed point of .

Proof. By the Theorem 18, we have is bounded. So, there exists a subsequence of -converges to some such that In the next step, we prove there exists a unique -limit in corresponding to each -convergent subsequence of . Consider has two subsequences and which are -convergent to and , respectively. Then, is bounded and which concludes that Let is a type function which is generated by Then, By Lemma 13, we infer as By uniqueness of element and definition of -convergence, we conclude that Similarly, one can easily show that By continuity of and definition of -convergence, we get which shows that ☐☐

Theorem 22. Let be a nonempty closed convex subset of a UCPOHMS and a monotone -nonexpansive mapping. Suppose there exists a sequence defined by (14) with and

Then, converges strongly to a fixed point of .

Proof. By Theorem 18, there exists a subsequence of which converges strongly to a point From Lemma 15, we get By Theorem 17, and is bounded, and Without loss of generality, we get On the other hand, by Lemma 13, we derive as By boundedness of , we have and hence, Therefore, which shows that By Theorem 18, exists, so This completes the proof.☐☐

Example 1. Let be a nonempty closed convex subset of a UCPOHMS and defined by be a monotone -nonexpansive mapping. Then, for the sequences , , and , there exists a sequence for which all the conditions of Theorem 22 are satisfied by , and hence, is the required fixed point.

4. Comparison of Iteration Processes

In this section, we are presenting some iterations [25–31] which we will be used in the numerical example.

Mann iteration process

In 1953, Mann proposed an iteration, namely, Mann iteration, for calculation of a fixed point for a nonexpansive mapping , defined as for each and .

Ishikawa iteration process

In 1974, Ishikawa proposed the two-step iteration process as follows: where and are in .

Noor iteration process

In 2000, Noor proposed the three-step iteration as follows: where , , and are in .

Agarwal iteration process

In 2007, Agarwal et al. introduced the three-step iteration as follows: where and are in .

Abbas and Nazir iteration process

In 2014, Abbas and Nazir introduced the three-step iteration as follows: where , , and are in .

Thakur iteration process

In 2016, Thakur et al. proposed the three-step iteration as follows: where and are in .

Two qualities fastness and stability play a vital role in iteration process to be performed. In [32], Rhoades mentioned that for the increasing functions, Ishikawa [27] iteration process is faster than Mann iteration process [26] but in the case of decreasing function, condition is reverse. In [29], Agarwal et al. proved that their iteration process was more stable than the previous ones. In [31], Thakur iteration process was considered faster convergent than all the abovementioned iteration processes.

Recently, Kalsoom et al. [25] introduced a new iteration process and proved it to be the fastest convergent than all. The following example is given to support this claim.

Example 2. Define for Then, is monotone nonexpansive mapping as well as is monotone -nonexpansive mapping.

Now we will compare abovementioned iteration processes to check the convergence of mapping By using MATLAB, we present graphs and table.

In Table 1, we discussed the convergence behavior of some iteration processes. It is clear that all iterations approach to which is the fixed point of mapping In this case, Figures 1–3 show that Kalsoom et al. iteration process converges faster to the fixed point as compared the other iterations.

5. Monotone Generalized -Nonexpansive Mappings

In this section, we define monotone generalized -nonexpansive mapping which generalizes the results of Pandey and Shukla [13] in hyperbolic spaces.

Now we will define monotone generalized -nonexpansive mappings in hyperbolic space with nontrivial example.

Definition 23. Let be a nonempty subset of an ordered hyperbolic metric space . A mapping is said to be monotone generalized -nonexpansive mapping, if is monotone and there exists such that for all with .

Proposition 24. Every monotone nonexpansive mapping is monotone generalized -nonexpansive mapping, but converse is not true.

Proof. By putting in , we have which shows that monotone generalized -nonexpansive mapping reduces to monotone nonexpansive mapping satisfying the condition (3). The following example will prove the converse statement does not hold.☐☐

Example 3. Let be a subset of endowed with usual order. Define by Then, for and , implies and so does not hold condition (3). Again, and and does not satisfy condition (3). Nevertheless, is generalized -nonexpansive mapping with

Proposition 25. Let be a nonempty subset of an ordered hyperbolic metric space and a monotone generalized -nonexpansive mapping which has a fixed point with Then, can be referred as a monotone quasi-nonexpansive mapping.

Proof. Let and ; then, by , where as which shows that is monotone quasi-nonexpansive mapping.☐☐

Proposition 26. Let be a nonempty subset of an ordered hyperbolic metric space . If is monotone generalized -nonexpansive mapping, then is closed. Furthermore, if is strictly convex, then is convex and is also convex.

Proof. Let be a sequence in which converges to . Since with the help of continuity of metric, we have Hence, is closed. Now, we assume that is strictly convex and is convex. Let with ; then, put . By similar argument, we get Therefore, From strict convexity of there exists such that By using value of , it gives and similarly, By putting values of and , we infer From the above two inequalities, it is concluded that Hence, is convex.☐☐

Lemma 27. Let be a nonempty subset of an ordered hyperbolic metric space and a monotone generalized -nonexpansive mapping. Then, for each with , (a)(b)Either or (c)Either or

Proof. Since implies Hence, part (a) is satisfied. Now, we will prove part (b); we argue with contradiction, and suppose By (a) and triangular inequality, which is contradiction to our supposition, hence proved. The proof of (c) is in a similar way, so we omit that.☐☐

Lemma 28. Let be a nonempty subset of an ordered hyperbolic metric space and a monotone generalized -nonexpansive mapping. Then, for each with ,

Proof. By the help of Lemma 27, we infer either or In the first case, we have and so In second case, and hence, Hence, we get the desired result.☐☐

Theorem 29. Let be a nonempty convex and closed subset of an ordered hyperbolic metric space and a monotone generalized -nonexpansive mapping. Then, iff is a sequence which is also bounded for some provides that for some and

Proof. Let be a bounded sequence for some . As we know that is monotone and so we get In the same manner, we get Define , Then, the asymptotic centre of w.r.t is where is unique and for all Now, we claim that is a nonincreasing sequence, that is, Since which gives that Now, we claim that To prove this, we consider the contradiction By using triangular inequality, which is not possible, so (102) is satisfied.
In the first case of (102), Putting on both sides, Similarly, in the second case, Putting on both sides, Conversely, ; then, there exists some and ; then, is a constant sequence, and hence, it is bounded and this completes the proof.☐☐

Example 4. Let where ; then, is defined as for any ; take , , and The fixed point of is and take initial point as Then, is monotone generalized -nonexpansive mapping.

In Table 2, we discussed the convergence behavior of some iteration processes. It is clear that all iterations approach to 4 which is the fixed point of . In this case, Figures 4–6 show that Kalsoom et al. iteration process converges faster to the fixed point as compared the other iterations.

6. Conclusion

It concludes that we have approximated fixed point results of monotone and generalized -nonexpansive mappings in hyperbolic spaces. Moreover, we proved some numerical applications and presented the graphical representations by using different iteration processes.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare to have no competing interests.

Authors’ Contributions

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

Acknowledgments

Authors are thankful to the editor and anonymous referees for their valuable comments and suggestions.

References

F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, no. 4, pp. 1041–1044, 1965.
View at: Publisher Site | Google Scholar
D. Göhde, “Zum prinzip der Kontraktiven abbildung,” Mathematische Nachrichten, vol. 30, no. 3-4, pp. 251–258, 1965.
View at: Publisher Site | Google Scholar
W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol. 72, no. 9, pp. 1004–1006, 1965.
View at: Publisher Site | Google Scholar
A. Dehici and R. Najeh, “Some fixed point results for nonexpansive mappings in Banach spaces,” Journal of Nonlinear Functional Analysis, vol. 2020, no. 1, article 36, 2020.
View at: Publisher Site | Google Scholar
B. Tan and S. Y. Cho, “An inertial Mann-like algorithm for fixed points of nonexpansive mappings in Hilbert spaces,” Journal of Applied and Numerical Optimization, vol. 2, no. 3, pp. 335–351, 2020.
View at: Publisher Site | Google Scholar
A. C. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004.
View at: Publisher Site | Google Scholar
J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.
View at: Publisher Site | Google Scholar
Y. Song, K. Promluang, P. Kumam, and Y. Je Cho, “Some convergence theorems of the Mann iteration for monotone -nonexpansive mappings,” Applied Mathematics and Computation, vol. 287-288, pp. 74–82, 2016.
View at: Publisher Site | Google Scholar
B. A. Bin Dehaish and M. A. Khamsi, “Browder and Göhde fixed point theorem for monotone nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2016, no. 1, Article ID 20, 2016.
View at: Publisher Site | Google Scholar
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
K. Aoyama and F. Kohsaka, “Fixed point theorem for -nonexpansive mappings in Banach spaces,” Nonlinear Analysis, vol. 74, no. 13, pp. 4387–4391, 2011.
View at: Publisher Site | Google Scholar
R. Pant and R. Shukla, “Approximating fixed points of Generalized -Nonexpansive mappings in Banach spaces,” Numerical Functional Analysis and Optimization, vol. 38, no. 2, pp. 248–266, 2017.
View at: Publisher Site | Google Scholar
S. P. Pandey and D. P. Shukla, “Fixed point theorem using monotone generalised -nonexpansive mappings in Banach space,” International Journal of Innovations in Engineering and Technology (IJIET), vol. 11, no. 2, p. 2319-1058, 2018.
View at: Google Scholar
T. C. Lim, “Remarks on some fixed point theorems,” Proceedings of the American Mathematical Society, vol. 60, no. 1, pp. 179–182, 1976.
View at: Publisher Site | Google Scholar
B. Tan and S. Li, “Strong convergence of inertial Mann algorithms for solving hierarchical fixed point problems,” Journal of Nonlinear and Variational Analysis, vol. 4, no. 3, pp. 337–355, 2020.
View at: Publisher Site | Google Scholar
W. A. Kirk and B. Panyanak, “A concept of convergence in geodesic spaces,” Nonlinear Analysis, vol. 68, no. 12, pp. 3689–3696, 2008.
View at: Publisher Site | Google Scholar
Y. Yao, L. Leng, M. Postolache, and X. Zheng, “Mann-type iteration method for solving the split common fixed point problem,” Journal of Nonlinear and Convex Analysis, vol. 18, no. 5, pp. 875–882, 2017.
View at: Google Scholar
Y. Yao, X. Qin, and J. C. Yao, “Projection methods for firmly type nonexpansive operators,” Journal of Nonlinear and Convex Analysis, vol. 19, no. 3, pp. 407–415, 2018.
View at: Google Scholar
F. Kohsaka and W. Takahashi, “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces,” SIAM Journal on Optimization, vol. 19, no. 2, pp. 824–835, 2008.
View at: Publisher Site | Google Scholar
M. Bachar and M. A. Khamsi, “On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces,” Fixed Point Theory and Applications, vol. 2015, no. 1, Article ID 160, 2015.
View at: Publisher Site | Google Scholar
U. Kohlenbach, “Some logical metatheorems with applications in functional analysis,” Transactions of the American Mathematical Society, vol. 357, no. 1, pp. 89–128, 2005.
View at: Publisher Site | Google Scholar
W. Takahashi, “A convexity in metric space and nonexpansive mappings. I,” Kodai Mathematical Journal, vol. 22, no. 2, pp. 142–149, 1970.
View at: Publisher Site | Google Scholar
K. Goebel and W. A. Kirk, “Iteration processes for nonexpansive mappings,” Contemporary Mathematics, vol. 21, pp. 115–123, 1983.
View at: Publisher Site | Google Scholar
F. Itoh, “Some fixed point theorems in metric spaces,” Fundamenta Mathematicae, vol. 102, no. 2, pp. 109–117, 1979.
View at: Publisher Site | Google Scholar
A. Kalsoom, M. Rashid, T. Sun et al., “Fixed points of monotone total asymptotically nonexpansive mapping in hyperbolic space via new algorithm,” Journal of Function Spaces, vol. 2021, Article ID 8482676, p. 10, 2021.
View at: Publisher Site | Google Scholar
W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, no. 3, pp. 506–510, 1953.
View at: Publisher Site | Google Scholar
S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, no. 1, pp. 147–150, 1974.
View at: Publisher Site | Google Scholar
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
R. P. Agarwal, D. O'Regan, and D. R. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings,” Journal of Nonlinear and Convex Analysis, vol. 8, pp. 61–79, 2007.
View at: Google Scholar
M. Abbas and T. Nazir, “A new faster iteration process applied to constrained minimization and feasibility problems,” Matematicki Vesnik, vol. 66, pp. 223–234, 2014.
View at: Google Scholar
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
B. E. Rhoades, “Some fixed point iteration procedures,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 1, 16 pages, 1991.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Amna Kalsoom 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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